Gregory C. November 5, 204 Two individuals have a regular task to complete that requires the effort of only one. Some days it is less costly for one of the individuals to complete the task, and some days it is less costly for the other individual. Each knows only their own cost. Taking turns is fair, but rigid turn-taking cannot account for changing costs. The person obligated might not be best suited for the job. A natural solution is to allow some flexibility - swapping turns when efficient. These arrangements are so familiar, we rarely think of them as economic mechanisms, but doing so provides interesting insights into their properties and performance. In this paper, I model flexible turn-taking as a simple dynamic mechanism (recurring rotation), derive theoretical properties of the mechanism and present results of an experiment designed to test these properties. Although the efficiency achieved by subjects is close to the expected efficiency, behavioral anomalies that cannot be explained by social preferences or strategic concerns suggest that subjects may be used to a different form of flexible turn-taking. An alternative form of flexible turn-taking (obligation takeover) retains familiar structure, is consistent with patterns of subject behavior, and can achieve approximately the same efficiency as an optimal mechanism using money transfers under uniformly distributed costs. Key Words: private information, repeated games, mechanism design, ex post incentive compatibility, volunteer s dilemma JEL Codes: C72, C73, D82 Gregory : Department of Economics, University of California-Santa Barbara, Santa Barbara, CA. Greg@gmail.com. I thank Ted Bergstrom, Rod Garratt, and Cheng-Zhong Qin, my dissertation committee, for their valuable guidance. I also thank Emanuel Vespa, Ryan Oprea, David Miller, my colleagues at UCSB and George Mason, and seminar participants at the 204 Public Economic Theory and 204 Economic Science Association conferences for helpful comments.
Introduction Turn-taking is a fundamental social behavior. The ability to take turns is a major developmental milestone for children (see Sheridan, Sharma and Cockerill, 204) and has also been observed in animal species. In settings where one individual s effort is needed repeatedly to complete a mutually beneficial task, taking turns is a fair way to ensure the task is always completed. Examples include a parent waking to calm a crying baby, a monitor keeping watch in a dangerous environment, or a doctor on-call for a late-night emergency. However, in these examples, the person obligated to complete the task is not always the most capable. The obligated parent or monitor may be particularly tired. The obligated doctor may have external obligations. Rigid turn-taking is blind to varying individual capabilities and may yield inefficient assignments. Further, the effort an individual must put forth to complete a task is best judged by that individual; it is a matter of private information. Extracting private information to make efficient collective decisions is not trivial. Without the right incentives, partners may find opportunity to exaggerate the inconvenience they will experience in completing the task. Getting these incentives right is the focus mechanism design. Why do we take turns doing tasks instead of paying each other? Why, in many cases, have traditional economic mechanisms, such as auctions, not supplanted the informal and money-free social arrangements, which we have developed over our long history with these problems? The reason is, some informal arrangements are economic mechanisms and very good ones. In this paper, I analyze a social arrangement that has never been formally studied: flexible turn-taking where partners may agree to swap turns. I formalize this arrangement as an economic mechanism referred to as recurring rotation. Recurring rotation uses no monetary transfers, requires no formal communication structure, and achieves impressive efficiency even relative to optimal mechanisms using money transfers. For the case of uniformly distributed costs, recurring rotation captures about three-quarters of the achievable efficiency and, in some environments, does substantially better. In a laboratory experiment, subjects achieved an efficiency close to what is predicted by theory. Yet, unpredicted behavior patterns, which cannot be explained by social preferences or strategic concerns, suggest that people may be used to turn-taking arrangements with a slightly different structure. An alternative, but still familiar turn-taking arrangement, which I call obligation takeover, naturally promotes the kind of behavior that appears as an anomaly under recurring rotation. Further, the obligation takeover mechanism achieves approximately optimal efficiency for patient players. Taken together, these results suggest that there may be little to gain from adopting formal mechanisms in place of familiar social arrangements. The following example demonstrates the recurring rotation mechanism. Suppose two partners, Alice and Bob, take turns completing some task for which they each have a private cost, drawn independently each period. The partner who did not complete the task last time is obligated to complete it in the current period unless there is a mutual decision to swap. All else equal, neither wants to be the one who has to complete the task. However, since completing the task in this arrangement results in becoming non-obligated in the next period, a partner with a low enough cost might prefer to complete the task immediately, delaying future obligation. On the other hand, a partner with a high cost might prefer to remain obligated rather than complete the task immediately. Suppose Alice is obligated and has a high enough cost to prefer putting off the task and remaining obligated. Bob has a low cost and prefers to complete the task immediately rather than be obligated next Birds flying in formation provide each other with energy-saving lift from wing-tip up-wash. One bird must fly at the lead-position where this advantage is reduced (Andersson and Wallander, 2004). In research related to study of formation flying published in Portugal et al. (204), the bald ibis engages in frequent pair-wise switches of the leading position (Portugal, 204). In an experimental setting, Stickleback fish feeding in pairs take turns visiting each other s preferred feeding locations (Harcourt et al., 200). 2
time. They can make a mutually beneficial swap. Further, if swapping turns is all that is at stake, they have no incentive to mislead each other about their preferences. Either partner s preference over completing the task immediately or putting it off in favor of being obligated next time is only determined by current private cost relative to the average cost of being obligated in the future. Both have preferences over the outcomes that are not affected by their partner s current cost. Alice has incentive to reveal her preference regardless of her beliefs about Bob s costs. Alice cannot attempt to get a better deal from Bob by misrepresenting her preference or by gaining some additional information about Bob s cost. In turn, Bob has no incentive to manipulate Alice s belief by lying about his costs. When it comes to swapping turns, the method by which the players communicate their preferences or their potential ability to spy on each other s costs does not affect the outcomes. Mechanisms that are robust in this way are appropriate for modeling interpersonal relationships where strict rules of communication might be difficult to enforce. For comparison, in some mechanisms, such as the sealed-bid first-price auction, outcomes depend heavily on communication structure and beliefs. In the first-price auction, a bidder would have incentive to submit a lower bid upon finding out, or somehow being influenced to believe, that others have submitted low bids as well. A benefit of modeling this arrangement in the framework of mechanism design is that incentives can be formalized, and the mechanism can be compared to alternatives that have similar desirable features. In this paper, I model recurring rotation as a Perfect Public Equilibrium (PPE) of a repeated, private cost version of the volunteer s dilemma game (Diekmann, 985). PPE act as a type of dynamic mechanism where incentives for revealing private information are provided by utility transfers within the repeated game. The robustness of the incentives in recurring rotation implies it is part of a particular PPE subclass known as Ex-post Incentive Compatible Perfect Public Equilibrium or EPPPE (Miller, 202). Comparing recurring rotation to other mechanisms that have similar robustness is a formal matter of comparing it to other EPPPE of the game. This comparison can even include mechanisms, like auctions, which use monetary transfers to provide incentives. I make this comparison in section 2 of the paper. Further, I compare two versions of recurring rotation. The first is the familiar version where partners must mutually agree to swap turns. The other version allows the non-obligated partner to demand a swap. Perhaps surprisingly, the less familiar version is always more efficient, though the difference is small in several numerical tests. To complement these theoretical results, in section 3 I present empirical results of an experiment designed to test the efficiency and properties of recurring rotation. Because of the importance of private information, an ideal environment to study these arrangements is one where information is private between partners but observable to the researcher. Such environments may be difficult to find in the real world but are achievable in the lab. In the experiment, achieved efficiency is close to what is predicted by theory. However, many subjects demonstrate a gap between their willingness to do the task when obligated and their willingness when non-obligated. Theory does not predict this gap. The results of a treatment designed to rule out social preference and strategic explanations for this pattern, by pairing subjects with computer partners, indicate that gap strategies might be due to behavioral heuristics developed under a flexible turn-taking arrangement with slightly different structure. In section 4, I consider an alternative, but still familiar, mechanism. This mechanism (obligation takeover), approximates the efficiency of an optimal robust mechanism for patient players. Further, this mechanism is consistent with the gaps that are prevalent in experimental data. In the obligation takeover mechanism, when the non-obligated partner fills in for the obligated partner, the non-obligated takes over an extra obligation in the future. In this way, partners can acquire a debt of turns. In recurring rotation, on the other hand, obligations are simply delayed, but partners do not accrue debt. The debt of turns is similar 3
to the debt of favors that appears models of favor trading (Mobius, 200). Though the nature of the favor trading environment is similar to that studied here, the favor trading literature does not discuss the kind of flexible turn-taking that is the focus of this paper. Further, while Athey and Miller (2007) and Miller (202) provide theoretical results on robust dynamic mechanisms (EPPPEs), they consider mechanisms that involve monetary transfers. Other authors construct mechanisms explicitly for dynamic settings without monetary transfers (see, for instance, the repeated Bertrand environment of Athey and Bagwell (200) and repeated auction collusion environments of Aoyagi (2003) and Skrzypacz and Hopenhayn (2004)). In contrast, this paper focuses on a natural and robust mechanism in an explicitly interpersonal environment. Section 5 contains a more detailed discussion on the relationship of this paper to the literature and concludes. 2 Recurring Rotation Mechanism In this section, I model flexible turn-taking as a simple dynamic mechanism called recurring rotation. In recurring rotation, one player is obligated to perform a task at each time period. The other is non-obligated. In each round, the players must decide whether to offer to do the task. I consider two main versions of the mechanism along with a hybrid. In the deferment version, the obligated must complete the task unless the non-obligated offers to do the task and the obligated does not offer. In the option version, the obligated must complete the task unless the non-obligated offers, regardless of the choice of the obligated partner. I derive several results on the incentive properties, equilibrium, and efficiency of the mechanism for general cost distributions. I also compare recurring rotation to other robust mechanisms, and provide more detailed numerical results for selected distributions. Any version of recurring rotation has a unique symmetric equilibrium in which players offer to do the task below some equilibrium threshold. The equilibrium threshold is less than half of the mean type for any cost distribution. For the case of uniformly distributed costs, recurring rotation captures 75% of the efficiency of a second-best optimal 2 mechanism for this environment, even with monetary transfers. Over the family of symmetric beta distributions the performance of recurring rotation is often even better, especially for the U-shaped distributions where the efficiency approaches first-best. 2. Environment Two players are engaged in a repeated game. Each discounts the future at rate β. In each period, a task must be completed. The cost of performing the task is given by θ i for player i. This cost is private information and is drawn independently in each period from identical, and commonly known distributions F (θ i ) on the domain [0, ]. The value of having the task completed is fixed and normalized to 0 for both players. The players engage in a mechanism which dictates who will complete the task in each period. To focus on the problem of assignment, while avoiding the complication of punishment, the value either experience when the task is not completed is assumed to be less than. Because of this, if a player knew for sure the other would not complete the task, that player would always be willing to complete it. This gives the stage game the structure of a Volunteer s Dilemma. The Volunteer s Dilemma, introduced by Diekmann (985) has asymmetric Nash equilibria in which any player completes the task with certainty. Mechanisms in this environment fix players beliefs about which Nash equilibrium should be played. Although a player has no incentive to deviate once assigned the 2 In this environment, first-best is not achievable. This is discussed in detail in section 2.5 4
task, I assume punishment for this involves repetition of the asymmetric Nash equilibrium in which that player completes the task in perpetuity. The players can communicate, but there is no formal authority to structure communication, prevent players from learning about each other s private information, and absorb budget imbalance. These assumptions capture the interpersonal nature of their relationship. This assumption does not affect the modeling of recurring rotation. As I will demonstrate, the mechanism is robust to these features. Rather, this assumption restricts the class of alternative mechanisms that is available to the players to those that are suitable for modeling arrangements in such informal situations. This is discussed further in section 2.5. 2.2 Recurring Rotation The mechanism has only two states that correspond to which player is obligated. From either player s perspective, the states are labeled o for I am Obligated and n for I am Not Obligated. The players communicate by offering to do the task. Since the players are ex-ante identical and other elements of the game are stationary, I focus on equilibria where players use identical Markov strategies. Here, a strategy maps the state and players type θ i into the action space {Offer,Do Not Offer}. There are several rules by which the task can be assigned, conditional on who is obligated and who offers. 2.2. Deferment Version In the deferment version, the obligated completes the task unless there is a mutual agreement to swap. When only one player offers, that player completes the task and becomes non-obligated in the next round. If both offer or both do not offer, the obligated completes the task and becomes non-obligated in the next round. A summary of this version of the mechanism is shown below in terms of which partner completes the task for any profile of actions. Obligated Action Non-Obligated Action Not Offer Offer Not Offer Obligated Non-Obligated Offer Obligated Obligated Figure 2.: Partner Completing Task by Joint Action Profile (Deferment) When should a player offer? A interesting feature of this mechanism is the symmetry of the decision problem faced in either state. A player s best response is a threshold strategy. Regardless of the state, when cost is below this threshold, the player should offer. When cost is above, the player should not offer. Let V (o) and V (n) be the present values associated with being in the obligated and non-obligated states respectively. These are stationary and not individual specific since it is assumed that the players implement identical Markov strategies. Note that, regardless of the state, the present value of the player who executes the task with cost θ i in the current period is βv (n) θ i. The present value of not executing the task is βv (o). Under the rules of recurring rotation, these are the only two possible outcomes. A player will want to do the task regardless of the state if and only if θ i β (V (n) V (o)). Whenever β (V (n) V (o)) [0, ] there is a well-defined threshold cost θ = β (V (n) V (o)) below which a player prefers to complete the task, becoming non-obligated in the next round, and above which the player prefers not to complete the task and become obligated in the next round. This preference completely determines the actions of a player. 5
Suppose a player has θ i < θ and so wants to complete the task. Refer again to figure 2.. If the player is obligated, it is weakly dominant to offer since it guarantees completing the task. It is weakly dominant for a non-obligated player to offer since it allows the possibility of completing the task if the other player does not offer. Now suppose a player has θ i > θ and so wants to not complete the task. If the player is obligated, it is weakly dominant to not offer since it provides the possibility the non-obligated will complete the task instead. If the player is non-obligated, it is weakly dominant to not offer since it excludes any chance of having to complete the task. Regardless of state, a player with cost below [above] threshold has a weakly dominant strategy (with respect to stage beliefs) to offer [not offer]. Thus, when players act according to symmetric Markov strategies, their actions are determined solely by the threshold θ. However, in equilibrium, the players actions must generate a discounted difference in state value β (V (n) V (o)) that is equal to the threshold they choose θ. If this is not the case, players have incentive to adjust their strategies. The calculation of β (V (n) V (o)) is dependent on the particular mechanism version and threshold θ. An expression for the value β (V (n) V (o)), conditional on mechanism version and threshold, is derived in section 2.2.4. The next section outlines a version of the mechanism in which the non-obligated player may demand to complete the task even if the obligated player does not prefer the swap. 2.2.2 Option Version In the option version, as in the deferment version, when only one player offers, that player completes the task and becomes non-obligated in the next round. When neither offers, the obligated completes the task. However, when both offer, the non-obligated completes the task. In essence, the non-obligated maintains an option to complete the task. A summary of this version is shown below. Obligated Action Non-Obligated Action Not Offer Offer Not Offer Obligated Non-Obligated Offer Obligated Non-Obligated Figure 2.2: Partner Completing Task by Joint Action Profile (Option) When offering, the non-obligated always completes the task. The obligated completes the task when the non-obligated does not offer. The action of the obligated does not affect the outcome. In this sense, this version of the mechanism may be thought of as a rotating dictator mechanism. The player who completed the task last round may choose who completes the task this round. However, as in the deferment version, the non-obligated chooses an action based on a single threshold. Suppose again that there is a well-defined threshold cost θ = β (V (n) V (o)). With a cost below threshold, the non-obligated prefers to complete the task. Offering guarantees this outcome. On the other hand, the non-obligated prefers not to complete the task with a cost above threshold. Not offering guarantees this outcome. Actions of the non-obligated are completely determined by the threshold θ. As before, in equilibrium, the actions of the players (specifically the non-obligated player) must produce a discounted difference in state values β (V (n) V (o)) equal to the threshold θ. This version of the mechanism may seem less natural than the deferment version since it is the nonobligated partner who can demand to complete the task. Because of this, only the non-obligated partner s costs determine assignment. In deferment, on the other hand, both players share information to determine 6
the assignment. However, the results of section 2.3 demonstrate that this version is actually more efficient than deferment regardless of the cost distribution. In determining this result, and solving for the equilibria of these mechanisms more generally it is also useful to define a hybrid version of option and deferment. 2.2.3 Hybrid Version The deferment and option versions differ only in how they handle the situation when both offer. This can be seen clearly in a comparison of figures 2. and 2.2. Deferment settles the disagreement on the side of the obligated, while option settles it on the side of the non-obligated. The hybrid version of the mechanism allows an arbitrary rule for deciding who completes the task when both offer. q will indicate this rule and corresponds to the probability that the obligated is assigned the task when both offer. It is noted that the hybrid version nests the two deterministic versions. q = 0 corresponds to option and q = corresponds to deferment. A summary of the mechanism in terms of which partner completes the task is given below in 2.3. For the case q (0, ), the argument for why offering according to the threshold θ = β (V (n) V (o)) is weakly dominant (with respect to stage beliefs) is nearly identical to the deferment case discussed above. Players maximize the probability of getting their favorite outcome by offering when θ i < θ and not offering when θ i > θ. Obligated Action Non-Obligated Action Not Offer Offer Not Offer Obligated Non-Obligated Offer Obligated Obligated Pr = q Non-Obligated Pr = q Figure 2.3: Partner Completing Task by Joint Action Profile (Hybrid) 2.2.4 Calculating V (n) V (o) In each version of the mechanism, the players actions depend only on the discounted difference in value between the obligated and non-obligated states. Equilibrium is achieved when the actions of players, according to some threshold, generate a discounted difference in the state continuation values that is equal to the threshold. Determining an explicit form for this value difference is necessary for finding equilibria of the mechanism. In this section, I derive an expression of this difference for the hybrid version with generic rule q, since it nests the deferment and option versions as well. The calculation of the difference in the value of the states is simplified by noting that whenever the two players are on opposite sides of the threshold θ, it does not matter who is obligated. Refer to figure 2.3. Regardless of who is obligated, if one player has a cost below threshold and the other has a cost above, only the player with the low cost will offer. That player will complete the task and move to the non-obligated state. Thus, the difference in the two state values only depends on what happens when both are on the same side of the threshold. With this simplification, the difference (V (n) V (o)) can be determined as follows: When both players are above the threshold, the obligated always carries out the task at an average cost E ( θ i θ i θ ) and becomes non-obligated, which carries discounted continuation value βv (n). The nonobligated does not carry out the task and becomes obligated, which has discounted value βv (o). Thus, the difference in value of being non-obligated and obligated, conditional on both being above the threshold, is given by [ E ( θ i θ i θ ) + βv (o) βv (n) ]. This event happens with probability ( F ( θ )) 2. 7
When both are below the threshold, the player chosen to do the task by the tie-breaking rule completes the task at average cost E ( θ i θ i θ ) and becomes non-obligated, which has discounted continuation value βv (n). The player who does not complete the task becomes obligated, which has discounted value βv (o). Taking in to account the tie-breaking rule q, the difference in the value of being non-obligated and obligated, conditional on both being below the threshold, is given by (2q ) [ E ( θ i θ i θ ) + βv (o) βv (n) ]. For instance, when q =, the obligated completes the task for sure and this term is: [ E ( θ i θ i θ ) + βv (o) βv (n) ]. When q = 0, the non-obligated completes the task for sure and this term is the negative of the previous. This event that both are below the threshold happens with probability ( F ( θ )) 2. Weighting the two differences above by the probabilities of occurrence, the entire difference in values is given in the equation below: V (n) V (o) = F ( θ ) 2 (2q ) [ E ( θi θ i θ ) + βv (o) βv (n) ] + (2.) ( F ( θ )) 2 [ E ( θi θ i θ ) + βv (o) βv (n) ] 2.3 Equilibrium In equilibrium, θ = β (V (n) V (o)). Imposing this relationship on equation 2. provides the following equilibrium condition: [ ] θ = β ( F (θ )) 2 [E (θ i θ i θ ) θ ] F (θ ) 2 (2q ) [θ E (θ i θ i θ )] (2.2) The equilibrium threshold θ solves the above fixed-point problem. Notice the right side of this relates weighted versions of the terms E (θ i θ i θ ) θ and θ E (θ i θ i θ ). These are respectively the difference in the average cost of execution and the continuation transfer associated with a deferment when both have cost above the threshold and the negative of the same difference when both have cost below the threshold. The term E (θ i θ i θ ) θ is familiar in survival analysis and is often referred to as mean residual lifetime. θ E (θ i θ i θ ) is a related term referred to as mean advantage over inferiors by Bagnoli and Bergstrom (2005). In solving for the equilibrium, it proves useful to convert the fixed-point problem into a root problem. Define: g ( θ ) = β θ + F ( θ ) 2 (2q ) [ θ E ( θ i θ i θ )] ( F ( θ )) 2 [ E ( θi θ i θ ) θ ] (2.3) θ is an equilibrium of recurring rotation if and only if g (θ ) = 0. Below, I demonstrate the function g ( θ ) has the three properties (for continuous cost distributions with positive density everywhere), which imply that the equilibrium must be unique, below half of the mean type, increasing in β, and decreasing in q. Lemma 2.. g (0) < 0. Proof. g (0) = E (θ i θ i 0) = E (θ i ) < 0. Lemma 2.2. g increases strictly over the interval [0, E (θ i )]. Proof in Appendix Section 6. 8
Lemma 2.3. g is strictly positive over the interval ( 2 E (θ i), ]. (Proof in Appendix) Proof in Appendix Section 6.2 These properties of g ( θ ) implied by these lemmas are demonstrated in the following figure. g( ~ )>0 g( ~ ) * g(0)<0 0 ½ E( i ) ~ Figure 2.4: Properties of g ( θ ). Proposition 2.4. For any continuous f with positive density everywhere and for any q [0, ] and β [0, ), the recurring rotation mechanism has a unique equilibrium that is below the half of the mean type. Proof. This follows from the combination of lemmas 2.,2.2,2.3. The equilibrium condition is g ( θ ) = 0. g (0) < 0 and g (E (θ i )) > 0. Since g increases [ strictly over [0, E (θ i )], it must cross 0 exactly once in the interval [0, 2 E (θ i)), and not again over 2 E (θ i ), ] since g remains strictly positive over this interval. Corollary 2.5. θ is increasing in β. Proof. By the implicit function theorem: δθ δβ = δg(θ,β) δβ = δg(θ,β) δθ θ β 2 (2.4) δg(θ,β) δθ This is positive since δg(θ,β) δθ 0 at the equilibrium by Lemma 2.2 and Proposition 2.4. Corollary 2.6. θ is decreasing in q. Proof. By the implicit function theorem: δθ δg(θ,q) δq = δq = 2F (θ ) 2 [θ E (θ i θ i θ )] (2.5) δg(θ,q) δg(θ,q) δθ δθ 9
This is negative since θ E (θ i θ i θ ) 0 and δg(θ,β) δθ Proposition 2.4. 0, at the equilibrium, by Lemma 2.2 and Corollary 2.5 is rather intuitive and due to the fact that players are less willing to take high costs in the current period to change the potential outcomes of future periods which are discounted more heavily. This result also implies that patient players swap more often in equilibrium than less patient players. As β increases, turn-taking becomes less rigid. Corollary 2.6 is less intuitive. Recall that the best-response threshold is equal to the discounted difference in the value of being non-obligated and obligated. Further, recall that the choice of q only matters in situations when both players want to complete the task. When q is small, the non-obligated is assigned the favorable outcome more often at the expense of the obligated player. This increases the difference between the non-obligated and obligated values for any fixed threshold. This in turn implies that g ( θ ) is smaller for any fixed threshold in option than in deferment. Because g ( θ ) is increasing to the equilibrium for both versions, the fact that g ( θ ) is smaller for the option version means it must cross 0 at a higher value than the deferment version. An implication of this result is that when swaps must happen by mutual agreement (deferment version), they happen less often then when swaps happen by the choice of the non-obligated only (option version). These results also have implications for the efficiency of the mechanism. Since no transfers are used, the joint efficiency can be measured by the average cost of execution. To derive an expression for average cost, recall that if only one player offers to do the task, that player will complete the task under any version of the mechanism. A player only offers if θ i θ. Thus, when only one player offers, the player chosen to do the task has a cost below θ. If both offer, the player chosen to do the task must also have a cost below θ regardless of who is chosen. Thus, whenever at least one player offers, the player chosen to do the task has cost below θ. When neither player offers, both have cost above θ. The obligated player is always chosen to do the task and so the chosen player (always the obligated) has cost above θ. Since the player s costs are independent, the average cost of the chosen player is E (θ i θ i θ ) when at least one offers, and the average cost of the chosen player (the obligated player) is E (θ i θ i θ ) when neither offer. The probability that neither offer is ( F (θ )) 2 and the probability that at least one offers is ( F (θ )) 2. Thus, the expression for average cost of execution is AC [ = ( F (θ )) 2] E (θ i θ i θ ) + ( F (θ )) 2 E (θ i θ i θ ). After simplifying: AC = E (θ i ) F (θ ) (E (θ i ) E (θ i θ i θ )) (2.6) This term is decreasing below the mean for any cost distribution since the derivative with respect to θ is f (θ ) E (θ i ) + f (θ ) θ, which is negative as long as θ E (θ i ). Since all equilibria occur below the mean, ordinal efficiency comparison can be achieved simply by comparing the equilibrium thresholds. Higher thresholds imply lower average cost and higher efficiency. With this, the corollaries 2.5 and 2.6 can be translated into results about efficiency. Corollary 2.7. Efficiency of recurring rotation is increasing in β and decreasing in the tie-breaking rule q (the option version (q = 0) of the mechanism is the most efficient for any cost distribution). This result implies higher efficiency is achieved by patient players and when swaps can be demanded by the non-obligated player. The fact that the seemingly less natural and less cooperative (in the sense of mutual agreement) version of the mechanism achieves higher efficiency is an interesting result, especially since assignments in the option version of the mechanism depend only by the cost of the non-obligated player, rather than on both costs. It is possible to put a bound on the difference between the equilibrium 0
threshold for the two mechanism versions (deferment, and option). Let θ d and θ o be the equilibria for the same parameters under deferment and option respectively. Claim 2.8. For any distribution where 2 E (θ i) is below the median, θ o θ d 2F ( θ d) 2 [ θ d E ( θ i θ i θ d)]. Proof in Appendix Section 6.3. While actual difference in the two will depend both on the location of the equilibrium and the particular shape of the distribution function, note that the term is increasing in θd. By proposition 2.4, θ d 2 E (θ i). Thus, an upper bound on θo θd is given by: ( ) θo θd 2 [ ( 2F 2 E (θ i) 2 E (θ i) E θ i θ i )] 2 E (θ i) (2.7) This bound applies for any symmetric distribution, since the median is E (θ i ). When f is uniform, this bound is θo θd 64. Notice that, for distributions with concave density, this term may be small. ( 2 [ ( )] 2F 2 E (θ )) is small due to low-tail weight, and 2 E (θ ) E θ i θ i 2 E (θ ) is small due to the fact that density is shifted towards the mean. For the beta distribution with both shape parameters equal to 2, the difference is: θo θd 892 35.0043. For distributions with convex density, the bound may be larger, indicating the possibility that the choice of tie-breaking rule might have a larger impact on play. For instance, the beta distributions with both shape parameters equal to 2 has a bound: θ o θd.0364. Although these results provide some general insight into the workings of the recurring rotation mechanism and the location of equilibria, the following sections offer a deeper analysis by focusing on the uniform distribution. Appendix section 6.4 contains results for symmetric beta distributions. 2.4 Example: Uniform Distribution In this section, I calculate the equilibrium thresholds and efficiency of recurring rotation for the environment where players costs are distributed uniformly. When θ i U (0, ), θ is the solution to the cubic 3 equation: θ = 2 β [( θ ) 3 (2q ) (θ ) 3]. For option (q = 0) and deferment (q = ), respectively these are: q = 0 : θ = 2 β [( θ ) 3 + (θ ) 3] (2.8) A graph of the solutions is show in figure 2.5. 3 At q = 0 the equation is quadratic. q = : θ = 2 β [( θ ) 3 (θ ) 3] (2.9)
* 0.25 0.20 0.5 0.0 Option Deferment 0.05 0 0 0.2 0.4 0.6 0.8 Figure 2.5: Equilibrium for Uniform Distribution by discount factor As predicted by the analytical results of 2.3, both versions have thresholds increasing in β and the equilibrium threshold in the option version is larger than the deferment version. As β, the equilibria approach.232 for option and.226 for deferment. Notice that the difference in these thresholds is well within the upper bound of 64 implied by equation 2.7. The equilibrium average stage costs are also close. Using equation 2.6 together with the calculated equilibrium thresholds, the resulting average stage costs are.4 for option and.43 for deferment. To put these costs in perspective, suppose players were to take turns in a rigid way, or flip a coin to determine assignments. The average stage cost of execution would simply be the mean of the distribution: 2. Since players can achieve this stage cost without sharing any information about their costs, it represents a of lower bound on the achievable efficiency 4. Compared to this lower bound, recurring rotation provides a substantial efficiency improvement. On the other hand, in a perfect information setting, the players could condition their play on the joint cost realization, assigning the task to the individual with lower cost. Such strategies would achieve an average stage cost of the expected value of the minimum of the partners costs. For instance, when the cost distribution is uniform, the average cost of the partner completing the task would be 3. Compared to this figure, recurring rotation appears to leave substantial welfare uncaptured. In this environment, recurring rotation falls about half-way between the lower-bound and first-best efficiency. These are plotted on the line segment below. 3 First Best 0.4 Recurring Rotation 2 Coin Flip Figure 2.6: Efficiency Comparison 4 They could do even worse, but systematically assigning the task to the player with a higher cost would require some information transfer, which could simply be ignored in favor of random assignment. 2
However, first-best is not be the most appropriate comparison on the top-end of the efficiency spectrum. Players do not have perfect information. Because of this, achieving first best would require a mechanism that perfectly reveals costs. However, in this environment where there is no planner to enforce communication rules, there is not a suitable mechanism that achieves first-best even with monetary transfers. An explanation of this requires some discussion of the mechanism design literature. 2.5 Efficiency Comparison Mechanism design is a field that attempts to create games in which player s actions in equilibrium reveal their private information so that the outcomes of the game have desirable properties. For instance, in an auction, players submit bids based on their private valuation of an item. In a well-designed auction, players with a higher valuation have incentive to submit higher bids. In a second price auction for instance, players have strong incentive to submit bids equal to their valuation regardless of what they believe others will bid. Because of this, the person that ends up winning the item will be the one with highest valuation - a desirable outcome. The problem studied here is a simply a repeated version of a mechanism design problem. In a repeated game, players may choose their actions based on the history of play. In recurring rotation, the relevant history is completely summarized by who is currently obligated. Player s choices in the equilibrium of recurring rotation depend only on who is currently obligated and on current cost realization. The only aspect of the history that is used is the part that is commonly known (who is obligated). There are other aspects of the history that are not used. For instance, a player could choose whether to offer based on both their current and their past cost realizations. Equilibria in which players use only publicly known aspects of the history to determine their actions are called perfect public equilibria (PPE) (Fudenberg, Levine and Maskin, 994). There is a rather elegant link between PPE and mechanism design. Generally, in one-shot mechanism design problems, players are incentivized to give up their private information through transfers. For instance, the transfers involved in an auction are the payments made to the auctioneer. In a PPE, various states of the equilibrium provide different amounts of expected utility to players. For instance, in recurring rotation, it is less desirable to be obligated than non-obligated. Because of this, a transition from one state to another in a PPE is a transfer - not of money but rather of continuation utility. Because of this, in a well-constructed PPE, players can be incentivized to reveal their private information to provide desirable outcomes just as in a one-shot mechanism that uses money. Because of this, a PPE may be considered a type of dynamic mechanism where incentives for revealing information are provided by continuation utility transfers rather than money (Miller, 202). There are many elegant results in one-shot mechanism design about what is possible and impossible under various constraints on the mechanism. These constraints include the robustness of players incentives to tell the truth, the feasible transfers that can be made, and the strength of players incentives to actually participate in the mechanism. For instance, a classic result by Myerson and Satterthwaite (983) implies that, for the trade of a single good between a buyer and seller (with private values), any efficient mechanism (all efficient trades are made) in which transfers balance cannot avoid the possibility of a player trading at a loss. In this case, the losing party would prefer not to have engaged in the mechanism at all. On the other hand, efficiency is possible when relaxing this participation constraint. In fact, there is a generalized mechanism known to provide efficiency under these conditions (d Aspremont and Gérard-Varet, 979). These possibility results of one-shot mechanism design have analogs in the dynamic mechanism literature. For instance, first-best efficiency is achievable by PPE in some settings (Fudenberg, Levine and 3
Maskin, 994; Athey and Bagwell, 200). However, in the interpersonal relationships considered in this paper, there is no authority to structure communication, prevent players from learning about each other s private information, and absorb budget imbalance. This provides constraints on the kind of dynamic mechanisms that are suitable, and first-best efficiency is not achievable by any of these suitable mechanisms. This is because these restrictions require a type of robust incentives known as ex-post incentive compatibility. One-shot mechanisms in which each player s incentive to tell the truth about their private information relies on their beliefs are said to be Bayesian incentive compatible. The incentives in these mechanisms require simultaneous communication to maintain strict control of player s beliefs about other s private information. For instance, the classic first-price auction relies on this type of incentives. The allocation of the good and the payment conditional on winning are both based on a player s bid. If a player knew, or at least believed, that others had submitted very low bids, that player would also have incentive to submit a low bid, just slightly higher than the others. The desirable outcome that the player with highest valuation receives the good in a first-price auction is not guaranteed if each player s beliefs cannot be controlled. Controlling these beliefs is the purpose of communication restrictions such as the sealed-bid. On the other hand, the second price auction demonstrates a stronger kind of incentives. In the second price auction, the transfers are partially independent of a players own bid. Because of this, players have incentive to bid exactly their values regardless of their beliefs about others bids. Bidding any higher can lead to winning the item, but at an unacceptable price. Bidding any lower can lead to losing the item in instances where the price would have been acceptable. This is known as dominant strategy incentive compatibility. Telling the truth is a weakly dominant strategy for each player. The fact that actions are independent of beliefs under this stronger incentive compatibility requirement means that the mechanisms are robust to communication structure and differences in interim beliefs. Because of this, dominant strategy incentive compatibility (or ex-post incentive compatibility in the case of interdependent values) serves as the appropriate incentive constraint in one-shot settings with restrictions on the ability to implement communication rules and prevent changes in beliefs (Chung and Ely, 2002). Hurwicz (975); Green and Laffont (979); Hurwicz and Walker (990) provide relevant results demonstrating the generic incompatabilty of first-best outcomes, dominant strategy incentive compatibility and a budget that must be balanced ex-post (for any realization of the action profile). Miller (202) extends ex-post incentive compatibility to the repeated environment, introducing the a subclass of PPE known as Ex Post Perfect Public Equilibrium (EPPPE). EPPPE, unlike the more general PPE, use ex-post incentive compatible mechanisms at each stage of a dynamic game. In recurring rotation, a player s incentive to report their cost truthfully subject to the threshold θ is weakly dominant, holding beliefs about future strategies fixed. Because of this, recurring rotation is an EPPPE. Further, (Miller, 202) provides an impossibility for this more restricted class of mechanisms. In an economically interesting class of environments, including the one studied here, first-best is not achievable by an EPPPE even with money transfers (under a no-subsidy condition on the ex-post budget) 5! This means that, even if money transfers are allowed, but no subsidy can be provided for the players, first-best is not achievable by a robust mechanism in this environment. This leaves open the question of what is achievable by an EPPPE either with or without money transfers. An EPPPE without money transfers uses instead transfers of utility within the repeated game to provide incentives. In an equilibrium, these transfers must come from within a self-generating set of consistent continuation values (Abreu, Pearce and Stacchetti, 986, 990). In more natural terms, the transfers of utility correspond to promises about future play, and these promises about the future are only valuable if they are believable - or, rather, consistent. Because of this, the values of transfers that can be made in a 5 Athey and Miller (2007) consider the repeated trade setting under weaker budget assumptions where first-best can sometimes be achieved, especially by patient players. 4
dynamic mechanism without money are limited. On the other hand, monetary transfers are not restricted in this way. Barring budget issues, players are free to transfer any amount. Consistency is not an issue. Because of this, the set of feasible transfers using money is at least as large as the set of feasible transfers in a dynamic mechanism without money. However, in a repeated setting where money transfers are allowed, all of the incentives can be handled by money without resorting to more complicated utility transfers. Because of this, the efficiency achievable by a one-shot ex-post incentive compatible mechanism with transfers under the no-subsidy condition is an upper-bound on achievable efficiency in each round of an EPPPE without money transfers since the one-shot mechanism can simply be repeated in each round Miller (202). This result simplifies the problem of characterizing optimal EPPPE to the problem of characterizing optimal one-shot ex-post incentive compatible mechanisms. Hagerty and Rogerson (987) demonstrate that in the static trade setting, a posted price mechanism must be optimal under ex-post incentive compatibility, ex-post individual rationality (participation constraint), and strong ex-post budget balance (transfers must balance exactly). However, the optimal ex-post incentive compatible mechanism in one-shot settings under the less restrictive no-subsidy condition (balance of transfers must be non-positive) is not a well-studied problem due to the overwhelming variety of available mechanisms. However, a recent paper by Shao and Zhou (203) provides a useful result for the one-shot allocation of a valuable good between two players with value uniformly distributed over the unit interval. They prove that the upper-bound on total welfare in that environment is 5 8. Compare this to, for instance, flipping a coin to determine allocation in which case the total welfare is 2 - the average of the individual values. The allocation problem may be thought of as the negative reflection of the assignment problem considered in this paper. Between two people, the problem of assigning a costly duty to do the task is the same as the problem of allocating the valuable right not to do the task. This implies that second-best optimal average stage cost in an EPPPE with monetary transfers is 8 3. This can be used as a lower-bound on the achievable average stage costs of an EPPPE without monetary transfers. However, it is also possible to directly approximate a bound in this case for patient players using the separating hyperplane methods of Fudenberg, Levine and Maskin (994). This procedure is also discussed in Miller (202, p. 792). This numerical exercise also yields a bound on the average stage cost of 3 8, and suggests that eschewing monetary transfers is not restrictive in this environment for patient players. In fact, in section 4, I discuss a mechanism that approximates this efficiency, and comes very close even for impatient players. This analysis provides an appropriate benchmark for comparing the recurring rotation mechanism to other suitable mechanisms in the uniform cost environment. Recurring rotation is plotted again on the line segment below with the addition of the lower bound on costs achievable by an EPPPE. 3 3 8 First Best EPPPE Lower Bound 0.4 Recurring Rotation 2 Coin Flip Figure 2.7: Efficiency Comparison 5
The recurring rotation mechanism achieves about 4 3 of the efficiency of an optimal ex-post6 incentive compatible mechanism with subsidy-free transfers. The efficiency lost over second-best comes from the fact that the threshold is too low. For comparison, if players used thresholds of 2, they would achieve second-best average stage cost of 3 8. However, the incentives provided by recurring rotation are smaller than necessary to get players to implement this optimal threshold. This results in an inefficient sorting of the players costs. This inefficiency is not a problem limited to the uniform case. In fact, the efficiency of recurring rotation is always bounded below second-best optimal. The following two lemmas are useful for demonstrating this. Lemma 2.9. Any threshold θ may be implemented by an ex-post incentive compatible mechanism using budget balanced money transfers. Proof. Consider a threshold mechanism using transfers of θ 2. The player completing the task will be paid θ 2 by the other. Both players report whether they are above or below the threshold. If both are above or both are below, one is chosen at random to execute the task. Otherwise, the player reporting a low cost completes the task. Note that a player prefers to complete the task and receive a payment of θ 2 rather than pay θ 2, as long as that player has a cost below θ. Otherwise, the player prefers not to complete the task and pay θ 2. Regardless of the other player s choice, offering to do the task maximizes a player s probability of doing the task. Not-offering maximizes the probability of not doing the task. Thus, a player with θ i θ has a weakly dominant strategy of offering to do the task while a player with θ i θ has a weakly dominant strategy of not-offering. Because of this, reporting costs according to the threshold is dominant strategy (ex-post) incentive compatible. Lemma 2.0. For any F (), the threshold mechanism that maximizes welfare (minimizes average stage cost) uses a threshold equal to the mean type. Proof. By Lemma 2.9, any threshold mechanism can be implemented by an ex-post incentive compatible mechanism with budget-balanced transfers. These transfers are welfare neutral, and so the mechanism s welfare is given by its average stage cost. The average stage cost under a threshold mechanism is given by: It s derivative is: Since δ AC( θ) δ θ AC ( θ ) = F ( θ ) E ( θ i θ i θ ) + ( F ( θ )) E (θ i ) (2.0) δac ( θ ) δ θ = f ( θ ) θ f ( θ ) E (θ i ) (2.) is negative below the mean and positive above the mean, reaches its global minimum at θ = E (θ i ). AC ( θ ) is quasi-convex and Proposition 2.. For distributions with positive density everywhere, the average stage cost of ( recurring ) rotation is bounded above second best optimal. The stage average welfare gap is at least AC E(θi ) 2 AC (E (θ i )). 6 Since ex-post and dominant strategy incentive compatibility are equivalent for the private valuation environment, they are used interchangeably in this paper. 6
Proof. By lemmas Lemma 2.9 and Lemma 2.0, the optimal threshold mechanism incurring average stage cost AC (E (θ i )) can be implemented by a robust mechanism. Thus, the second-best optimal average stage cost must be at least as small as AC (E (θ i )). By proposition 2.4, the equilibrium threshold in recurring rotation is always below half of the mean type. The average stage cost associated with using a threshold θ is AC ( θ ) = E (θ i ) F ( θ ) ( E (θ i ) E ( θ i θ i θ )). This is strictly ) decreasing over [0, E (θ i )] and so the average cost of recurring rotation must be larger than AC. ( E(θi ) 2 Together, this implies that the gap in welfare (in terms of ) average stage cost) between recurring rotation and a second-best optimal mechanism is at least AC AC (E (θ i )) > 0. ( E(θi ) 2 Using the result above for the uniform distribution predicts a gap of at least 32.0278. In fact, the gap for the option version of the mechanism as β is approximately.4.375 =.036 since 8 3 =.375 is known to be the second-best optimal cost. I do not claim that the mean-threshold mechanism provides a tight bound on the welfare achievable by robust mechanisms for any distribution except uniform. In fact, (Miller, 202, example 4) provides a counter-example under which the mean-threshold mechanism is not optimal under a particular asymmetric distribution where density is 2 5 for θ i 2 and 8 5 for θ i > 2. However, several results suggest that the best threshold mechanism may still be a good benchmark. General results showing that threshold mechanisms are optimal among the class of deterministic mechanisms for type distributions with monotonic hazard rate are given in Drexl and Kleiner (202); Shao and Zhou (203). In addition, Hagerty and Rogerson (987) prove that a threshold, specifically a posted price mechanism, is optimal in a trade setting under dominant strategy incentive compatibility, ex-post individual rationality and strong ex-post budget balance. Further, in Athey and Miller (2007) the authors suggest that in numerical tests where the threshold mechanism was not optimal, the computed optimal mechanism improved efficiency very little. Using the best threshold rather than the true optimal for comparison has the benefit that the best threshold mechanism is well defined by Lemma 2.9 and Lemma 2.0 as the mean threshold mechanism and has an average stage cost that is easy to calculate. This provides the flexibility to make efficiency comparisons in environments with cost distributions that are not uniform. Despite the fact that recurring rotation is suboptimal relative to the best threshold mechanism, in appendix section 6.4 I demonstrate that there are some environments where the welfare loss is small. To compliment these results, in the next section I present results from an experiment designed to test the empirical properties of recurring rotation. 3 Experiment Experimental methods are ideal for studying the empirical properties of mechanisms. Since mechanism design problems are characterized by private information, traditional empirical analysis may be difficult. Determining the efficiency of a mechanism is impossible without observing private information. Yet, it may be hard to find environments where information is private between players, but observable by the researcher. However, in the laboratory, we can assign private information. The experiment put subjects in the scenario of playing an abstracted version of the repeated task game with uniformly distributed costs. The subjects used the deferment version of the recurring rotation mechanism to determine assignments. On average, subjects chose threshold strategies higher than the predicted 7
equilibrium threshold. Further, many subjects choose higher thresholds in the obligated state than the non-obligated state ( gap strategies). Using subject data and theoretical results from the previous section, I estimate the long-run efficiency of recurring rotation under the empirical distribution of play. Despite the variance in subject behavior, efficiency is close to what is expected in equilibrium. However, the use of gap strategies had a substantial impact on the efficiency of the mechanism relative to what would have been achieved if subjects played non-gap strategies equal to the average of what they chose in the obligated and non-obligated states. Further, the gaps remain in a treatment of the experiment designed to control strategic and pro-social variation by pairing subjects with computer partners. This suggests that the gaps may be due to heuristics that may have been developed under a turn-taking arrangement with slightly different structure. 3. Design Details Data was collected at the University of California, Santa Barbara Experimental and Behavior Economics Laboratory using ZTREE (Fischbacher, 2007). The Online Recruitment System for Economic Experiments was used to recruit subjects (Greiner, 2004). A total of 99 subjects participated in the experiment, including pilot sessions. Each session of the experiment lasted about 50 minutes. No subjects participated in more than one session. The experiment consisted of two key treatments. In the first, subjects were partnered with other subjects in their session. In the second, subjects were paired with a computer playing a known strategy. The computer offered to do the task with probability 2 in both states. In both treatments, the environment was the same and the subjects were aware of all of the details of the game. In the person/person treatment there were four sessions. Three session used 6 subjects and one session used 8 subjects for a total of 66. Subjects were paired randomly with another subject from the session. One subject from each pair was randomly selected to start as the obligated partner. For the first 20 rounds, players found out their random cost, which was distributed uniformly between $0 and $3 in penny increments, before choosing an action. The obligated player chose whether to ask his/her partner to do the task. The non-obligated chose whether to agree to do the task should his/her partner ask. The obligated partner carried out the task and became non-obligated on the next round unless the obligated asked and the non-obligated agreed. After round 20, subjects instead submitted a threshold strategy before learning about their private costs - the obligated choosing above what cost to ask and the non-obligated below what cost to agree. The language used in the experiment is slightly different than the way the actions are described in theory section. The obligated choose whether to ask the non-obligated to do the task and the non-obligated chose whether to agree conditional on being asked. In a pilot study, this language made the mechanism more transparent to subjects over having each player offer to do the task (or not) regardless of state as in theory section. An obligated partner asking the non-obligated to do the task is equivalent to not offering to do the task in theoretical game. Similarly, the non-obligated partner agreeing to do the task is equivalent to offering in theoretical game. Below, I refer to the situation where the obligated asks the non-obligated to complete the task as the obligated partner asking for a swap. After each round, subjects found out whether the obligated had asked and whether the non-obligated agreed or would have agreed (if the obligated did not ask). In each case, they also found out how much they paid in the round, if anything, and who would be obligated on the next round. Cumulative payoffs were not shown throughout the experiment, though no effort was made to prevent subjects from recording data with pen and paper (this was rare). The probability of continuing with a partner in each round was 0 9 to generate an effective discount rate 8
of β =.9. At the end of each round, players found out whether they would continue with the same partner ( 9 0 probability ) ( ) or whether they had been paired with a new partner 0 probability for the next round. Whenever a new partnership was formed, the software randomly chose one partner to be obligated. After 48 rounds (the subjects did not know this cutoff, but rather that there was some specified cutoff), the game ended on the next partnership termination. 48 was chosen based on the average play time of groups in pilot sessions. Although 58 is the expected number of rounds, actual average duration, in rounds, of the sessions were shorter. The sessions lasted 49, 5, 52, 58 rounds respectively. Subjects were paid for every round of the game. This scheme was chosen for simplicity and to induce far-sighted behavior. The rationale for this choice is based on the experimental comparison of repeated game payment schemes in Sherstyuk, Tarui and Saijo (203). Subjects started the experiment with $5 and earned $3.20 every 4 rounds. Subjects were reminded after each round that they would receive $3.20 every 4 rounds but did not know explicitly when this occurred. This was chosen to avoid focal-points and strategy distortion on the paying period. Pilot-study data and theoretical efficiencies of the recurring rotation mechanism were used to select these payment parameters along with the range of the cost distribution to target an average subject payment of $5 for an hour experiment while providing as much curvature on the payoff function as possible without risking many subjects ending up below the $5 lab minimum guaranteed payment. Subjects that suspected they might be below the minimum could face distorted incentives. Under these parameters, for 58 rounds, rigid turn-taking would result in an average payoff of $7.90 for the experiment while second-best play would result in an average of $8.78 and equilibrium play in roughly $5.2. The parameter choice was relatively accurate to the intended targets. Under these parameters, only 2 subjects ended up below $5 in the person/person treatment. Both earned about $3.50 in the experiment (and had payment rounded to $5). At this level, even if the two subjects had been aware their cumulative earnings were below the $5 limit, the round incentives were still meaningful since moving above the threshold might have taken only a few rounds. On average, players in the person/person experiment earned just over $5. In the computer treatment, all parameters were the same. However, instead of being paired with a new subject after a partnership termination, the partnership was instead restarted. Each time this happened, the software chose randomly whether the subject or the computer would start as the obligated partner. Between the person/person and the person/computer treatments, partnership lengths were matched within sessions. That is, session one of the person/person treatment had the same partnership lengths and ultimate session length as session one of the person/computer treatment. Again, the sessions lasted 49, 5, 52, 58 rounds respectively. A total of 63 subjects participated over 4 sessions. Three sessions included 6 subjects and one session included 5. Earnings in the computer treatment were higher with average of about $20 per subject. This is not surprising. The computer s strategy of offering half of the time is quite favorable to the subjects. 3.2 Results: Person/Person Experiment This section presents an analysis of play over rounds 2 48 7. On average, selected thresholds were larger than those expected in a symmetric Markov equilibrium. Over all four sessions, the average chosen threshold was $0.99 while equilibrium prediction is $0.64. There was a great deal of variance in threshold choice. 7 Recall that in rounds -20, subjects chose an action rather than a threshold strategy. Although all sessions went at least until round 49, it is useful to split up the rounds into several large chunks to look for learning/convergence. Restricting the sample to rounds 2 49 provides a prime number of rounds. While the rounds 2 48 can be split into 4 blocks of 7 rounds. 48 was also the last round that was guaranteed to be played in a session, although the participants did not know this. 9
Appendix figure 6.2 displays a histogram of average thresholds by subject in total and split by obligated and non-obligated states. Average play did not vary much over these rounds, but there was a striking tendency for players to choose different thresholds in the obligated and non-obligated state. Appendix figure 6. provides CDF of average thresholds over 7-round blocks by subject and split by obligated and non-obligated states. Changes in play appear to have settled down by round 35. More formal testing confirms this pattern and is provided in appendix section 6.6. This motivates focusing on rounds 35 48 for calculating steady-state properties of the mechanism. Appendix figure 6.3 shows average threshold histograms, by subject, restricted to rounds 35 48. Over these rounds, the average threshold conditional on being obligated was $., and $0.78 when non-obligated. From a strategic standpoint, one of the most obvious questions to ask about subject behavior is how it compares to best-response. Due to the variance in subject behavior, a full best-response analysis would be difficult. However, the following thought experiment provides a rough idea of how strategies correspond to best response on the aggregate level. Two subjects are partnered who both play strategies consistent with the average choices of subjects in the experiment. The threshold they choose when obligated is $. and $0.78 when non-obligated. By simulating their partnership it is possible to approximate the difference in continuation value of being obligated and non-obligated. This discounted difference is roughly $0.63. Since the best response, regardless of state, is to set a threshold equal to this discounted difference, this suggests that, when two average subjects are paired, they offer swaps too often when non-obligated and, to a greater extent, reject swaps too often with obligated. From an aggregate perspective, it appears subjects biggest mistake relative to best-response was not asking for deferment often enough when obligated. The gap in thresholds appears to be an important part of subject behavior. However, this aggregate analysis does not provide insight into the extent that gap strategies were pervasive on a subject level. In fact, gaps are pervasive. I use the following model to test the significance of the threshold gap between states for each subject. A player i s threshold θ i,t in round t is estimated by an individual constant α i plus some individual-specific change δ i when the player is obligated (o i,t = when subject i is obligated in period t). θ i,t = α i + δ i o i,t + ɛ i,t (3.) 59% of subjects have a significantly positive difference between obligated and non-obligated threshold at the 5% level (against one-sided alternative), 9% have a significantly negative difference and 32% have an insignificant difference. This result suggests that subjects were biased in favor of playing larger thresholds when obligated. The next section presents an analysis of how these patterns in subject behavior would affect the implied empirical efficiency of recurring rotation. 3.3 Efficiency Estimation The previous section suggests that players choices in the experiment were not strongly consistent with equilibrium predictions. On average, subjects chose thresholds higher than equilibrium. Important as well is the variance in choices between subjects and the tendency for subjects to choose gap strategies. These results suggest the calculated equilibrium efficiency of recurring rotation might not correspond to the implied efficiency under subjects actual behavior. In this section, I attempt to estimate the empirical efficiency of recurring rotation using subject behavior. To do this, it is first necessary to determine the efficiency two players achieve under disequilibrium strategies. To begin, and although this is not consistent with subject behavior, consider strategies that are the same in both states. 20
Equation 2.0 gives the average stage cost associated with symmetric strategies. If strategies are allowed to be asymmetric between players (θ i is player i s strategy), the equation for average cost in the state that i is obligated is given by: AC ( θ, i ) = F ( ) ( ) ( ) [ ( )] ( ) θ i E θi θ i θ i + F θ j F θ i E θj θ j θ j + [ F ( )] [ ( )] ( ) θ j F θ i E θi θ i θ i (3.2) Since the average stage cost depends on the state (who is obligated), the joint welfare in the long-run depends on the how often each state occurs. This was not the case when both players use the same strategy since the average stage cost is the same in both states and, in the long-run, each state is equally likely. The joint welfare of disequilibrium play with asymmetric strategies can be calculated by weighting the average cost in each state by the long-run visiting probabilities of the states. The state transition matrix is given below. Entry i, j gives the probability of moving from the state where i is obligated to the state where j is obligated. [ ( F ( θ )) F ( θ 2 ) β ( F ( )) ( ) θ F θ 2 ( F ( )) ( ) ( ( )) ( ) θ 2 F θ F θ 2 F θ [ α α For a 2-state Markov chain with transition matrix β β α+β and π 2 = is: ] (3.3) ] the steady-state distribution is π = α+β α. Thus, the probability player is obligated in the long-run under asymmetric strategies π ( θ ) = The long-run average stage cost under strategy profile θ is given by: F ( θ ) F ( θ 2 ) F ( θ ) + 2F ( θ ) F ( θ 2 ) F ( θ ) F ( θ 2 ) + 2 (3.4) π ( θ ) AC ( θ, ) + π 2 ( θ ) AC ( θ, 2 ) (3.5) A contour plot of equation (3.5) is given in figure 3.. Contour lines are drawn at.46 (equilibrium efficiency for β =.9),.4,.39, and.38. Notice that much of the joint strategy space yields efficiency better than equilibrium. 2
Figure 3.: Disequilibrium Efficiency for Recurring Rotation (Deferment) Suppose players in a population are randomly paired and play strategies that are the same in both states with a distribution corresponding to the distribution of average thresholds (normalized to (0, )) chosen by subjects in the experiment. The average efficiency achieved by players in this population can be estimated by averaging over the efficiency achieved by every possible pair (with replacement) of subjects using the value of equation 3.2 and each pair s average thresholds. This procedure provides an estimate of an average cost of.399, a substantial improvement over predicted equilibrium average cost of.46. This is because, as discussed above, the threshold choice of players tended to be larger than equilibrium prediction. Figure 3. demonstrates visually why such a scenario leads to improved efficiency - most pairs tend to fall in the darker blue area of the figure. However, this calculation is made under the assumption that players use strategies that are symmetric in the states. Given that most play involved a gap between obligated and non-obligated thresholds, calculating the average population efficiency under this assumption may provide a biased estimate of the implied empirical efficiency. In order to account for different strategies in each state, it is necessary to use a modified version of the equations above. Letting θ i,o be player i s threshold strategy when player o is obligated, the modified state transition matrix is given in below: [ ( ( )) ( ) F θ, F θ 2, ( F ( )) ( ) ] θ, F θ 2, ( F ( )) ( ) ( ( )) ( ) (3.6) θ 2,2 F θ,2 F θ 2,2 F θ,2 The probability player is obligated in the long-run is given by: π ( θ ) = F ( θ,2 ) F ( θ 2,2 ) F ( θ,2 ) + 2 + F ( θ, ) F ( θ 2, ) F ( θ 2, ) + F ( θ,2 ) F ( θ 2,2 ) F ( θ,2 ) (3.7) 22
The average stage cost under gap strategies θ is then given by: π ( θ ) AC ( θ, ) + π 2 ( θ ) AC ( θ, 2 ) (3.8) In the equation above, note that it is necessary to account for the gap strategies when calculating the average cost terms AC ( θ, ) and AC ( θ, 2 ) by using the appropriate thresholds. To determine the distribution of θ i,o, players strategies must be estimated separately for the obligated and non-obligated states. Using this distribution, I apply the same procedure of averaging over all possible pairs under the modified average cost equation. The resulting average cost estimate is.42. This is higher than the previous estimate and closer to the equilibrium prediction.46. This also corresponds closely to the actual achieved average cost for subjects in the experiment. Over rounds 35 48 the partner who ended up completing the task in each pair paid an average of $.26 or.42 when normalized to [0, ]. For comparison, the difference between an average cost of.399 and.42 corresponds to an average of about $0.06 under the parameters faced by subjects. Over 50 rounds this corresponds to about $3 per pair left on the table due to the use of gap strategies over the hypothetical strategies created by averaging over the states. These estimates suggest the gap strategies are not just an interesting and unexpected phenomenon relative to theoretical predictions, but also an important factor in the implied empirical efficiency achieved by recurring rotation, and potentially other flexible turn-taking type arrangements. What assumptions in theory could account for the failure to capture these gaps? Perhaps subjects are responding to incentives that extend beyond the monetary incentives of the game. They may be pro-social. Perhaps the gaps are a strategic response to subject s beliefs about their opponents strategies. Perhaps they are non-markov. I treat these two possibilities by analyzing data from this treatment in appendix section 6.7. The evidence for either explanation is limited. However, a full analysis of these aspects could be very complex, and this treatment is not designed to test these possibilities directly. On the other hand, the computer partner treatment controls for both pro-social and strategic aspects of the game since the computer s strategy is known, and explicitly Markov. The fact that gap strategies are prevalent in the computer treatment suggest that gaps are not likely to be due to pro-social behavior or strategic response. What s left over? Perhaps the gap strategies are due to heuristics used to deal with the rather complex decision problem underlying this game. 3.4 Results: Person/Computer Treatment In this treatment, subjects faced a much more straightforward decision problem since belief about their partner s strategy was fixed. Subject were paired with a computer known to offer to do the task with probability 2 in both states. From a strategic point of view, this is equivalent to having the computer use a threshold of $.50, or 2 when normalized to [0, ]. The histograms in appendix figure 6.4 give player s choices for rounds 2 48. Appendix figure 6. displays cumulative distribution plots of average thresholds by subject separated by obligated and non-obligated states and split over the 7 round groups. As in the person/person treatment, changes in play appear to have settled down by round 35. A formal test of this is provided in appendix section 6.6. As in the person/person treatment, the results suggest focusing on rounds 35 48 for stead-state results. Average subject threshold histograms for these rounds are plotted in appendix figure 6.5. 23
3.4. Best Response Analysis As in the person/person treatment, subjects set average thresholds higher than predicted by best-response. The best response of player can be derived from a modification of equation 2.2 allowing different threshold values for two players θ and θ 2. This is given below: β θ = F ( ) ( ) [ ( ) ] ( ( )) ( ( )) [ ( ) ] θ F θ 2 E θ θ θ θ + F θ F θ 2 E θ θ θ θ (3.9) When normalized to the uniform [0, ] environment, since the computer implements threshold θ 2 = 2 and the discount rate is.9, the best response for player is the solution to the following quadratic equation: [ θ = 9 ( ) 2 θ θ ] 2 (3.0) 0 4 4 The solution is 58 9.56 in the normalized game or about $0.47 for the parameters faced by subjects. For rounds 35 48 the average threshold chosen by subjects in the obligated state was $0.83 and $0.66 in the non-obligated state. Both are significantly different from the best response at better than the % level. From these averages, it is apparent that a gap remains in average threshold choices between the states on an aggregate level. 3.4.2 Gap Strategies To check that the gap strategies remain an individual phenomenon and not just an average phenomenon, I estimate each subjects difference in threshold when obligated and non-obligated for rounds 35 48 in the same way as in the person/person treatment. 49% of subjects have a significantly positive difference at the 5% level (against one-sided alternative), 3% have a significantly negative difference and 48% have an insignificant difference. Slightly fewer subjects show a significantly positive gap than in the person/person treatment where the figure was 59%. However, this still dwarfs the percentage with a significantly negative gap. The persistence of these gaps, even against a computer that has a known strategy, suggests they might be due to complexities in the underlying decision problem facing subjects in the recurring rotation game. Although it is not clear precisely why gap strategies are prevalent, one explanation is that the experience subjects are drawing on to make choices in the lab comes from situations in which gaps are a more natural part of the arrangement. For instance, in the mechanism described in the next section, gaps appear as part of equilibria. It may be that recurring rotation, as a model of flexible turn-taking, is exceptional in terms of this feature relative to the kinds of arrangements people are used to in these settings. Further study into the precise arrangements people use to solve these problems may provide a deeper insight into this pattern. 4 Obligation Takeover Mechanism In recurring rotation, there are only two states. The worst-off a partner can be is to be obligated. Recurring rotation does not allow a partner to acquire debt. This simplicity is why it provides a natural starting point for modeling flexible turn-taking. Yet, people are not averse to using informal and non-monetary debts. The prevalence of the idiom I owe you one is evidence of this. The idea that a partner would acquire multiple periods of obligation in a row by deferring their duty is not at all foreign. In this section I 24
present a mechanism, obligation takeover, which allows this. In this mechanism, unlike in recurring rotation, players can accrue multiple periods of obligation in a row. This simplicity of recurring rotation comes at the cost of efficiency, recall that by proposition 2., there is always an efficiency gap to second-best optimal. The inefficiencies of recurring rotation result from the fact that the tradeoff between being obligated and non-obligated is not large enough to get players to sort their types in an efficient way. Obligation takeover provides more favorable incentives for sorting efficiently. This section starts with a theoretical motivation to provide an intuition for why obligation takeover can achieve these improved incentives. I then formalize the mechanism and derive equilibrium conditions. Equilibria of this mechanism are more complicated than in recurring rotation due to the larger number of states (who is obligated and how many times in a row). Because of this, I use recursive methods to solve equilibria numerically. The computational results suggest that as players become patient, the efficiency of obligation takeover approaches second best for uniformly distributed costs and approach best threshold efficiency for symmetric beta distributions. Further, the mechanism performs very well even for moderate patience levels. Further, gaps in obligated and non-obligated strategies naturally appear in the equilibria of obligation takeover. If gaps in the strategies of subjects in the experiment are due to heuristics built from their experience with flexible turn-taking, it may be that the kind of flexible turn-taking they are used to has a similar structure to obligation takeover. 4. Theoretical Intuition for Improved Efficiency The following provides intuition for why obligation takeover can achieve improved efficiency. Suppose in the uniform cost environment, players use the following mechanism with money transfers. One player is obligated. If the non-obligated executes the task, that player will be paid 2 by the obligated. Otherwise no transfer is made. To execute the mechanism, both say whether their costs are above or below 2. If one is below and the other above, the player with the low cost executes the task. If both are below or both are above, the obligated player executes the task. In this mechanism, truth-telling is a dominant strategy. Notice that an obligated player with a cost of 2 is indifferent between doing the task and paying 2 not to do the task. Similarly a non-obligated player with a cost of 2 is indifferent between doing the task and receiving a transfer of 2 while not doing the task. A player with a cost below 2 strictly prefers to execute the task while a player with cost above 2 strictly prefers to not execute the task under these transfer rules. Telling the truth maximizes a player s chance of getting their preferred outcome regardless of their belief about the other s strategy. The ex-ante net expected cost for an obligated player is 6 7 while the net expected gain for the nonobligated is 6. The difference in value of being obligated and non-obligated is thus precisely 2. Suppose players face this problem repeatedly and use this mechanism each time. They start with some plan of obligation. Notice that a patient player is nearly indifferent between a money transfer of 2 and an extra round of non-obligation at some point in the near future. That is, if the obligated player takes over one of the non-obligated player s future obligations, the 2 money transfer is unnecessary. In fact, this works for any symmetric distribution. Claim 4.. Under a mechanism where one partner is obligated and must complete the task unless a transfer of E (θ i ) is paid to the non-obligated, the difference in stage value of being non-obligated and obligated is E (θ i ). Proof in Appendix Section 6.8 25
Claim 4. provides the intuition that mechanisms using the take over of obligations to incentivize swaps can achieve nearly the same efficiency as the best threshold mechanism (for symmetric distributions). I formally construct the obligation takeover mechanism in the following sections. 4.2 Obligation Takeover Mechanism Assume the players start with a plan to alternate who is obligated. Let p t i=0 {, 2} be a vector that is the plan of obligation. At time t, the player identified by the first element of the vector p t is obligated. Letting p t + indicate pt with the first element of the vector truncated, then when the obligated completes the task, the plan continues: p t+ = p t +. If, on the other hand, the non-obligated completes the task, the plan proceeds but with an additional period of obligation added at the nearest position for the obligated player. Letting a ( p t) be the operation that adds one period of obligation, if the non-obligated individual completes the task, p t+ = a ( p t +). For instance if the plan is alternating so that p t =, 2,, 2,, 2,, 2... and the obligated individual completes the task, then p t+ = 2,, 2,, 2,, 2,... On the other hand, if the non-obligated completes the task then p t+ =,, 2,, 2,, 2,... In this example the difference between the two resulting plans is precisely one extra period of obligation for player. Under this arrangement, p t will always include some repetition of either or 2 followed by alternation. The state of the mechanism can thus be represented by a single variable z Z/0, which represents the number of repetitions of obligation and sgn (z) represents for whom that repetition pertains. For instance, p t =,,,, 2,, 2,... can be represented by state variable z = 4. While p t = 2, 2, 2,, 2,, 2... can be represented by z = 3. In a sense, z represents the debt owed by one partner to the other. Assuming a positive z represents extra obligation for, then it is the case that player will have to complete z more executions of the task than player 2 before 2 can even begin to accrue debt. Further, because a state is characterized by a repetition and an alternation, the plan can be split in two parts. The near-term repetition of obligation returns the players to an equal position, making up for recent imbalance in the number of executions of the task. In fact, not only does the absolute value of the state represent the number of near-term repetitions of obligation and the debt owed, it also represents the number of extra executions carried out by the non-obligated player over the entire history of play! Because of this, the mechanism has an elegant way of bringing about fairness in history of task-execution. For computational reasons, I will assume that there is a limit on the number of obligations a player can accrue so that z z. In a state where z = z, the player currently obligated is forced to complete the task without the possible intervention of the non-obligated 8. When this limit is imposed, the mechanism is similar to the chips mechanism, in the integer accounting of debt and obligation, as used by Mobius (200) in the favor trading environment. In the next section, I derive equations for calculating equilibria in the obligation takeover mechanism. 4.2. Equilibria - Obligation Takeover Mechanism Assume available actions are to offer and not offer to do the task and that players implement symmetric Markov strategies. Let V i (z) be the discounted ex-ante utility for player i in state z. For any V function, there is a unique Markov strategy that is weakly dominant with respect to stage beliefs in each round. As 8 In equilibrium, the effect of the limit on efficiency is practically inconsequential as long as the limit is not very small. This is because the equilibrium strategies of the players make the probability of visiting larger states exponentially rare. A computational analysis of efficiency with limited states is provided in appendix section 6.0. 26
in recurring rotation, there are only two alternatives in each round. If the obligated completes the task, the state decreases. If the non-obligated completes the task, the state increases. Consider player in a state z =. Regardless of whether player is obligated, completing the task provides utility θ i + βv (z ). When not completing that task, player s utility is βv (z + ). Player prefers to complete the task if and only if θ < β (V (z ) V (z + )) and prefers not to complete the task otherwise. Thus, β (V (z ) V (z + )) is the cost threshold that determines player s preference over completing the task in state z. This threshold will be written θ (z). z = : θ (z) = β (V (z ) V (z + )) (4.) For z =, a slight modification is needed since there is no state 0. z = : θ () = β (V ( ) V (2)) (4.2) z = : θ ( ) = β (V ( 2) V ()) (4.3) Although the thresholds are written for player, by symmetry, player 2 s thresholds are simply the reverse ( θ 2 (z) = θ ( z) ). As in recurring rotation, I focus on symmetric equilibria and drop the player index. For simplicity, the assignment rule is assumed to be identical to the deferment version of the recurring rotation mechanism. It is summarized below: Obligated Action Non-Obligated Action Not Offer Offer Not Offer Obligated Non-Obligated Offer Obligated Obligated Figure 4.: Partner Completing Task by Joint Action Profile (Obligation Takeover) To see that player s actions are weakly dominant with respect to stage beliefs in state z around the threshold θ (z), suppose a player has θ i < θ (z) and so wants to complete the task. Refer again to figure 4.. If the player is obligated, it is weakly dominant to offer since it guarantees completing the task. It is weakly dominant for the non-obligated player to offer since it allows the possibility of completing the task in the case the player s partner does not offer. Now suppose a player has θ i > θ (z) and so wants to not complete the task. If the player is obligated, it is weakly dominant to not offer since it provides the possibility the non-obligated will complete the task instead. If the player is non-obligated, it is weakly dominant to not offer since it excludes any chance of having to complete the task. Regardless of state, a player with cost below [above] that state s threshold has a weakly dominant strategy (with respect to stage beliefs) to offer [not offer]. Thus, when players act according to symmetric Markov strategies, their actions are determined solely by the vector of thresholds: θ (z). However, just as in recurring rotation, the value function depends on the strategies of players. In equilibrium, the vector of threshold strategies must generate discounted state value differences equal to the thresholds. Unlike in recurring rotation, the dimensionality of the problem requires the use of numerical methods in solving for the equilibria. Expressions for the continuation values for each state are derived in appendix section 6.9. It is possible to derive equations that relate only the equilibrium thresholds by taking the difference of the value functions for states that are two-apart. However, these terms are somewhat cumbersome and numerical approximations of equilibria can be accomplished by iteration of the value functions from appendix section 27
6.9 through the following algorithm: Algorithm Calculate Equilibrium Value Function. Initiate a random guess for V 0 (noting V0 2 is the reverse). 2. Calculate the weakly dominant threshold strategies θ 0 (z) and θ0 2 (z) under V0 and V0 2. 3. Calculate V and V 2 using equations 6.44, 6.48, 6.49, 6.50 4. Repeat steps 2 and 3 updating the strategies and value function until the value function converges in the sup. norm returning the resulting approximate equilibrium value function V. Unlike in recurring rotation, a proof that there is a unique equilibrium for this mechanism has been elusive. However, I was never able to find multiple equilibria in numerical experiments using random starting points for the same parameters. Even if there are multiple equilibria, the efficiencies of the equilibria computed below still represent an approximate lower-bound on the achievable equilibrium efficiency of this mechanism. In the next section, I present an example equilibrium to provide insight into the workings of the mechanism, approximate the efficiency achieved by the mechanism for the uniform cost distribution over different discount rates. Appendix section 6.2 contains results on the efficiency achieved by the mechanism for a range of symmetric beta distributions. 4.3 Numerical Example and Efficiency Calculations Obligation takeover achieves very impressive efficiency even for moderately patient players. Under uniform (0, ) costs with β =.9 and a state limit z = 0 (a player can accrue up to 0 periods of obligation) the average cost of execution in the approximated equilibrium is.38, only.006 higher than the cost of an optimal mechanism with money transfers. The details of this equilibrium are plotted below in figure 4.2 9. The plot on the left of figure 4.2 shows the equilibrium thresholds. In the negative states (those in which player 2 is obligated), player s willingness to offer is decreasing in the number of obligations (larger negatives). This is an intuitive outcome since, as obligations accrue, additional obligations are added to the end of the current chain. Since the effect is discounted, the further out the state swaps occur, the less valuable they are to the non-obligated. Once a long chain of obligations has accrued, the efficiency of the stage mechanisms is lower since both implement lower thresholds (see figure 6.6 for the average costs associated with arbitrary thresholds in a single stage). This distortion is clear in the third plot, which adds player and 2 s continuation values in each state and so measures total welfare of each state (since these are costs, higher is better). The effect of distortion weighs more heavily on the obligated player who will more often be stuck executing the task at high costs. Because of this, the obligated is willing to take higher costs to avoid additional obligations than the non-obligated is to accept them. This causes the strategic gaps between the threshold implemented in an obligated state and the threshold in the analogous non-obligated state. The following equation is derived from 4.2 and 4.3 and gives an expression for the gap between obligated and non-obligated thresholds in state : 9 The strategy chosen in states 0 and 0 is plotted at 0, but since the mechanism is completely deterministic in those states, strategy plays no role. 28
θ () θ ( ) = V () + V β ( ) V (2) V ( 2) (4.4) V () + V ( ) and V (2) + V ( 2) represent the total efficiency of the mechanism in state and 2 respectively. The gap between obligated and non-obligated thresholds in state are due to the fact that the mechanism is more efficient in state than in state 2. This extends to the larger states as well. 0.0 0. 0.2 0.3 0.4-3.5-3.0-2.5-2.0 -.5 -.0-0.5 -.4 -.40 -.39 -.38-0 - 0 State -0-0 State -0-0 State Figure 4.2: Equilibrium Thresholds (Left), Value Function (Center), and Efficiency (Right). β =.9, z = 0 4.3. Uniform Distribution, Unlimited States In order to determine the efficiency of the mechanism under fixed discount factor β, but without limit on the states, I utilize the following procedure: for each β, the equilibrium is approximated under a a limited number of states and the efficiency is calculated for the computed equilibrium. After this, the efficiency is again calculated for the computed equilibrium with the number of states doubled. This process is repeated until the difference in efficiency of the two most recently computed equilibria converges to 0. The result is an approximation of the limit of the efficiency of the obligation takeover mechanism under fixed discount factor β with unlimited states 0. The result of this computation is plotted in figure 4.3. It is striking how much efficiency is generated even for relatively low discount factors. Further, for large β, the approximate efficiency is nearly identical to second best. Though these were calculated for unlimited states, computations in appendix section 6.0 suggests the distortion associated with finite limits tends to be small. Little is gained by allowing for more than about 5 obligations regardless of the discount rate. These results suggest that not only are gap strategies, such as those seen in the experimental results, consistent with equilibria in obligation takeover, the mechanism achieves approximately optimal efficiency. 0 Since the uniqueness of these equilibria is not known, the approximation actually represents a lower bound on the approximate achievable efficiency. 29
0.38 0.39 0.40 0.4 0.5 0.6 0.7 0.8 0.9.0 Figure 4.3: Average Cost in β of Obligation Takeover with Unlimited States 5 Discussion 5. Comparison to Existing Research This work is related to several areas of mechanism design. In the area of robust mechanism design with transfers, Drexl and Kleiner (202); Shao and Zhou (203) consider robust allocation of a valuable good in a one-shot environment with transfers, and construct optimal mechanisms. Athey and Miller (2007) consider a repeated trade setting and demonstrate that first-best efficiency can be achieved by robust mechanisms under relaxed budget balance conditions. Miller (202) focuses on robust mechanisms for collusion of two firms in repeated settings with transfers. Like these papers, the mechanisms described here do not require a planner to enforce information structure. However, unlike in these, I explicitly analyze mechanisms where players do not use money transfers. A procedure for constructing the set of achievable payoffs under EPPPE without transfers is discussed in Miller (202, p. 792) and used in section 2.5. However, this exercise provides little insight into the potential structure of such mechanisms. A primary contribution of this paper is to show that substantial efficiency can be achieved by robust and familiar turn-taking mechanisms. Several papers also characterize or explicitly construct mechanisms using only continuation transfers in repeated settings without money transfers, but without focusing on robust mechanisms. Athey and Bagwell (200) consider a repeated Bertrand environment with discrete cost-types and demonstrate that first-best profits can be achieved by impatient firms without money transfers through the use of promises about future market-share. The conclusion of their paper discusses the potential extension to interpersonal relationships that are the focus of this paper. In contrast to Athey and Bagwell (200) however, I focus on a characterizing simple, robust mechanisms, in an environment with continuous type-space. Several papers consider collusion in repeated auctions. Aoyagi (2003) considers repeated auctions with a type-space on the unit interval, and constructs highly efficient collusion mechanisms. However, these mechanisms require a coordinating institution. Skrzypacz and Hopenhayn (2004) also consider a repeated 30
auction environment, but where communication and monitoring are restricted. Although the specifics of the environment are quite different, the mechanism for bid-rotation developed there is similar to the obligation takeover mechanism presented in section 4. Although these papers provide a great deal of insight into the details of using continuation transfers to incentivize mechanisms, the specifics of the environments are quite different from those considered here. The literature on favor-trading also has several features similar to the environment considered here, especially in terms of the interpersonal context. Mobius (200) considers a model where two players can offer each-other favors. A favor is an opportunity for one player at a fixed cost c to offer another a fixed benefit of b. The ability for one player to offer the favor is only privately known and arrives at random times. Though the type of private information is a different from that considered here, the mechanism Mobius considers (the chips mechanism) is similar to the obligation takeover mechanism discussed in section 4 and in Skrzypacz and Hopenhayn (2004). Hauser and Hopenhayn (2008) consider alternative mechanisms with improved efficiency in this favor trading environment. Lau (20) extends the favor trading environment to random cost and benefits but with one-sided private information and comes closest in the favor-trading literature to the kind of private information in the task assignment environment. In addition to these theoretic papers, Roy (202) discusses an experimental implementation of the Mobius (200) environment. Other experiments are also related, especially those on turn-taking style cooperation in games without private information. Cason, Lau and Mui (203) study a common-pool resource allocation game in which turn-taking is the efficient cooperative outcome. They find that subjects who successfully engage in turntaking are able to teach this strategy to future partners. Kuzmics, Palfrey and Rogers (202) 2 studies repeated allocation games in the laboratory and also finds that turn-taking is a prevalent form of behavior 3. Kaplan and Ruffle (202) study a repeated entry game with private information but where communication is not permitted. They demonstrate that the form of cooperation depends on the particular distribution of private information. With a low variance in private value, rigid turn-taking emerges. However, with larger variance, cut-off strategies based on private value emerge. In contrast, the mechanisms considered here combine both of these elements in an environment where the parties are permitted to communicate. In addition to these, the current paper may be seen as a repeated extension of the volunteer s dilemma literature. See for instance (Diekmann, 985; Weesie, 994; Weesie and Franzen, 998; Goeree, Holt and Moore, 2005) and a similar model related to the volunteers dilemma (Bliss and Nalebuff, 984). 5.2 Conclusion This work represents, to my knowledge, the first formal analysis of flexible turn-taking. Even a simple model of this familiar arrangement achieves substantial efficiency. Further, the empirical efficiency is similar to that expected in theory. However, anomalies in subject behavior suggest they may be used to a form of flexible turn-taking with slightly different structure. The gap strategies subjects implement in the experiment are consistent with the kind of gaps that appear theoretically in the obligation takeover mechanism. Further, the obligation takeover mechanism can achieve approximately optimal efficiency. Similar to these papers in its use of auctions, Guo, Conitzer and Reeves (2009) develops a mechanism for repeated allocation without transfers using auctions of a fiat currency in a binary valuation environment 2 Kuzmics, Palfrey and Rogers (202) also gives an interesting theoretical interpretation of the Thue-Morse sequence in terms of symmetry of the continuation values for allocation orderings. A similar result is given in Cooper and Dutle (203) for the example of structuring a fair duel. 3 In a computational model, Neill (2003) shows that turn-taking can be achieved in a noisy environment, even when agents use limited memory strategies. 3
Together, these results suggest that whatever the form of flexible turn-taking people use, it is likely to provide substantial efficiency. It is possible that people naturally implement a form of flexible turntaking that achieves nearly optimal efficiency. If this is the case, it is not surprising that such familiar social arrangements persist. There are no formal mechanisms that would do much better. However, this research only scratches the surface. Work confirming the precise type of flexible turn-taking that is common among people is needed to make more progress in the comparison between social arrangements and formal mechanism. Further, there are several interesting theoretical extensions possible including analysis of arrangements for larger groups, the effect of cost distributions that are asymmetric between players, and the potential for players to condition on partially observable information. Turn-taking is not limited to assigning tasks. While there are many informal situations that have the structure of the repeated assignment problem, the results of this paper should provide insight into analogous environments such as the repeated allocation, trade, and collusive environments discussed above. The results here extend immediately to indivisible resource allocation problems when the task that is being assigned is the task of going without the resource for a period. The mechanism design literature has produced sophisticated and successful solutions to many private information problems. But, private information problems have also been part of our long history. Our experience with these problems has provided some elegant social solutions such as flexible turn-taking. While we now know how to design alternative solutions, it is not clear that we would be much better off replacing familiar social arrangements with formal mechanisms. 32
6 Appendix 6. Proof of Lemma 2.2 Lemma 2.2: g increases strictly over the interval [0, E (θ i )]. g ( θ ) = β θ + (2q ) F ( θ ) 2 [ θ E ( θ i θ i θ )] ( F ( θ )) 2 [ E ( θi θ i θ ) θ ] (6.) The following two relationships are useful in re-writing this equation: F ( θ ) E ( θ i θ i θ ) = F ( θ ) θ ˆ 0 ζ f (ζ) dζ θ F ( θ ) = ζ f (ζ) dζ (6.2) 0 ( ( )) F θ E ( θ i θ i θ ) = ( F ( θ )) θ ζ f (ζ) dζ F ( θ ) = Using these relationships, equation 6. can be rewritten: g ( θ ) ( = θ β + (2q ) F ( θ ) 2 ( ( )) ) ( 2 + F θ (2q ) F ( θ ) ˆ θ 0 ˆ θ ζ f (ζ) dζ (6.3) ζ f (ζ) dζ + ( F ( θ )) ˆ ) ζ f (ζ) dζ ( The derivative of the term θ β + (2q ) F ( θ ) 2 ( ( )) ) 2 + F θ with respect to θ is: ( β + (2q ) F ( θ ) 2 ( ( )) 2 + F θ The derivative of the term θ (6.4) ) + 2 (2q ) θ f ( θ ) F ( θ ) 2 θ f ( θ ) ( F ( θ )) (6.5) ( (2q ) F ( θ ) θ 0 ζ f (ζ) dζ + ( F ( θ )) θ ζ f (ζ) dζ ) with respect to θ is: (2q ) f ( θ ) ˆ θ ζ f (ζ) dζ (2q ) θf ( θ ) f ( θ ) + f ( θ ) ˆ ζ f (ζ) dζ + θ ( F ( θ )) f ( θ ) (6.6) Together: g ( θ ) = 0 ( β + (2q ) F ( θ ) 2 ( ( )) ) 2 + F θ + 2 (2q ) θ f ( θ ) F ( θ ) 2 θ f ( θ ) ( F ( θ )) (6.7) (2q ) f ( θ ) ˆ θ ζ f (ζ) dζ (2q ) θf ( θ ) f ( θ ) + f ( θ ) ˆ ζ f (ζ) dζ + θ ( F ( θ )) f ( θ ) 0 One of the two (2q ) θ f ( θ ) F ( θ ) from the top of the above equation offsets (2q ) θf ( θ ) f ( θ ) from the bottom: θ θ g ( θ ) = ( β + (2q ) F ( θ ) 2 ( ( )) ) 2 + F θ + (2q ) θ f ( θ ) F ( θ ) 2 θ f ( θ ) ( F ( θ )) (6.8) (2q ) f ( θ ) F ( θ ) E ( θ i θ i θ ) + f ( θ ) ( F ( θ )) E ( θ i θ i θ ) + θ ( F ( θ )) f ( θ ) 33
Similarly, one of the two θ f ( θ ) ( F ( θ )) from the top offsets with the θ ( F ( θ )) f ( θ ) on the bottom: g ( θ ) = ( β + (2q ) F ( θ ) 2 ( ( )) ) 2 + F θ + (2q ) θ f ( θ ) F ( θ ) θ f ( θ ) ( F ( θ )) (6.9) (2q ) f ( θ ) F ( θ ) E ( θ i θ i θ ) + f ( θ ) ( F ( θ )) E ( θ i θ i θ ) Terms with f ( θ ) ( F ( θ )) or (2q ) f ( θ ) F ( θ ) multipliers are gathered: g ( θ ) = ( β + (2q ) F ( θ ) 2 ( ( )) ) 2 + F θ + (2q ) f ( θ ) F ( θ ) [ θ E ( θ i θ i θ )] (6.0) + f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] Terms with (2q ) multipliers are gathered: g ( θ ) [ = (2q ) F ( θ ) 2 ( ) + f θ F ( θ ) [ θ E ( θ i θ i θ )]] (6.) + β + ( F ( θ )) 2 + f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] Bound the derivative below by choosing β = and q = 0 since the term multiplying 2q must be positive. g ( θ ) 2 ( F ( θ )) f ( θ ) F ( θ ) [ θ E ( θ i θ i θ )] + f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] (6.2) Thus, g () is increasing if: 2 ( F ( θ )) + f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] f ( θ ) F ( θ ) [ θ E ( θ i θ i θ )] (6.3) Since 2 ( F ( θ )) 0, the following is sufficient: f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] f ( θ ) F ( θ ) [ θ E ( θ i θ i θ )] (6.4) Since we have assumed density is positive everywhere, it is possible to eliminate f ( θ ) from both sides. The result is then simplified in the following steps: ( F ( θ )) [ E ( θ i θ i θ ) θ ] F ( θ ) [ θ E ( θ i θ i θ )] (6.5) F ( θ ) E ( θ i θ i θ ) + ( F ( θ )) E ( θ i θ i θ ) ( F ( θ )) θ + F ( θ ) θ (6.6) F ( θ ) E ( θ i θ i θ ) + ( F ( θ )) E ( θ i θ i θ ) θ (6.7) Noting that ( F ( θ )) E ( θ i θ i θ ) + F ( θ ) E ( θ i θ i θ ) = E (θ i ) (refer to 6.2 and 6.3): E (θ i ) θ (6.8) 34
6.2 Proof of Lemma 2.3 Proof. g is strictly positive for the interval ( 2 E (θ i), ]. g ( θ ) = β θ + (2q ) F ( θ ) 2 [ θ E ( θ i θ i θ )] ( F ( θ )) 2 [ E ( θi θ i θ ) θ ] > 0 (6.9) Since [ θ E ( θ i θ i θ )] 0 and β θ is decreasing in β, let q = 0 and β =. This results in the following sufficient condition: 2 ( F ( θ )) θ + F ( θ ) [ E ( θ i θ i θ )] > ( F ( θ )) [E (θ i )] (6.20) When θ =, this simplifies to E (θ i ) > 0, which is true by assumption that density is positive everywhere. Over the interval ( 2 E (θ i), ), drop the positive term ( F ( θ, α )) [ E α ( θi θ i θ )] to yield yet another sufficient condition: 2 ( F ( θ )) θ > ( F ( θ )) [E (θ i )] (6.2) Since density is positive everywhere, F ( θ ) > 0 over the interval ( 2 E (θ i), ). Eliminating ( F ( θ )) from both sides: This is true over the entire interval ( 2 E (θ i), ). 6.3 Proof of Claim 2.8 θ > 2 E (θ i) (6.22) Define g o ( θ ) and g d ( θ ) to be the option and deferment versions of the equilibrium root equation 2.3. From equation 2.3, the difference g d ( θ ) g o ( θ ) is given by: 0. ( g ) ( d θ g ) o θ = 2F ( θ ) 2 [ θ E ( θ i θ i θ )] (6.23) Evaluate this expression at the unique equilibrium point for the deferment version θd, at which g ( ) d θ d = g o (θd ) = 2F (θ d )2 [θd E (θ i θ i θd )] (6.24) The slope of g o is greater than 2 ( F ( θ )) over the interval [0, E (θ i )] by the results in 6.: g ( θ ) 2 ( F ( θ )) f ( θ ) F ( θ ) [ θ E ( θ i θ i θ )] + f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] (6.25) And for θ E (θ i ): f ( θ ) ( F ( θ )) [ E ( θ i θ i θ ) θ ] f ( θ ) F ( θ ) [ θ E ( θ i θ i θ )] (6.26) Thus, over the interval [0, E (θ i )]: g ( θ ) 2 ( F ( θ )) (6.27) 35
Further, for any distribution in which 2 E (θ i) is below the median, 2 ( F ( θ )) [ ] [ ] for any θ 0, 2 E (θ i). When this is true, the slope of g o is greater than over 0, 2 E (θ i). By proposition 2.4, all [ ] equilibria are in the interval 0, 2 E (θ i). Together, this implies: ( g ) ] o ˆθ = 0 for some ˆθ [θd, θ d + 2F (θ d )2 [θd E (θ i θ i θd )] (6.28) Alternatively: And thus, θo θd + 2F (θ d )2 [θd E (θ i θ i θd )] (6.29) θo θd 2F (θ d )2 [θd E (θ i θ i θd )] (6.30) 6.4 Recurring Rotation Under Symmetric Beta Distributions In this section, I assume costs are beta distributed with equal shape parameters both set to γ. I focus on the symmetric beta distributions for their ability to represent a wide variety of spread patterns with a single parameter while nesting the uniform distribution, and maintaining the same mean E (θ i ) = 2. γ = is the uniform distribution. For γ <, the distribution is U-shaped and for γ >, the distribution is unimodal and concave. The table below reports the equilibrium threshold and associated average costs for recurring rotation for various parameter levels γ as β using equation 2.2. This is accomplished by taking approximations of the equilibrium threshold over an increasing sequence of β until that sequence converges. Deferment q = Option q = 0 Other Mechanisms γ θ Equilibrium AC θ Equilibrium AC Best Threshold AC First Best AC 300.68.253.250.25.25.250 00.69.255.248.254.253.250 0.86.290.240.286.279.255 5.98.38.235.33.30.266 3.207.345.232.34.322.28 2.25.369.23.366.34.297.226.42.232.4.375 3 2.236.45.237.45.406.37 3.240.470.24.469.422.392 5.245.486.245.486.438.44 0.249.498.249.498.456.438 50.250.500.250.500.480.472 Table 6.: Computed Equilibrium Values for Beta Distribution It is noted that the thresholds diverge most sharply among the U-shaped distributions (small γ). This was predicted to some extent by the bound given in equation 2.7. Despite this, the difference in average 36
cost of the two mechanism versions remains small over the whole interval. The mechanism performs well relative to both best threshold and first-best at the extremes. When γ is large, the distribution becomes nearly a mass point at the mean. In this case, the problem of private costs disappears and any assignment results in nearly the same cost of execution. On the other hand, when γ is small, the distribution becomes nearly a binary distribution with either high or low cost. Private information is still a problem. However, any threshold mechanism will result in nearly first-best efficiency. When both are on the opposite side of the threshold, the efficient outcome is achieved. When both are on the same side, the two costs are nearly identical and so any assignment inefficiencies are small. In this family, the recurring rotation mechanism compares least favorably to first-best and best threshold under cost distributions that have a relatively high entropy (relatively flat). However, even in these cases recurring rotation offers substantial welfare improvement over rigid turn-taking, which results in an average cost of.5 for any distribution in the symmetric beta family. 6.5 Experiment Figures Probability Probability 0.2.4.6.8 0.2.4.6.8 Obligated Non-Obligated Person/Person Treatment Rounds 2-27 Rounds 28-34 Rounds 35-4 Rounds 42-48 0.2.4.6.8 0.2.4.6.8 0.2.4.6.8 0 00 200 300 0 00 200 300 0 00 200 300 0 00 200 300 Threshold Choice (Average by Subject) Person/Computer Treatment Rounds 2-27 Rounds 28-34 Rounds 35-4 Rounds 42-48 0.2.4.6.8 0.2.4.6.8 0.2.4.6.8 0 00 200 300 0 00 200 300 0 00 200 300 0 00 200 300 Threshold Choice (Average by Subject) Median Median Figure 6.: CDF of Average Threshold Choices (By Subject) Over 7 Round Blocks 37
Percent of Subjects 0 5 0 5 20 25 0 50 00 50 200 Mean Threshold By Subject (Bin Width 20) Percent of Subjects 0 0 20 30 0 50 00 50 200 Mean Threshold By Subject in Obligated State (Bin Width 20) Percent of Subjects 0 5 0 5 20 0 50 00 50 200 250 Mean Threshold By Subject in Non Obligated State (Bin Width 20) Figure 6.2: Histograms of average threshold choice by subject. Rounds 2-48. (Person/Person Treatment) Percent of Subjects 0 5 0 5 20 0 50 00 50 200 250 Mean Threshold By Subject (Bin Width 20) Percent of Subjects 0 5 0 5 20 25 0 50 00 50 200 250 Mean Threshold By Subject in Obligated State (Bin Width 20) Percent of Subjects 0 5 0 5 20 0 00 200 300 Mean Threshold By Subject in Non Obligated State (Bin Width 20) Figure 6.3: Histograms of average threshold choice by subject. Rounds 35-48. (Person/Person Treatment) Percent of Subjects 0 5 0 5 20 0 50 00 50 200 Mean Threshold By Subject (Bin Width 20) Percent of Subjects 0 5 0 5 0 50 00 50 200 Mean Threshold By Subject in Obligated State (Bin Width 20) Percent of Subjects 0 5 0 5 20 25 0 50 00 50 200 Mean Threshold By Subject in Non Obligated State (Bin Width 20) Figure 6.4: Histograms of average threshold choice by subject. Rounds 2-48. (Person/Computer Treatment) 38
Percent of Subjects 0 5 0 5 20 0 50 00 50 200 Mean Threshold By Subject (Bin Width 20) Percent of Subjects 0 5 0 5 20 0 50 00 50 200 Mean Threshold By Subject in Obligated State (Bin Width 20) Percent of Subjects 0 0 20 30 0 50 00 50 200 Mean Threshold By Subject in Non Obligated State (Bin Width 20) Figure 6.5: Histograms of average threshold choice by subject. Rounds 35-48. (Person/Computer Treatment) 6.6 Experiment Time Trend Analysis Subjects thresholds are estimated by an individual constant α i plus some individual-specific change δ i when the player is obligated (o i,t = when subject i is obligated in period t). Trends in average obligated and non-obligated thresholds are estimated using dummy variables for the same 7-period blocks in figure 6.. The estimates of these trend coefficients are provided in the tables below. θ i,t = α i + δ i o i,t + I (28 t 34) (γ(28 t 34) n + o i,tγ(28 t 34) o ) I (35 t 4) (γ(35 t 4) n + o i,tγ(35 t 4) o + (6.3) ) I (42 t 48) (γ(42 t 48) n + o i,tγ(42 t 48) o + ɛ i,t In the person/person treatment, the average thresholds chosen by non-obligated players in rounds 35 4 and 42 48 were both significantly different from those chosen in rounds 2 28. Further, obligated players in rounds 42 48 chose thresholds significantly lower than obligated players in rounds 2 28. However, neither obligated nor non-obligated players average thresholds differ significantly over any of the 7 round groupings after round 27. This suggests any learning and adjustment in the experiment happened over the action rounds 20 and the early threshold submission rounds, but had settled down by round 35. In the person/computer treatment, average non-obligated thresholds in rounds 28 4 were significantly different than in 2 27. While the non-obligated thresholds in rounds 42 48 were not significantly different, they were also not significantly different than those in rounds 28 34 and 35 4. Obligated thresholds in rounds 35 48 were significantly different than in rounds 2 27. Though the estimates of the obligated difference coefficients in rounds 35 4 and 42 48 are significantly different, the average threshold chosen by obligated players in those rounds measured by the sums γ(35 t 4) n + γo (35 t 4) and γ(42 t 48) n + γo are not significantly different. This does not provide any overwhelming evidence (42 t 48) against focusing on rounds 35 48, as in the person/person treatment. ) 39
Coefficient Point Estimate Standard Error 4 t-statistic γ(28 t 34) n 8.69 5.24.66 γ(35 t 4) n 2.7 4.39 2.77 ( ) γ(42 t 48) n 0.43 5.02 2.08 ( ) γ o (28 t 34).29 8.37 0.5 γ o (35 t 4) 2.58 7.39 0.35 γ o (42 t 48) 4.58 7.85 0.58 Table 6.2: Coefficient Estimates for Model 6.3: person/person treatment. Coefficient Point Estimate Standard Error 5 t-statistic γ(28 t 34) n 9.6 4.02 2.28 ( ) γ(35 t 4) n 4.24 6.09 2.34 ( ) γ n (42 t 48) 8.32 5.5.62 γ o (28 t 34).73 5.74 0.30 γ o (35 t 4) 0.62 5.62 0. γ o (42 t 48) 8.98 5.3.69 Table 6.3: Coefficient Estimates for Model 6.3: person/computer treatment. 6.7 Analysis of Potential Explanations for Gap Strategies in Person/Person Treatment Social Preferences The level of average thresholds may be consistent with subject altruism. As discussed above, when two average subjects are paired, the non-obligated partner offers swaps slightly more than the individually optimal level. However, subject altruism does not fit well with the fact that the subjects tended to select higher thresholds in the obligated state. Suppose player j has a utility function that weighs personal outcomes equally with player i s outcomes and that i plays according to some known threshold θ i in both states. Conditional on i asking for a swap when obligated, player j will believe player i s expected cost is E ( ) θ i θ i θ i. With a personal cost below E ( ) θ i θ i θ i, player j should agree to a swap. On the other hand, when i is non-obligated and agrees to a swap, j believes i has cost E ( ) θ i θ i θ i and should ask for a swap with a personal cost below this amount. Together, player j uses a threshold E ( ) ( ) θ i θ i θ i when non-obligated and E θi θ i θ i when obligated. In this way, gaps are potentially consistent with altruism. Further, these gaps can provide improved efficiency, even over second-best optimal. The efficiency of a single stage where player i is obligated and 40
j is non-obligated is given in equation 3.2. For instance, in the uniform 0, environment, if both players choose threshold 3 when obligated and 3 2 when non-obligated, the average joint stage cost is.352 - an improvement over second best.375. Equation 3.2 is plotted below for the uniform 0, environment. Grey points on this graph represent the average thresholds chosen by subjects (normalized to [0, ]) in the obligated (x-axis) and non-obligated (y-axis) states for rounds 35 48. The value associated with the location of each point can be thought of as the efficiency that would be achieved if a subject was to be paired with partner playing the exact same strategy. Figure 6.6: Average Stage Cost Under Asymmetric Strategies Notice that greater efficiency (blue area) is achieved above the 45-degree line. In this area, the threshold is higher in the non-obligated state. This is also the direction of the gap implied by the example given above. Player j chooses a threshold that is higher in the non-obligated state ( E ( θ i θ i θ i )) than in the obligated state ( E ( θ i θ i θ i )). Instead, most subjects chose strategies in which their thresholds were higher in the obligated state. This explains why the gap strategy is costly in terms of efficiency relative to symmetric strategies as calculated in section 3.3. While gaps may be due to some other social preference, this casual analysis provides some evidence against an altruism explanation. However, the second treatment of this experiment controls completely for social preferences by pairing subjects with computers. In that experiment, presented in section 3.4, gap strategies remain prevalent, providing evidence against any social preference explanation. Strategic Aspects 4