Peer Transparency in Teams: Does it Help or Hinder Incentives?

Transcription

1 Peer Transpareny in Teams: Does it Help or Hinder Inentives? Parimal Kanti Bag Nona Pepito June 19, 010 Abstrat In a joint projet involving two players who an ontribute in one or both rounds of a two-round effort investment game, transpareny, by allowing players to observe eah other s efforts after the first round, ahieves at least as muh, and sometimes more, olletive and individual efforts relative to a non-transparent environment in whih efforts are not observable. Without transpareny multiple equilibria an arise and transpareny eliminates the inferior equilibria. When full ooperation arises only under transpareny, it ours gradually: no worker sinks in the maximum amount of effort in the first round, preferring instead to smooth out ontributions over time. The benefit of transpareny, demonstrated both for exogenous rewards and in terms of implementation osts (with rewards optimally hosen by a prinipal to indue full ooperation), obtains under a general omplementary prodution tehnology. If the players efforts are substitutes, transpareny makes no differene to equilibrium efforts. JEL Classifiation: D0; J01. Key Words: Transpareny, team, omplementarity, substitution, free-riding, weak dominane, neutrality, implementation osts. Department of Eonomis, National University of Singapore, AS Level 6, 1 Arts Link, Singapore ; esbpk@nus.edu.sg Department of Eonomis, National University of Singapore, AS Level 6, 1 Arts Link, Singapore ; n.pepito@nus.edu.sg

2 1 Introdution Joint projets in teams based on voluntary ontributions of efforts are vulnerable to free-riding. In formulating inentives, an organization may try to influene its members effort deisions by designing the struture of ontributions. In partiular, the organization may be able to determine how muh the members know about eah other s efforts. This type of knowledge an be failitated by an appropriate work environment, suh as an open spae work-floor or regular reporting of team members atual working hours. We aim to show how transpareny in effort ontributions within a team may (or may not) help to mitigate shirking and foster ooperation. Empirial evidene ertainly point to the relevane of this kind of transpareny as a key determinant of produtive effiieny (Teasley et al., 00; Heywood and Jirjahn, 004; Falk and Ihino, 006). When efforts are observable during a projet s live phase (i.e., in a transparent environment), team members play a repeated ontribution game. On the other hand, when efforts annot be observed (i.e., a non-transparent environment), the projet is a simultaneous move game. The repeated ontribution game expands the players strategy sets relative to a simultaneous move game beause later period ations an be onditioned on the history. The additional strategies an reate new equilibria that are not available under the simultaneous move game, or remove existing equilibria of the simultaneous move game by introduing strategies that lead to profitable deviations. By enlarging or shrinking the equilibrium set or by simply altering it, does observability of interim efforts indue more overall efforts or less efforts? Whih game form is better? We will show two main results. First, if the prodution tehnology exhibits omplementarity in team members efforts, transpareny is benefiial. On the other hand, if the tehnology involves substitutability in efforts, transpareny is mostly neutral in its impat on individual and olletive team efforts. In teams, repeated games and dynami publi good settings, the general issue of transpareny (i.e., observability/dislosure of ations) and its inentive impliations have been studied by several other authors. See Che and Yoo (001), Lokwood and Thomas (00), Andreoni and Samuelson (006) et. in the ontext of dynami/repeated games, Winter (006a), and Mohnen et al. (008) in the ontext of sequential and repeated ontribution team projets, and Admati and Perry (1991), Marx and Matthews (000), et. in dynami voluntary ontribution pure publi good settings. 1 1 There is also a growing literature on tournaments with more reent ontributions by Gershkov and Perry (009), Aoyagi (010), et. where the fous is on interim performane evaluations (or feedbaks) as a way of inentivizing ompeting players to exert greater efforts. Transpareny in teams, as an issue, is very different from the feedbak idea for two reasons: (i) beause of the publi good nature of the players rewards, in ontrast to tournaments where the reward is of the winner-take-all variety; (ii) interim efforts do not diretly 1

3 Our paper is loser to the peer transpareny problems of Mohnen et al. (008) and Winter (006a). Mohnen et al. onsider a team of two workers exerting efforts over any (or both) of two rounds, with the total output equaling the sum of efforts by the two workers (i.e., the tehnology is one of perfet substitutes). The workers are paid idential remunerations a fixed wage plus bonus with the latter being a positive fration of the team output. When eah worker is averse to inequality of efforts (relative to o-worker s effort), allowing the ontribution game to be transparent by making eah other s first-round efforts observable improves the overall ontribution and output relative to when the workers annot observe the first-round efforts. Further, if the workers utility funtions are modified by dropping the inequity aversion omponent, then transpareny makes no differene to the equilibrium efforts (and output). Thus in their model the benefits of transpareny are realized largely due to the workers distaste for inequity. In the ontext of a team projet, Winter (006a) asks when more information among peers about eah other s efforts (IIE or internal information about effort measuring transpareny) makes it easier for the prinipal to provide inentives so that all agents exert effort (alled the INI outome). The agents an either exert effort or shirk as a one-off effort investment deision, and eah agent s effort hoie is made at different points of time although an agent may or may not observe the past deisions by the earlier agents. With an ayli binary order, k, on the agents refleting an IIE, 3 if any two IIEs, say k 1 and k, an be ompared in the manner k 1 is riher than k, 4 then k 1 is said to be more transparent than k. Then, defining a projet to exhibit omplementarity (substitution) if an agent s effort is marginally more (less) effetive in improving the projet s probability of suess as the set of other agents who also exert effort expands, the paper makes several interesting observations: (i) if a projet satisfies omplementarity, then it is less ostly to indue INI the more transparent the IIE; (ii) a sequential arhiteture in whih eah agent observes the effort deision of his immediate predeessor is the most transparent IIE; and (iii) if the projet exhibits substitution, transpareny is no longer important, i.e., neutral, in induing INI ; et. We omplement and extend the analysis of Mohnen et al. (008) and Winter (006a), by translate into rewards whereas in tournaments rewards are a funtion of interim performane. Winter (006b) analyzes the problem of inentive provision in a team where its members exert efforts sequentially towards a joint projet but does not analyze the transpareny issue, whereas Winter (004) studies another team efforts problem where the agents move simultaneously (rather than sequentially). On inentive design with omplementarities aross tasks but in a prinipal-agent setting (rather than team setting), see MaDonald and Marx (001). 3 An ordering of peers in the form of i 1 k i k...k i r indiates that peer i 1 knows peer i s effort, i knows i 3 s effort, and so on. 4 I.e., i k j would imply i k 1 j but not neessarily the other way around; see the previous footnote.

4 studying a team setting with some plausible and important model features not onsidered by these authors. There is a projet onsisting of two tasks. Two workers work over two rounds on one task eah, and in eah round a worker may hoose to put in zero, one or two units of effort with total efforts over two rounds not exeeding two units. The suess or failure of the projet materializes only at the end of the seond round. The projet s suess probability is inreasing in the total efforts invested in eah task. The projet exhibits omplementarity (substitutability) if the inremental suess probability due to additional efforts in a task is inreasing (dereasing) in the efforts invested in the other task. Following suessful ompletion of the projet eah worker reeives a (ommon) reward v > 0 and reeives zero if the projet fails; rewards annot be onditioned on efforts as the latter might not be verifiable. Two alternative work environments are onsidered: in a transparent (or open-floor) environment first-round efforts are publily observed by eah worker before eah hooses respetive seond-round efforts; in a non-transparent (or losed-door) environment efforts are not observed. Among the modeling differenes, ours onsider more general tehnologies than the one analyzed by Mohnen et al. (general omplementary/substitution tehnologies vs. perfet substitution tehnology) but the agents preferenes are standard utilitarian without any onern for equity. Different from Winter (006a), we allow for repeated efforts by the players and thus transpareny in our setting not only allows a player to influene another player s future play through his own ation today but also by onveying how he himself might again play/respond in a future round. 5 This intertemporal oordination in players ations through publi observation of all players past ations demands more ompliated strategi onsiderations ompared to the one-off effort investment deision model of Winter. So the relationships between transpareny, tehnologies and inentive provision need further srutiny. We show the following results. Under omplementary tehnology, with exogenous player rewards, the transparent environment is weakly better than the non-transparent environment (Propositions and 3) in the following sense: the best Nash equilibrium efforts pair in the non-transparent environment entailing partial or full ooperation by the players an be supported in a unique subgame-perfet equilibrium in the transparent environment, by eliminating any other inferior Nash equilibrium (or equilibria); in addition, we show that under ertain onditions the maximal efforts equilibrium, (, ), obtains only under transpareny. It should be noted that when suh an equilibrium ours under transpareny (but not so 5 In Winter (006a) the struture of IIE rules out mutual knowledge of efforts as there is a fixed timing struture aording to whih the agents make their investment deisions (formally, any binary order k refleting IIE is ayli). 3

5 in the absene of transpareny), it involves eah worker putting in one unit of effort in the first round followed by another unit of effort in the seond round. Thus, full ooperation is ahieved at best gradually transpareny allows workers to make observable partial ommitments in the first round and omplete the projet suessfully by supplying the remaining efforts in the seond round (Proposition ). 6 These results we obtain assuming effort osts are linear. For inreasing marginal osts, similar results (weak-dominane and gradualism) obtain exept that now the uniqueness of equilibrium may not be guaranteed under transpareny. Based on the weak-dominane result in Proposition 3 we further show that, when the prinipal determines the rewards optimally, ompared to non-transpareny the prinipal an ahieve weak or unique implementation of full ooperation at no more and possibly lower overall osts in a transparent environment (Proposition 4). Finally we show that if the tehnology exhibits substitutability in efforts and effort osts are linear, transpareny is neutral in terms of equilibrium efforts indued (Propositions 5 and 6). 7 The weak-dominane property of transpareny in our setup, while similar to the main theoretial result of Mohnen et al., is due to different underlying reasons. First, as our results show, the workers inequity aversion is not neessary for explaining why organizations may favor transpareny; in our setup the dominane (of transpareny) obtains mainly due to the omplementary nature of the prodution tehnology. 8 This enrihes the possibilities under whih organizations may favor a transparent work arrangement beyond the environment studied by Mohnen et al. The ontrast between omplementary and substitution tehnologies with their differing impliations (for transpareny) is similar to Winter s (006a) result. But unlike in Winter s paper the players in our setting reeive idential rewards, so there is no disrimination among team members (aording to one s position in the sequential efforts hain). Another related point may be noted here. In a pure publi good setting, Varian (1994) made the observation that if agents ontribute sequentially, rather than simultaneously, the free-riding problem gets worse total ontribution in a sequential move game is never more and possibly less than in a simultaneous move game. 9 As Winter (006a) has shown, if 6 Besides a number of papers mentioned earlier, some of the other works on gradualism are Bagnoli and Lipman (1989), Fershtman and Nitzan (1991), and Gale (001). 7 Elsewhere Pepito (010) has shown that for inreasing marginal osts of effort, transpareny is harmful (i.e., indues stritly lower efforts). 8 Knez and Simester (001) and Gould and Winter (009) doument the positive impat of peer efforts due to omplementarity between team members roles the former is a ase study on the performane of Continental Airlines in 1995, and the latter is a panel data analysis of the performane of baseball players. Gould and Winter also show negative peer effet when the players are substitutes. 9 Bag and Roy (008) show that if agents ontribute repeatedly to a publi good and have inomplete information about eah other s valuations, expeted total ontribution may be higher relative to a simultaneous ontribution game. 4

6 an external authority an give disriminatory rewards to the ontributors of a joint projet (unlike in voluntary ontribution publi good models), then even though suh projets exhibit publi good features, sequential game performs better than a simultaneous move game when player efforts are omplementary. And we show that, in joint projets, the domination over the simultaneous move format an be extended to the repeated ontributions format. So unlike in the sequential move game of Varian, observability of ontributions is distintly a positive aspet for omplementary prodution tehnology. The model is presented next. In setions 3 and 4, we derive our main results on transpareny. Setion 5 onludes. The proofs not ontained in the text appear in the Appendix. A separate Supplementary materials file ontains some additional results. The Model A team of two idential risk-neutral members, heneforth players, engage in a joint projet involving two tasks, with one player eah separately responsible for one of the tasks. The probability of the projet s suess depends on the players aggregate effort profile over a horizon of two rounds. In eah round, players simultaneously deide on how muh effort to put in. Denote player i s sequene of effort hoies by {e it } t=1, i = 1, and his overall effort t=1 e it by e i E i = {0, 1, }. Let p(e i, e j ) be the projet s suess probability. The ost to player i of performing his task is per unit of effort, > 0. If the projet sueeds, both players reeive a ommon reward v > 0; otherwise, they reeive nothing. The payoff to player i (= 1, ), given his overall effort e i and player j s overall effort e j (j i, j = 1, ), is: u i (e i, e j ) = p(e i, e j )v e i. (1) The efforts are irreversible: shirking by player i (e i = 0) means {e it } t=1 = {0, 0}, partial ooperation by player i (e i = 1) means either {e it } t=1 = {1, 0} or {e it} t=1 = {0, 1}, and full ooperation by player i (e i = ) implies any of the following: {e it } t=1 = {, 0}, {e it} t=1 = {0, }, or {e it } t=1 = {1, 1}. So a player an hoose full ooperation either by making a single ontribution of two units of effort early or late in the game or by ontributing gradually, one unit of effort in eah round. The suess probability funtion p(e i, e j ) has the following properties: A1. p(, ) = 1 and p(0, 0) > 0; A. Symmetry: p(e i, e j ) = p(e j, e i ); 5

7 A3. Monotoniity: For given e j, p(e i, e j ) is (stritly) inreasing in e i ; and A4. General Complementarity: For any e j {0, 1}, p(1, e j) p(0, e j) > p(1, e j ) p(0, e j ) and p(, e j) p(1, e j) > p(, e j ) p(1, e j ), where e j > e j. In other words, while the projet sueeds for ertain if and only if both players exert the maximum amount of effort, there is, however, still some hane of suess if players shirk or ooperate only partially. We have speified omplementarity in a general form, requiring only that any additional effort by player i is more effetive (in terms of inremental probability of suess) the more ooperative player j is. This formulation admits perfetly omplementary tehnology, p(e i, e j ) = p(e i )p(e j ), where p(e i ) and p(e j ) are the individual tasks suess probabilities. Also note that symmetry and monotoniity are very natural and weak assumptions; further, for omplementary tehnology to be analyzed in setion 3, we do not require any further urvature restrition on the suess probability funtion: p(.,.) an be onave or onvex in eah effort omponent (i.e., inremental probability of suess is dereasing or inreasing). 10 Finally, v an be interpreted in two ways as the players valuation for the projet, or their ompensation as set by a prinipal, with v being ommon knowledge. The prinipal an ondition the rewards only on the outome and not diretly on the efforts; in fat, the prinipal need not neessarily observe the efforts. Sine players are idential, v 1 = v = v. The paper s main insights do not depend on the idential players assumption. Most of the analysis will be arried out assuming v to be exogenous. Later on v will be solved to minimize the prinipal s osts of induing full (or partial) ooperation. We will onsider two versions of the effort investment game. In one version, players are able to observe first-round effort hoies in an interim stage before the seond-round effort hoies are made, while in the other version players are unable to observe ations taken in the first round. Observability of efforts (or the lak of it) may be due to the prinipal designing a suitable work environment or beause of diret reporting. Following others studying similar environments, we term the observable effort ase transparent and the one with non-observable ations non-transparent. Most of our analysis in this paper will be arried out under the assumption of onstant per-unit ost of effort, as speified above. Towards the end we disuss briefly how hanging to inreasing marginal osts (of effort) might alter the results. 10 However, in setion 4 with players efforts ating as substitutes, p(.,.) will be stritly onave. 6

8 3 Benefit of Transpareny: Complementary Efforts Unobservable ontributions. When a player is unable to observe the amount of effort exerted by the other player before the end of the projet s ative phase, the overall efforts are determined by the Nash equilibrium (or NE) of the following simultaneous move game: Player 1 Player p(0, 0)v, p(0, 0)v p(0, 1)v, p(0, 1)v p(0, )v, p(0, )v 1 p(1, 0)v, p(1, 0)v p(1, 1)v, p(1, 1)v p(1, )v, p(1, )v p(, 0)v, p(, 0)v p(, 1)v, p(, 1)v v, v Figure 1: Simultaneous move game G Denote this one-shot game by G, any strategy profile (e 1, e ) of G by e G, and a pure-strategy NE, (e 1, e ) of G, by e G. Lemma 1. Suppose suess probability p(.,.) satisfies A1-A4. Then the game G has no asymmetri pure strategy Nash equilibrium. In view of Lemma 1, in the one-shot game we fous on the symmetri pure strategy Nash equilibrium (or equilibria): Proposition 1 (One-shot Nash equilibrium). In the one-shot game G (i.e., with unobservable ontributions), the pure strategy Nash equilibrium (or equilibria) an be haraterized as follows: Equilibrium (e 1, e ) = (0, 0) obtains if and only if max{(p(1, 0) p(0, 0))v, [(p(, 0) p(0, 0))v]/}; Equilibrium (e 1, e ) = (1, 1) obtains if and only if (p(, 1) p(1, 1))v (p(1, 1) p(0, 1))v; Equilibrium (e 1, e ) = (, ) obtains if and only if min{(1 p(1, ))v, [(1 p(0, ))v]/}. 7

9 Note that the above is a haraterization result. In the Appendix we show that there always exists a pure strategy Nash equilibrium. This existene result (and we will show a similar result in setion 4 with player efforts as substitutes, rather than omplements) is noteworthy, although in a speifi ontext, beause while there are general existene results on Nash equilibrium (e.g., Dasgupta and Maskin, 1986), to our knowledge there is no existene result available on pure strategy Nash equilibrium in games with finite number of players and finite ation sets. Observable ontributions. The effort investment game proeeds as follows: Round 1 : Players simultaneously hoose their efforts e i1 {0, 1, }, i = 1,. Interim period: Players first-round deisions are revealed. Denote the set of possible observed effort levels e 1 = (e 11, e 1 ) by Ê 1. Clearly, Ê 1 = {(0, 0), (0, 1), (0, ), (1, 0), (1, 1), (1, ), (, 0), (, 1), (, )}. Round : Players make their effort deisions simultaneously, having observed eah other s first-round effort hoies. Denote player i s set of admissible seond-round effort hoies by Ê i. Sine overall effort e i annot exeed, {0,1,} if e i1 = 0; Ê i = {0,1} if e i1 = 1; {0} if e i1 =. At the end of Round, the projet onludes. Both players reeive reward v if the projet is suessful. If the projet fails, they both reeive 0. With observability, the joint projet indues a repeated ontribution game in whih players move simultaneously in eah round. The extensive form appears in Fig.. The payoffs in eah ontinuation game are in terms of the seond-round inremental gains relative to those yielded by the pair of observed effort levels e 1 that gives rise to the ontinuation game. For example, suppose that both players hoose one unit of effort in the first round. This restrits the set of admissible ations for players 1 and to Ê 1 = Ê = {0, 1}, resulting in a ontinuation game with the strategy spae S = {0, 1} {0, 1}. (In general, the strategy spae of any ontinuation game is S = Ê 1 Ê.) Denote player i s interim payoff, i.e., payoff generated by observed effort levels e 1 = (e 11, e 1 ), by û i1 (e i1, e j1 ), 11 and inremental gains following seond-round ations (e i, e j ) by û i (e i, e j e 1 ) = u i (e i1 +e i, e j1 +e j ) û i1 (e i1, e j1 ). () 11 Interim payoffs are alulated assuming as if the players will exert no further effort in Round. 8

10 ❶ ❷ v-, v- ❷ ❷ ❷ 0 1 ❷ ❶ 0 0 0, 0 ❶ ❷ 1 1 (p(1,)-p(0,))v -, (p(1,)-p(0,))v 0 1 0, 0 (1-p(,1))v, (1-p(,1))v ,0 (p(,1)-p(,0))v, (p(,1)-p(,0))v - (1-p(,0))v, (1-p(,0))v- 0 1 (1-p(0,))v-, (1-p(0,))v 0 0,0 (p(0,1)-p(0,0))v, (p(0,1)-p(0,0))v - (p(0,)-p(0,0))v, (p(0,)-p(0,0))v- 1 (p(1,0)-p(0,0))v -, (p(1,0)-p(0,0))v (p(1,1)-p(0,0))v -, (p(1,1)-p(0,0))v - (p(1,)-p(0,0))v-, (p(1,)-p(0,0))v- (p(,0)-p(0,0))v -, (p(,0)-p(0,0))v (p(,1)-p(0,0))v-, (p(,1)-p(0,0))v- (1-p(0,0))v-, (1-p(0,0))v - ❶ ❷ ❶ ,0 (p(1,1)-p(1,0))v, (p(1,1)-p(1,0))v- (p(1,)-p(1,0))v, (p(1,)-p(1,0))v - 0 0, 0 1 (1-p(1,))v-, (1-p(1,))v 1 (p(,0)-p(1,0))v-, (p(,0)-p(1,0))v (p(,1)-p(1,0))v-, (p(,1)-p(1,0))v- (1-p(1,0))v-, (1-p(1,0))v - ❶ ❷ , 0 (p(0,)-p(0,1))v, (p(0,)-p(0,1))v- 1 (p(1,1)-p(0,1))v -, (p(1,1)-p(0,1))v (p(1,)-p(0,1))v-, (p(1,)-p(0,1))v- ❶ ❷ ,0 (p(1,)-p(1,1))v, (p(1,)-p(1,1))v - 1 (p(,1)-p(1,1))v -, (p(,1)-p(1,1))v (1-p(1,1))v -, (1-p(1,1))v - (p(,1)-p(0,1))v -, (p(,1)-p(0,1))v (1-p(0,1))v -, (1-p(0,1))v - Figure : Extensive-form game Ĝ

11 Therefore, player i s payoffs in the ontinuation game following e 1 = (1, 1) are 0 if e i = 0, e j = 0; (p(1, ) p(1, 1))v if e i = 0, e j = 1; û i (e i, e j (1, 1)) = (p(, 1) p(1, 1))v if e i = 1, e j = 0; (1 p(1, 1))v if e i = 1, e j = 1. Payoffs for the other ontinuation games are omputed in the same way. One speifi ontinuation game is worth noting here: the game following (0, 0) efforts in the first round. This ontinuation game is same as the one-shot game G exept that all the payoffs are subtrated by p(0, 0)v. For later use, we will desribe these two games as idential, given that the players strategi deisions will be the same. Denote the extensive-form game by Ĝ, and any subgame-perfet equilibrium (or SPE) strategy (e 11, e 1; e 1(e 11, e 1), e (e 11, e 1)) of this game by e Ĝ.1 Given the extensive-form representation in Fig., we an evaluate how the overall equilibrium efforts hange when efforts are made transparent. In partiular, take an equilibrium (or equilibria) that arises in the one-shot game; from Proposition 1 we see that this equilibrium (or equilibria) results if and only if ertain onditions hold. Taking these onditions as given, we then examine the setting with repeated, observable ontributions, and determine whih overall efforts result (or do not result) in an SPE under these onditions. Below we start with some preliminary results hoping to demonstrate, at the end, how transpareny an sometimes be ritial to ahieving full ooperation and ensure the projet s suess. Lemma. Assume A1-A4. (i) If, without observability, full ooperation is not an equilibrium, then the only way full ooperation an arise with observability is through gradual ooperation, i.e., (1, 1; 1, 1). (ii) If, without observability, partial ooperation is an equilibrium while full ooperation is not, then full ooperation annot arise with observability. Lemma 3. Assume A1-A4. Suppose, without observability, shirking is the unique equilibrium. Then full ooperation may arise with observability and an only be through gradual 1 To be preise, equilibrium seond-round strategies should be more general funtions of any first-round effort deisions and not just of (e 11, e 1). Our equilibrium analysis uses the formal definition of SPE. 10

12 ooperation. A set of suffiient onditions that guarantee full ooperation, and whih an be onsistent with shirking as the unique equilibrium without observability, is as follows: and p(0, )v > v p(1, )v p(0, 1)v > p(0, )v v p(0, 1)v. Moreover, if shirking is the unique equilibrium without observability and (3) hold, shirking remains an equilibrium with observability. Fig. 3 illustrates Lemma 3 for the perfetly omplementary tehnology, p(e 1, e ) = p(e 1 )p(e ), where for i = 1,, α if e i = 0; p(e i ) = β if e i = 1; 1 if e i =. (3) (4) Payoffs βv - 1 β v αv αβv (α=0.) -0.5 v - b βv - αβv αv - - Figure 3: (0, 0) is the unique e G and (, ) is supported in subgame-perfet equilibrium, for p(e 1, e ) = p(e 1 )p(e ) with p(0) = α, p(1) = β, and p() = 1. Given this speifiation, p(0, ) = α, p(1, ) = β, p(0, 1) = αβ, and p(1, 1) = β. The 11

13 figure plots the payoffs against β and identifies the values of β suh that the payoffs satisfy onditions (3) for a profile of the remaining parameters, (α = 1, v =.4, = 1).13 Further, 5 e G = (0, 0) sine for all β (0, 1), α v > 0, αβv < 0, and αv < 0 (i.e., p(0, 0)v > 0, p(1, 0)v < 0, and p(, 0)v < 0). To verify uniqueness of e G = (0, 0), first note that (1, 1) is not an NE sine p(0, 1)v > p(1, 1)v (beause αv > β v ), and (, ) is not an NE beause p(0, )v > v (follows from (3)), and there is no other pure strategy equilibrium (by Lemma 1). Let us now denote the value of β at whih v = βv by β 1. In this example, β 1 = 7, and we see that, for the given parameter values of (α, v, ), all the onditions (i.e., 1 (3) as well as uniqueness of e G = (0, 0)) are simultaneously satisfied for β ( 1, 7 5 1]. It is lear from the first and the third onditions in (3) above that p(0, )v > v > p(0, 0)v. In other words, full ooperation Pareto-dominates shirking, though the latter prevails when there is no way to observe the ongoing ontributions. There is mutual interest in ooperating, but it is not in any player s individual interest to ooperate. In this setting, making efforts observable enourages full ooperation. However, sine efforts are irreversible, sinking two units of effort in the first round is risky, as the other player an exert zero effort in both rounds, get p(0, 0)v > v, and go unpunished. (The only way to punish him would be for the ooperating player to move bak to shirking, whih is not possible.) Therefore, while transpareny indues ooperation, it an only do so using partial ommitments, i.e., gradually. The result is similar to the gradualism result of Lokwood and Thomas (00). Lemma and Lemma 3, together, yield the following behavioral predition for one type of full ooperation equilibrium under observability: Proposition (Gradualism). Suppose a joint projet involves two tasks satisfying a general form of omplementarity as defined in A1-A4 in setion. If full ooperation does not arise when transpareny is laking, then transpareny an ahieve full ooperation only through gradual reiproity. Moreover, in this ase full ooperation obtains under transpareny only if under non-transpareny partial ooperation fails to realize (along with full ooperation not being an NE), and if onditions (3) hold. Thus gradualism is one way to make transpareny make a differene when, without it, the worst (i.e., shirking) would have realized. This may lead to a distint ost advantage for a prinipal who wants to design reward inentives to uniquely implement full ooperation, as we will see in Proposition The figure has been generated in Mathematia. 1

14 In Proposition we assumed full ooperation not being an equilibrium under nontranspareny. It is possible that sometimes shirking or partial ooperation is not an equilibrium under non-transpareny. Then, a similar outome also fails to realize under transpareny: Lemma 4. (i) If (0, 0) e G, then overall efforts of (0, 0) annot arise in an SPE of the extensive-form game Ĝ. (ii) If (1, 1) e G, then overall efforts of (1, 1) annot arise in an SPE of the extensive-form game Ĝ. Finally, full ooperation being an equilibrium under non-transpareny has the following impliations for the transpareny regime: Lemma 5. Suppose full ooperation is an NE in the one-shot game. Then: (i) Full ooperation obtains in an SPE in the transparent environment. Speifially, all strategy profiles in the extensive-form game Ĝ that orrespond to full ooperation are SPE. (ii) Partial ooperation, i.e. (1, 1), annot arise in an SPE of the extensive-form game Ĝ. While Lemmas 4 and 5 (and other lemmas to be reported) may not offer a very lean piture of their standalone eonomi impliations/motivations, these should be seen as neessary steps to develop our main results on the performane of transpareny vis-à-vis nontranspareny for implementation of better effort profiles and the related optimal inentive osts. We begin with the laim that by allowing players to observe eah other s efforts during the projet s ative phase, the prinipal would do no worse and possibly do better. For example, if full ooperation is an equilibrium in the one-shot game but not neessarily unique, then full ooperation must be the only equilibrium in the extensive-form game. Define the set of outomes inferior to e G = (e 1, e ) by I eg = {(ẽ 1, ẽ ) ẽ 1 < e 1 or ẽ < e }. Note that by this definition, (, 0) and (0, ) are inferior to the effort pair (1, 1). We now look at two ases: when partial ooperation is a one-shot equilibrium, and when full ooperation is a one-shot equilibrium. 13

15 Lemma 6. Suppose that e G = (1, 1) (not neessarily unique). Then under transpareny overall efforts that entail shirking by any player annot arise in an SPE. Lemma 7. Suppose that e G = (, ) (not neessarily unique). Then under transpareny overall efforts where any player exerts less than two units of effort annot arise in an SPE. Thus, making efforts observable eliminates all outomes inferior to the best one-shot equilibrium possible where best is interpreted in terms of total team efforts. But still elimination does not establish superiority of transpareny. We must show that the best one-shot equilibrium, or perhaps a better effort profile, an be supported as a pure-strategy SPE of the extensive-form game under transpareny. The following proposition ahieves this objetive. Proposition 3 (Benefiial Transpareny). Suppose a joint projet involves two omplementary tasks as defined in A1-A4. Then transpareny dominates over nontranspareny in the following sense: Equilibrium (or equilibria) in the non-transparent environment entailing partial or full ooperation by both players is weakly improved upon in a unique equilibrium in the transparent environment by retaining the best equilibrium and at the same time by eliminating all inferior effort profiles (i.e., ones in whih at least one player exerts lower effort). Moreover, under appropriate onditions, when shirking (i.e., (0, 0)) is a unique equilibrium under non-transpareny, with transpareny it is possible to ahieve full ooperation by both players. Thus, when there are multiple one-shot equilibria, the weak dominane of transpareny is ahieved through (i) preservation of the best one-shot equilibrium and (ii) the elimination of all potential inferior outomes (inluding inferior one-shot equilibria). When the one-shot equilibrium is unique and involves ooperation (partial or full), overall equilibrium efforts under transpareny oinide with the efforts under non-transpareny. Finally, when shirking is the unique one-shot equilibrium, transpareny improves upon non-transpareny by making full ooperation possible (under ertain onditions) through partial ommitments. At this stage it may be appropriate to add ouple of remarks. First, as already mentioned in the Introdution, relative to non-transpareny the expanded strategies under transpareny has the potential to result in additional equilibria and equally it ould eliminate some oneshot equilibrium. Proposition 3 onfirms both these preditions to be true but what is interesting is the uniform impat of the two effets to make transpareny superior in terms of effort inentives (not only inferior outomes are eliminated, stritly superior outome may 14

16 emerge). We will see later on that the omplementarity between team members efforts is quite important for this dominane; if, instead, the efforts are substitutes, transpareny is either neutral or sometimes may even be harmful. Seond, it is easy to see that when there are multiple equilibria under non-transpareny, the one with the highest aggregate efforts Pareto-dominates the others. While equilibrium seletion using the riterion of Pareto domination may seem a valid reason not to worry about the inferior equilibria, the problem of misoordination in team settings is a very reasonable onern whih gets worse as the team size beomes large. And with the introdution of slight risk aversion on the part of the players (in our treatment players are risk neutral in monetary rewards), non-transpareny is likely to tilt the balane towards lower efforts equilibria. Transpareny fully resolves this oordination problem by eliminating the inferior equilibria. 14 In Table 1 we provide (see detailed formal derivations in the Appendix), for a omplete breakdown of the ost parameter in an asending order (for any given value of v and the projet tehnology p(e 1, e )), the list of various equilibria under the two arrangements, nontranspareny and transpareny. 15 It demonstrates leanly the value of mutual observability of team members interim efforts. The ase of inreasing marginal osts. So far our analysis has been based on the assumption of linear effort osts. We now briefly disuss possible modifiation to the main result if effort osts are onvex: the ost of exerting the seond unit of effort within the same round is + δ, δ > 0, i.e., the marginal ost of effort is inreasing within a round. With the hange in effort osts, our previous intuition in favor of transpareny gets somewhat weakened. After all, due to inreasing marginal osts players are strongly disouraged against sinking in two units of effort within a single round. This gives fewer options to ontribute two units of effort in both the transparent and the non-transparent environments, as the players should like to spae out their effort ontributions over the two rounds. In the non-transparent environment this lak of options is of no real onsequene, beause the players an shift their ontributions aross the two rounds privately. But in the transparent environment, this reates a perverse inentive among the players to withhold individual 14 For example, in the ase where e G = (0, 0), e G = (1, 1), and e G (, ), transpareny allows any player to onfidently sink in one unit of effort early on regardless of whether the other player hooses zero effort or one, beause when the other player observes his move it will be in his best interest to math it (if he has not already done so). Sine this deision by any player will always be mathed by the other player, a situation where one player partially ooperates and the other player shirks annot arise with observability. 15 In Table 1 and later on in Table and for the supporting derivations for Table 1 in the Appendix, we will slightly abuse the notation e to refer to overall efforts pair in the two-round game that an be supported Ĝ in SPE. 15

17 Table 1: Improved outome possibilities with transpareny Parameter Configuration e G e Ĝ Main ondition Additional onditions (a) (p(, 0) p(1, 0))v < (p(1, 0) p(0, 0))v (, ) (, ) (b) (p(1, 0) p(0, 0))v and < (p(,0) p(0,0))v (, ) and (0, 0) (, ) () (p(1, 0) p(0, 0))v and (p(,0) p(0,0))v (1 p(0,))v (, ) and (0, 0) (, ) (d) (p(1, 0) p(0, 0))v and (1 p(0,))v < (0, 0) (0, 0) (e) (p(, 0) p(1, 0))v < < (p(, 1) p(1, 1))v < (p(1, 0) p(0, 0))v (, ) (, ) (f) (p(1, 0) p(0, 0))v (1 p(0,))v (, ) and (0, 0) (, ) (g) (p(1, 0) p(0, 0))v and (1 p(0,))v < (1 p(0,1))v (0, 0) (, ) and (0, 0) (h) (p(1, 0) p(0, 0))v and (1 p(0,1))v < (0, 0) (0, 0) 16

18 Table 1: Improved outome possibilities with transpareny, ontd. Parameter Configuration e G e Ĝ Main ondition Additional onditions (i) (p(, 1) p(1, 1))v (1 p(1, ))v 1 p(0,) v and < (p(1, 0) p(0, 0))v (, ) and (1, 1) (, ) (j) 1 p(0,) v and (p(1, 0) p(0, 0))v (p(1, 1) p(0, 1))v (, ), (1, 1) and (0, 0) (, ) (k) (p(1, 1) p(0, 1))v < and (1 p(0,))v (, ) and (0, 0) (, ) (l) (1 p(0,))v < (1 p(0,1))v and main ond. add. onds < (p(1, 0) p(0, 0))v (m) (1 p(0,))v < (1 p(0,1))v and (p(1, 0) p(0, 0))v (p(1, 1) p(0, 1))v (1, 1) and (0, 0) (1, 1) (n) (1 p(0,))v < (1 p(0,1))v and (p(1, 1) p(0, 1))v < (0, 0) (, ) and (0, 0) (o) (1 p(0,1))v < and (p(1, 1) p(0, 1))v (1, 1) and (0, 0) (1, 1) (p) (1 p(0,1))v < and (p(1, 1) p(0, 1))v < (0, 0) (0, 0) (q) (1 p(1, ))v < < (p(1, 0) p(0, 0))v (1,1) (1,1) (r) (p(1, 0) p(0, 0))v (p(1, 1) p(0, 1))v (1, 1) and (0, 0) (1, 1) (s) (p(1, 1) p(0, 1))v < (0, 0) (0, 0) 17

19 Table : Improved outome possibilities with transpareny: the ase of rewards (a) (b) Ranges of the reward v e G e Ĝ Main ondition Additional onditions v p(,0) p(1,0) < v (, ) (, ) p(1,0) p(0,0) < v p(,0) p(0,0) (, ) and (0, 0) (, ) p(1,0) p(0,0) () v and p(1,0) p(0,0) 1 p(0,) p(,0) p(0,0) (, ) and (0, 0) (, ) (d) v p(1,0) p(0,0) and v < 1 p(0,) (0, 0) (0, 0) (e) < v p(,1) p(1,1) p(,0) p(1,0) p(1,0) p(0,0) (, ) (, ) (f) v 1 p(0,) (, ) and (0, 0) (, ) p(1,0) p(0,0) (g) v and p(1,0) p(0,0) v < (0, 0) (, ) and (0, 0) 1 p(0,1) 1 p(0,) (h) v and p(1,0) p(0,0) v < (0, 0) (0, 0) 1 p(0,1) (i) v v and 1 p(1,) p(,1) p(1,1) 1 p(0,) < v (, ) and (1, 1) (, ) p(1,0) p(0,0) (j) v 1 p(0,) p(1,1) p(0,1) v and p(1,0) p(0,0) (, ), (1, 1) and (0, 0) (, ) (k) v < p(1,1) p(0,1) 1 p(0,) and v (, ) and (0, 0) (, ) (l) (m) (n) main ond. add. onds v < 1 p(0,1) 1 p(0,) and p(1,0) p(0,0) < v v < and 1 p(0,1) 1 p(0,) v (0, 0) and (1, 1) (1, 1) p(1,1) p(0,1) p(1,0) p(0,0) and v < (0, 0) (, ) and (0, 0) p(1,1) p(0,1) v < 1 p(0,1) 1 p(0,) (o) v < 1 p(0,1) p(1,1) p(0,1) and v (0, 0) and (1, 1) (1, 1) (p) v < 1 p(0,1) and v < p(1,1) p(0,1) (0, 0) (0, 0) (q) v < < v 1 p(1,) p(1,0) p(0,0) (1,1) (1,1) (r) v p(1,1) p(0,1) (1, 1) and (0, 0) (1, 1) p(1,0) p(0,0) (s) v < p(1,1) p(0,1) (0, 0) (0, 0)

20 ontributions in the first round, thereby redibly onveying to the other player that pushing up ontribution in a later round would be unlikely (this effet is the prinipal reason why transpareny is potentially harmful in the substitution tehnology ase). So players may well end up in a bad oordination under transpareny with redued first-round efforts and lower aggregate efforts. We show that, in our three efforts setup, suh harmful effet never arises and transpareny ontinues to be (weakly) better than non-transpareny. The main differene, ompared to the linear effort osts ase, is that we an no longer guarantee the uniqueness of the overall equilibrium efforts in the extensive-form game. The formal analysis is developed in a separate Supplementary materials file. Optimal rewards. So far we did not onsider the question of optimal inentives: what should be the minimal rewards to indue a partiular pair of aggregate efforts, with and without transpareny? Table 1 provides an exhaustive summary of the various equilibria possible as the effort ost parameter,, is varied. We then onstrut Table by rearranging the same information given in Table 1 but now in terms of the ranges of v. It should be lear from Table how to determine the optimal v: for any given effort implementation target, identifiation of the required minimal v would minimize the implementation osts. Below we demonstrate the proedures for unique implementation of full ooperation; similar methods apply for weak implementation of full ooperation. Suppose the objetive is to uniquely implement full ooperation under non-transpareny. From Table, we know that the optimal reward, all it vnt u, is either in (a) or in (e) (by optimal reward we mean the lower bound (i.e., the infimum) of the reward, v, induing any target efforts pair). p(,0) p(1,0) v = p(1,0) p(0,0) Suppose that the set of v-values defined by (e) is empty, i.e.,. Then p(1,0) p(0,0) vu NT =. Now under transpareny, aside from p(1,0) p(0,0), any v that satisfies any of the onditions in Λ = {(b), (), (e), (f), (i), (j), (k)} would uniquely implement full ooperation. Let be a typial ondition enumerated in the first olumn in Table, and denote the lower bound of any set of v values defined by, when non-empty, by m. Clearly, m is equal to either the lower bound of v satisfying the main ondition or the lower bound of v satisfying the additional ondition(s) (under ), whihever is greater. Note that m, when it is well-defined for any Λ, will be no greater than. p(1,0) p(0,0) Then it must be that the least-ost reward that uniquely implements full ooperation under transpareny, all it v u T, is equal to the min{m } with being the elements from Λ for whih m s are well-defined. By onstrution vt u = min{m } < vnt u, whenever m is well-defined for at least one Λ; otherwise, vt u = vu NT. On the other hand, suppose the set of v s defined by (e) is non-empty, i.e.,. Then m p(,0) p(1,0) (a) =, while m p(,0) p(1,0) (e) < 19 < p(1,0) p(0,0). Thus in this ase, p(,0) p(1,0) vu NT =

21 m (e) = max{, }. By onstrution p(,1) p(1,1) p(1,0) p(0,0) vu T = min{m } < m (e) = vnt u, whenever m is well-defined for at least one {(f), (i), (j), (k)}; otherwise, vt u = vu NT. More generally, we an make the following observation: Proposition 4 (Implementation osts). Suppose a joint projet involves two omplementary tasks as defined in A1-A4. Then full ooperation by both players, i.e. overall efforts (, ), an be uniquely (or weakly) implemented under transpareny for a reward that is no more and possibly less than the minimal reward needed for unique (respetively, weak) implementation under non-transpareny. 4 Substitution Tehnology: A Neutrality Result In this setion, we onsider team projets with player efforts primarily as substitutes. The main objetive is to see whether the hange from omplementary to substitution tehnology alters how transpareny impats on team members efforts. We hope to onvine that muh of the benefits of transpareny will be lost as a result, and transpareny may even prove rather unhelpful. To formalize, let the projet s suess probability, denoted by ρ(e 1, e ), inherit properties A1-A3 from the previous setion and satisfy the following property: A4. General Substitutability: For any e j {0, 1}, ρ(1, e j) ρ(0, e j) < ρ(1, e j ) ρ(0, e j ) and ρ(, e j) ρ(1, e j) < ρ(, e j ) ρ(1, e j ), where e j > e j. That is, the inremental probability of projet suess due to an extra unit of effort by a player is dereasing in the other player s effort. 16 We ontinue to assume linear effort osts. At the end we disuss the likely hanges in results if one assumes inreasing marginal osts. Unobservable ontributions. When efforts are unobservable, the indued effort ontribution game is essentially a simultaneous move game although the efforts are exerted over two rounds. The normal form, denoted by G S, is as follows: 16 It is easy to hek that in the perfet substitution ase, ρ(e 1, e ) = ρ(e 1 +e ), the general substitutability property is satisfied: ρ(1) ρ(0) > ρ() ρ(1) > ρ(3) ρ() > ρ(4) ρ(3) > 0. 0

22 Player ρ(0, 0)v, ρ(0, 0)v ρ(0, 1)v, ρ(0, 1)v ρ(0, )v, ρ(0, )v Player 1 1 ρ(1, 0)v, ρ(1, 0)v ρ(1, 1)v, ρ(1, 1)v ρ(1, )v, ρ(1, )v ρ(, 0)v, ρ(, 0)v ρ(, 1)v, ρ(, 1)v v, v Figure 4: Simultaneous move game G S Denote the NE of this game by e G S. In the Appendix we show that there always exists a pure-strategy NE in G S. We also establish the following result: Lemma 8. In the normal-form game G S, multiple symmetri pure strategy Nash equilibria annot arise. That is, any e G S = (e, e) must be a unique equilibrium. While for omplementary tehnology one-shot equilibrium is neessarily symmetri, for substitution tehnology one-shot equilibrium an be asymmetri. Moreover, an asymmetri equilibrium an arise along with a symmetri one-shot equilibrium. 17 Observable ontributions. When first-round efforts are observable, the extensive form is as in Fig. 5. Denote the extensive-form game by ĜS, any SPE of this game by e, and the ontinuation game following e Ĝ 1 = (e 11, e 1 ) in the extensive-form game ĜS by S G S (e11,e 1 ). With player efforts as substitutes (as opposed to omplementary efforts), free-riding beomes a more serious problem under either ontribution format, with and without transpareny, beause one player s slak an be more easily piked up by another player. But then a player annot easily free ride by simply putting in low effort in the first round beause this effort redution an be made up for by the same player by putting in more effort in the seond round, given linear osts of effort. So how substitutability in efforts affets the players overall effort inentives under the two formats, transpareny and non-transpareny, is not a priori lear. Our next result shows that unlike in the omplementary tehnology ase, when efforts are substitutes, transpareny annot eliminate inferior efforts equilibrium if there are multiple equilibria under non-transpareny. 17 For example, suppose that v > p(1, )v and v = p(0, )v, suh that e G S = (, ). By Lemma 8, we know that e G S (1, 1) and e G S (0, 0). However, v > p(1, )v and v = p(0, )v imply that, using A4 and A, p(0, )v > p(, 1)v and p(0, )v > p(0, 0)v. Together with the fat that v = p(0, )v, these onditions imply that e G S = (0, ). 1

23 ❶ ❷ v-(+δ), v-(+δ) ❷ ❷ ❷ 0 1 ❷ ❶ 0 0 0, 0 ❶ ❷ 1 1 (ρ(1,)-ρ(0,))v -, (ρ(1,)-ρ(0,))v 0 1 0, 0 (1-ρ(,1))v, (1-ρ(,1))v ,0 (ρ(,1)-ρ(,0))v,( ρ (,1)- ρ (,0))v - (1- ρ(,0))v,(1-ρ(,0))v-(+δ) 0 1 (1-ρ(0,))v-(+δ), (1-ρ(0,))v 0 0,0 (ρ(0,1)-ρ(0,0))v,(ρ(0,1)-ρ(0,0))v - (ρ(0,)-ρ(0,0))v,(ρ(0,)-ρ(0,0))v-(+δ) 1 (ρ(1,0)-ρ(0,0))v -,(ρ(1,0)-ρ(0,0))v (ρ(1,1)-ρ(0,0))v -,(ρ(1,1)-ρ(0,0))v - (ρ(1,)-ρ(0,0))v,(ρ(1,)-ρ(0,0))v-(+δ) (ρ(,0)-ρ(0,0))v (+δ),(ρ(,0)-ρ(0,0))v (ρ(,1)-ρ(0,0))v (+δ),(ρ(,1)-ρ(0,0))v (1-ρ(0,0))v (+δ),(1-ρ(0,0))v (+δ) ❶ ❷ ❶ ,0 (ρ(1,1)-ρ(1,0))v,(ρ(1,1)-ρ(1,0))v- (ρ(1,)-ρ(1,0))v,(ρ(1,)-ρ(1,0))v (+δ) 0 0, 0 1 (1-ρ(1,))v-, (1-ρ(1,))v 1 (ρ(,0)-ρ(1,0))v-,(ρ(,0)-ρ(1,0))v (ρ(,1)-ρ(1,0))v-,(ρ(,1)-ρ(1,0))v- (1-ρ(1,0))v-,(1-ρ(1,0))v (+δ) ❶ ❷ , 0 (ρ(0,)-ρ(0,1))v, (ρ(0,)-ρ(0,1))v- 1 (ρ(1,1)-ρ(0,1))v -, (ρ(1,1)-ρ(0,1))v (ρ(1,)-ρ(0,1))v-, (ρ(1,)-ρ(0,1))v- ❶ ❷ ,0 (ρ(1,)-ρ(1,1))v,(ρ(1,)-ρ(1,1))v - 1 (ρ(,1)-ρ(1,1))v,(ρ(,1)-ρ(1,1))v (1-ρ(1,1))v -,(1-ρ(1,1))v - (ρ(,1)-ρ(0,1))v-(+δ), (ρ(,1)-ρ(0,1))v (1-ρ(0,1))v-(+δ), (1-ρ(0,1))v - Figure 5: Extensive-form game ĜS

24 Proposition 5. Suppose a joint projet involves effort substitution as defined by A1-A3 and A4. Any NE efforts pair (η 1, η ) under non-transpareny an be supported as an SPE of the effort ontribution game under transpareny with the strategy profile eĝs = (η 1, η ; 0, 0). The next result shows that any overall effort profile ahievable under transpareny an also be repliated in the one-shot game under non-transpareny: Proposition 6. Suppose a joint projet involves effort substitution as defined by A1-A3 and A4. If under transpareny eĝs = (e 11, e 1; e 1(e 11, e 1), e (e 11, e 1)) is an SPE, then the aggregate efforts pair e GS = (η1, η), where η1 = e 11 + e 1 and η = e 1 + e, is an NE of the effort ontribution game under non-transpareny. Substitutability in efforts thus takes away from transpareny the distintive advantage of gradualism noted previously: under omplementary tehnology sometimes full ooperation ould be supported mainly by gradualism that might fail to materialize otherwise. To summarize, Propositions 5 and 6 together establish, in ontrast to our findings in setion 3, a form of neutrality of transpareny when player efforts are broad substitutes in team output and effort osts are linear: observability of efforts is neither gainful nor harmful for induing efforts. The result further implies that if one were to expliitly design inentives to implement full ooperation (or partial ooperation), the optimal reward v will be idential with and without transpareny. Our neutrality result ontrasts with Varian (1994), who showed that total ontribution in a two-player voluntary ontribution publi good game under observability of ontributions is often less than (and never exeeds) the total ontribution under non-observability. Note that in Varian s setup, due to sequential struture of ontributions, an early mover has the opportunity to free ride on the late mover by ommitting to low ontribution; in our setup, the fat that in the last round both players get to move simultaneously, ombined with the fat that marginal ost of effort is onstant, ompletely nullify the extra freeriding opportunity assoiated with an early move and observability makes no differene. But if marginal ost of effort is inreasing, low ontribution in the early round will have a ommitment value similar to Varian s setup beause to make it up in the seond round will push up the player s effort osts at an inreasing rate, making observability of efforts harmful (from the organization s point of view). 18 This result is demonstrated elsewhere in a 18 A similar ontrast an be found between the dynami ontribution game of Admati and Perry (1991), whih assumes sequential ontributions, and the repeated ontribution game of Marx and Matthews (000), whih assumes simultaneous ontributions within eah round. 3