Bargaining Under Strategic Uncertainty

Transcription

1 Bargaining Under Strategic Uncertainty Amanda Friedenberg September 2, 2013 Extremely Preliminary Abstract This paper provides a novel understanding of delays in reaching agreements based on the idea of strategic uncertainty i.e., the idea that a Bargainer may face uncertainty about her opponent s play, even if there is no uncertainty about the structure of the game. It considers a particular form of strategic uncertainty, called on path strategic certainty. On path strategic certainty is the assumption that strategic uncertainty can only arise after surprise moves in the negotiation process. The paper shows that Bargainers who engage in forward induction reasoning can face strategic uncertainty after surprise moves. Moreover, rational Bargainers who engage in forward induction reasoning and satisfy on path strategic certainty may experience delays in reaching agreements, even if they engage in forward induction reasoning. The paper goes on to characterize the behavioral implications of rationality, forward induction reasoning, and on path strategic certainty. Bargaining is an important feature of many economic and political phenomena. Understanding how negotiators form agreements is instrumental to understanding employment contracts, legislative outcomes, sovereign debt, etc. In each of these applications, we observe an important behavioral feature: at times, parties fail to reach an immediate agreement. Such failures lead to strikes, holdouts, legislative stalemates, delays in renegotiating debt contracts, and so on. Each of these situations have important (and sometimes long-term) economic consequences. What is the source of such negotiation failures? Many answers have been put forward in the literature. One natural answer is that such failures reflect incomplete information, i.e., reflect the fact that bargainers face uncertainty about some structural aspect of the game. This can be uncertainty about how bargainers value the object (see, e.g., Admati and Perry, 1987; Sobel and Takahashi, 1983; Cramton, 1984; Fudenberg, Levine and Tirole, 1985; Grossman and Perry, Arizona State University, amanda.friedenberg@asu.edu. I am indebted to Ethan Bueno de Mesquita, Martin Cripps, Eddie Dekel, Asher Wolinski and, especially, Marciano Siniscalchi, for helpful conversations. I also thank audiences at the Paris Game Theory Seminar, Northwestern University, University of Texas, University of British Columbia, the Game Theory Society World Congress, the Belief Change in Social Context Workshop, the Games Interactive Reasoning and Learning Workshop, the Tsingua Economic Theory Conference, and the Society for Advancement of Economic Theory. Much of this paper was written while visiting University College London; I am indebted to UCL for their continued hospitality. 1

2 1986; Feinberg and Skrzypacz, 2005), uncertainty about the bargainers cost of waiting (see, e.g., Rubinstein, 1985; Watson, 1998), uncertainty about the bargainers posture (see, e.g., Abreu and Gul, 2000; Abreu and Pearce, 2007; Abreu, Pearce and Stacchetti, 2012; Wolitzky, 2012), uncertainty about the ability to make future offers (see, e.g., Yildiz, 2004, 2011; Ortner, 2013), etc. To understand why incomplete information is a natural source at least, natural from the perspective of theory refer to Rubinstein s (1982) canonical alternating offers bargaining model. Suppose there is delay in reaching an agreement. Under a standard equilibrium analysis, Bargainers have correct beliefs about the bargaining process. When there is delay, the responder rejects an offer because she correctly anticipates that she will fare better in the future. This rejection is anticipated by the proposer. So, the proposer has an incentive to, instead, offer a proposal that is better from the perspective of the responder. If there is a failure to reach immediate agreement, it must be that the proposer does not understand that the responder will necessarily reject the offer. Under an equilibrium analysis, this is obtained by introducing uncertainty about the structure of the game, e.g., uncertainty about how the other bargainer values the object. The premise of this paper is that incomplete information cannot provide a full picture of why Bargainers experience delays in reaching agreements. There are important cases where incomplete information does not appear to be a significant source of delay. For example, wars can be seen as a failure by nations to reach agreements. But, understanding war as an artifact of incomplete information leads to some peculiar conclusions. In particular, Fearon (2004, page 290) and Powell (2006, page 172) point out that it leads to the conclusion that, after prolonged battle, parties retain significant informational asymmetries. Likewise, in the case of athletic contracts, it is difficult to argue that incomplete information is a significant source of delay: Often, the athleticproduct is well-known, i.e., the athlete s statistics, traits, outside options, etc., are commonly understood by the parties. And, similarly, often, the traits of the teams are well-known. This paper provides a novel understanding of delay in reaching agreements based on the idea of strategic uncertainty: Even in the extreme case where the game is transparent to the Bargainers, they may nonetheless face residual uncertainty about how others negotiate (or play the game). It is a trivial statement that strategic uncertainty can lead to bargaining impasses. In particular, a Bargainer may offer a split of the pie with the expectation that the offer will be accepted; under strategic uncertainty, this expectation need not be correct and the offer may be rejected. This simple observation might suggest that there is nothing interesting to be gained from an exercise based on strategic uncertainty it might suggest an anything goes result. Instead, the analysis imposes two important restrictions on the analysis. These restrictions explicitly rule out trivial forms of strategic uncertainty and, so, trivial forms of delay. The output is not an anything goes result. Preview of Approach The starting point is that each Bargainer faces a direct form of strategic uncertainty that is, each Bargainer faces uncertainty about how the other plays the game, where 2

3 the uncertainty is not an artifact of other features of the environment (e.g., incomplete information or player randomization). Epistemic game theory provides a formalism in which to analyze strategic interaction, when Bargainers face a direct form of strategic uncertainty. Epistemic game theory formally changes the definition of a game to include Bargainers hierarchies of beliefs about the play of the game. In so doing, epistemic game theory can be used to specify conditions on the Bargainers reasoning; these are referred to as epistemic conditions. We now preview the key epistemic conditions underlying the analysis. A background requirement is that each Bargainer maximizes his conditional subjective expected utility. That is, at each information set, the Bargainer maximizes his expected utility given his belief about how the other Bargainer plays the game. Two key epistemic conditions restrict the Bargainer s beliefs: (a) Forward Induction Reasoning, and (b) On Path Strategic Certainty. Forward induction reasoning is the idea that Bargainers rationalize past behavior, when possible. (The idea goes back to Kohlberg, 1981.) On path strategic certainty limits the nature of the strategic uncertainty. It requires that, along the path of play, Bargainers correctly anticipate how the bargaining process will unfold. Strategic uncertainty only arises in the event of surprise offers or surprise rejections. (The idea is similar in spirit to self-confirming equilibrium, i.e., as in Fudenberg and Levine, 1993; Dekel, Fudenberg and Levine, 1999.) We now discuss the importance of these two criteria. Begin with forward induction reasoning. It will be useful to draw an analogy to the standard equilibrium analysis. We pointed to the fact that, if a proposer correctly anticipates that his offer will be rejected, he has an incentive to offer a proposal that is better from the perspective of the responder. The implicit presumption was that the proposer correctly anticipates that the responder would accept such a mutually beneficial offer; indeed, a threat to reject such an offer would not be credible. A standard Nash equilibrium analysis does not rule out such an incredible threat and, indeed, allows for delays in reaching agreements. The implicit argument was one based on subgame perfection where no Bargainer uses a strategy that involves an incredible threat and, as a consequence, no player thinks the other player uses such a strategy. Forward induction reasoning provides an analogue for the case of strategic uncertainty. One natural conjecture is that, if the Bargainers beliefs are consistent with forward induction reasoning, then we will return to the conclusion of immediate agreement. Turn to on path strategic certainty. Here, both Bargainers begin the bargaining process with an understanding that they will, say, reach an agreement on a 50 : 50 split of the pie in period 10; both bargainers understand that the other understands this, etc. And, indeed, this is the correct outcome. The strategic uncertainty arises when one Bargainer, say Bargainer 1 (B1), deviates from the believed path of play, e.g., by making a, say, mutually beneficial offer. At that point, Bargainer 2 (B2) can face uncertainty about which future offers B1 will accept/reject. We will see 3

4 that on path strategic certainty rules out a trivial form of delay, i.e., delay based on uncertainty about how others resolve their indifferences. The Main Theorem characterizes the set of outcomes consistent with forward induction reasoning under on path strategic certainty. As will become clear, there are outcomes ruled out by these assumptions that is, the result is not one of anything goes. But, delay is consistent with these assumptions. Let us preview the mechanism by which delay arises: The two Bargainers begin the game understanding that the outcome will be an x : 1 x split of the pie in period n; they each understand that they each understand this, etc. This outcome involves delay. So, there is a mutually beneficial offer to be made upfront. But, no Bargainer makes such a mutually beneficial offer. The reason is that each Bargainer faces uncertainty about how the other Bargainer will react to the unexpected. In particular, each Bargainer fears that, by making a better than expected offer upfront, the other Bargainer will become more optimistic about her future prospects and this will cause the other Bargainer to reject the mutually beneficial offer specifically, holding out for an even better offer. It should be evident that this mechanism is consistent with on path strategic certainty. The difficulty is in understanding why it is consistent with forward induction reasoning. This is explained in Section 3. This is not the first paper to draw a connection between optimism and delay. See, e.g., Farber and Katz (1979); Shavell (1982); Yildiz (2004, 2011); Ortner (2013). Here, optimism reflects a belief about the future offers the other Bargainer will accept/propose. By contrast, in the previous literature, optimism reflects beliefs about other aspects of the bargaining problem, e.g., optimism about outside options or optimism about the likelihood of making future offers. 1 Implications for Improving Efficiency Delays in reaching agreements are a source of economic inefficiency. Strikes and holdouts are detrimental to both workers and firms. Legislative stalemates have caused government shutdowns. Delays in renegotiating debt contracts can have negative macroeconomic consequences. And so on. A first step to moving past these inefficiencies is a better understanding of their cause. Consider the case where the source of the inefficiency is uncertainty about surprise moves in the negotiation process. This can be thought of as a case where Bargainers are trapped in a situation with a bad set of beliefs (about play of the game). There are mutually beneficial outcomes; each Bargainer fears making such a mutually beneficial offer, uncertain how the other will react to the unexpected. One might conjecture that a mediator can be particularly effective in such a situation helping the parties to overcome fears based on strategic uncertainty. This is not to suggest that mediation is only helpful when the source of bargaining impasse is 1 Also note that, here, the driving force appears to be a notion of second-order optimism, i.e., Bargainers not deviating out of fear that it will cause the other Bargainer to become more optimistic. However, it should be emphasized that, as of now, there is no proof that the driving force is such a second-order optimism. 4

5 strategic uncertainty. Rather, the claim is that identifying the source of impasse is an important step toward implementing effective mediation. In particular, the source may influence the type of mediation that would be most effective. For instance, if the primary source of inefficiency is private information about a Bargainer s valuation, the mediator would need to find ways to overcome the informational asymmetry. On the other hand, if the primary source of inefficiency is strategic uncertainty (in the sense put forward here), the mediator would need to find ways to help the parties overcome fears of putting good offers on the table. The paper proceeds as follows. Section 1 describes the strategic situation, i.e., the bargaining game and the Bargainers type structure. Section 2 formalizes the key epistemic conditions. Section 3 provides the main theorem, a characterization of the set of outcomes consistent with rationality, forward induction reasoning and on path strategic certainty. The result is further explored in Section 4. Section 5 explores implications for delay and Section 6 discusses comparative statics. Section 7 revisits the assumption of on path strategic certainty. 1 The Description The epistemic game describes the strategic situation. It consists of a Bargaining game, denoted B, and a type structure, denoted T. The type structure describes the Bargainers strategy uncertainty it gives an implicit description of Bargainers hierarchies of conditional beliefs about how the other bargainer plays the game. We now describe these two components. Bargaining Game The game is the canonical alternating offers Bargaining model of Ståhl (1977); Rubinstein (1982): Two bargainers, viz. B1 and B2, negotiate on how to split a pie [0, 1]. We will refer to B1 as she and B2 as he. Write i for a particular bargainer and i for the other bargainer. In each bargaining phase, some Bargainer i (henceforth, Bi) takes on the role of the proposer and the other Bargainer (henceforth, B( i)) takes on the role of the responder. In particular: Bi makes a proposal x [0, 1]. B( i) chooses to Accept (A) or Reject (R) the proposal. If B( i) chooses A, the game is over: Bi gets x and B( i) gets 1 x. If B( i) chooses R, then a new bargaining phase begins. In the new bargaining phase B( i) is in the proposer s role. Period 1 begins with B1 in the proposer role; if the game does not conclude, Period 2 continues with B2 in the proposer role, etc. The game can last for at most N N + { } periods. The game has a deadline if and only if N is finite. 5

6 An outcome of the game is some (x 1, x 2, n), where x i denotes Bi s share of the pie and n denotes the period in which x 1 : x 2 are determined. There are two types of outcomes: An agreement outcome (x 1, x 2, n) = (y, 1 y, n) is associated with a division of the pie, viz. (x 1, x 2 ) = (y, 1 y), and a period at which the Bargainers agree to the division, viz. n. A disagreement outcome, viz. (x 1, x 2, N) = (0, 0, N), results if all offers are perpetually rejected. At times we will refer to (x 1, x 2, n) as an n-period outcome. Each Bargainer discounts the future; the discount factor is δ (0, 1). Thus, a utility function for Bargainer i is given by Π i (x 1, x 2, n) = δ n 1 x i if n is finite and Π i (x 1, x 2, n) = 0 if n is infinite. A history, viz. h, is a sequence of moves; a history can be identified with a node of the game and so with an information set. We will say that h is an n-period history if it occurs in a n th -bargaining phase. The Bargainer who moves at h either takes on the proposer s role (and chooses an element of [0, 1]) or takes on the responder s role (and chooses an element of {A, R}). Write H P i for the set of histories at which Bargainer i takes on the proposer s role and H R i for the set of histories at which Bargainer i takes the responder s role. Then, H i = H P i H R i is the set of histories at which i moves and H = H 1 H 2 is the set of non-terminal histories. Write Z for the set of terminal histories. A strategy s i maps each information set into a choice available at that information set, i.e., s i : H i [0, 1] {A, R} with s i (h) [0, 1] for h Hi P and s i (h) {A, R} for h Hi R. Write S i for the set of strategies of Bi. Say a strategy s i allows history h H Z if there is some strategy s i so that the path induced by (s i, s i ) passes through h. Write S i (h) for the set of strategies of Bi that allow information set h H Z. Each strategy profile (s 1, s 2 ) S 1 S 2 induces a terminal history z Z. Write ζ : S 1 S 2 Z for the mapping from strategy profiles to terminal histories. Each terminal history z Z induces an outcome; write ξ for the mapping from terminal histories to outcomes. Then, the strategicform payoff function for Bargainer i is π i = Π i ξ ζ. Note, there are two parameters of the Bargaining game B: the horizon N and the discount factor δ. At times, we will want to emphasize that we are looking at a Bargaining game with particular parameters N and δ. In that case, we will write B[N, δ]. Type Structure The premise is that each Bargainer faces uncertainty about how the other plays the game. Thus, we will want to specify Bi s belief about S i, i.e., about the strategy that B( i) chooses. But, note, Bi may be forced to revise his beliefs during the course of play. For instance, B2 may begin the game assigning probability one to B1 offering x = 1 4 upfront, only to find her, instead, offering a larger share x = 1 2. At that point, B2 will need to form a new assessment about B1 s future play, e.g., which future offers will she accept or reject. Thus, we will describe Bargainers as having beliefs at each information set; this system should satisfy the rules of conditional probability when possible. 6

7 We have just described a so-called first-order systems of conditional beliefs, i.e., a system of conditional beliefs about the play of the game. Whether a strategy is rational vs. irrational may depend on the Bargainer s first-order system of conditional beliefs. For instance, it may be rational for B1 to offer a particular split of the pie upfront, e.g., x = 1 2, if she has particular beliefs about B2 s (current and future) play; but it may be irrational to do so if she has other beliefs. We will not only want to capture the idea that each Bargainer is rational, but the idea that each Bargainer thinks the other Bargainer is rational. To do so, we cannot only specify a belief about the other Bargainer s play of the game after all, whether a strategy is rational vs. irrational for B2 may depend on what B2 believes about B1 s play of the game. Thus, we will need to specify B1 s so-called second-order system of conditional beliefs, i.e., a system of conditional beliefs about both B2 s play and B2 s belief about B1 s play. Continuing further along these lines, we will want to specify B1 s hierarchy of conditional beliefs about B2 s play of the game. We will implicitly describe the hierarchies of conditional beliefs by a type structure. The type structure will be a modification of Harsanyi s (1967) type structure model, now modified to give beliefs at each information set. (The modification was introduced in Battigalli and Siniscalchi, 1999.) We now proceed to give the formal definition. Some mathematical preliminaries will be of use: Fix a metric space Ω and the Borel sigmaalgebra thereof. Endow the product of metric spaces with the product topology, unless otherwise explicitly stated. Write P(Ω) for the set of Borel probability measures on Ω. Endow P(Ω) with the weak topology. A conditional probability space is some (Ω, E), where each element of E is a Borel subset of Ω. The collection E is a set of conditioning events. An array of conditional measures on (Ω, E) is some µ where, for each conditioning event E E, µ( E) P(Ω) with µ(e E) = 1. Write A(Ω, E) for the set of arrays of conditional measures on (Ω, E), or simply A(Ω) when E is clear from the context. Note, A(Ω) = E E {µ P(E) = 1} is the product of Borel sets. (See Aliprantis and Border, 2007, Lemma ) Endow A(Ω) with the product sigma-algebra. Definition 1.1. A countable conditional probability system (CPS) on (Ω, E) is some µ A(Ω, E) satisfying the following properties: (i) For each E E, µ( E) P(Ω) has countable support (ii) For any E F G, E is Borel, and F, G E, then µ(e G) = µ(e F )µ(f G). Note, a countable CPS ensures that the array of beliefs satisfy the rules of conditional probability when possible. Of course, an array may satisfy the rules of conditional probability, but need not be a countable CPS. Write C (Ω, E) for the set of countable conditional probability systems on (Ω, E). When the set E is clear from the context, simply write C (Ω). We can now formally describe the Bargainers strategic uncertainty. Bi is uncertain about which strategy B( i) will choose. Thus, she will have a belief about S i at the beginning of the game and at each history at which she moves. Her belief at history h assigns probability one to 7

8 the history h being reached. Write S i = {S i (h) : h H i {φ}}, where φ represents the initial history. The set S i will, in a sense, correspond to the set of i s conditioning events. Endow S i with the uniform metric, so that each element of S i is Borel. (See Lemma A.1.) Definition 1.2. A B-based type structure is some T = (B; T 1, T 2 ; S 1, S 2 ; β 1, β 2 ) so that: (i) T i is a metrizable type set for Bi; (ii) S i T i = {S i (h) T i : h H i {φ}} is the set of conditioning events for Bi; (iii) β i : T i A(S i T i ; S i T i ) is a measurable belief map for Bi. A B-based conditional type structure is a B-based type structure T = (B; T 1, T 2 ; S 1, S 2 ; β 1, β 2 ) where, for each i = 1, 2, β i (T i ) C (S i T i ; S i T i ). We will abuse notation and write β i,h (t i ) for the measure β i (t i )( S i (h) T i ). Remark 1.1. A technical remark: Fix a B-based type structure T, where T may not be a conditional type structure. Consider histories (h, x) H1 R and (h, x, R) HR 1, i.e., that differ only in that B1 rejected the proposal on the table. Note S 2 (h, x) T 2 = S 2 (h, x, R) T 2, i.e., there is a single conditioning event that corresponds to both histories (h, x) and (h, x, R). So, each type of B1 is constrained to have the same belief at each of these histories. This fact will be used in the analysis. Note, a type structure induces hierarchies of conditional beliefs about the strategies played: Each type t i has a system of conditional beliefs on the strategies and types of the other Bargainers, viz. β i (t i ). By marginalizing onto S i, each type t i then has a system of first-order beliefs on the strategies of the other Bargainer. Since, each type t i has such a system of first-order beliefs, this induces each type t i s system of second-order beliefs, i.e., on the strategies and first-order beliefs of the other Bargainer. And so on. 2 Epistemic Game An epistemic game is a pair (B, T ), where T is a B-based type structure. A conditional epistemic game is an epistemic game (B, T ) where T is a B-based conditional type structure. An epistemic game induces a set of states, viz. S 1 T 1 S 2 T 2. A state (s 1, t 1, s 2, t 2 ) describes the Bargainers play, viz. s 1 and s 2, and the Bargainers beliefs, viz. β 1 (t 1 ) and β 2 (t 2 ). 2 This is, of course, an informal argument. A related formal argument appears in Battigalli and Siniscalchi (1999). However, that argument does not quite apply here, as the set of conditioning events is uncountable. We do not attempt formalize this argument within the context of this paper. 8

9 2 Epistemic Conditions Throughout this section, we will fix an epistemic game (B, T ). We will impose restrictions on the set of states, which correspond to forward induction reasoning and on path strategic certainty. We now review the basic ingredients. Rationality The basic starting point will be to consider the requirement that each Bargainer is rational. Recall, a strategy of Bi may be rational given some belief about B( i) s play and irrational given some other belief. Since types specify beliefs (via the belief map), rationality is a property of a strategy-type pair. The idea will be that (s i, t i ) is rational if s i maximizes Bi s expected payoff under t i, at each history allowed by s i. Definition 2.1. Fix a strategy s i and an array of conditional measures µ i A(S i ; S i ). Say s i is sequentially optimal under µ i if, for each information set h H i with s i S i (h), the following hold: (i) for each r i S i (h), π i (r i, ) : S i R is µ i ( S i (h))-integrable and (ii) for each r i S i (h), S i [π i (s i, ) π i (r i, )]dµ i ( S i (h)). Remark 2.1. Refer to Condition (i) of Definition 2.1: For each h H i and each r i S i (h), Bi must be able to compute her expected payoff under µ i ( S i (h)). This will necessarily hold if µ i ( S i (h)) has countable support or if each π i (r i, ) is measurable. 3 Fix a pair (B, T ). For each µ i A(S i T i ; S i T i ), write marg S i µ i for the marginal array of measures. This is an array of measures ν i A(S i ; S i ) with ν i ( S i (h)) = marg S i µ i ( S i (h) T i ). Definition 2.2. A strategy-type pair (s i, t i ) S i T i is rational if s i is sequentially optimal under the marginal array marg S i β i (t i ). Write R i for the set of rational strategy-type pairs for i and R = R 1 R 2 for the set of states at which each Bargainer is rational. Rationality is a requirement about maximizing subjective conditional expected utility given the Bargainer s beliefs. It does not impose any requirements on what the Bargainer s belief is. The next step imposes restrictions on the Bargainer s beliefs. There are two types of restrictions: one that will arise from forward induction reasoning and the second that will arise from on path strategic certainty. Forward Induction Reasoning Forward induction is an idea that goes back to Kohlberg (1981). It is the idea that Bargainers rationalize past behavior whenever possible. 3 A future draft of this paper will deal with this in a more appropriate manner. 9

10 If a type t 1 of B2 rationalizes B1 s past behavior when possible, the type should assign probability one to the event B2 is rational, whenever she reaches an history consistent with this event. The key ingredient is the concept of strong belief: Definition 2.3 (Battigalli and Siniscalchi, 2002). A type t i strongly believes an event E i S i T i if E i [S i (h) T i ] implies β i,h (t i )(E i ) = 1. Write SB i (E i ) = S i {t i : t i strongly believes E i }. Forward induction reasoning is then captured by the requirements of: strong belief of rationality, strong belief of rationality and strong belief of rationality, etc. More formally, set Ri 1 = R i. Inductively define sets Ri m, for each m N+, so that Ri m+1 = Ri m SB i (R i m ). Set Ri = m N + Rm i. The set Rm = R1 m Rm 2 is the set of states at which there is rationality and (m 1) th -order strong belief of rationality. The set R = R1 R 2 is the set of states at which there is rationality and common strong belief of rationality (RCSBR). The set of states at which there is RCSBR corresponds to rationality plus forward induction reasoning. On Path Strategic Certainty On path strategic certainty is that idea that, along the path of play, no Bargainer faces uncertainty about the terminal node that will be reached even though, along the path of play, the Bargainers may very well face uncertainty about the actual strategy the other Bargainer employs. The path of play is determined by a state. Specifically, a given state (s 1, t 1, s 2, t 2 ) induces a path through the tree to a particular terminal node z. At (s 1, t 1, s 2, t 2 ), there is on path strategic certainty if, at each history h along the path induced by that state, type t 1 (resp. t 2 ) assigns probability one to the event that the terminal node z will be reached. To formalize this idea, write Z i [s 1, s 2 ] for the set {r i : ζ(s i, r i ) = ζ(s i, s i )}. This is the set of strategies of i that will induce the same terminal node as (s i, s i ), viz. z = ζ(s i, s i ), when Bi plays s i. This set is closed. (See Corollary A.1.) At the state (s 1, t 1, s 2, t 2 ) there is on path strategic certainty if each t i strongly believes the event Z i [s 1, s 2 ] = {r i : ζ(s i, r i ) = ζ(s i, s i )} T i S i T i. Write C = (s 1,s 2 ) S 1 S 2 [({s 1 } T 1 ) SB 1 (Z 2 [s 1, s 2 ]))] [({s 2 } T 2 ) SB 2 (Z 1 [s 1, s 2 ]))]. Then, C is the set of states at which there is on path strategic certainty. If there is on path strategic certainty at (s 1, t 1, s 2, t 2 ), then at each history allowed by (s 1, s 2 ), t 1 and t 2 have correct beliefs about the terminal node that will obtain. It is important to note that, at histories precluded by (s 1, s 2 ), types t 1 and t 2 cannot have correct beliefs about the path of play. (This is by definition. ) And, in fact, they may very well have different beliefs about the path. 10

11 At times, we will want to study the assumption that, at a state, Bargainers reason about on path strategic certainty. For instance, at (s 1, t 1, s 2, t 2 ) C, t 1 may strongly believe {(s 2, t 2 ) : there is some (r 1, u 1, s 2, t 2 ) C}, i.e., at each information set consistent with some state at which there is on path strategic certainty, t 1 assigns probability one to the event that B2 plays a strategy-type pair that is consistent with on path strategic certainty. 4 With this in mind, set C 1 = C and inductively define C m so that C m+1 = C m [ SB 1 (proj S2 T 2 C m ) SB 2 (proj S1 T 1 C m ) ]. Then the set C = m=1 Cm is the set of states at which there is on path strategic certainty and common strong belief of on path strategic certainty. Remark 2.2. The conditions of rationality, strong belief of rationality, and on path strategic certainty are related to by distinct from the equilibrium dominance criterion discussed in Battigalli and Siniscalchi (2002, Section 6.1). The equilibrium dominance criterion looks at states at which there is rationality and strong belief of rationality and the correct path of play. Here we look at states at which there is rationality, strong belief of rationality, and strong belief of the correct path of play. This implies the equilibrium dominance criterion. However, the converse does not hold. In particular, strong belief of rationality and the correct path of play allows a player to give up on the other s rationality, once the other has deviated from the given path of play. Here, if a player can continue to believe the other is rational, she does so even if the other has deviated from the proposed path of play. 3 Characterization Theorem Fix a Bargaining game, viz. B[N, δ]. This section characterizes the set of outcomes consistent with rationality, forward induction reasoning, and on path strategic certainty, i.e., across all epistemic games based on B[N, δ]. Theorem 3.1. Fix a Bargaining Game B[N, δ]. For each finite n N, there exists an interval [x n, x n ] so that the following are equivalent: (i) There is an epistemic game (B[N, δ], T ) and a state thereof (s 1, t 1, s 2, t 2 ) that induces the outcome (x 1, x 2, n) so that, at (s 1, t 1, s 2, t 2 ), there is rationality, common strong belief of rationality, and on path strategic certainty. (ii) x [x n, x n ]. 4 This is one way to formalize reasoning about on path strategic certainty. As will become clear, the particular choice will not be important for our purposes. 11

12 In fact, we will show a stronger result than that presented above. There will be two halves of the proof. The first is to show necessity, i.e., that part (i) implies that the outcome (x 1, x 2, n) necessarily has x 1 in the set specified by part (ii). For that, we will be able to weaken the premise in part (i). We will assume that there is a state at which there is rationality, strong belief of rationality and on path strategic certainty. We will show that then the outcome will necessarily satisfy the requirement in part (ii). The second is to show sufficiency, i.e., that part (ii) implies part (i). For that we will be able to strengthen the conclusion in part (i). Specifically, we will fix some x [x n, x n ] and show that we can construct a conditional epistemic game and an associated state at which there is rationality, common strong belief of rationality, on path strategic certainty, and common strong belief of on path strategic certainty. It is important to note that there is no presumption that the sets [x n, x n ] are necessarily nonempty. We will be able to calculate these sets based on parameters of the Bargaining game (that is, the deadline and discount factor); whether the set is empty versus non-empty will depend on these parameters plus the particular period n in question. Specifically, for each finite n N, take x n max{ 1 δ, δ N n } δ = n 1 1 δ δ n 1 if N < is odd otherwise and x n min{1 δ(1 δ), 1 δ N n } if N < is even, δ = n 1 1 δ(1 δ) otherwise. δ n 1 Take x = 1 and x = 0. Take the case of a two-period deadline, i.e., N = 2. If there were delay in this case, i.e., an outcome (x 1, x 2, 2), then x 1 would be required to be in [x2, x 2 ] = [ 1 δ δ, 0], an impossibility. Thus, in the case of a two-period deadline, there cannot be delay (under the assumptions of Theorem 3.1). Let us understand why: Two-Period Deadline: No Delay Fix an epistemic Bargaining game and a state (s 1, t 1, s 2, t 2 ) at which there is rationality, strong belief of rationality, on path strategic certainty. Suppose, at the state, there is delay in reaching agreement, i.e., in the second-period, B1 and B2 get shares x 1 and x 2 of the pie. We will show that this cannot be the case. Consider the path of play induced by (s 1, s 2 ). Along the path, there is an information set h at which B2 proposes. At that information set, B2 can continue to maintain the hypothesis that B1 is rational. (The state allows h and is consistent with the event B1 is rational. ) Thus, when B2 proposes at h, t 2 must assign probability one to the event B1 accepts any offer y < 1. Since, at this state, B2 is rational, it follows that s 2 must offer y = 1 at h. Thus, irrespective of whether s 1 accepts or rejects the offer, B1 gets 0 at this state and, so, by on path strategic certainty, t 1 must expect to get 0 at the initial node. But, t 1 also strongly believes the event B2 is rational. 12

13 At the initial node, t 1 can maintain a hypothesis that B2 is rational and so, at the initial node, must assign probability one to the event B2 accepts any offer x < 1 δ. (Such an offer would t 1 give B2 a share 1 x > δ, which is better than any possible future agreement.) By offering some x (0, 1 δ), t 1 can improve her expected payoff over s 1. Thus, we conclude that: When bargaining with a two-period deadline, the Bargainers must agree immediately, under the assumptions of rationality, strong belief of rationality and on path strategic certainty. Now we turn to bargaining with a three-period deadline. Three-Period Deadline: Limits on Delay Fix an epistemic Bargaining game and a state (s 1, t 1, s 2, t 2 ) at which there is rationality, strong belief of rationality, on path strategic certainty. Suppose, at the state, there is delay in reaching agreement. Repeating the argument for the two-period deadline, we can conclude that, at this state, the players must agree in the second period, i.e., the outcome is (x 1, x 2, 2). We will show that there are limits on this sort of delay; in particular, x 1 = δ. Note, along the path of play induced by (s 1, s 2 ), there is an information set h at which B2 proposes x 2 and subsequently the offer is accepted. By on path strategic certainty, when B2 proposes x 2, the type t 2 believes B1 will accept the offer, i.e., expects his payoffs to be δx 2. Moreover, since t 2 strongly believes B1 is rational and h is consistent with the event that she is rational, t 2 must assign probability one to the event B1 accepts any offer y < 1 δ. (Such an offer would give B1 a share 1 y > δ, which is larger than the discounted third-period pie.) Thus, by rationality, δx 2 δy for all y < 1 δ. From this it follows that δ x 1. Now turn to B1. At h, B2 makes the offer of x 2. At that point, B1 can continue to maintain the hypothesis that B2 is rational. So, using the fact that t 1 strongly believes that B2 is rational, must believe that B2 will accept any third period offer z < 1. Since, at the state, B1 is rational, t 1 δx 1 δ2 z, for all z < 1 or x 1 δ. In sum, we have two requirements: B2 does not have an incentive to make a particular offer upfront, i.e., the first time he can make an offer. B1 does not have an incentive to wait for the deadline. Taken together, these two requirements say that x 1 = δ. There are two things to take note of in the above argument. First, we used on path strategic uncertainty to conclude that B2 must expect his offer x 2 to be accepted by B1. We could alternatively obtain this conclusion by assuming that B2 is rational and strongly believes B1 is rational and strongly believes I am rational. We will discuss this in Section 7. (See Proposition 7.1.) Second, the above requirements only provide part of the picture they are necessary requirements. Notice, for instance, they are silent about requiring that B1 not have an incentive to make 13

14 an alternate offer upfront. And, indeed, she might: By on path strategic certainty, at the start of the game, t 1 anticipates her payoffs will be δ2. At that point, she also continues to maintain the hypothesis that B2 is rational and, so, she anticipates that, if she makes an offer of x < 1 δ upfront, B2 will accept the offer. (In that case, B2 gets 1 x > δ which is more than tomorrow s discounted pie.) If t 1 is to wait for an agreement in the second period, δ2 1 δ. In light of the above, there is a third requirement: B1 does not have an incentive to make a particular offer upfront, i.e, the first time she can make an offer. This is a requirement that the discount factor must be sufficiently high to make waiting profitable. Three-Period Deadline: Possibilities for Delay Suppose that δ 2 1 δ. We will focus on a particular strategy profile (s 1, s 2 ), that results in agreeing on a δ : 1 δ split in the secondperiod. We will informally argue that we can construct a type structure so that there is a state at which there is rationality, forward induction reasoning, on path strategic certainty, and (s 1, s 2 ) is played. The strategy profile (s 1, s 2 ) has the following features: B1 s Strategy, s 1 : At the initial node, B1 offers to take the full pie for herself, i.e., x = 1. If this offer is rejected and, subsequently, B2 offers to take y for himself, B1 Accepts if and only if y 1 δ. In the third period, B1 offers to take the full pie for herself, i.e., z = 1, irrespective of history. B2 s Strategy, s 2 : If, at the initial node, B1 offers to take x for herself, B2 Accepts if and only if x < 1 δ. If B1 s initial offer was x = 1, B2 Rejects and subsequently makes an offer to take y = 1 δ for himself. If B1 s initial offer was x [1 δ, 1), B2 Rejects and subsequently makes an offer to take the full pie for herself, i.e., y = 1. At each third-period information set, B2 Accepts an offer of z if and only if z 1, irrespective of history. The idea will be to construct a type structure with one type for each Bargainer, viz. T 1 = {t 1 } and T 2 = {t 2 }. The belief of type t 1 (resp. t 2 ) will assign probability one to (s 2, t 2 ) (resp. (s 1, t 1 )), at any information set allowed by s 2 (resp. s 1 ). Thus, there will be on path strategic certainty, at the state (s 1, t 1, s 2, t 2 ). At an information set h inconsistent with s 2 (resp. s 1 ), t 1 (resp. t 2 ) will assign probability one to a so-called accommodating strategy of B2 (resp. B1). B2 s accommodating strategy is one that allows h and subsequently accommodates B1 by accepting any third period offer. B1 s accommodating strategy is one that allows h and subsequently accommodates B2 by accepting any offer (if allowed by h) and proposes to take zero share of the pie (i.e., z = 0). Notice, at (s 1, t 1, s 2, t 2 ), each Bargainer is rational. This may seem peculiar, at first: For instance, take δ =.7. At the start of the game, both t 1 and t 2 expect payoffs of.49 and.21. Thus, they would both strictly prefer to accept a.6 :.4 split in the first period. But offering 14

15 x =.6 upfront would give t 1 a strictly lower expected payoff. Type t 1 expects that if she were to make this offer, then B2 would respond by rejecting the offer and proposing to take the full pie. And, indeed, type t 1 would be correct about this presumption; at (s 1, t 1, s 2, t 2 ), B2 would reject such a mutually beneficial offer. While B2 rejects such a mutually beneficial offer at this state, B2 is nonetheless rational at this state. In particular, when such a beneficial offer is made, t 2 is forced to update his belief; now t 2 expects B1 to accept an offer that gives B2 the full pie. More loosely, when such a beneficial offer is made, t 2 becomes more optimistic about his future prospects and so accepting what appears to be a better offer is not a best response for t 2. How can t 2 s belief be consistent with forward induction reasoning after all, conditional upon a mutually beneficial offer being made, he thinks that B1 will accept a zero-share of the pie? The key is that (under the construction) when B1 offers, say, x =.6 upfront, B2 must maintain a hypothesis that B1 is irrational: At the initial node, every type of B1 believes that B2 rejects such a mutually beneficial offer and responds by offering a zero-share of the pie (to B1). Thus, conditional upon B1 making such an offer, B2 must believe that B1 has not maximized her expected payoffs. At this point, he may reason that B1 will again fail to maximize her expected payoffs in the future. 5 4 Revisiting the Characterization Theorem The Characterization Theorem can be seen as a consequence of two Propositions: one showing necessity and the second showing sufficiency. Each of these Propositions will provide an important strengthening of one half of the Characterization Theorem. 4.1 Necessity Here we establish that the behavioral predictions of rationality, strong belief of rationality, and on path strategic certainty must be contained in some set [x n, x n ]. Proposition 4.1. Fix an epistemic game (B[N, δ], T ) and a state (s 1, t 1, s 2, t 2) R [SB 1 (R 2 ) SB 2 (R 1 )] C. The outcome induced by (s 1, s 2 ) is (x 1, x 2, n), where x 1 [xn, x n ]. We now provide the intuition for the result. To do so, fix a state at which there is rationality, strong belief of rationality and on path strategic certainty. Suppose, at this state, the Bargainers agree in period n on an x 1 : x 2 split of the pie. Under our epistemic assumptions, there are two constraints: an upfront constraint and a deadline constraint. 5 Of course, there exists a different epistemic game where, conditional upon B1 making such a mutually beneficial offer upfront, B2 updates his belief but, nonetheless, concludes that B1 will maximize her expected payoffs in the future. That structure will have different implications for forward induction reasoning. 15

16 Upfront Constraint At the start of the game, the Bargainers anticipate that they will be able to reach agreement on a x 1 : x 2 split in period n. (This is by on path strategic certainty.) Upfront, they also each believe that the other Bargainer is rational. So, upfront, they each anticipate the other Bargainer will accept any upfront offer that gives the other Bargainer more than the discounted total pie, i.e., more than δ. Thus, each Bargainer must prefer an x 1 : x 2 split in period n to making an offer that gives the other Bargainer a δ share of the pie upfront. Note, the idea of an upfront offer is implemented differently for the two Bargainers. For B1, an upfront offer involves an offer in the first bargaining phase. For B2, an upfront offer involves an offer in the second bargaining phase. (In the case of immediate agreement, B2 is certain of the offer she accepts and, at the same time by strong belief of rationality maintains a hypothesis that she can induce B1 to accept a sufficiently good offer in period 2.) As such, the upfront constraint requires x 1 1 δ and x δ n 1 2 δ(1 δ). δ n 1 Deadline Constraint Take the case of a deadline N < and suppose Bi is the proposer in the last bargaining phase N. There is an n-period history at which either (i) Bi accepts a x 1 : x 2 split, or (ii) Bi proposes a x 1 : x 2 split. In either case, at the given n-period history, he expects the outcome to be a x 1 : x 2 split. (This is by on path strategic certainty.) Note, at that point, he continues to maintain the hypothesis that the other Bargainer is rational. Thus, he maintains the hypothesis that, if the final bargaining phase is reached, the other Bargainer will accept any strictly positive share of the pie. Thus, Bi must prefer a x 1 : x 2 split in period n to waiting for (essentially) the full pie in period N, i.e., x i δn n. This is Bi s deadline constraint. Of course, for any given bargaining game, there is at most one Bargainer for which the deadline constraint is active, i.e., the Bargainer who proposes in the final bargaining phase. If Bi is the proposer in the final bargaining phase, we will say that Bi has deadline bargaining power. Note, carefully, the deadline bargaining power of i does not arise because, under our epistemic assumptions, Bi will get the full pie in the final period. (If our epistemic assumptions are met, the final period will never be met. If they fail, Bi need not get the full pie in the final period.) Instead, it arises because, under our epistemic assumptions, at the point of reaching an agreement, Bi anticipates that he would be able to obtain the full pie in the final period. Of course, if an agreement is not reached, Bi may very well rethink such an assessment. 4.2 Sufficiency Here we establish that x [x n, x n ] is sufficient to guarantee that (x, 1 x, n) is an outcome consistent with rationality, common strong belief of rationality, on path strategic certainty and common strong belief of on path strategic certainty, in a conditional epistemic game. Proposition 4.2. Fix some n N and some x [x n, x n ]. There is a conditional epistemic game, viz. (B, T ), and a state (s 1, t 1, s 2, t 2 ) thereof, so that (i) (s 1, t 1, s 2, t 2 ) induces the outcome (x 1, x 2, n) = (x, 1 x, n), and 16

17 (ii) (s 1, t 1, s 2, t 2 ) R C. Throughout the exposition, we fix some finite time period n N and some x [x n, x n ]. We begin by constructing particular strategies s 1 and s 2, so that (s 1, s 2 ) induces the outcome (x 1, x 2, n) = (x, 1 x, n). The strategy s i satisfies the following properties. For any history h H P i, set s i (h) = x i if h = h, and s i (h) = 1 if h h. For any history h H i P, set s i (h, x) = A if and only if one of the following hold: x < 1 δ, (h, x) = (h, x i ), and h is an N-period history and x < 1. Note, the strategy profile (s 1, s 2 ) induces each Bargainer to propose 1, reject, propose 1, reject, etc., up until the n th -bargaining phase. In the n th -bargaining phase, the Proposer makes an offer which is accepted. This offer is x 1 = x if B1 is the Proposer and x 2 = 1 x if B2 is the Proposer. In either case, in the n th -bargaining phase, the Bargainers come to an agreement, with B1 getting x and B2 getting 1 x. The construction of the epistemic game will be analogous to the example with a three period deadline (pages 14-15): The idea will be that Bi begins the game with a hypothesis that B( i) plays the strategy s i. If Bi observes a deviation from this behavior, she/he updates his belief and subsequently believes B( i) will act in an accommodating manner. Thus, it will be useful to have the concept of the accommodating strategy. The accommodating strategy for Bargainer i, written a i, is a strategy so that a i (h) = 0 for all h H P i and a i (h, x) = A for all h H P i. The h-accommodating strategy for Bargainer i, written a h i otherwise agrees with the accommodating strategy. Now, construct where, for each Bi, T i = {t i } and β i,h (t i )(s i, t i ) = 1, if s i S i(h), and β i,h (t i )(ah i, t i ) = 1, if s i S i(h) Begin with some observations: Observation 4.1. T = (B; T 1, T 2 ; S 1, S 2 ; β 1, β 2 ), (i) The epistemic game (B, T ) is a conditional epistemic game. is a strategy that allows h but 17

18 (ii) At the state (s 1, t 1, s 2, t 2 ), there is on path strategic certainty and common strong belief of on path strategy certainty. Part (i) follows from the following fact: If S i (h ) T i S i (h) T i and a h i S i(h ), then the h -accommodating strategy a h i is the h-accommodating strategy ah i. Part (ii) follows immediately from the construction. To show Proposition 4.2 it suffices to show that, at (s 1, t 1, s 2, t 2 ), there is RCSBR. The key is the following Lemma: Lemma 4.1. (i) For each i, (s i, t i ) is rational. (ii) For each i and h H 1 H 2, R i [S i (h) T i ] implies s i S i(h). The proof of Lemma 4.1 can be found in the Appendix. Proof of Proposition 4.2. By Lemma 4.1(i), (s 1, t 1, s 2, t 2 ) R 1 R 2. Fix an information set h H i {φ} with R i [S i (h) T i ]. By Lemma 4.1(ii), s i S i(h). So, by construction, t i strongly believes the event R1 i. This delivers that R2 i = R1 i. Proceeding inductively, Rm i for all m, and so (s 1, t 1, s 2, t 2 ) Rm 1 Rm 2 for each m. = R 1 i Remark 4.1. The construction displays the following no indifference property: Fix an information set h along the path of play. If r i S i (h) and r i is a best response at h under marg S i β i,h (t i ), then r i (h) = s i (h). More loosely, along the path of play, no Bargainer is indifferent between any two actions. Lemma C.1 shows that a stronger no indifference property holds: If t i does not have a uniquely optimal action at some h H i allowed by s i, then h is a history at which Bi is in the receiver role and has received an offer of 1 δ. (This cannot happen along the path of play induced by (s 1, s 2 ).) We constructed s i to reject such an offer. This choice was made to simplify the construction; the choice was not instrumental to the conclusion. 6 5 Implications for Delay Section 3 fixed a bargaining game and provided a Characterization Theorem: a characterization of the set of outcomes consistent with rationality, forward induction reasoning, and on path strategic certainty (across all associated epistemic games). This section uses the Characterization Theorem to point to both the possibilities for and bounds on delay. We begin by pointing out that there are limits to delay. A preliminary observation: 6 Put differently: We could have chosen s i to reject such an offer. But, then, constructing a state at which there is RCSBR would have involved a significantly more delicate argument. A proof is available upon request. 18

19 Observation 5.1. Fix a Bargaining game B[N, δ] with N 2. There exists some finite ˆn(δ) with N ˆn(δ) 2 so that [x n, x n ] = if and only if n ˆn(δ). Fix a Bargaining game. If there is rationality, strong belief of rationality, and on path strategic certainty, there are bounds on delay that are determined by the given deadline (if there is one) and the discount factor. The next proposition establishes that, in the case of a deadline, there are bounds on delay that exist independent of the discount factor. Proposition 5.1. Fix a Bargaining game B[N, δ], with a deadline N. (i) [x N, x N ] =. (ii) If [x N 1, x N 1 ], then N 3. To understand part (i), refer to the case of a two-period deadline. If there is rationality, strong belief of rationality, and on path strategic certainty, then the Bargainers reach immediate agreement. Taken together, Propositions 4.1 and 5.1(i) provide an analogue for the case of an N-period bargaining game. At any state at which there is rationality, strong belief of rationality, and on path strategic certainty, the Bargainers agree prior to the deadline. To understand part (ii), refer to the case of a three-period deadline. Fix a state at which there is rationality, strong belief of rationality, and on path strategic certainty. Suppose, at that state, there is delay until the penultimate period, viz. n = N 1 = 2. B1 s deadline constraint implies that, at that state, she must get at least a δ share of the pie; when the Bargainers agree on (x 1, x 2, 2), B1 maintains the hypothesis that B2 is rational, and so she thinks she can get the full share of the pie in the final period. B2 s upfront constraint implies that, at the state, he must get at least a 1 δ share of the pie (and so B1 s share can be at most δ); when he makes a second period proposal of x 2, he maintains the hypothesis that B1 is rational and so B1 will accept any offer of x [0, 1 δ). Put together, this constrains x 1 = δ. Now suppose the deadline is N 4 and Bi is the proposer in the last period. Fix a state at which there is rationality, strong belief of rationality, and on path strategic certainty. Suppose, at that state, there is delay until the penultimate period, viz. n = N 1 2. Bi s deadline constraint continues to imply that, at the state, she must get at least a δ share of the pie. But, we will argue, that B( i) s upfront constraint implies that he must get strictly more than 1 δ. Thus, the two Bargainers cannot come to an agreement in period n = N 1. To see that B( i) s upfront constraint implies that he must get strictly more than 1 δ: Since N 4, there must be some earlier period, viz. k N 3, at which B( i) is in the proposer role. In the k th bargaining phase that occurs along the path of play, B( i) can continue to maintain the hypothesis that Bi is rational. So, there, B( i) thinks that Bi will accept any share of the pie that gives her, i.e., Bi, strictly more than δ. Thus, B( i) can secure 1 δ upfront. This is worth strictly more than 1 δ in period n = N 1. By on path strategic certainty, B( i) correctly anticipates his n = N 1 period share of the pie. Since he is rational and willing to wait, it must be the case that, in period n = N 1, he gets strictly more than a 1 δ share of the pie. 19

20 Nonetheless, for each n N 2, there is the possibility for delay, provided the discount factor is sufficiently large. More precisely: Proposition 5.2. Fix n with N 2 n 2. There exists δ[n, n] ( 1 2, 1) so that [xn, x n ] if and only if δ δ[n, n]. Taken together, Propositions 4.2 and 5.2 give the following: Given some N period deadline and some period n with N 2 n 2. There exists a conditional epistemic game (B[N, δ], T ) and a state at which there is rationality, common strong belief of rationality, on path strategic certainty, common strong belief of on path strategic certainty, and agreement in the n th period. Implicit in this last statement is a requirement that the discount factor δ be sufficiently high specifically, sufficiently high requires that the discount factor is greater than 1 2. It is important to note that this still allows for delay when the discount factor is far from 1. For instance, in the case of delay until the second period (n = 2), it suffices for the discount factor to be at least 5 (irrespective of the deadline) It is of interest to note that, for a given deadline N, we can choose a sufficiently high discount factor δ so that the above holds for all n N 2. Specifically: Remark 5.1. Fix some N. There exists δ[n] ( 1 2, 1) so that, for each δ δ[n] and n N 2, [x n, x n ]. 6 Comparative Statics How does changing the exogenous parameters of the bargaining game change the resulting behavioral predictions? When the description of the strategic situation is given by an epistemic game, there are (at least) two ways to ask this question. We will refer to the first as an within comparative static analysis and refer to the second as an across comparative static analysis. Within Comparative Statics The idea here is to analyze comparative statics within an epistemic game. It begins with an epistemic game and changes an exogenous parameter holding all other features of the epistemic game fixed including the type structure. It applies the epistemic conditions to the new and old epistemic game and asks how the behavioral predictions have changed. It is important to note that this sort of comparative statics exercise may not be meaningful. Consider increasing the deadline N. This necessitates changing the type structure, since the information sets of the game change. There is no unique or obvious way to map the original type structure to the new type structure. That is, by changing the deadline, it is impossible to identify a single description of the strategic situation. By contrast, this sort of comparative statics exercise is meaningful when we vary the discount factor varying the discount factor changes the payoff function and does not influence the 20

21 type structure. However under the assumptions of rationality, forward induction reasoning and on path strategic certainty the analysis does not yield qualitatively interesting answers. The following example makes this point. Example 6.1. Consider the case of a three-period deadline. Fix discount factors δ > δ > δ, so that 1 δ δ 2 > 0 > 1 δ δ. 2 By Proposition 4.2, there exists a conditional epistemic game (B[N, δ ], T ) and a state thereof, viz. (s 1, t 1, s 2, t 2 ), at which there is RCSBR, on path strategic certainty, and the outcome is (δ, 1 δ, 2). But, referring to analysis on pages 14-15, in each of the epistemic games (B[N, δ ], T ) and (B[N, δ ], T ), there is no state at which there is RCSBR and on path strategic certainty. Thus, when we increase the discount factor from δ to δ we strictly increase the set of predictions and when we increase the discount factor from δ to δ we strictly decrease the set of predictions. Across Comparative Statics The idea here is to analyze comparative statics across epistemic games. It begins with a game and changes an exogenous parameter holding all other features of the game fixed. It then applies the epistemic conditions to the class of all epistemic games associated with the new and old games and asks how the behavioral predictions have changed. Note, now, the behavioral predictions are not defined within a given epistemic game, but across a class of epistemic games. To perform this exercise, we will make use of the Characterization Theorem it provides the behavioral predictions of RCSBR and on path strategic certainty, across all epistemic (or conditional epistemic) games. The predictions, given by the sets [x n, x n ], depend on the parameters of the bargaining game, viz. N and δ. Thus, it will be convenient to write [x (n,n,δ), x (n,n,δ) ] to emphasize that we are computing the n-period interval in the Bargaining game B[N, δ]. Increasing the Deadline Recall, under the assumptions of rationality, strong belief of rationality, and on path strategic certainty, the Bargainer who proposes in the last period has deadline bargaining power. Increasing the deadline has two potential effects. First, it can transfer the deadline bargaining power from one Bargainer to the next. Second, it can diminish the deadline bargaining power (i.e., without transferring the deadline bargaining power from one Bargainer to the next). First consider the case where increasing the deadline shifts the deadline bargaining power from one Bargainer to the next. Here, the analysis does not appear to yield qualitatively interesting answers. This is illustrated by the following example. Example 6.2. Fix a discount factor δ > 1 δ 2. In B[3, δ], B1 has the deadline bargaining power and, in B[4, δ], B2 has the deadline bargaining power. Consider outcomes associated with agreement in period 2. Note, [x (2,3,δ), x (2,3,δ) ] = {δ} and [x (2,4,δ), x (2,4,δ) ] = [ 1 δ δ, 1 δ2 ]. Using the fact that δ > 1 δ 2, [x (2,3,δ), x (2,2,δ) ] [x (2,4,δ), x (2,4,δ) ] =. 21

22 Let us review Example 6.2. We began by increasing the deadline from N = 3 to N = 4, thereby shifting the deadlin bargaining power from B1 to B2. We studied the behavioral implications of rationality, forward induction reasoning and on path strategic certainty. By Proposition 5.1, there cannot be delay until period n = 3 (or, in the case of N = 4, until n = 4). So, focus on the case of delay until the second period. This is possible in both cases, provided the discount factor is large (i.e. 0 > 1 δ δ 2 ). But, in this case, the behavioral implications are distinct. Now consider increasing the deadline, without shifting deadline bargaining power. That is, but Bi proposes in the last period, both before and after an increase in the deadline. This increase in the deadline diminishes Bi s deadline bargaining power and, in so doing, relaxes Bi s deadline bargaining constraint. This leads to a (weakly) larger set of outcomes consistent with RCSBR (across all type structures). Proposition 6.1. For N N : (i) [x (n,2n,δ), x (n,2n,δ) ] [x (n,2n,δ), x (n,2n,δ) ] [x (n,,δ), x (n,,δ) ] and (ii) [x (n,2n +1,δ), x (n,2n +1,δ) ] [x (n,2n +1,δ), x (n,2n +1,δ) ] [x (n,,δ), x (n,,δ) ]. Increasing the Discount Factor Increasing the discount factor makes waiting for the future more profitable. This serves to diminish each Bargainer s upfront constraint. However, because it makes waiting more profitable, it also tightens the deadline constraint. 1 U 1 D 1 U Figure 6.1: n = 2 and N 5 Odd Refer to Figure 6.1 which depicts the case where B1 has deadline bargaining power, n = 2 and N 2 2. B2 s upfront constraint, viz. U 2, is strictly increasing in the discount factor. B1 s upfront constraint, viz. U 1, is strictly decreasing in the discount factor. If the Bargainers agree on an x 1 : x 2 split in period n = 2, x 1 must lie between U 1 and U 2. So, indeed, the upfront constraints are relaxed, as stated. Also notice that B1 s deadline constraint, viz. D 1, is strictly 22