Otimism, Delay and In)Efficiency in a Stochastic Model of Bargaining Juan Ortner Boston University Setember 10, 2012 Abstract I study a bilateral bargaining game in which the size of the surlus follows a stochastic rocess and in which layers might be otimistic about their bargaining ower. Following Yildiz 2003), I model otimism by assuming that layers have different beliefs about the recognition rocess. I show that the unique subgame erfect equilibrium of this game might involve inefficient delays. I also show that these inefficiencies disaear when layers can make offers arbitrarily frequently. JEL Classification Codes: C73, C78. Keywords: Bargaining, Otimism, Stochastic games, Dynamic games. Address: Deartment of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, United States. E-mail: ortner@bu.edu.
1 Introduction Exerimental and field evidence shows that, when bargaining, eole tend to form otimistic beliefs about how future uncertainty will be resolved e.g. Babcock and Loewenstein, 1997). This evidence also shows that there is a ositive correlation between the otimism of the arties and the robability that bargaining ends in an imasse. In other words, this evidence identifies otimism as a ossible exlanation for bargaining delays. However, Yildiz 2003) showed that otimism by itself cannot cause delays in bargaining. Yildiz 2003) studied a comlete information bilateral bargaining game a la Rubinstein 1982) in which layers have otimistic beliefs about their future bargaining ower. The main result of his aer is that agreement will always be immediate whenever otimism is ersistent and the number of bargaining rounds is sufficiently large. In this aer, I extend Yildiz s 2003) model by allowing the size of surlus that the layers are bargaining over to follow a stochastic rocess. The game has an infinite horizon and layers have otimistic beliefs about their bargaining ower. The model in this aer can also be thought of as an extension of the stochastic bargaining model of Merlo and Wilson 1995, 1998), allowing agents to have different beliefs about the recognition rocess. One of the main goals of the current aer is to show that ersistent otimism can lead to costly delays in this stochastic environment. 1 The following is a descrition of this aer s model and an overview of its main results. Let S be a finite set of states and let P be a transition matrix over elements in S. At each eriod until layers reach an agreement, nature selects a state s according to the one eriod ahead distribution imlied by P. The state s determines the size of the surlus. After agents learn the state, one layer is recognized to make an offer. Players may have otimistic beliefs about the recognition rocess. The recognized layer can either make a feasible offer or ass. In the first case, the resonder can either accet or reect the offer. If she accets the roosal the game ends. If the rooser chooses to ass or if the resonder reects the current offer, the game moves on to the next bargaining eriod. By adating arguments in Merlo and Wilson 1995, 1998), I show that this game has a unique subgame erfect equilibrium. The unique equilibrium satisfies the following roerty: there exists a set of states S a such that agents come to an agreement in eriod t if and only if s t S a. Therefore, whether there is agreement or not at any given eriod deends only 1 Cris 1998) also considers a bargaining game in which the size of the surlus follows a Markov rocess. However, there are no differences in beliefs in his model, and inefficiencies can only arise in equilibrium when the buyer and the seller have different discount factors. 1
on the size of the surlus, and is indeendent of the rooser s identity that eriod. Suose an agent is endowed with a surlus whose size follows the Markov rocess above. At each eriod, the agent must decide whether to consume the surlus or to wait. roblem is an otimal stoing roblem, whose solution is given by an otimal stoing region S. At every state in S it is otimal for the agent to sto the rocess and consume the surlus, while at any other state it is otimal to wait. This Say that the outcome of the stochastic bargaining game is efficient if the set of states S a at which there is agreement is equal to S. Merlo and Wilson 1998) showed that these two sets are identical when agents have common beliefs about the recognition rocess. In contrast, in this aer I show that S a is always a subset of S when agents hold otimistic beliefs. Imortantly, this set inclusion might be strict. That is, there might be states at which it is otimal to consume the surlus but at which layers fail to reach an agreement, as the following examle shows. Examle 1 Suose the surlus can take two ossible values, h = 1 and l < h. Let δ < 1 be the common discount factor. State h is an absorbing state, while Prs t+1 = l s t = l) = P ll. Both layers believe that they will make offers next eriod with robability 1 regardless of the state next eriod). In this setting it is always efficient to consume the surlus at state h. Moreover, consuming the surlus at state l is efficient if and only if l δ P ll l + 1 P ll )) l δ 1 P ll) 1 δp ll. 1) Since h is an absorbing state, the results in Yildiz 2003) imly that layers will reach an agreement the fist time s t = h: the rooser will obtain a ayoff of 1/ 1 + δ) and the resonder will obtain δ/ 1 + δ). h: Let V i l) be layer i s ayoff at state l from delaying an agreement until the state reaches V i l) = δ P ll V i l) + 1 P ) ll V i l) = 1 + δ δ 1 P ll ) 1 + δ) 1 δp ll ). 2) With robability P ll the state will be l next eriod, in which case layer i again gets V i l). With robability 1 P ll the state will be h next eriod, in which case layer i exects to obtain 1/ 1 + δ) since she believes she will make offers with robability 1. If V 1 l)+v 2 l) > l layers will delay at state l, since there is no agreement that can satisfy both layers exectations. Since V 1 l) + V 2 l) > δ 1 P ll ) / 1 δp ll ), there are values of δ, P ll and l for which it is efficient to consume the surlus at state l but for which layers delay at this state. The intuition behind these inefficiencies is as follows. When layers reach an agreement, 2
the layer who makes the offer extracts a non-informational rent. If the surlus is exected to grow in the near term, the non-informational rent that the rooser gets in the future might be large relative to the current size of the surlus. When layers are otimistic about the recognition rocess, they both exect to extract this large non-informational rent with high robability. In this case, the sum of what the layers exect to get from delaying might be larger than the size of the surlus today, thereby making it imossible for them to reach an agreement. As Examle 1 shows, this might occur even at states at which it is otimal to consume the surlus right away. The ultimate reason for this is that otimism about bargaining ower has more imact in this stochastic environment, since now layers can be otimistic about their bargaining ower at states at which the surlus is large. Examle 1 shows how inefficient delays can arise when otimistic layers bargain over a stochastic surlus. However, this aer also shows that these inefficiencies can only occur when there are non-negligible frictions in the bargaining rocess, and that they disaear when layers can make offers arbitrarily frequently. The intuition behind this result is as follows. As the time between bargaining rounds goes to zero, the non-informational rent that the rooser gets also vanishes to zero: in the continuous-time limit the right to make an offer becomes extremely transient, so layers get the same share of the surlus when they are making offers and when they are not. Therefore, when offers are arbitrarily frequently, the differences in beliefs about the recognition rocess no longer have an effect on the sum of the layers exected continuation ayoffs from delaying, and the outcome becomes fully efficient. The following examle illustrates this. Examle 2 Examle 1 continued) Suose that the surlus can take two ossible values, h = 1 and l < h. Let measure the time between offers, with δ = e r the discount factor. Suose h is an absorbing state, while Pr s t+ = l s t = l) = e λ for some λ 0, ). Both layers believe that they will make offers next eriod with robability 1. By equation 1), it is efficient to consume the surlus at state l if and only if l e r 1 e λ ) 1 e r e λ as 0 λ r + λ. By equation 2), the ayoff layer i gets by always delaying an agreement at state l is V i l) = so V 1 l) + V 2 l) λ r+λ e r 1 e λ ) h 1 + e r ) 1 e r e λ ) as 0 1 λ 2 r + λ, as 0. In the continuous-time limit layers will delay at state 3
l only if λ r+λ > l; that is, only if it is inefficient to consume the surlus at state l. Related literature: This aer relates rimarily to the literature of bargaining with otimism, starting with Yildiz s 2003) immediate agreement result. In a different aer, Yildiz 2004) shows that otimism about bargaining ower may roduce inefficient delays if layers udate their beliefs as the game goes on. Ali 2006) shows that otimism may lead to disagreement and delays in multilateral bargaining settings. Simsek and Yildiz 2009) consider a model in which the layer s bargaining ower evolves as a stochastic rocess, and show how otimism can lead to delays when layers exect bargaining ower to become more durable at some future date. 2 More broadly, this aer also relates to the literature on bargaining imasses. Delays in bargaining can arise when layers have rivate information Kennan and Wilson, 1993), when layers can commit not to accet oor offers Fershtman and Seidmann, 1993) or when layers may receive new information while bargaining Avery and Zemsky, 1994). Inefficiencies may also arise when layers can build a reutation for being irrational Abreu and Gul, 2000), when outside otions are history-deendent Comte and Jehiel, 2004) or when layers have higher order uncertainty Feinberg and Skrzyacz, 2005). 2 Model Let N = {1, 2} be the set of layers, with tyical elements i and, and let S be a finite set of states. The realization of the state s S each eriod determines the size of the surlus that eriod: there exists a function c : S R + such that c s) is the size of the surlus when the state is s. Let s t denote the realization of the state at time t. For simlicity, I assume that the random variable that determines the state at time zero is degenerate, assigning robability one to s 0 S. From eriod zero onwards the state evolves according to an exogenous timehomogeneous Markov rocess. Let P denote the transition matrix over elements in S, and let P ss denote the robability of moving from s to s. At each date t > 0 until layers reach an agreement, nature draws a state s t S according to the distribution { } P st 1 s. s S At each date t = 0, 1, 2,... before layers reach an agreement, and after the realization of state s S, one layer is randomly recognized to make an offer. The recognized layer can either make an offer u C s t ) := {x R 2 + : x 1 + x 2 c s t )} or ass. In the first case, the other layer must choose to either accet or reect the offer. If she accets the offer the game ends and layers receive their ayoffs. If the rooser chooses to ass or if the resonder 2 See Yildiz 2011) for a comlete review of the literature on bargaining with otimism. 4
reects the current offer the game moves on to the next eriod. At eriod t + 1 nature draws a state s t+1 S according to the distribution {P sts} s S, and then a new layer is recognized to make an offer. The rocess continues until a layer accets an offer, so there is no limit on the number of bargaining rounds. 3 Let δ < 1 be the common discount factor. The difference between the model I study in this aer and the one studied by Merlo and Wilson 1995, 1998) is that I allow agents to hold different beliefs about the recognition rocess. Let i s be the robability with which layer i believes she will be recognized to make an offer when the state is s S. I only consider the case in which layers are otimistic, so 1 s + 2 s 1 for all s S. If this inequality is strict for some s, then layers hold divergent beliefs at s. Define y s) := 1 s + 2 s 1 to be the level of otimism in state s, with y s) 0 for all s S. Note that the layers beliefs about the recognition rocess deend only on the state s, and are indeendent of time. This difference in beliefs is common knowledge. Let Z = S N. The set Z is an exanded set of states, with z = s, i) Z denoting the state in which the surlus size is c s) and agent i makes offers. Given the transition matrix P over elements of S and given the layers beliefs about the recognition rocess, I can calculate for each layer i a transition matrix P i over elements in Z. Since layers may have different beliefs about the recognition rocess, it might be that P 1 P 2. Let E i z] denote the conditional exectation associated with transition matrix P i, and let E s] denote the conditional exectation associated with transition matrix P. An outcome of this bargaining game is a air u, τ), where τ is a stoing time and u = u 1, u 2 ) is a 2-dimensional random variable, measurable with resect to the σ-algebra generated by z τ = z 1,..., z τ ), with u s τ, τ ) C s τ ) if τ < and u = 0 if τ =. That is, an outcome u, τ) is given by a time τ at which layers reach an agreement and a random variable u which determines the share of the surlus that each agent gets. I assume that layers are risk-neutral exected utility maximizers: given an initial state z, layer i s ayoff from outcome u, τ) is given by E i δ τ u i z τ ) z]. An outcome u, τ) is stationary if there exists a subset Z a Z and a function η : Z a R 2 such that i) τ := inf {t 0 : s t, i t ) Z a } and ii) u s τ, i τ ) = η s τ, i τ ) C s τ ). That is, under a stationary outcome layers reach an agreement the first time the state reaches the set Z a, in which case the division of the surlus is η z τ ). Given an initial state z, agent i s 3 Although I assume that the game has an infinite horizon, the setu can incororate games with a finite horizon or games with an exogenous risk of breakdown. To do this, I need to add an absorbing state s such that c s) = 0 and in the case of a finite horizon I also need to exand the set of states S accordingly, so that a state now reresents the size of the surlus and the time left until the final bargaining round). 5
utility from stationary outcome u, τ) is { η i z) if z Z a, V i z) = E i δ τ η i z τ ) z 0 = z] if z / Z a. In what follows, I will denote a stationary outcome by Z a, η). Finally, a stationary outcome Z a, η) is anonymous if Z a = S a N for some S a S. Thus, in an anonymous stationary outcome layers reach an agreement the first time the Markov rocess enters the set S a, indeendently of which layer makes an offer that eriod. Given an initial state s, k), layer i s ayoff from outcome S a N, η) is { η i s, k) if s S a, V i s, k) = E δ τ i s τ η i s τ, i) + ) 1 i s τ ηi s τ, i) ) s 0 = s ] if s / S a. Note that in this case I can dro the subscrits in the exectations, since now the stoing time τ deends only on the realization of s 0, s 1, s 2,...; i.e., in this case τ = inf{t 0 : s t S a }. For each layer, a strategy secifies a feasible action at every history at which that layer must act. A strategy rofile is a air of strategies, one for each layer. A strategy rofile σ induces an outcome u, τ), and hence an exected ayoff for each layer at every history of the game. A strategy rofile σ is a subgame erfect equilibrium SPE) if for every layer i N, σ i is a best resonse to σ i at every history. 3 SPE ayoffs In this section, I show that this bargaining game has unique SPE ayoffs and that the game also has a unique SPE outcome. This unique SPE outcome is both stationary and anonymous. For any f : S R, let E s f s )] = s S P ss f s ) denote the exectation of f s t+1 ) conditional on s t = s. Theorem 1 There exists unique SPE ayoffs. For every s, k) Z, let V i s, k) denote the ayoff of layer i when the state is s, k). These ayoffs satisfy: V i s, ) = δe s i s V i s, i) + ) 1 i s Vi s, ) ], 3) { } c s) δe s i s V s, ) = max V i s, i) + 1 i s ) V i s, )], δe s s V s, ) + 1. 4) s )V s, i)] 6
Proof. See Aendix A.1. Let V : Z R 2 be the unique SPE ayoffs. Suose the state is s, ) Z and consider the equilibrium ayoff of layer i at that state. If layer chooses to ass or if layer makes an offer that layer i reects, then layer i s utility at state s, ) is given by her continuation ayoff. On the other hand, if layer makes an offer that layer i accets, this offer must give layer i her continuation ayoff, since in equilibrium layer i will accet any offer that gives her this much. Therefore, the equilibrium ayoff of layer i at state s, ) is given by her continuation value, regardless of whether there is agreement at that state or not; this is the content of equation 3). Consider next the equilibrium ayoff of layer at state s, ). If layer is to make an offer that layer i will accet, she will offer to give layer i her continuation ayoff. In this case, layer gets a ayoff equal to the size of the surlus c s) minus the continuation ayoff of layer i. However, layer can also choose to ass on her right to make an offer, in which case she gets her own continuation ayoff. Therefore, layer will make an offer that layer i will accet only if the first quantity is larger than the second, and will choose to ass otherwise; this is the content of equation 4). It follows from equation 4) that layers will reach an agreement at state s, ) only if c s) δe s i s V i s, i) + 1 i s ) Vi s, ) ] δe s s V s, ) + 1 s ) V s, i) ], 5) and will delay if the reverse inequality holds. Moreover, since continuation values are indeendent of the layer who is making offers, if there is agreement at state s, ) there will also be agreement at state s, i). If equation 5) holds with equality then equilibrium is consistent with both agreement and delay at state s, ). From now on I will ignore this trivial source of multilicity and assume that layers always come to an agreement in state s, ) if the inequality in equation 5) holds with equality. Define the function φ : S R as φ s) = δe s 1 s V 1 s, 1) + 1 1 s ) V1 s, 2) + 2 s V 2 s, 2) + 1 2 s ) V2 s, 1) ]. Let S a := {s S : c s) φ s)}. It then follows that the set of states at which there is agreement is S a N. That is, the SPE outcome of this stochastic bargaining game is stationary and anonymous. Remark 1 In Aendix A.1, I show that the unique SPE ayoffs in Theorem 1 can be 7
derived via iterated conditional dominance Proosition A1). Therefore, as in Yildiz 2003), the analysis in the aer is immune to the critique of Dekel, Fudenberg and Levine 2004) to games in which layers have heterogeneous beliefs. 4 Delay and inefficiency Let w : S R be the solution to w s) = su E δ τ c s τ ) s 0 = s], 6) τ T where T is the set of stoing times. Problem 6) is an otimal stoing roblem. This roblem arises when one agent with discount factor δ has comlete control over the rocess c s) and must decide at each time eriod whether to consume the surlus or wait. Using standard dynamic rogramming arguments one can show that w is the unique solution to w s) = max {c s), δe s w s )]}. 7) Moreover, the stoing time that solves roblem 6) is given by τ = inf {t 0 : s t S }, with S := {s S : w s) = c s)}. Definition 1 An anonymous stationary outcome S N, η) is efficient if and only if S = S. Let S a N, η) be the SPE outcome. The main result of this section shows that it is always the case that S a S. That is, the set of states at which there is agreement is a subset of the set of states at which it is otimal to sto the rocess and consume the surlus. This imlies that the only tye of inefficiency that can arise in equilibrium is that layers wait for too long before coming to an agreement. Let V : Z R 2 be the unique SPE ayoffs of the bargaining game. Equations 3) and 4) imly that V 1 s, k) + V 2 s, k) = max{c s), φ s)} for every s S and k = 1, 2. Since V 1 s, 1) + V 2 s, 1) = V 1 s, 2) + V 2 s, 2) for all s S, it follows that φ s) = δe s V 1 s, 1) + V 2 s, 1) + y s ) V 1 s, 1) V 1 s, 2))], where y s ) = 1 s + 2 s 1 for all s S. For all s S, let ṽ s) = y s) V 1 s, 1) V 1 s, 2)) = y s) c s) φ s)) 1 {s S a }, 8
where 1 { } denotes the indicator function. Note that ṽ s) = 0 for all s / S a. Moreover, since V 1 s, 1) V 1 s, 2) for all s S a and since y s) 0 for all s S, it follows that ṽ s) 0 for all s S. Therefore, φ s) = δe s V 1 s, k) + V 2 s, k) + ṽ s )], and V 1 s, k) + V 2 s, k) = max {c s), δe s V 1 s, k) + V 2 s, k)] + δe s ṽ s )]}. 8) Let W : S R be given by W s) = V 1 s, k) + V 2 s, k). Equation 8) imlies that W s) = max {c s), δe s W s )] + δe s ṽ s )]}. 9) Finally, note that S a = {s S : φ s) c s)} = {s S : W s) = c s)}. Theorem 2 Let S a N, η) be the SPE outcome. Then S a S. Proof. Recall that S = {s S : w s) = c s)} and S a = {s S : W s) = c s)}. To rove Theorem 2 it suffices to show that W s) w s) for all s S. To see that these inequalities hold, let T : F S) F S) be the oerator T f) s) = max {c s), δe s ṽ s )] + δe s f s )]}. Oerator T is a contraction with modulus δ and T W ) = W is its unique fixed oint. The oerator T is increasing. That is, if f, g F S) are such that f s) g s) for all s S, then T f) s) T g) s) for all s S. To see this, note that there are two ossible cases: i) T g) s) = c s) and ii) T g) s) = δe s ṽ s )] + δe s g s )]. In case i), T f) s) c s) = T g) s). In case ii), T f) s) δe s ṽ s )] + δe s f s )] δe s ṽ s )] + δe s g s )] = T g) s), where the first inequality follows from the definition of T and the second one from the assumtion that f s) g s) for all s S. Define the sequence {f r } with f 1 = w and f r = T f r 1 ) for all r > 1, where w is the solution to 7). Note that f 2 s) = T f 1 ) s) f 1 s) = w s) for all s S. Since T is increasing, it follows by induction that f r s) f r 1 s) for all s S and for all r 2. Finally, the fact that T is a contraction imlies that {f r } W. I conclude that W s) w s) for all s S, which in turn imlies S a S. Corollary 1 If layers have common beliefs about the recognition rocess so y s) = 0 for all s S), then S a = S. 9
Proof. Note that y s) = 0 for all s S imlies that ṽ s) = 0 for all s S. In this case equation 9) becomes W s) = max {c s), δe s W s )]}. 9 ) Note that the solution to 9 ) is equal to the solution to 7), so W s) = w s) for all s S. Thus, S a = {s : W s) = c s)} = {s : w s) = c s)} = S. Corollary 1 is a known result, which I include here only for comleteness. Indeed, when y s) = 0 for all s S the model in this aer becomes a secial case of the stochastic bargaining game with transferable utility studied by Merlo and Wilson 1998), and they showed that in their model there can only be efficient delays. 4 Theorem 2 and Corollary 1 highlight which are the sources of inefficiencies in this stochastic bargaining model with differences in beliefs. When agents have otimistic beliefs about the recognition rocess, the sum of what the layers exect to get if they delay an agreement might be larger than the value of waiting in the single erson roblem; i.e., W s) w s). Therefore, there might be states at which it is otimal to sto the single erson roblem but at which there is no division of the surlus that satisfies both arties exectations as in Examle 1 in the Introduction). In other words, layers might wait for too long before reaching an agreement, leading to inefficient delays. Remark 2 In this aer, I assume that layers have otimistic beliefs about their bargaining ower and common beliefs about the evolution of the stochastic rocess that drives the size of the surlus. The assumtion of common beliefs about the stochastic rocess allows me to define efficiency in terms of the solution to the otimal stoing roblem 6). One of the main insights of this aer is that otimism about bargaining ower may lead to inefficient delays in this stochastic environment; i.e., that layers may delay an agreement at states at which it would be otimal to consume the surlus. One could also allow layers to have divergent beliefs about the evolution of the underlying stochastic rocess driving the surlus. However, in such a setting it would not be ossible to make statements about the efficiency of the bargaining outcome without making assumtions regarding the structure of the true stochastic rocess governing the size of the surlus. 4 The efficiency result in Merlo and Wilson 1998) deends crucially on the assumtion that utility is erfectly transferable between layers. Indeed, Merlo and Wilson 1995) show that inefficiencies may arise in stochastic bargaining games with non-transferable utility. In this setting layers may reach an agreement at states at which it would be otimal to delay, since they are not able to commit to future divisions of the surlus. 10
4.1 Conditions for inefficiencies In this subsection, I study conditions under which inefficient delays will arise in the unique SPE. I start by analyzing a simle setting in which the surlus can take two ossible values. Then, I consider the general setting with any finite number of states. Suose that there are two ossible states, S = {h, l}, with c h) > c l). In this setting, it is always efficient to consume the surlus at state h, so h S. It also true that h S a, so layers will always reach an agreement at state h. To see this, suose by contradiction that h / S a. Since S a, it follows that S a = l. 5 Then, it must be that c h) < W h) = δ P hh W h) + 1 P hh ) c l)) + δy l) 1 P hh ) c l) φ l)). 10) Since φ l) δc l), it follows from 10) that W h) δ 1 P hh) c l) 1 + y l) 1 δ)) 1 δp hh δ 1 P hh) c l) 2 δ) 1 δp hh < c l) < c h), a contradiction. Thus, it must be that h S a. This imlies that inefficient delays can only arise in this setting if S = {h, l}; otherwise, {h} = S S a h, so S a = S. Proosition 1 Suose S = {h, l}, with c h) > c l). Then, the unique SPE is inefficient if and only if δ 1 P ll ) 1 + y h)) c h) 1 δp ll ) 1 + δy h) P hh ) + δ 2 1 P ll ) y h) 1 P hh ) > c l) δ 1 P ll) c h). 11) 1 δp ll Proof. Suose that the SPE is inefficient. By the discussion above, it must be that S = {h, l} and that S a = {h}. The fact that S = {h, l} imlies On the other hand, l / S a imlies c l) δ 1 P ll) c h) 1 δp ll. W l) = δ P ll W l) + 1 P ll ) c h)) + δ 1 P ll ) y h) c h) φ h)) > cl), 12) φ h) = δ P hh c h) + 1 P hh ) W l)) + δy h) P hh c h) φ h)). 13) 5 If S a = then the ayoffs of both layers would be zero, which cannot occur in a SPE: in this case the rooser could always offer her oonent a share ε > 0 of the current surlus an offer her oonent would accet) and earn c s) ε. 11
Combining equations 12) and 13), it follows that W l) = δ 1 P ll ) 1 + y h)) c h) 1 δp ll ) 1 + δy h) P hh ) + δ 2 1 P ll ) y h) 1 P hh ) > c l). On the other hand, if 11) does not hold then either S a = S = {h} or S a = S = {h, l}. Proosition 1 rovides necessary and sufficient conditions for inefficiencies in settings in which the surlus can take two ossible values. From equation 11) it follows that inefficient delays can only arise whenever layers are otimistic about their bargaining ower at state h: the inequalities in equation 11) can never hold if y h) = 0, since in this case the left hand side of 11) would be equal to the right hand side. That is, inefficiencies can only arise when layers are otimistic about their bargaining ower at the state at which the surlus is large. On the other hand, the conditions in 11) do not deend on y l), so whether or not there are inefficient delays in this setting is indeendent on the level of otimism at state l. The intuition behind these results is as follows. When layers reach an agreement at state h, the layer who gets to be rooser extracts a non-informational rent: she obtains a larger ayoff than what she would obtain if she was resonder. If layers are otimistic about their relative bargaining ower at state h, then they both exect to extract this non-informational rent with high robability. In this case, the sum of what the layers exect to get from delaying an agreement at state l will be strictly larger than the value of waiting in the single erson otimal stoing roblem. Given this large erceived value of waiting, layers might delay an agreement at state l even in situations in which it would be efficient to agree. Note that the erceived value of waiting at state l deends only on the level of otimism at state h the state at which layers exect to extract the non-informational rent), and is indeendent of the level of otimism at state l. Thus, inefficiencies can only arise in this environment whenever y h) is large enough. Consider next the general setting in which S is any finite set of states. Let c = max s S c s) and c = min s S c s) denote the largest and smallest size of the surlus, resectively. Let y = max s S y s) denote the highest level of otimism. Proosition 2 Suose there exists a state s S such that δe s w s ) 1 + y s ))] δ 2 E s y s )] c + y c δc)) > c s). 14) Then, s / S a, so the SPE outcome of the bargaining game is inefficient. 12
Proof. Suose there exists a state s S such that 14) holds. Note then that ] φ s) = δe s W s )] + δe s y s ) c s ) φ s )) 1 {s S a } = δe s W s )] + δe s y s ) W s ) φ s ))] δe s W s ) 1 + y s ))] δ 2 E s y s ) c + y c δc))] δe s w s ) 1 + y s ))] δ 2 E s y s ) c + y c δc))] > c s), where the second equality follows from the fact that W s ) = c s ) for all s S a and W s ) = φ s ) for all s / S a, the first inequality follows from the fact that φ s ) δ c + y c δc)) for all s, the second inequality follows since W s ) w s ) for all s and the last inequality follows from 14). Therefore W s) φ s) > c s), so s / S a. Proosition 2 gives sufficient conditions for inefficiencies based on the rimitives of the game: the level of otimism y s), the ossible sizes of the surlus, and the value w s) of the single erson otimal stoing roblem which itself deends on the transition matrix P and the ossible sizes of the surlus). By equation 14), inefficiencies are more likely to arise in settings in which layers are otimistic about their bargaining ower at states at which the value function w s) is larger; i.e., at states at which the surlus over which layers are bargaining is larger. I stress however that, unlike equation 11) in Proosition 1, the conditions in Proosition 2 are not necessary for inefficient delays to arise in equilibrium; they are only sufficient. 4.2 The benefits of otimism In this subsection, I study what haens to equilibrium ayoffs when one of the agents becomes more otimistic. Let i 0, 1] S be the vector secifying the beliefs of layer i about the robability with which she exects to make offers at each state s S. For i, i 0, 1] S, I write i i whenever i s i s for all s S, with at least one strict inequality. Proosition 3 Assume i i, and let V and Ṽ be the SPE ayoffs when beliefs about the recognition rocess are given by i, ) and i, ), resectively. Then, Ṽi z) V i z) and Ṽ z) V z) for all z Z. Proof. See Aendix A.2. 13
Proosition 3 shows that there is a gain for a layer who becomes more otimistic and whose new level of otimism becomes common knowledge). For instance, if there was an initial stage rior to the game at which layers could choose their beliefs, then by Proosition 3 it would be a weakly dominant strategy for both layers to choose extremely otimistic beliefs i.e., to choose i s = 1 for all s S). 5 Frequent offers and efficiency In this section, I analyze the roerties of equilibrium when layers can make offers arbitrarily frequently. As usual, I can write the discounting between bargaining rounds as δ = e r for some r, > 0, where r measures the rate of time reference and is the time interval between bargaining rounds. The assumtion that > 0 catures the frictions in the bargaining rocess. The obective of this section is to study how the equilibrium outcome changes as these frictions disaear i.e., as 0). Fix an interval between offers > 0. For each s S, the robability that s t+ = s when s t = s is equal to P ss = e λs for some > λ s 0. 6 The robability that s t+ = s s when s t = s is equal to P ss = α ss 1 e λ s ) for some α ss 0, with s s α ss = 1. With this arametrization, the seed at which the Markov rocess moves remains roughly constant as I change the time between rounds. Let P = { Pss denote the transition }s,s S matrix when the interval between bargaining rounds is. Finally, for any f : S R, let E s f s )] = s S P ss f s ) denote the exectation of f s t+ ) conditional on s t = s. Let S ) := {s S : w s) = c s)}, where w s) = max{c s), δ Es w s )]}. That is, S ) is the set of states at which it is otimal to sto the rocess in the single erson roblem when the interval between eriods is. Let V be the SPE ayoffs when the interval between offers is. For each s S, let W s) = V 1 s, k) + V 2 s, k). Hence, S a ) := {s S : W s) = c s)} is the set of states at which there is agreement when the interval between bargaining rounds is. By Theorem 2, S a ) S ) for all > 0. The main result of this section is that S a ) converges to S ) as 0. In words, the outcome of the bargaining game converges to the efficient outcome when layers can make offers arbitrarily frequently. For any ε > 0 and any interval between offers, let S ε ) := { s S : c s) δ E s w s )] + ε }. 6 The restriction that λ s < for all s S imlies that for any > 0 there is ositive robability that the state of the Markov rocess next eriod will be the same as the state today. 14
That is, S ε ) is the set of states at which it is strictly otimal by an amount weakly larger than ε) to sto the single erson rocess and consume the surlus. Note that S ε ) S ε ) for any ε > ε 0, and that S 0 ) = S ). Theorem 3 For every ε > 0, there exists ε) > 0 such that S ε ) S a ) S ) for all < ε). Proof. See Aendix A.3. Theorem 3 shows that inefficiencies disaear when layers can make offers arbitrarily frequently. In other words, this model can only deliver inefficient delays in cases in which there are non-negligible frictions in the bargaining rocess. In this sense, Theorem 3 extends the results of Yildiz 2003) to a stochastic bargaining game. As discussed in the Introduction, Yildiz 2003) showed that excessive otimism about the recognition rocess cannot be by itself a cause of inefficient delays. Similarly, in this aer s model inefficiencies vanish when layers can make offers arbitrarily frequently. As a way to illustrate the content in Theorem 3, consider a setting in which the surlus can take two ossible values, c h) and c l) < c h). Let Pll = e λl and Phh = e λh, with λ s 0, ) for s = h, l. By equation 11) in Proosition 1, for any > 0 the outcome of this stochastic bargaining game will be inefficient if and only if e r 1 e λ l ) 1 + y h)) c h) 1 e r e λ l ) 1 + e r y h) e λ h ) + e 2r 1 e λ l ) y h) 1 e λ h ) > c l) e r 1 e λ l ) c h) 1 e r e λ l 15) One can check that the left hand side and the right hand side of 15) both converge to λ l c h) / r + λ l ) as 0, so there is no room for inefficiencies when layers can make offers arbitrarily frequently. P ss I assumed throughout this section that λ s < for all s S, where λ s is such that = Prs t+ = s s t = s) = e λs. This restriction imlies that there are no instantly transient states; that is, there are no states such that once the Markov rocess gets into them the robability of moving away from them in the next bargaining round is one, regardless of how short the interval between rounds is. The roof of Theorem 3 relies strongly on this restriction. Indeed, inefficient delays may ersist even as 0 if P ss = 0 for all > 0 for some s S, as the following examle shows. Examle 3 Suose S = {h, l}, with c h) > c l). Assume that P hh = 0 for all and 15
P ll = e λ l for some λ l 0, ). By equation 11) in Proosition 1, for any > 0 the outcome of this stochastic bargaining game will be inefficient if and only if e r 1 e λ l ) 1 + y h)) c h) 1 e r e λ l ) + e 2r 1 e λ l ) y h) ) e r 1 e λ l c h) > c l). 16) 1 e r e λ l The left-hand side of 16) converges to λ l 1 + y h)) c h) /r+λ l 1 + y h))) as 0, while the right-hand side of 16) converges to λ l c h) / r + λ l ) as 0. Therefore, the limiting equilibrium outcome will dislay inefficiencies if and only if λ l 1 + y h)) c h) r + λ l 1 + y h)) > c l) λ lc h) r + λ l. Note that there is a non-emty set of arameters for which these inequalities hold. Therefore, inefficient delays may ersist in the continuous-time limit when some states are instantly transient. To understand the intuition behind Theorem 3 and Examle 3, consider first the case in which there are no instantly transient states i.e., 0 λ s < for all s S). In this case, the difference between the utility layers get when they make offers and what they get when they don t i.e., Vi s, i) Vi s, )) converges to zero as the time between eriods goes to zero. Indeed, when λ s < the robability that the state will be the same in the next bargaining round becomes arbitrarily close to one as 0. Furthermore, a short time between bargaining rounds also imlies essentially no discounting from one round to the next. Since a new layer is recognized to be rooser at each round, the cost in terms of discounting that the resonder incurs by waiting until she becomes rooser converges to zero as 0, so Vi s, i) Vi s, ) 0 as 0. In the continuous-time limit each layer obtains the same ayoffs when she rooser and when she is resonder, so the differences in beliefs about the recognition rocess no longer have an imact on the sum of what the layers exect to gain by delaying an agreement. As a result, efficiency is restored. Consider next the case in which P ss = 0 for all > 0 for some s S such that c s) > c s ) for all s s as in Examle 3). In this case, if the current state is s the layers know that with robability one the state will be different from s in the next bargaining round, and that it may take a long time for the Markov rocess to come back to state s. Since c s) > c s ) for all s s, the layer who is making offers at state s has a great deal of bargaining ower, because if the other layer reects her offer the size of the surlus in the next bargaining round will be significantly smaller. Moreover, even if the time interval between offers is 16
small, there might still be a substantial cost of delay in waiting for the Markov rocess to return to state s. In this case, the difference between the ayoff a layer gets at state s when she makes offers and what she gets when she is resonder i.e., Vi s, i) Vi s, )) will remain bounded away from zero even as 0. Therefore, as Examle 3 shows, in this case inefficiencies may ersist even as 0. 6 Conclusion This aer studies an infinite horizon bargaining game in which the size of the surlus follows a stochastic rocess and in which layers might have otimistic beliefs about their future bargaining ower. In a setu with a constant surlus, Yildiz 2003) showed that otimistic layers will always come to an immediate agreement rovided otimism is ersistent and the number of bargaining rounds is large enough. In contrast, this aer shows that ersistent otimism about bargaining ower can generate otentially long lasting inefficient delays in settings in which layers bargain over a stochastic surlus. The ultimate reason for this is that otimism about future bargaining ower has more imact when the surlus is stochastic, since now layers can be otimistic about their relative bargaining ower at states at which the surlus is large. However, this aer also shows that these inefficiencies can only occur when bargaining frictions are significant. As Theorem 3 shows, inefficiencies disaear when layers can make offers arbitrarily frequently. The key reason behind this is that the right to make roosals becomes extremely transient when the time between bargaining rounds converges to zero, since a new layer is selected to make offers at each round. This imlies that the noninformational rent that a layer obtains when she is rooser goes to zero when offers are frequent: in the continuous-time limit, layers get the same ayoff when they are making offers and when they are not. As a result, when offers are arbitrarily frequently otimism about future bargaining ower no longer affects the sum of the layers continuation ayoffs, and the outcome of the game becomes fully efficient. The reasoning in the revious aragrah suggests that, in settings in which there is some ersistence in the right to make roosals, otimism about bargaining ower could otentially generate inefficient delays even when layers can make offers arbitrarily frequently. following examle illustrates that this intuition is correct. Examle 4 Suose S = {h, l}, with c h) = 1 > c l). Assume further that P hh = 1 for all > 0 and that P ll The = e λ l for some λ l 0, ). Let δ = e r denote the common 17
discount factor. At each round before agreement is reached, the layer making offers last eriod retains the right to make offers with robability q = e γ with γ 0, )). With robability 1 q natures selects a new layer to be rooser. The arameter γ measures the ersistence of roosal ower: γ 0 corresonds to the case in which roosal ower is erfectly ersistent, while γ corresonds to the case in which roosal ower is transient. For simlicity, suose that both layers believe that they will be recognized to make offers with robability 1 every time nature selects a new rooser. One can show that, for any > 0, layers will reach an immediate agreement in any subgame that starts at state h: the rooser will obtain a ayoff of V = 1 δ q )/1 2δ q + δ ) and the resonder obtains V = δ 1 q )/1 2δ q + δ ). 7 Let Vi l, k) denote the ayoff layer i gets from always delaying an agreement when the size of the surlus is l and layer k is making offers: V i l, i) = δ V l, i) = δ P ll P ll q Vi + 1 P q V l, i) + 1 q ) Vi l, i) ) ) ) q V + 1 q ) V ), 17) ll + 1 P ll l, i) + 1 q ) V l, ) ) ) ) q V + 1 q ) V ). 18) Let W l) = V 1 l, 1) + V 2 l, 1) = V 1 l, 2) + V 2 l, 2). Note that layers will delay at state l whenever W l) > c l), since in this case there is no agreement that can satisfy both layers exectations. Using 17) and 18), it follows that W l) = δ P ) ll 1 q 1 δ Pll ) + δ 1 Pll 1 δ P ll δ ) 1 Pll q 1 δ P ll q V V ) q + 1 q ) 2V ). In the continuous-time limit, it is efficient to consume the surlus at state l if and only if c l) λ l /r + λ l ). On the other hand, layers will delay at state l in the continuous-time limit if lim W l) = 0 λ l rγ r + λ l ) r + 2γ) r + γ + λ l ) + λ l r + λ l > c l). Therefore, for any γ 0, ) there is a non-emty set of arameters for which there will be inefficient delays even as layers can make offers arbitrarily frequently. 7 Indeed, when the size of the surlus is constant the model in this examle is a secial case of a model studied by Simsek and Yildiz 2009); and they show that in their more general model layers will always come to an immediate agreement regardless of the level of otimism. 18
A Aendix A.1 Proof of Theorem 1 Let F 2 Z) be the set of bounded functions on Z taking values in R 2. Let denote the su norm on R 2. For any f F 2 Z), let f Z = su { f z) : z Z}. For any h : S R, let E s h s )] = s S P ss h s ) denote the exectation of h s t+1 ) conditional on s t = s. Define the oerator A : F 2 Z) F 2 Z) as A i f) s, i) = max { c s) δe s s f s, ) + 1 s ) f s, i) ], δe s i s f i s, i) + 1 i s ) f i s, )] A i f) s, ) = δe s i s f i s, i) + 1 i s ) fi s, ) ]. Lemma A1 The oerator A : F 2 Z) F 2 Z) is a contraction. Proof. Let f, g F 2 Z). For any state s, i) Z, A f) s, i) A g) s, i) = δes s f s, ) g s, )) + 1 s ) f s, i) g s, i)) ] δe s s f s, ) g s, )) + 1 s ) f s, i) g s, i)) ) δ f g Z. Next I show that A i f) s, i) A i g) s, i) δ f g Z. Assume wlog that A i f) s, i) A i g) s, i). There are two ossible cases: 1) A i f) s, i) = c s) δe s s f s, ) + 1 s )f s, i)], or 2) A i f) s, i) = δe s i s f i s, i) + 1 i s )f i s, )]. In case 1), A i f) s, i) A i g) s, i) = c s) δes s f s, ) + ) 1 s f s, i) ] A i g) s, i) δes s g s, ) f s, )) + ) 1 s g s, i) f s, i)) ] On the other hand, in case 2), δ f g Z. A i f) s, i) A i g) s, i) = δes i s f i s, i) + ) 1 i s fi s, ) ] A i g) s, i) δes i s f i s, i) g i s, i)) + ) 1 i s fi s, ) g i s, )) ] δ f g Z. Hence, A f) A g) Z δ f g Z, so A is a contraction. } 19
For any air M, m F 2 Z) and for = 1, 2, i, define H M, m i ) s, k) = max { c s) m i s, k), δe s s M s, ) + 1 s ) M s, i) ]}. Define the oerator H : F 2 Z) F 2 Z) F 2 Z) as H M, m) = H 1 M 1, m 2 ), H 2 M 2, m 1 )). Proof of Theorem 1. To rove Theorem 1, I start out assuming that the set of SPE ayoffs is non-emty. At the end of the roof I show that the game indeed has a SPE. Let u, τ) be a SPE outcome and let f i z) = E i δ τ u i z 0 = z] be the ayoff that layer i gets from this SPE when the initial state is z Z. Let M = M 1, M 2 ) and m = m1, m 2 ) denote the suremum and infimum SPE ayoffs of layers 1 and 2 so that M i z) f i z) m i z) for all z Z, i = 1, 2). At any state z = s, i), layer i can guarantee herself a ayoff of at least A m, M i ) s, i) = δes s m s, ) + 1 s )m s, i)] by reecting the offer. Similarly, at state z = s, ) layer s ayoff is at least as large as A m, M i ) s, ) = max { c s) δe s i s M i s, i) + 1 i s ) M i s, ) ], δe s s m s, ) + 1 s ) m s, i) ] as layer i never reects an offer of δe s i s M i s, i) + 1 i s ) M i s, )]. This imlies that, for all z Z, f z) A m, M i ) z) for = 1, 2, i. For any air M, m F 2 Z) and for = 1, 2, i, let G M, m) = A m, M i ). Define the oerator G : F 2 Z) F 2 Z) F 2 Z) as G M, m) = G 1 M, m), G 2 M, m)). Suose next that the state is s, ). If layer makes an offer that layer i accets, then f s, ) + f i s, ) c s). Since f i s, ) m i s, ), it follows that f s, ) c s) m i s, ). If layer asses or if layer i does not accet her offer, then f s, ) δe s s M s, ) + 1 s )M s, i)], since the right hand side of this inequality is layer s highest ossible continuation ayoff at state s, ). On the other hand, f s, i) δe s s M s, ) + 1 s )M s, i)] when the state is s, i), as layer will always accet any offer that gives her this much. It then follows that f s, k) H M, m i ) s, k) for = 1, 2, i where H, ) was defined above). Next, let M, M and m, m all bounded functions on Z taking values on R 2 ) be such that M i z) M i z) for all z Z, i = 1, 2 and m i z) m i z) for all z Z, i = 1, 2. One can check that this imlies that H i M, m ) z) H i M, m ) z) for all z Z, i = 1, 2 and G i M, m ) z) G i M, m ) z) for all z Z, i = 1, 2. It what follows, for any air f, g F 2 Z) I will write f g if f i z) g i z) for all z Z, i = 1, 2. }, 20
Define the sequences {M r } and {m r } as follows. Let M 1, m 1 ) = M, m ), and for all r 2 let M r, m r ) = HM r 1, m r 1 ), GM r 1, m r 1 )). Note that M 2 = H M 1, m 1 ) M 1 and m 2 = G M 1, m 1 ) m 1. It follows then by induction and using our observation in the revious aragrah that {M r } is an increasing sequence and {m r } is a decreasing sequence. Moreover, it must be that m r 0 for all r and M r b for all r, where b is such that b c s) for all s S such a b always exists, since S is a finite set). Thus, both {M r } and {m r } are bounded and monotonic sequences, so there exists M and m such that {M r } M and {m r } m. Finally, since both H and G are continuous functions it follows that H M, m ) = M and G M, m ) = m. Therefore, m m 1 = m M = M 1 M. The next ste is to show that Mi = m i for i = 1, 2. Since m m M M, this will imly that m = M, so there are unique SPE ayoffs. The strategy to rove this is to show that both M1, m 2) and m 1, M2 ) are fixed oints of the oerator A, and to note that by Lemma A1 the oerator A is a contraction and therefore has a unique fixed oint. Consider M1, m 2). By the analysis above, H 1 M1, m 2) = M1 and G 2 M, m ) = m 2. By definition of the oerator G, m 2 = G 2 M, m ) = A 2 m 2, M1 ). Thus, to show that M1, m 2) is a fixed oint of A it suffices to show that A 1 M1, m 2) = M1. To see this, consider first states z = s, 1) in which layer 1 is the rooser. For those states it must be that m 2 s, 1) = A 2 m 2, M 1 ) s, 1) = δe s 2 s m 2 s, 2) + 1 2 s ) m 2 s, 1) ]. A.1) Equation A.1) together with the definition of oerator A imlies A 1 M 1, m 2) s, 1) = max { c s) m 2 s, 1), δe s 1 s M 1 s, 1) + 1 1 s ) M 1 s, 2) ]} = H 1 M 1, m 2) s, 1) = M 1 s, 1). Consider next states z = s, 2). There are two ossible cases. The first case is m 2 s, 2) = c s) δe s 1 s M 1 s, 1) + 1 1 s ) M 1 s, 2) ] δe s 2 s m 2 s, 2) + 1 2 s ) m 2 s, 1) ]. The second case is one in which m 2 s, 2) = δe s 2 s m 2 s, 2) + 1 2 s ) m 2 s, 1) ] > c s) δe s 1 s M 1 s, 1) + 1 1 s ) M 1 s, 2) ]. 21
In either case, δe s 1 s M 1 s, 1) + 1 1 s ) M 1 s, 2)] c s) m 2 s, 2). Therefore, H 1 M 1, m 2) s, 2) = max { c s) m 2 s, 2), δe s 1 s M 1 s, 1) + 1 1 s ) M 1 s, 2) ]} = δe s 1 s M 1 s, 1) + 1 1 s ) M 1 s, 2) ] = A 1 M 1, m 1) s, 2). Hence, A M1, m 2) = M1, m 2). A symmetric argument shows that A m 1, M2 ) = m 1, M2 ). Since A is a contraction with a unique fixed oint, it must be that M1, m 2) = m 1, M2 ). So far I showed that if the set of SPE is not emty, then all SPE must be ayoff equivalent. Now I show that the set of SPE is indeed non-emty. To see this, let V be such that A V ) = V. I first construct an anonymous stationary outcome S a N, η) with ayoffs equal to V. Define φ : S R as in the main text, and let S a = {s S : φ s) c s)}. Let η s, ) = V s, ) C s, ) for all s, ) S a N. One can check using standard dynamic rograming arguments that the layers ayoffs from outcome S a N, η) are equal to V. To show that S a N, η) is the outcome induced by a SPE, consider the following strategy rofile. At any state z = s, i), agent i rooses division η z) = V z) if z S a N and asses otherwise. Resonder i accets any roosal which gives her a ayoff of no less than V s, i) = δe s s V s, ) + 1 s )V s, i)), and reects any roosal that gives her a lower ayoff. Note that this strategy rofile induces outcome S a N, η). Moreover, no layer can gain by unilaterally deviating from her strategy at any state z Z. For i = 1, 2, let Γ i denote the set of strategies of layer i that survive iterated elimination of conditionally dominated strategies. Let Γ = Γ 1 Γ 1. Since Γ contains all the SPE and since the set of SPE is non-emty, it follows that Γ is non-emty. The following result generalizes arguments in Yildiz 2003) to show that the unique SPE ayoffs derived in Theorem 1 can also be derived via iterated elimination of conditionally dominated strategies. Proosition A1 Let V 1, V 2 ) be the unique SPE ayoffs derived in Theorem 1. Then, for i = 1, 2, for each z Z and for each strategy rofile in Γ, layer i s ayoffs at time t with z t = z are equal to V i z). Proof. For eriod t with z t = z, let V i,t z) and V i,t z) be the infimum and suremum ayoffs of layer i at time t, resectively, given that the set of remaining strategy rofiles is Γ with the infimum and suremum taken over all ossible histories of lay and all strategies of layer in Γ ). To establish the claim, I will show that V i,t z) = V i,t z). As a first ste, note that the stationarity of the game imlies that V i,t z) = V i,s z) and V i,t z) = V i,s z) for any t and s such that z t = z s = z. Hence, for every state z, let V i z) and V i z) be the 22