Can Learning Cause Shorter Delays in Reaching Agreements?

Transcription

1 Can Learning Cause Shorter Delays in Reahing Agreements? Xi Weng 1 Room 34, Bldg 2, Guanghua Shool of Management, Peking University, Beijing 1871, China, Abstrat This paper uses a ontinuous-time war of attrition model to investigate how learning about private payoffs affets delays in reahing agreement At eah point in time, players may reeive a private Poisson signal that ompletely reveals the onession payoff to be high (good-news learning) or low (bad-news learning) In the good-news model, the expeted delay is always non-monotoni in the learning rate: an inrease in the learning rate prolongs delay in agreement if the learning rate is suffiiently low In the bad-news model, numerial examples suggest learning prolongs delay as well Keywords: Learning; War of Attrition; Delay; Inomplete Information JEL D8 C7 I would like to thank Sven Rady (the editor), and four anonymous referees for omments and suggestions that greatly improved the artile I am espeially grateful to Jan Eekhout, Hanming Fang, George Mailath and Andrew Postlewaite for their insightful disussions and omments I aknowledge finanial support from the National Natural Siene Foundation of China (Grant No ) and Guanghua Leadership Institute (Grant No 12-2) I also aknowledge support from Spanish Ministry of the Eonomy and Competitiveness (Projet ECO ) and Key Laboratory of Mathematial Eonomis and Quantitative Finane (Peking University), Ministry of Eduation, China address: wengxi125@gsmpkuedun (Xi Weng) 1 Department of Applied Eonomis, Guanghua Shool of Management, Peking University Preprint submitted to Journal of Mathematial Eonomis June 22, 215

2 1 Introdution This paper develops a ontinuous-time war of attrition model with learning to investigate how learning affets delays in reahing agreement In the model, two risk-neutral players deide when to onede A player reeives a high deterministi payoff if her opponent onedes first, while there is unertainty about her payoff if she onedes first As long as no player onedes, eah player has a hane of privately learning whether her onession payoff is high or low (normalized to ) For tehnial tratability, learning is modeled in a very stylized way: at eah point in time, eah player may reeive a private Poisson signal that ompletely reveals the onession payoff This paper fouses on two different ways of interpreting the Poisson signal In the good-news (resp bad-news) model, the signal reveals a high (resp low) onession payoff, whih inreases (resp dereases) the reeiver s inentive to onede For further simpliity, we also assume full symmetry between the two players; hene, we an fous on symmetri equilibrium of the game as in [3] In the presene of learning, the war of attrition starts as a omplete information game, but due to learning, inomplete information about the payoffs may develop over time Due to the Poisson signal struture, a player an be either informed or uninformed about her onession payoff at any point of time, and one type of player is more willing to onede than the other This enables us to fully haraterize the unique symmetri equilibrium of the game The paper first ompares the uninformed benhmark, in whih there is no revelation of the onession payoff, with the full-information benhmark, in whih there is immediate revelation of the onession payoff It is shown that the expeted delay is always shorter in the full-information benhmark Sine these two benhmarks orrespond to the speial ases in whih the learning rate is and infinity, respetively, one may onjeture that an inrease in the learning rate always leads to shorter delays in reahing agreements This onjeture is shown to be inorret: the expeted delay an be non-monotoni in the learning rate, and learning an ause longer delays in reahing agreements In the good-news model, we show that, at any point in time, equilibrium play falls into one of three possible ases In the first ase, an uninformed player randomizes between oneding and staying, while an informed player stritly prefers oneding immediately In the seond ase, an uninformed player stritly prefers staying, while an informed player randomizes between oneding and staying In the third ase, an uninformed player stritly prefers staying, while an informed player stritly prefers oneding immediately The equilibrium haraterization depends on the expeted learning rate When the expeted learning rate is relatively small, the game always stays in the first ase When the expeted learning rate is intermediate, the game is initially in the third ase, and swithes to the first ase if no player has reeived the Poisson signal for a suffiiently long time When the expeted learning rate 2

3 is very high, the game starts out in the seond ase and eventually swithes to the first ase Compared to the uninformed benhmark, learning auses longer (resp shorter) delays when the learning rate is suffiiently low (resp high) In the good-news model, we ondut omparative statis analysis with respet to the learning rate When the learning rate is very low, the average expeted onession rate is determined by the indifferene ondition of the uninformed players, who beome less optimisti about the onession payoffs as the learning rate goes up Although an inrease in the learning rate results in more informed players, who will onede immediately, the expeted onession rate has to be lower in equilibrium in order to make the less optimisti uninformed players indifferent between oneding and staying As a result, the expeted delay is inreasing in the learning rate Due to this negative effet, an inrease in the learning rate has no impat on welfare (in terms of a player s expeted equilibrium payoff at the start of the game) when the learning rate is suffiiently low In ontrast, when the learning rate is very high, a higher learning rate dereases the expeted delay, and thus inreases the welfare Here, an inrease in the learning rate has two opposite effets on the average expeted onession rate First, it leads to a higher average expeted onession rate by inreasing the hane of getting informed, sine the informed players have the highest expeted onession rate Seond, it leads to a lower average expeted onession rate by making the uninformed players more relutant to onede When the learning rate is suffiiently high, a higher learning rate leads to a higher average expeted onession rate by making the distribution of posterior beliefs more dispersed In the bad-news model, there is always one unique equilibrium pattern beause the uninformed are more willing to onede than the informed players who have reeived bad-news signals Initially, an uninformed player randomizes between oneding and staying, while an informed player stritly prefers staying If both players reeive the Poisson signal before oneding, the game eventually swithes to a war of attrition between the informed players Different from the good-news model, the uninformed players obtain an additional value from reeiving the Poisson signal in the bad-news model In the good-news model, the ontinuation value after reeiving the Poisson signal is always the high onession payoff, sine the informed players always onede first However, in the bad-news model, this ontinuation value is stritly larger than zero (the low onession payoff), sine the informed benefit from the onession of the uninformed players Beause of this positive learning value, the uninformed players are more relutant to onede We show by example that in the bad-news model, learning leads to a longer expeted delay ompared to the uninformed benhmark 2 Moreover, an inrease in the learning rate has no impat on welfare in the 2 In an earlier version of the paper, we show the opposite result when the positive learning value is absent 3

4 bad-news model, sine the uninformed players are always indifferent at the beginning of the game In the literature, ontinuous-time wars of attrition have been studied under both omplete information ([8]) and inomplete information ([1], [6, 7], and [12]) 3 This paper extends the literature by onsidering a situation in whih inomplete information is endogenously aused by learning The game starts as a omplete information game, and inomplete information about the payoffs develops over time The remainder of this paper is organized as follows Setion 2 presents the onession game Setion 31 and 32 analyze the uninformed benhmark and the full-information benhmark, respetively Setion 4 haraterizes the symmetri equilibria of this war of attrition under the good-news model, and Setion 5 disusses how the learning rate affets expeted delay and welfare Setion 6 ontains the equilibrium haraterization and omparative statis results under the bad-news model Setion 7 onludes the paper 2 The Game 21 Model Setting Consider a ontinuous-time war of attrition with two risk-neutral players (i = 1, 2) without disounting Both players deide when to onede As long as neither player onedes, the game ontinues with eah player inurring a flow ost, whih reflets the ost of delay The game ends when one of the players onedes The players lump-sum payoffs in this event are speified in the following matrix: Stay 2 Conede 1 Stay, v H, v 2 Conede v 1, v H M, M If player i stays while her opponent i onedes, then player i is the winner of the game and gets a winning payoff of v H If player i onedes first, then she is the loser and gets a onession payoff v i The payoff when both players onede simultaneously is M It is ommon knowledge that M < v H, but there is inomplete information about onession payoffs v 1 and v 2 In partiular, we assume that v 1 and v 2 are independently and identially drawn from a binary distribution: v i an be either a positive number v L < v H or zero Eah player i initially does not know the exat value of v i It is ommon knowledge that v i = v L with prior probability p 3 The game is a speial ase of the more general quitting games in [14] and timing games in [1] 4

5 Following [9] and [15], we introdue learning by assuming that as long as no player onedes, eah player reeives an exogenous private signal whih is distributed as the first arrival time of a Poisson proess The Poisson proesses are independent aross players, and for simpliity, we assume that the arrival of the Poisson signal ompletely resolves unertainty about v i In the good-news model, the arrival rate is λ if v i = v L and zero otherwise After reeiving this good-news signal, player i assigns probability one to the event that v i = v L Absene of the signal will make the player inreasingly pessimisti about the probability that v i = v L In the opposite bad-news model, the arrival rate is λ if v i = and zero otherwise After reeiving this bad-news signal, player i assigns probability one to the event that v i = Absene of the signal will make the player inreasingly optimisti about the probability that v i = v L 22 Strategies and Equilibrium At any point in time, a player is either informed or uninformed about her onession payoff We refer to the former as an informed player, and the latter as an uninformed player A strategy for the uninformed player i is a umulative distribution funtion X i : R + [, 1], where X i (t) denotes the probability that player i onedes to her opponent i by time t (inlusive) 4 Given X i (t), let γ i t be the joint probability of player i being uninformed at time t and not having oneded by time t Obviously, γ i = 1, and take the good-news model for example, γ i t = ( p e λt + 1 p ) ( 1 X i (t) ), (1) where p e λt + 1 p is the probability that player i has not reeived the good-news signal Hene, the evolution of γ i t is fully determined given X i (t) In partiular, if X i (t) has a jump at time t, then γt i drops at t; if X i (t) is ontinuously differentiable at time t, then uninformed player i s onession rate at t is omputed as x i t = dxi (t)/dt The probability of uninformed 1 X i (t) player i oneding to her opponent in the interval [t, t+dt) is then x i tdt Equation (1) implies that γ i t follows the law of motion γ i t = x i tγ i t λq i tγ i t, (2) where q i t is the posterior probability that uninformed player i has the type to reeive the Poisson signal In the above good-news example, q i t = p e λt p e λt +1 p is the posterior probability that v i = v L onditional on no signal being reeived; in the bad-news model, q i t = (1 p )e λt p +(1 p )e λt is the posterior probability that v i = onditional on no signal being reeived γ i t delines 4 This definition of strategy avoids the well-known problem in ontinuous-time games in whih well-defined strategies may not lead to well-defined outomes, as shown by [2] and [13] 5

6 over time through two hannels: onession at rate x i t, and arrival of the private signal at rate λq i t A strategy for the informed player i is a olletion (Y i ( ; τ)) τ of distribution funtions Y i ( ; τ) : [τ, ) [, 1], where τ is the time when the first Poisson signal is reeived 5 Both X i () and Y i (τ; τ) are allowed to be stritly positive so that player i onedes to player i immediately We use Z i to denote the overall strategy of player i: Z i = (X i, (Y i ( ; τ)) τ ) Z i determines F i (t), whih is the probability player i assigns to player i oneding by time t The umulative probability, F i (t), omes from onession by both informed and uninformed player Take the good-news model for example, and we have: 1 F i (t) = γt i + p λe λτ (1 X i (τ))(1 Y i (t; τ))dτ (3) The interpretation of equation (3) is as follows If player i has not oneded by time t, she an be either uninformed or informed The probability of the former event is γ i t, while the probability of the latter event is p λe λτ (1 X i (τ))(1 Y i (t; τ))dτ Notie that the term (1 X i (τ)) is needed in the expression beause an uninformed player has to stay in the game in order to reeive the good-news signal From equation (1), equation (3) an be rewritten as: F i (t) = 1 γ i t λq i τγ i τ ( 1 Y i (t; τ) ) dτ If X i (t) is ontinuously differentiable, then equation (2) implies F i (t) = [ ] x i τγτ i + λqτγ i τy i i (t; τ) dτ, (4) where xi τγτdτ i is the probability player i assigns to uninformed player i oneding by time t and λqi τγτy i i (t; τ)dτ is the probability player i assigns to informed player i oneding by time t If F i (t) is ontinuously differentiable at time t, then ft i is player i s expeted onession rate at time t onditional on no onession before time t Upon reahing = df i (t)/dt 1 F i (t) time t, player i then expets player i to onede in the interval [t, t + dt) with probability f i t dt 5 There is a ontinuum of informed players that is indexed by the arrival time of the first Poisson signal The informed players are not required to use the same strategy at any time t There might be a ontinuum of equilibria by assigning different informed players different onession rates However, all of the equilibria are outome equivalent in terms of the expeted onession rate 6

7 Given Z i, an informed player i s expeted payoff by oneding at time t is W i (t; τ, Z i ) = τ s<t (v H (s τ)) df i (s τ) + (M (t τ)) ( F i (t τ) F i (t τ) ) + (v i (t τ)) ( 1 F i (t τ) ), (5) where τ is the time when the first Poisson signal is reeived, F i (t τ) is the probability player i assigns to player i oneding by time t onditional on time τ having been reahed, F i (t τ) = lim s t F i (s τ), and v i is informed player i s onession payoff 6 Given Z i, an uninformed player i s expeted payoff from oneding at time t is [ ] V i (t; Z i ) = E τ Pr(τ t)vi i (τ; Z i ) + Pr(τ > t)vu(t; i Z i ), (6) where VI i is player i s expeted payoff if she gets informed at τ t, and VU i i s expeted payoff if she has not beome informed by t V i I is player depends on τ but not on t, beause it is not neessarily the best response for an informed player to onede at t Let Ŵ i (τ; Z i ) = sup t τ W i (t; τ, Z i ), and p t be uninformed player i s posterior belief that v i = v L Then, VI i (τ; Z i ) = (v H s) df i (s)+(v H τ) ( F i (τ) F i (τ ) ) +Ŵ i (τ; Z i ) ( 1 F i (τ) ), s<τ and VU(t; i Z i ) = s<t (v H s) df i (s)+(m t) ( F i (t) F i (t ) ) +(p t v L t) ( 1 F i (t) ) A strategy profile (Z 1, Z 2 ) is an equilibrium if the expeted payoffs of both the uninformed and the informed players are maximized at any time t given the opponent s strategy Throughout this paper, we will fous on symmetri equilibrium in whih X 1 (t) = X 2 (t) and Y 1 (t; τ) = Y 2 (t; τ) for all t and τ t The supersript i will be omitted in the future if no onfusion arises 3 War of Attrition without Learning We will disuss two benhmarks in this setion In the first benhmark, both players are uninformed of their value of oneding until they atually onede In the seond benhmark, both players are privately informed of this value from the outset These two benhmarks 6 In the good-news model, v i = v L, and in the bad-news model, v i = 7

8 an be viewed as the limiting ases where λ = and, respetively 31 Benhmark I: Both Players Uninformed In the uninformed benhmark, eah player i assigns probability p to v i = v L throughout the game As a result, no matter when onession ours, the winner gets v H while the loser of the game gets the expeted payoff p v L The unique symmetri equilibrium is haraterized by the following proposition 7 The proofs of the following and all subsequent results an be found in the appendix Proposition 1 In the uninformed benhmark, the unique symmetri equilibrium is suh that at any time t [, ), eah player onedes at the onstant rate v H p v L In a symmetri equilibrium, no player onedes with a positive probability at time, and eah player is always indifferent between oneding and staying at any point in time By oneding, eah player reeives p v L for ertain, whereas, by staying, eah player pays flow ost and expets her opponent to onede at rate f t Therefore, the best responses of the players are determined by the sign of f t (v H p v L ) Eah player i stritly prefers to stay if the expeted onession rate f t is stritly larger than v H p v L ; is indifferent between oneding and staying if f t is exatly v H p v L ; and stritly prefers to onede if f t is stritly smaller than is v H p v L v H p v L In the unique symmetri equilibrium, the equilibrium onession rate to make eah player always indifferent between oneding and staying Although the game may ontinue for an arbitrarily long period of time, the expeted delay is finite in equilibrium, and equals v H p v L 2 32 Benhmark II: Both Players Informed In the seond benhmark, there is immediate revelation of payoffs Eah player i knows exatly what v i is, but her opponent does not The initial beliefs are suh that eah player has high onession payoff v L with probability p, and low onession payoff with probability 1 p This game is similar to the war of attrition with inomplete information analyzed by [1], [7], and [12] The next result shows that, in this benhmark, the unique symmetri equilibrium starts with the high-value player randomizing and the low-value player stritly preferring to stay; eventually, as the posterior probability of v L reahes zero, the low-value player randomizes until the end of the game Let t = ( ) ( log 1 1 p ) 7 In a omplete-information war of attrition, there always exists a degenerate equilibrium in whih player i onedes immediately, while player i never onedes, as shown by [11] However, as shown by [1], the equilibrium is unique if there is unertainty about the players types The symmetri equilibrium in a omplete-information war of attrition orresponds to the limit of the unique equilibrium in an inompleteinformation war of attrition as the degree of type unertainty onverges to 8

9 Proposition 2 In the full-information benhmark, the unique symmetri equilibrium has the following properties: (1) For t [, t ), the high-value player onedes at a rate, while the lowvalue player never onedes; v H ; /( ) t 1 (1 p ) (2) for t [t, ), only the low-value players stay, and they onede at a onstant rate (3) for any p (, 1), the expeted delay is smaller than in the uninformed benhmark The equilibrium analysis follows that in [1], [7], and [12] In a symmetri equilibrium, no player will onede with a positive probability at any time t, and F i (t) is ontinuous and stritly inreasing The best responses of the high-value (resp low-value) players are determined by the sign of f t (v H v L ) (resp f t v H ) Hene, high-value (resp lowvalue) player i stritly prefers to stay if the expeted onession rate f t is stritly larger than (resp v H ); is indifferent between oneding and staying if f t is exatly (resp v H ); and stritly prefers to onede if f t is stritly smaller than (resp v H ) Obviously, the high-value player is more willing to onede; hene, in the unique symmetri equilibrium, the game starts as a war of attrition between the high-value players, while the low-value player stritly prefers to stay The prior of being a high-value player is p, and the t v posterior, 1 (1 p )e H v L, is dereasing over time Eventually, as the high-value player has oneded with probability one (t t ), the expeted onession rate drops to v H game swithes to a war of attrition between the low-value players and the Proposition 2 also shows that the expeted delay is smaller than that in the uninformed benhmark This result has a very intuitive explanation In the full-information benhmark, with probability p, v i = v L and the expeted onession rate is ; with probability 1 p, v i = and the expeted onession rate is v H Sine v H pv L is onvex in p, we have p + (1 p ) v H > v H p v L, so the average expeted onession rate is higher in the full-information benhmark This effet is strong enough to outweigh the fat that in the full-information benhmark the uninformed type starts oneding only after some fixed amount of time 4 War of Attrition with Good-News Learning In this setion, we will first haraterize equilibrium in the good-news model In this model, at any time t >, the informed player believes v i = v L for ertain, while the uninformed player is still unsure about her onession payoff Reall that p t denotes the posterior probability whih an uninformed player assigns to the onession payoff v L at time 9

10 p t From Bayes rule, p t = e λt p e λt +1 p, sine the high-value player reeives no signal before time t with probability e λt while the low-value player reeives no signal for sure The game starts as a war of attrition with omplete information However, inomplete information develops over time due to learning Any andidate equilibrium shares the main features of a war of attrition with inomplete information studied in [1], [7], and [12] Therefore, as in the full-information benhmark, in a symmetri equilibrium, no player will onede with positive probability at any time t, and F i (t) is ontinuous and stritly inreasing This enables us to fous on analyzing the expeted onession rate f i t For both the informed and uninformed players, there are three possibilities: stritly prefer oneding, stritly prefer staying, or indifferene Thus, there are nine different ombinations in total The next lemma shows that only three of them an our in equilibrium Lemma 1 In any symmetri equilibrium, at any time t, only one of the following three ases is possible: (1) the uninformed player is indifferent between oneding and staying and the informed player stritly prefers oneding; (2) the informed player is indifferent between oneding and staying and the uninformed player stritly prefers staying; (3) the informed player stritly prefers oneding but the uninformed player stritly prefers staying We shall refer to ase (1) as uninformed indifferene, to ase (2) as informed indifferene, and to ase (3) as strit preferenes The above lemma has very intuitive interpretations Sine the informed player is more optimisti about the private payoff state than the uninformed player, the informed player has a higher inentive to onede As a result, if the uninformed player is indifferent between oneding and staying, the informed player must stritly prefer oneding; if the informed player is indifferent between oneding and staying, the uninformed player must stritly prefer staying In the full-information benhmark, at eah point in time, either the high-value or the low-value player is indifferent between oneding and staying Here, however, it is possible that neither the uninformed nor the informed player is indifferent Based on Lemma 1, Setions will fully haraterize symmetri equilibrium behavior in the good-news model Throughout these setions, we assume that v H 2p v L, whih guarantees that in the absene of good news, p t (v H p t v L ) dereases monotonially over time Setion 44 will briefly disuss the ase where v H < 2p v L 1

11 41 Slow Learning We will first haraterizes the unique symmetri equilibrium when the expeted learning rate is suffiiently low Proposition 3 If λp v H p v L, the unique symmetri equilibrium has the following properties: An uninformed player onedes with probability zero at time and at a positive rate x t = v H p t v L λp t for any t > An informed player onedes with probability one upon reeiving the first Poisson signal For suffiiently small λ, the unique symmetri equilibrium thus always features uninformed indifferene In fat, the low expeted arrival rate of good news makes it impossible for uninformed players to stritly prefer staying: even if informed players onede immediately, the expeted rate λp t at whih a player beomes informed is too low for an uninformed opponent to stritly benefit from staying in the game For uninformed indifferene, the expeted onession rate must be f t = v H p t v L provide a detailed proof of this result in the appendix Intuitively, the uninformed player s value of staying, V (p t ), must equal her value of oneding, p t v L, and it must satisfy the differential equation We + λp t (v L V (p t )) + f t (v H V (p t )) λp t (1 p t )V (p t ) = (7) Indeed, as long as an uninformed player pays ost to stay, her opponent onedes at the rate f t ; in this event, her ontinuation value jumps to v H With arrival rate λp t, the uninformed player reeives good news and then stritly prefers oneding, so in this event her ontinuation value jumps to v L If the opponent does not onede and there is no news, finally, the uninformed player s value of staying dereases deterministially; this is aptured by the term λp t (1 p t )V (p t ) 8 anel, leaving f t (v H p t v L ) = 9 Inserting V (p t ) = p t v L, we see that the last two terms Consequently, the onession rate of an uninformed player is the differene between f t and λp t, the onession rate generated by the arrival of good news Even if the ondition λp v H p v L Therefore, for any λ, the inequality λp t is not satisfied, as t tends to, p t dereases to zero v H p t v L will hold for t suffiiently large Consequently, uninformed indifferene and behavior as desribed in Proposition 3 will eventually arise in any symmetri equilibrium 8 As shown by [9], the law of motion for p t satisfies: ṗ t = λp t (1 p t ) 9 Intuitively, this is beause the value V (p t ) is linear in p t, whih is a martingale 11

12 42 Intermediate Speed of Learning For λp somewhat higher than assumed in Proposition 3, uninformed indifferene beomes impossible at the start of the game: the expeted rate at whih a player beomes informed (and then immediately onedes) makes it stritly benefiial for an uninformed player to stay On the other hand, from the analysis of the full-information benhmark, we know that for informed indifferene, the expeted onession rate must be ( v H p v L, ], there are strit preferenes initially Let T 1 satisfy: T 1 = 1 [ ] λ log p (1 p T1 ), (1 p )p T1 So, if λp where p T1 = λv H λ 2 v 2 H 4λv L 2λv L Proposition 4 If λp ( v H p v L, ], the unique symmetri equilibrium has the following properties: An informed player always onedes immediately, while an uninformed player never onedes prior to time T 1 For t T 1, players behave as desribed in Proposition 3 For expeted learning rates in an intermediate range, the expeted onession rate equals the expeted arrival rate of the first Poisson signal, λp t Strit preferenes further imply that if the game has not ended yet, it is ommon knowledge among the players that neither of them is informed When T 1 is reahed, therefore, ontinuation play an indeed be the one desribed in Proposition 3, started at the belief p T1 43 Fast Learning For λp even higher than assumed in Proposition 4, informed players have to be indifferent at the start of the game The high expeted arrival rate of good news makes it impossible for informed players to stritly prefer oneding, beause otherwise the expeted onession rate is so high that the opponent stritly benefits from staying in the game Let T be impliitly defined by Proposition 5 If λp > the following properties: 1 T p (1 e λt ) = 1 e (8), all symmetri equilibria are outome-equivalent and have 1 There is an indeterminay in the informed player s strategy X(t; τ) Any strategy an be an equilibrium as long as equation (4) is satisfied 12

13 (1) Up to time T, the informed player is indifferent between oneding and staying, while the uninformed player never onedes The expeted onession rate is (2) If v H 2v L, there exists a unique ˆλ > p ( ) λ ˆλ, then λp T λ < ˆλ, then λp T < v H p T v L (3) ˆλ is dereasing in p suh that the following holds: when and players behave as desribed in Proposition 4 for t T ; when v H p T v L and players behave as desribed in Proposition 3 for t T 11 In equilibrium, an informed player randomizes until the posterior belief of being an informed player reahes zero An informed player onedes with probability one by time T, whih is determined by equation (8) The left-hand side of equation (8), p (1 e λt ), is the probability that a player has reeived good news by time T ; the right-hand side, 1 e T, is the probability that a player has oneded until time T, given that the expeted onession rate is In summary, a symmetri equilibrium an be ompletely haraterized in terms of expeted rates of learning λp Players behave as desribed in Proposition 3 (resp 4) for small (resp intermediate) expeted rates of learning For high expeted rates of learning, any symmetri equilibrium features informed indifferene until T defined by equation (8) If the game has not ended yet at T, play ontinues under ommon knowledge that no player is informed Depending on the value of λ, the ontinuation play swithes to either strit preferenes or uninformed indifferene In the former ase (when λ is smaller than ˆλ), there exists T 1 suh that the symmetri equilibrium features strit preferenes for t (T, T + T 1 ); and uninformed indifferene for t (T + T 1, ) In the latter ase (when λ larger than ˆλ), the symmetri equilibrium always features uninformed indifferene after T The expeted onession rate may not be ontinuous in time (as shown by the full-information benhmark), but it is non-inreasing over time for any value of λ: while it is relatively easy to reah an agreement initially, it beomes inreasingly diffiult over time 12 The equilibrium haraterization is more ompliated if v H < 2p v L, beause there may exist two different roots < p 2 < p 1 < 1 solving p(v H pv L ) = Therefore, in a symmetri λ equilibrium, it is possible that a symmetri equilibrium features uninformed indifferene when p < p 2 or p > p 1, and strit preferenes when p (p 2, p 1 ) Although the equilibrium an be haraterized similarly, we skip the analysis sine it provides no new insights into the 11 From the proof in the appendix, v H 2v L is atually stronger than what is needed It is enough to assume v H (1 + p )v L 12 From the equilibrium haraterizations (Propositions 3-5), the expeted onession rate is a onstant for t T, and there must be a drop in the expeted onession rate if the game has not ended at T The expeted onession rate dereases ontinuously for t > T Therefore, the expeted onession rate is non-inreasing over time 13

14 omparative statis 5 Comparative Statis with Respet to the Learning Rate Based on the equilibrium haraterization in the previous setion, this setion will disuss how the learning rate λ affets expeted delay and welfare in equilibrium 51 Expeted Delay In a symmetri equilibrium, the expeted onession rate hanges over time As p t dereases, it beomes inreasingly diffiult to reah an agreement The non-stationary path of beliefs and onession rates makes it diffiult to derive expliit expressions for the expeted delay However, we show that an inrease in the learning rate inreases the expeted delay when λ is relatively low, but dereases the expeted delay when λ is relatively high Proposition 6 The expeted delay in symmetri equilibrium is non-monotoni in λ λp v H p v L, the expeted delay is inreasing in λ Moreover, the expeted delay is dereasing in λ when λ is suffiiently large The non-monotoniity result is driven by the fat that learning rate λ affets the expeted onession rate in two opposite ways On the one hand, faster learning makes it more likely for an uninformed player to reeive good news, and this inreases the expeted onession rate, sine an informed player is more willing to onede On the other hand, faster learning makes an uninformed player more willing to wait for both good news and her opponent s onession, and this strategi response leads to a lower expeted onession rate For small expeted rates of learning, this strategi effet dominates, beause in equilibrium, the expeted onession rate is determined to make an uninformed player indifferent between oneding and staying, and an inrease in λ indues the uninformed player to wait longer For intermediate expeted rates of learning, an uninformed player will always stay while an informed player will always onede Hene, faster learning leads to a quiker onession For high expeted rates of learning, an inrease in λ does not affet the expeted onession rate in equilibrium, whih is always to make an informed player indifferent Naturally, when λp v H p v L, the equilibrium will always feature uninformed indifferene Although an inrease in λ leads to a larger measure of informed players who will onede immediately, this inrease in the onession rate is ompletely rowded out by the derease in the onession rate of uninformed players, so as to make the less optimisti uninformed players indifferent between oneding and staying It is diffiult to derive an expliit omparative stati result for λp > v H p v L, beause of the oexistene of the opposite effets However, Proposition 6 shows that the expeted If 14

15 Expeted delay in the uninformed benhmark expeted delay Expeted delay in the full information benhmark λ Figure 1: Expeted delay as a funtion of λ (v H = 2, v L = = 1, p = 5) delay is dereasing in λ when λ is large Then, the equilibrium onsists of two phases: one features informed indifferene and the other features uninformed indifferene An inrease in λ has three effets on the average expeted onession rate in this situation First, when λ beomes higher, the game has a higher probability to stop before T defined by equation (8) Seond, a higher λ leads to a lower posterior belief p T : the uninformed players beome less optimisti Finally, a higher λ leads to a longer delay for t T The omparison of the two benhmarks suggests that the expeted delay is dereasing in λ if the third effet does not exist As v H pv L is onvex in p, a higher λ may lead to a higher expeted onession rate by making the distribution of p more dispersed 13 When λ approahes, the third effet beomes a seond order effet, beause as p T goes to zero, the possible redution in the expeted onession rate for t T is negligible As a result, the expeted delay will derease in λ when λ is suffiiently large 13 To be more speifi, assume that there is no learning one the game reahes T, and the players onede with probability x before T Then the expeted onession rate an be written as: x + (1 x) v H v L v H p x 1 x v, L where p x 1 x is the posterior belief p T The above term is dereasing in x, whih in turn is inreasing in λ 15

16 Based on Proposition 6, it is natural to onjeture that the expeted delay is first inreasing and then dereasing in λ Figure 1 onfirms this for a numerial example Compared to the uninformed benhmark, learning does not neessarily lead to shorter delays, and this ours only when the learning rate λ is suffiiently high 52 Welfare Based on the above equilibrium analysis, we ompute the expeted equilibrium payoff of eah player First, for small expeted rates of learning, if the ommon belief is that v = v L ours with probability p t, then the uninformed player i s value is V (p t ) = p t v L sine she is indifferent between oneding and staying Seond, for intermediate expeted rates of learning, the uninformed player s value V (p t ) satisfies the differential equation 2λp t V (p t ) = + λp t (v H + v L ) λp t (1 p t )V (p t ) By paying flow ost to stay, an uninformed player reeives good news with arrival rate λp t, and her ontinuation value jumps to v L in this event With arrival rate λp t her opponent reeives good news and onedes; in this event, her ontinuation value jumps to v H The differential equation has the general solution V (p) = 1 2 (v L + v H λ ) λ (1 p p)2 log( 1 p ) λ (1 p) + D 1(1 p) 2, where the oeffiient D 1 is determined by the boundary ondition that V (p) = pv L when λp(v H pv L ) = Finally, for intermediate expeted rates of learning, the uninformed player s value V (p t ) satisfies the differential equation, (f t + λp t )V (p t ) = + λp t v L + f t v H λp t (1 p t )V (p t ) This equation differs from the previous one in that the opponent s onession rate is f t rather than λp t From the indifferene ondition of the informed player, f t v H = f t v L, the differential equation is rewritten as with the general solution ( + λp t )V (p t ) = ( + λp t )v L λp t (1 p t )V (p t ) v H v L v H v L V (p) = v L + D 2 ( 1 p p ) λ() (1 p) The boundary ondition is given by value mathing at p T, where T is defined by equation 16

17 v(p ) λ Figure 2: Welfare as a funtion of λ (v H = 2, v L = = 1, p = 5) (8) Based on the above expressions for the value funtions, it an be shown that welfare is inreasing in λ when λ is suffiiently high Corollary 1 Suppose λ is suffiiently high and, therefore, the equilibrium is as desribed in Proposition 5 Then, V (p ;λ) λ > Figure 2 provides an illustration of the equilibrium value V (p ) as a funtion of λ As expeted, inreasing λ has a positive effet on V (p ) only for intermediate and high expeted rates of learning; for small expeted rates of learning, inreasing λ has no impat on the equilibrium value V (p ) = p v L, beause faster learning has both a positive effet by shortening the expeted time of reeiving good news and a negative effet of inreasing delay It turns out that these two opposite effets anel eah other out at time ; hene, inreasing λ has no impat on the expeted equilibrium payoff In ontrast, when λ is so high that the expeted delay is dereasing in λ, the negative effet is overturned and naturally, welfare inreases in λ 6 War of Attrition with Bad-News Learning In this setion, we will onsider the bad-news model in whih reeiving a Poisson signal immediately reveals v = In this model, the posterior probability whih an uninformed 17

18 p player assigns to the onession payoff v L at time t satisfies p t = p +(1 p : an uninformed )e λt player is more optimisti about the onession payoff than an informed player, and, hene, is more willing to onede As a result, the unique symmetri equilibrium always features two phases for any λ > : in the first phase, the uninformed players are randomizing between oneding and staying, while the informed players stritly prefer staying; in the seond phase, only the informed players are left in the game and they are randomizing between oneding and staying Let T be impliitly defined by F (T )p T T λp t (1 p t )Σ(t)dt = p, (9) where and T F (T ) = Γ(t)dt T λv Lp t (1 p t )Γ(t)dt + (v H p T v L )Γ(T ), (1) Σ(t) = Γ(s)ds F (T ) λv Lp s (1 p s )Γ(s)ds, (11) (v H p t v L )Γ(t) Γ(t) = e t v H [ (vh v L )p e λt + (1 p )v H v H p v L ] v L λv H ( ) (12) Proposition 7 The unique andidate symmetri equilibrium in the bad-news model has the following properties: (1) Eah uninformed player onedes with probability zero at time and at a positive rate between time and time T An informed player never onedes before T (2) After time T, only informed players stay and they onede at rate v H In ontrast to the good-news model, there is one unique equilibrium pattern in the badnews model Uninformed players always onede before informed players For any t < T, an uninformed player s value of staying, V (p t ), must equal her value of oneding, p t v L, and it must satisfy the differential equation + λ(1 p t )(W (p t ) V (p t )) + f t (v H V (p t )) + λp t (1 p t )V (p t ) = (13) Reall that in the good-news model, an informed player s value, W (p t ), is v L sine the informed player onedes immediately However, in the bad-news model, W (p t ) is not zero sine the informed player never onedes before T In partiular, W (p t ) must satisfy the differential equation + f t (v H W (p t )) + λp t (1 p t )W (p t ) = (14) 18

19 Using V (p t ) = p t v L, we solve W (p t ) = F (T ) F (t) p 1 F (t) t v L > This implies that the equilibrium onession rate is f t = λp t(1 p t )v L (F (T ) F (t))/(1 F (t)) v H p t v L (15) The equilibrium onession rate onsists of two parts The first part, v H p t v L is inreasing over time: as an uninformed player beomes inreasingly optimisti about the onession payoff, she is more willing to onede However, reeiving bad news also brings positive learning value W (p t ), whih makes an uninformed player relutant to onede This is aptured by the negative part of equation (15) The negative part gradually onverges to zero as t approahes T We derive F (t) by solving the differential equation (15) The utoff time T is pinned down by the fat that the uninformed players onede with probability one by time T Equation (9), whih determines T, is in turn derived from equation p = T p t F (t)dt = p T F (T ) T λp t (1 p t )F (t)dt (16) p is the measure of high-value players; while T p tf (t)dt is the measure of high-value players who have oneded until time T, whih is the integral of the density of high-value players oneding at time t T These two terms have to be the same sine only the low-value players stay after time T Compared to the uninformed benhmark, learning affets expeted delay through three different hannels First, for t < T, learning redues expeted delay by making the uninformed players optimisti about the onession payoff Seond, for t < T, learning inreases expeted delay due to the positive learning value Finally, for t T, learning inreases expeted delay sine the low value players have the lowest onession rate Figure 3 illustrates a numerial example using the same parameter values as for Figure 1 For these parameters, the expeted delay is atually monotonially inreasing in λ As to welfare, sine in equilibrium an uninformed player is always indifferent between oneding and staying, the equilibrium value V (p ) is always p v L Therefore, in the badnews model, although the learning rate would affet the expeted delay, the equilibrium value V (p ) is not affeted by the learning rate at all On the one hand, faster learning makes it more likely to reeive bad news, whih redues the expeted payoff to On the other hand, faster learning makes the uninformed opponent more willing to onede, and this leads to a higher expeted payoff In equilibrium, these two opposite effets anel out at time ; thus the equilibrium value does not hange with the learning rate 19

20 86 84 Expeted delay Expeted delay in the uninformed benhmark λ Figure 3: Expeted delay as a funtion of λ (v H = 2, v L = = 1, p = 5) 7 Conlusion Delay is a pervasive phenomenon in many realisti situations inluding bargaining or oordination The paper onsiders a ontinuous-time war of attrition to investigate how exogenous learning about private payoffs affets delays in reahing agreements In the model, the onession payoff an be either high or low, and at eah point in time, players may reeive a private Poisson signal that ompletely reveals the onession payoff to be high (good-news learning) or low (bad-news learning) In the good-news model, learning an ause shorter delays in reahing agreements, but not always In partiular, an inrease in the learning rate auses longer delays when the learning rate is suffiiently low In the bad-news model, we show by example that learning auses longer delays, but has no impat on welfare The model studied in this paper an be extended in several different ways First of all, it would be interesting to explore a speifiation in whih the signal follows a Brownian motion with drift v i Then the posterior belief follows a diffusion proess, as shown in [4] It is natural to onjeture an equilibrium in whih there exists a utoff p t at time t suh that players with p t p t (p t < p t ) will onede (stay) However, analyzing the utoff p t is hallenging Seond, in the model, it is assumed for simpliity that the two players are faing the same learning rate This enables us to onsider symmetri equilibria, in whih no player onedes with a positive probability at time It is more ompliated to analyze equilibria 2

21 for differential learning rates In this ase, as shown by [1], it is still possible to onstrut a non-degenerate equilibrium in whih onession may ontinue for an arbitrarily long period of time However, one of the two players will onede with a positive probability at time Finally, endogenous information aquisition an be added into the model, by assuming that the players an inrease the learning rate through ostly effort as modeled by [5] At every instant of time, both players have to deide whether or not to onede, and the learning rate if they hoose to stay In the working paper version of this artile, it is shown that in the good-news model, no player will aquire information if the maximum ahievable learning rate is lower than p (v H p v L More generally, one would expet that there is insuffiient ) inentive to aquire information in this setting 21

22 Appendix A1 Proof of Proposition 1 Suppose there is a symmetri equilibrium in whih eah player s strategy is denoted by X(t) Notie that in this model, X(t) is the same as the assoiated distribution over stopping times, F (t) Any andidate symmetri equilibrium in this game shares the following properties: (i) At time, none of the two players onedes with a positive probability; (ii) X(t) is ontinuous and stritly inreasing for any t; (iii) The game has an infinite horizon: there does not exist any t suh that X(t) = 1 Properties (i)-(iii) are proven by a series of sublaims: (a) If X 1 has a jump at t, then X 2 does not jump at t If X 1 jumps at t, then player 2 reeives a stritly higher utility by oneding an instant after t than by oneding exatly at t This in turn implies that in a symmetri equilibrium, X(t) has to be ontinuous (b) There does not exist any t suh that X 1 (t) = X 2 (t) = 1 Suppose not and let ˆt = sup{t : X(t) < 1} < be the earliest time by whih the two players onede with probability one From the ontinuity of X( ), there is no mass point at ˆt and there exists t suh that X(t) is ontinuous and stritly inreasing on [ t, ˆt] Therefore, for any t [ t, ˆt], = t (v H s)dx(s) + (p v L t)(1 X(t)) (p v L t)(1 X( t)) We hene obtain the following differential equation: whih implies: X (t) = v H p v L (1 X(t)), log(1 X(t)) log(1 X( t)) = (t t) v H p v L For ˆt <, X(ˆt) < 1 but this ontradits the definition of ˆt () There is no interval (t 1, t 2 ) suh that both X 1 and X 2 are onstant on (t 1, t 2 ) Assume the ontrary and without loss of generality, let ˆt be the supremum of t 2 for whih (t 1, t 2 ) satisfies the above property Fix any t (t 1, ˆt) Sine X( ) is onstant on (t, ˆt), eah player prefers to onede at t than to onede at ˆt: ˆt t (v H s)dx(s) + (p v L ˆt)(1 X(ˆt)) (p v L t)(1 X(t)) < Furthermore, from the ontinuity of X( ), the expeted payoff from stopping at ˆt + ϵ will beome arbitrarily lose to the expeted payoff from stopping at ˆt, as ϵ > onverges to zero Therefore, for ϵ > suffiiently small, eah player prefers to onede at t than to onede at ˆt + ϵ: ˆt+ϵ t (v H s)dx(s) + (p v L (ˆt + ϵ))(1 X(ˆt + ϵ)) (p v L t)(1 X(t)) < But this ontradits the definition of ˆt 22

23 In models with learning, the equilibrium distribution over stopping times in any andidate symmetri equilibrium should also satisfy Properties (i)-(iii) Sine the proofs are similar, we will apply these properties without reproving them in the subsequent analysis Properties (i)-(iii) imply that in any symmetri equilibrium, eah player randomizes between oneding and staying at any time t The expeted payoff for a player from stopping at t is V (t; X) = (v H s)dx(s) + (p v L t)(1 X(t)) A player is willing to randomize only if her payoff is onstant on [, ) Thus, the derivative of her expeted payoff with respet to t must be zero: whih implies (v H t)x (t) (1 X(t)) (p v L t)x (t) =, x t X (t) 1 X(t) = v H p v L A2 Proof of Proposition 2 The proof proeeds in three steps 1 We onstrut a andidate symmetri equilibrium satisfying the equilibrium onditions 2 We verify that the onstruted equilibrium is indeed the unique symmetri equilibrium of the game 3 We ompute the expeted delay in the unique symmetri equilibrium Step 1 Suppose there is a symmetri equilibrium in whih the strategy for the high-value player has full support on an interval [, t ), and the low-value player never onedes before t With a little abuse of notation, we use X( ) to denote the strategy of the high-value player The expeted payoff for a high-value player from stopping at t < t is V (t; X) = p (v H s)dx(s) + (v L t)(1 p X(t)) A high-value player is willing to randomize over the entire interval [, t ) only if her payoff is onstant on [, t ) As a result, p X (t)(v H t) (1 p X(t)) p X (t)(v L t) =, t < t After simplifying the last equation, we obtain and hene p X (t) 1 p X(t) = v H v L, p X(t) = 1 e t 23

24 By definition, the onession rate of the high-value player is x t X (t) 1 X(t) = /(v H v L ) 1 (1 p ) t In equilibrium, the threshold t must have the property that the high value player onedes with probability one before t (ie, X(t ) = 1) This yields ( ) ( ) t vh v L 1 = log 1 p After t, only low-value players stay, and the equilibrium onession rate is v H to make them indifferent between oneding and staying Step 2 We will verify the strategies desribed above indeed onstitute a symmetri equilibrium First, by onstrution, the high-value player is indifferent between oneding and staying for all t t, and the low-value player is indifferent between oneding and staying for all t > t Seond, by oneding at t t, the low-value player s expeted payoff is (v H s)d (1 e s ) ( te t = v L 1 e t ) whih is stritly positive and inreasing in t Therefore, the low-value player stritly prefers staying until time t Similarly, by oneding at t > t, the high-value player s expeted payoff is ( (v H s)d (1 e s )+(1 p ) (v H s)d 1 e (s t t ) ) v H, +(v L t)(1 p )e (t t ) v H, whih is stritly dereasing in t Therefore, the high-value player stritly prefers oneding before time t Any symmetri equilibrium must have the form desribed above (uniqueness), beause by Lemma 1, in any symmetri equilibrium, the high-value player must onede before the low-value player Step 3 In the unique symmetri equilibrium, with probability p 2, both players onede before time t, and the expeted delay is td 1 ( 1 1 e p t with probability 2p (1 p ), one player onede before t while the other player stays after t, and the expeted delay is ) t (1 e t td ; and with probability (1 p ) 2, both players stay after t, and the expeted delay is t + v H 2 24 p ) 2 ;