Are Obstinacy and Threat of Leaving the Bargaining Table Wise Tactics in Negotiations?

Transcription

1 Are Obstinacy and Threat of Leaving the Bargaining Table Wise Tactics in Negotiations? Selçuk Özyurt Sabancı University Very early draft. Please do not circulate or cite. Abstract Tactics that bargainers use to build reputation during negotiations would have asymmetric impacts on negotiation outcomes, and the precautions that should be taken to ease this asymmetry are formally investigated in this paper. Two widely observed tactics in bargaining will be investigated. These are obstinacy and the threat of leaving negotiation table. For his purpose, bilateral negotiation is modelled as a non-cooperative game. Then, the equilibrium analyses of the game is executed and the optimal strategies are determined. The conditions under which use of these tactics are advantageous or disadvantageous are analyzed. Moreover, the measures that could be taken to neutralize the bargaining power of the negotiator who uses these tactics are examined. Faculty of Arts and Social Sciences, Sabancı University, 34956, Istanbul, Turkey. ozyurt@sabanciuniv.edu) (

2 2. The Model Two players, and 2, bargain over the division of a surplus. At time t = 3, players simultaneously demand a share of the surplus; α i [0, ] i =, 2. If the players make compatible demands (α, α 2 ), that is α +α 2, then one of the two divisions of surplus (α, α ) or ( α 2, α 2 ) is implemented with probability /2 each and the game ends. The game continues if the players initial demands are incompatible. At time t = 2, after observing his opponent s demand, player has the option of threatening his opponent by announcing a particular time indicating that he would leave the bargaining table at that time unless his demand is accepted. More formally, player announces K [0, ). Both players believes that nature independently and privately sends one of two messages {c, d} to each player at time t =. If a rational player receives the message d don t commit, he will continue to play the game. A player who receives the message c is replied with his behavioral counterpart before the game proceeds to t = 0. Behavioral type player never accepts α 2 and leaves the bargaining game at time K if player 2 does not accept α before this time. Behavioral type player 2 never accepts α but he never leaves the bargaining game. Each rational player i {, 2} receives the message c with probability z i (0, ). Therefore, after t =, each player knows his own type, but not his opponents true types. However, the initial priors, i.e. z and z 2, are common knowledge. No discounting applies before time t = 0. At time t = 0, players immediately begin to play the following continuous-time concession game: At any given time t 0, a player either accepts his opponent s initial demand, waits for his concession or exit the bargaining table. Concession of a player marks the completion of the game. If a player leaves the bargaining table, then he receives his outside option, which is normalized to zero. At time t = K, there occurs some measure theoretic pathologies associated with behaviors of player s rational and behavioral types. We resolve this in the manner introduced in Abreu and Pearce (2007) and later used by Abreu, Pearce and Stacchetti (202): corresponding to the conventional time K, there are two logically consecutive stages K and K 2 in the model. In both of these stages a player either accepts his opponent s initial demand or waits for his concession. Player (rational or not) leave the bargaining table at time K (the first stage of time K) not at time K 2. No discounting applies between these two stages. Rational players are assumed to maximize the expected discounted value of their shares. If agreement is never reached, they both receive 0 payoffs. If rational player leaves the bargaining table, then he receives payoff of 0. If the game finishes at time t 0 with player i s acceptance of j s demand, then the payoffs to the players i and j 2

3 are ( α j )e rit and α j e rjt, respectively. Rational player i discounts time with a rate r i > 0. I denote this infinite horizon, continuous-time bargaining game by G. In what follows, I am interested in investigating the case where nature sends the message d to both players. In addition to this, since payoffs of the behavioral type players are irrelevant for equilibrium calculations, I left them unspecified. A rational player who receives the message c leaves the game at time t =, and so the game ends for him before it proceeds to time t = 0. However, replaced players do not receive payoff of zero. I suppose that, in equilibrium, the payoff of a rational player when he receives the message c is identical to his continuation value when he receives the message d. That is, a rational player who is forced to leave the game at time t = will be compensated, and the amount of this compensation in equilibrium is such that the rational player s demand selection at time t = 3 is independent of what specific message he would receive at time t =. Three simplifying assumptions of the model deserve explicit clarification here. First, the model adopts a war of attrition protocol after t = 0, disallowing counteroffers and permitting buyers only two choices: concede or wait. Second, the assumption that players who receive the message c are replaced and are compensated is rather unconventional. Third, rational player can leave the bargaining table only at time K. First, The bargaining game is modeled as a war of attrition game where each rational player has to choose the timing of acceptance. Alternatively, for example, we could suppose that players can modify their offers at times {, 2,...} in alternating orders, but can concede to an outstanding demand at any t [0, ). Assuming that a behavioral type of player i always offer α i, accepts any price offer greater or equal to α i and rejects all smaller offers, modifying his offer would reveal a player s rationality, and in the unique equilibrium of the continuation game he should concede to the opponent s demand immediately (see Section 8.8 of Myerson (99), Proposition 4 of Abreu and Gul (2000) and Lemma of Abreu and Pearce (2007)). Thus, in equilibrium, rational players would never modify their demands. The second assumption each rational player is replaced with his behavioral counterpart with some positive probability at time t = and compensated for his loss is a critical one, and it is imposed mainly for technical reasons. Denseness of the choice set [0, ] is problematic. This was a valid problem in other papers of the bargaining and reputation literature. Abreu and Gul (2000), Abreu and Pearce (2007) and Abreu, Pearce and Stacchetti (202) handled this problem by considering a finite set of obstinate types. According to their models, an obstinate type of a player is identified by a number In case of simultaneous concession one of the two divisions of surplus (α, α ) or ( α 2, α 2 ) is implemented with probability /2 each. 3

4 α [0, ]. Therefore, a type α player always demands α, accepts any price offer greater or equal to α and rejects all smaller offers. According to this way of thinking, behavioral types are born with their demands. However, to make their model workable, they had to assume that the set of behavioral types, call it C, is a finite subset of [0, ]. Finiteness of C in their paper is vital and here is why: Suppose for a moment that ) nature does not send a message at time t =, 2) the set of obstinate types C is equal to the unit interval [0, ], and so π is a continuous distribution (density) function over C, and 3) if player i is rational and demanding α C at time t = 3, then this is his strategic choice, and if he is the behavioral type, then he merely declares the demand corresponding to his type. Then, given the rational player i pure equilibrium strategy for time t = 3 (say posting α ), the posterior probability that player i is the behavioral type is simply zero if player i actually demands α. This is true because the probability that player i is obstinate is z i whereas the conditional probability that player i is behavioral of type α, i.e. π(α ), is zero. That says, in equilibrium, if a rational player announces his demand according to his prescribed strategy, then his rationality will be revealed. However, if he deviates and demands some other share, then his opponent will believe that he is the behavioral type. This conclusion is fairly absurd, and most importantly it makes the model impracticable. If I adopt the framework of Abreu and Gul (2000) and assume that C is a finite set instead of using the time t = assumption then this would cause some other technical difficulties. For example, in equilibrium the players have to mix between certain demands at time t = 3. More importantly, rational player would have to mix his exit time from the bargaining game, which complicates the equilibrium computation without adding much of an insight. On the other hand, Kambe (999) solved the problem that arises due to denseness of the choice set by assuming that behavioral players can choose their demand to commit. Therefore, Kambe (999) allows the rational players to choose their demands from the set [0, ] before the war of attrition stage begins, but their choices are conditioned on the uncertainty that they may be forced to commit their prices during the negotiation phase. Although, this approach can be justified in some grounds, I prefer to follow the approach that reputations are dispositional. That is, reputations stem from underlying differences among people, firms, or states, and it is at least partly an enduring characteristics of the players (see Schelling 960, Kreps and Wilson 982, Myerson 99). For example, Kreps and Wilson (982) argue that monopolists may be strong or weak and that they wish to develop reputations for strength in order to convince new firms not to enter the market. In this regard, the role of time t = assumption is vital to overcome the technical 4

5 problems with the dense set of demands. Nevertheless, it is possible to provide a more conventional assumption for time t = that yields the same equilibrium behaviors. One can still suppose that nature independently and privately sends one of two messages {c, d} to each player at time t =, a player receiving the message c is replaced with his obstinate counterpart, and that a rational player who receives the message d proceeds to bargaining phase. Regarding the players beliefs about the occurrence of the messages, however, we can alternatively suppose that each rational player believes before the game starts that he will receive the message d at time t = with probability one, but is uncertain about the messages his opponents would receive. In particular, rational player i believes that his opponent will receive the message c with probability z j. Finally, we could suppose that all of this information is common knowledge between all two rational players. Thanks to the heterogeneous initial priors assumption of this alternative treatment, we can get rid of the assumption that the players who receive the message c will be compensated. To sum up, with the time t = assumption, regardless of player i s equilibrium strategy, his initial demand at time t = 3, player j s posterior belief about the rationality of player i is z i. Likewise, regardless of player s choice of exit time at time t = 2, player 2 s posterior belief about the rationality of player is z. I am not in the position to criticize the argument that the players posterior beliefs should (or should not) depend on players strategies and previous actions. Nevertheless, one may generalize the current setting and assume that both z and z 2 are functions of demands that the players announce at time t = 3, and so players may perceive their opponents more likely to be rational under certain demand selections. Finally, the model allows rational player to leave the bargaining table only at time K. One can easily extend the model and suppose that player can leave the bargaining game at any time he wants. Since rational player is assumed to receive zero payoff if he leaves the negotiation, rational player never leaves the bargaining game in equilibrium. However, player 2 s belief that player may be the behavioral type, who certainly leaves the game at time K if his demand is not accepted, provides rational player strong incentive to manipulate his opponent s belief and build false reputation regarding his type. Strategies of the Rational Players: At time t = 3, a strategy for rational player i is a pure action alpha i (0, ). Since the subsequent analysis is quite involved, I restrict the players to play pure strategies at time t = 3. Given the players demand selections, a strategy for rational player at time t = 2 is a pure action K [0, ). Although K 5

6 depends on α, α 2, this connection is omitted for notational simplicity. In the war of attrition phase, strategy for player i =, 2 is a right-continuous distribution function F i : [0, ] [0, ] representing the probability of rational player i conceding to player j by time t (inclusive). Let B i : [0, ] [0, z i ] denote the probability of player i conceding to player j by time t inclusive. That is, B i indicates player j s belief about i accepting α j and finishing the war of attrition game at or before time t. Therefore, we have B i (t) = F i (t)( z ). Note that rational player receives the payoff of zero if he leaves the bargaining table. Hence, it is never optimal for rational player to leave the bargaining table. Therefore, I ignore the strategies of rational player, prescribing him to leave the bargaining table at any time of the game. Given F j, rational player i s expected payoff of conceding to player j at time t is U i (t, F j ) := α i ( z j ) t 0 + ( α j )[ B j (t)]e r it e r iy df j (y) + 2 ( + α i α j )[B j (t) B j (t )]e r it () with B j (t ) = ( z j ) lim y t F j (y). 3. Main Results In this section I will characterize the (sequential) equilibrium strategies of the game G. I first characterize the equilibrium strategies of the players in the war of attrition game. Therefore, I take α, α 2 and K(α, α 2 ) as given. Since the game finish at time t = 3 if α + α 2, I suppose that α + α 2 > ; Propositions -6. Then, I will characterize the optimal strategy K(α, α 2 ) of player at time t = 2 (Theorem ). Finally, I will characterize the optimal demand selection of the players at time t = 3 (Proposition 7 and Theorem 2). It is important to note that behavioral type of player certainly leaves the game at time K if his demand is not accepted before. Furthermore, since rational player receives zero payoff if he leaves the bargaining table, in equilibrium, player s rationality will certainly be revealed at time K 2. Thus, in equilibrium, if the game reaches time t = K 2, player accepts α 2 and finalizes the game. This is a direct implication of Myerson (99) and Abreu and Gul (2000); if player s rationality is revealed but player 2 is not known to be rational, then the unique equilibrium in he continuation game is that player immediately accepts player 2 s demand α 2. 6

7 CASE A: K = 0 Behavioral type of player leaves the game at time K = 0. If the game does not end at time 0, rational player accepts α 2 at time 0 2 with probability one. The reason for this simply follows arguments of Myerson (99), Abreu and Gul (2000). Player 2 s reputation at time 0 2 is either z 2 or higher. There is no way he would reveal his rationality at this time. Hence, rational player will immediately accept 2 s offer at time 0 2. Hence, in equilibrium, the game will end at time zero (either at time 0 or 0 2 ) for sure and F 2 (0) = 0. Proposition. In equilibrium where K = 0 and α + α 2 > we have;. If z > z = α +α 2 α 2, then rational player 2 concedes at time 0 with probability one and rational player accepts α 2 only if the game proceeds time 0 2. That is, F (0 ) = 0, F (0 2 ) = and F 2 (0 ) =. 2. If z < z, then rational player 2 never concedes and rational player is indifferent between conceding ) at times 0 and 0 2. That is, F 2 (0 ) = F 2 (0 2 ) = 0, 0 F (0 ) 2 and F (0 2 ) =. ( z z z ( z ) 3. If z = z, then rational player concedes only at time 0 2 and rational player 2 is indifferent between conceding at time 0 and 0 2. That is, F (0 ) = 0, F (0 2 ) =, 0 F 2 (0 ) and F 2 (0 2 ) = F 2 (0 ). CASE B: K T 0 = min{ ln z λ, ln z 2 λ 2 } Proposition 2. In equilibrium where K [T 0, ) and α + α 2 > we have { z F i (t) = i ( c i e λit ), if t [0, T 0 ], otherwise, such that c i = z i e λ it 0, T 0 = min{ ln z λ, ln z 2 λ 2 } and λ i = r j( α i ) α +α 2. Proof. The proof directly follows from Hendricks, Weiss and Wilson (988) and is analogous to the proof of Lemma in Abreu and Gul (2000). CASE C: 0 < K < T 0 Consider a history where no player concedes before time t = K. Given the players equilibrium strategies, let ẑ and ẑ 2 denote the posterior probabilities that player and 2, respectively, are behavioral types. It must be the case that ẑ i is either greater than or equal to z i. Therefore, with probability ẑ, player will leave the game at time K in which case player 2 receives the payoff of zero. However, with probability ẑ player is rational and hence accepts player 2 s demand either at time K or time K 2 (depending on his equilibrium strategy) for sure, which implies payoff of 7

8 α 2 to player 2. Thus, in equilibrium, rational player 2 will concede at time K with a positive probability if and only if ẑ is high enough. Note that, player 2 never concedes after time K. Given the first player s equilibrium strategy F, let U 2 (K, F ) and U 2 (W ait, F 2 ) denote player 2 s expected payoff of conceding and waiting, respectively, at time K. For notational simplicity suppose that these payoffs are calculated at time K. Thus, U 2 (K, F ) U 2 (W ait, F 2 ) if and only if ẑ ( α ) + ( ẑ )p (K )( +α 2 α 2 ) + ( ẑ )( p (K ))( α ) ( ẑ )α 2 where p (K ) indicates the probability that rational player accepts α 2 at time K. If p (K ) = 0 is true, then U 2 (K, F ) is equal to U 2 (W ait, F 2 ) and ẑ = z. Moreover, if ẑ = z, then U 2 (K, F ) U 2 (W ait, F 2 ) is equal to p (K )( α α 2 )( α +α 2 ) which is non-negative for all values of p 2 (K ) 0. The last observations imply that, in equilibrium, if player s reputation ẑ at time K reaches z and if rational player concedes at time K with some positive probability, then rational player 2 should concede at time K with certainty. However, in an equilibrium strategy where rational player 2 concedes at time K with a positive probability, rational player prefers to wait at time K. The last argument is true because U (K, F 2 ) = ( ẑ 2 )p 2 (K )( +α α 2 2 ) + [ẑ 2 + ( ẑ 2 )( p 2 (K )]( α 2 ) is strictly smaller than U (K 2, F 2 ) = ( ẑ 2 )p 2 (K )α + [ẑ 2 + ( ẑ 2 )( p 2 (K ))]( α 2 ) whenever p 2 (K 2 ) > 0. Hence, in equilibrium, if player s reputation at time K reaches z, then player does not concede at time K. Then, the question is whether player prefers to build his reputation to z before time K. The next sequence of results investigate the answer of this question. Proposition 3. In any equilibrium where K (0, T 0 ) and α + α 2 >, the followings must be true:. Equilibrium strategies F and F 2 of the concession game are continuous and strictly increasing over [0, K ]. In particular, for all t [0, K ], F i (t) = z i ( c i e λ it ) where c i [0, ] with ( c )( c 2 ) = 0 and λ i = r j( α i ) α +α 2. Moreover, F 2(t) = F 2 (K ) for all t > K and F (K 2 ) =. 2. Player s reputation at time K, ẑ, reaches exactly z as long as the second player s reputation at time K, ẑ 2, is strictly less than. If ẑ 2 =, then ẑ is. According to Proposition 3, if player does not make a positive probabilistic concession at time zero, then the second condition of the same proposition implies that K = ln(z /z ) λ := T z must hold. This is true because ẑ (t) =, implying z +( z )( F (t)) that = z since c =. Solving the last equality yields the desired relation. z e λ K 8

9 Proposition 4. In equilibrium where 0 < K < min{t, T 0 } and α + α 2 > we have (i) F (t) = z ( z z e λ (K t) ) for all t [0, K ] and F (t) = for all t K 2. (ii) F 2 (t) = z 2 ( e λ 2t ) for all t [0, K ] and F 2 (t) = F 2 (K ) for all t K 2. Proof. Player cannot build his reputation to z at time K if he does not make a positive probabilistic concession. Therefore, we have c 2 = must hold in equilibrium. The last observation, together with Proposition 3, yields F 2. In order to find F (t), we need to solve z z +( z )( F (K )) = z. Proposition 5. In equilibrium where 0 < T < K < T 0 and α + α 2 > we have (i) F (t) = z ( e λ t ) for all t [0, K ] and F (t) = for all t K 2. (ii) F 2 (t) = z 2 ( z 2 e λ 2(K t) ) for all t [0, K ] and F 2 (t) = for all t K 2. Proof. Since T < K, player has enough of time to build his reputation to z, reputation of player at time K, i.e. ẑ will be strictly higher than z (since F is strictly increasing on [0, K ]). Hence, player 2 strictly prefers to concede at time K. However, in equilibrium P 2 (K ) = 0 must hold. These arguments imply that we must have F 2 (K ) =.The last equality implies the value of F 2 (t). Since c 2 <, in equilibrium, we have c =, yielding the value of F. Proposition 6. In equilibrium where 0 < K = T < T 0 and α + α 2 > we have (i) F (t) = z ( e λ t ) for all t [0, K ] and F (t) = for all t K 2, and (ii) F 2 (t) = z 2 ( c 2 e λ 2t ) for all t [0, K ] and F 2 (t) = F 2 (K ) for all t K 2 where c 2 [z 2 ( z z ) λ 2/λ, ]. Proof. Here, if F (0) > 0, then ẑ (K ) > z. But then player 2 strictly prefers conceding at time K, but then we must have F 2 (K ) = z 2. But, since K < T 0, the last equality is possible only if F 2 (0) > 0. However, in equilibrium, F (0)F 2 (0) = 0 must hold. Hence, we must have F (0) = 0, yielding the value of F. Since equilibrium implies that F 2(K ) z 2, we must have c 2 z 2 e λ 2K, implying the set of possible values of c 2. Theorem. In equilibrium where α + α 2 >, (i) if z z, then player chooses K = 0, rational player 2 accepts α at time 0 with certainty and rational player accepts α only at time 0 2 if the game has not ended yet, 9

10 (ii) if zz λ /λ 2 2 < z < z, then player chooses K = ln(z /z ), concession game strategies F and F 2 are as characterized in Proposition 6, where c 2 = z 2 ( z z ) λ2 /λ is uniquely determined, and (iii) if z z z λ /λ 2 2, then player chooses any K [T 0, ) and concession game strategies F and F 2 are as characterized in Proposition 2. For the fixed values of the primitives z, z 2 (0, ) and r, r 2, define the set { { r α + α 2 2 ( α } } ) r B = α (0, ) sup z ( α 2 ) < z α 2 [ α,) and let sup α (0,) B = ᾱ. Note that ᾱ is well-defined because the set B is non-empty. This is true because for any { values of z and z 2, we can find α sufficiently close to r 2 ( α } ) r zero so that sup α2 ( α,) z ( α 2 ) is also close to zero, and thus less than z. α +α 2 α 2 2 Furthermore, for the fixed values of the primitives and for any α, α 2 (0, ) such that α + α 2 > define the set { D(α, α 2 ) = z (0, ) α + α 2 α 2 α 2 2 λ r 2 ( α ) r z ( α 2 ) 2 < z α } + α 2 α 2 The next result shows, in equilibrium, that rational player cannot choose a demand higher than ᾱ, player 2 is weak and player is strong in equilibrium. Proposition 7. Given z and z 2 and the other primitives, in any equilibrium where players demands α and α 2, we must have z D(α, α 2) and α 2 α ᾱ Theorem 2. In equilibrium the players demands α and α2 must satisfy ( ) α arg max α z 2 (α + α2 α + α2 r ( α 2 ) r 2 ( α ) ) z α2 α [ α 2,ᾱ ] z D(α,α 2 ) As a direct implication of Theorems and 2, in equilibrium rational player will select the exit time positive, as a function of initial demands, α and α 2, K = ln(z /z ) λ = ( ) α +α 2 ln z α 2 r 2 ( α ) α +α 2. Thus equilibrium exhibits delay. For fixed values of z 2, ᾱ and the first player s equilibrium payoff increase with z. However, for fixed values of z, ᾱ the rational player s equilibrium payoff decrease with z 2. Example : An Illustration of the Equilibria: Equilibrium demand selections of the players are not unique. Here I provide a numerical example. I consider the following parameter values; z = z 2 = 0. and r = r 2 = 0.7. The optimization problems are solved 0

11 Figure : Best response correspondences of the players for z = z 2 = 0. and r = r 2 = 0.7. by Matlab. In figure BR i (α j ) is the best response function of player i given the value of α j. Player s best response function is found by choosing α such that z (zz λ 2 2, z] and it maximizes u = ( z 2 ( z z ) λ2 λ )α + z 2 ( z z ) λ2 λ ( α 2 ). Similarly for each α (0, ), BR 2 (α ) is found by choosing α 2 (0, ) such that λ λ λ 2 zz λ 2 2 z and maximizes u 2 = ( z z2 )α 2 + z z2 ( α ). If there is no α 2 (0, ) λ such that zz λ 2 2 z, then BR 2 (α ) = (0, ) since 2 s expected payoff is independent of α 2. Thus, the red curve in the figure indicates the equilibrium demand selections. One equilibrium demand profile is such that: α = and α 2 = In λ that case zz λ 2 2 = and z = 0.337, so z (zz 2, z) as required. Then, in equilibrium K = T = Expected payoffs are u = and u 2 = Since u + u 2 = <, the amount of inefficiency is Example 2: Figure 2 underlines the fact that for fixed values of z 2, ᾱ decreases as z decreases to zero. In the figure, the dashed lines show ᾱ for each z and the solid lines gives equilibrium demands for each z. We find the numerical results and create the figure λ λ 2 λ λ 2 λ

12 Figure 2: Equilibrium demands for different values of z. by using Matlab program. In this example parameters are such that z 2 = 0., r = 0.7 and r 2 = 0.7 while z takes four different values; 0., 0.0, 0 6 and Possible Extensions In the appendix, I characterize the conditions that determine the equilibrium prices when players sequential choose their demands. In the first case, I suppose that player 2 chooses his demand first, and then player chooses his demand and the exit time. In the second case, player determines his price, and then player 2 chooses his price. Finally, player chooses his exit time. Since the players payoff functions are complicated, determining the set of equilibrium prices is not possible. Another possible extension is the case where both players (either sequentially or simultaneously) choose an exit time. Although this case would be interesting from analytical perspective, it does not make much of a sense in practical sense. When a bargainer makes a threat of leaving the bargaining table unless his demand is not accepted, the 2

13 other bargainer is supposed to make a decision between accepting his demand, making an acceptable counter offer or letting him leave the negotiation. Making a counter threat by choosing an earlier time that the opponent declare is usually not what the other player is expected to do. On the other hand, from analytical point of view, investigating the case where two players can choose some time K to exit the negotiation leads to significant amount of complication. Suppose for example that players simultaneously declare some exit time K i for player i. If K < K 2 for example, it is not clear if rational player will immediately accept the second player s demand if the game does not finish before time K. As we know, player s rationality will be revealed if he does not leave the bargaining table at time K. However, if K 2 is not topi far from the time K 2, then rational player would prefer to wait until time K 2 depending on the outcome of the war of attrition game that is likely to follow time K 2 between two rational players. One first need to investigate the equilibrium of the war of attrition game between two rational players. As we know from Hendricks, Weiss, and Wilson (988) there are multiple equilibrium of this game (depending on the players declared demands at the beginning of the game.) Now, I consider the case where the negotiators are not compensated for their loss that may occur due to commitment. In this case, the players expected payoff at the beginning of the game changes. Supposing that the equilibrium strategies are such that players choose α, α 2 and K in stage and play according to F and F 2 in the war of attrition game, then the expected payoff of player is K ( z )[( α 2 ) + ( z 2 )F 2 (0)(α + α 2 )] + z e ry df 2 (y) The first term that is multiplied by z is the expected payoff of player if he does not commit to his demand for the rest of the game. The second term is his expected payoff when he is forced to commit. The first observation is that Propositions -6 must still hold because these results characterize the uncommitted (rational) players equilibrium strategies. One can easily check the proofs of Theorem and Proposition 7 that these results will continue to hold. However, Theorem 2 will not hold. It will change. For any α (0, ), the best response correspondence of player 2 is as follows: α, if α ᾱ BR 2 (α ) = arg max Ū 2, otherwise. α 2 [ α,) z z zλ /λ 2 2 ( ) where Ū2 = ( z 2 ) α 2 z (α + α 2 )z λ /λ z 2 ( z ) K 0 e r 2y df (y). In the best response correspondence of the second player, the critical change is that player 2 3 0

14 prefers to choose a compatible demand ( α ) and finish the game at the very beginning of the game. The reason for this is the following. When player chooses a price α which is less than ᾱ, that means that player 2 cannot choose a demand α 2 to make himself strong. However, the weak player 2 s expected payoff is strictly less than α 2 since he has to bear the possible costs of commitment. The optimality implies that player 2 will finish the game by choosing a compatible demand at the very beginning of the game. Given that player 2 prefers to choose a compatible demand, then the unique equilibrium is that player chooses α = ᾱ and the second player chooses α 2 = ᾱ and the game finishes in stage. No price α > ᾱ can be sustained in equilibrium because the second player prefers to deviate to some other α 2 > α and become strong. No price α < ᾱ can be supported in equilibrium because player can increase his expected payoff by increasing his price to some α + ɛ. If player chooses α < ᾱ, then in equilibrium the second player will choose α. However, by choosing α + ɛ, the first player guarantees that z > z = ɛ α 2 and so increases his expected payoff over α. Hence, the unique equilibrium is that first player chooses ᾱ. Hence, if negotiators are not compensated for their possible losses, they eliminate all the equilibria where there will be some delay and inefficiency. Recall from Theorem 2 that equilibrium prices are multiple and most of them are inefficient because they have some delay. Multiplicity is also eliminated by not compensating the negotiators. 4

15 Appendix A Proof of Proposition. Let U (t, F 2 ) denote rational player s expected payoff of conceding at time t {0, 0 2 } given F 2. Therefore, U (0, F 2 ) = F 2 (0 )( z 2 )( +α α 2 2 )+ z 2 ( α 2 ) + ( z 2 )( F 2 (0 ))( α 2 ) and U (0 2, F 2 ) = ( z 2 )F 2 (0 )α + [( z 2 )( F 2 (0 )) + z 2 ]( α 2 ). Similarly, let U 2 (t, F ) denote rational player 2 s expected payoff of conceding at time t given F. Therefore, U 2 (0, F ) = F (0 )( +α 2 α 2 )( z ) + z ( α ) + ( z )( F (0 ))( α ). Let U 2 (F ) denote rational player 2 s expected payoff of not accepting player s demand. Thus, U 2 (F ) = ( z )F (0 )α 2 + ( z )( F (0 ))α 2.. Suppose now that z > z. To show F (0 ) = 0, F (0 2 ) = and F 2 (0 ) = are the equilibrium strategies, first show that F 2 (0 ) = is a best response to F (0 ) = 0, F (0 2 ) =. For this reason, we need to show U 2 (0, F ) U 2 (F ). The last inequality implies that z ( α ) + ( z )( α ) ( z )α 2, which is true if and only if z > α +α 2 α 2 = z. Similarly, to show that F is a best response to F 2 (0 ) =, we need to show U (0 2, F 2 ) = α ( z 2 ) + z 2 ( α 2 ) > U (0, F 2 ) = z 2 ( α 2 ) + ( z 2 )( +α α 2 2 ). The last inequality holds if and only if α + α 2 >. Thus, these strategies constitute an equilibrium. To establish uniqueness, first observe that U (0 2, F 2 ) > U (0, F 2 ) for all values of F 2 (0 ) > 0. Thus, as long as F 2 (0 ) > 0 we have F (0 ) = 0. But, if F (0 ) = 0, then player 2 prefers to accept at time 0. Hence, we need to show that F 2 (0 ) = 0 cannot hold in equilibrium: Suppose for a contradiction that F 2 (0 ) = 0. Then, we must have that U 2 (F ) U 2 (0, F ). However, the last inequality holds if and only if F (0 ) 2( z z z ( z ) ), which is never true since z > z. 2. Given that F (0 ) 2( z z z ( z ) ) we have U 2(F ) U 2 (0, F ). That is, player 2 s strategy is a best response to any F (0 ) satisfying the above inequality. Likewise, if F 2 (0 ) = F 2 (0 2 ) = 0, then U (0, F 2 ) = U (0 2, F 2 ) = α 2, and so F (0 ) is a best response. 3. If z = z and F (0 ) = 0, then U 2 (0, F ) = U 2 (F ), and so any F 2 (0 ) [0, ] is a best response to F. Moreover, as F 2 (0 ) [0, ], we have U (0 2, F 2 ) U (0, F 2 ). Q.E.D. for the proof of Proposition. Proof of Proposition 3. First, I will study the properties of the equilibrium strategies (distribution functions) in the concession game, i.e. F and F 2. For this purpose, given 5

16 α, α 2 and K where α +α 2 > and K (0, T 0 ), consider a pair of equilibrium distribution functions (F, F 2 ) defined over the domain [0, ) where F i (t) z i for all i {, 2} and t < K. Proofs of the following results directly follow from the arguments in Hendricks, Weiss and Wilson (988) and are analogous to the proof of Lemma in Abreu and Gul (2000), so I skip the details. Lemma A.. If a player s strategy is constant on some interval [t, t 2 ] [0, K ), then his opponent s strategy is constant over the interval [t, t 2 + η] for some η > 0. Lemma A.2. F and F 2 do not have a mass point over (0, K ). The idea behind the proof of Lemma A.2 is that if player i concedes with a positive probability at some time t (0, K ), then j prefers to wait during [t ɛ, t] for some ɛ > 0 and concede right after time t. This implies that player j s strategy is constant on interval [t ɛ, t]. Therefore, by Lemma A. player i s equilibrium strategy is also constant on this interval, contradicting with the initial assumption that i concedes with a positive probability at time t. Lemma A.3. F (0)F 2 (0) = 0. Therefore, according to Lemma A. and A.2, both F and F 2 are strictly increasing and continuous over [0, K ]. To prove this claim, first note that there is no interval (t, t ) with 0 t < t < K such that both F and F 2 are constant. Assume on the contrary that t < K is the supremum of the upper bounds of t s such that both F and F 2 are constant. However, through lemma A., if F i is constant on (t, t ) for i {, 2}, then F j where j {, 2}, j i is constant on (t, t + η) for some small η > 0. Hence, both F and F 2 are constant on this later interval, contradicting the definition of t. Hence, if 0 t < t 2 K, then we have F i (t 2 ) > F i (t ) for i =, 2. Moreover, Lemma A.2 implies that both F and F 2 are continuous over [0, K ). Finally, to show that both F i s are continuous on [0, K ], suppose for a contradiction that F 2 has jump at time K. But, then player prefers to wait for some time before K and concede at time K 2. However, this contradicts the fact that F is strictly increasing over [0, K ]. Likewise, F cannot have a jump at time K. Suppose for a contradiction that F (K ) F (K ) = p > 0 where F (K ) = lim t K F (t). Then let U 2 (t, F ) denotes rational player 2 s expected payoff of waiting until time t and accepting α at that time. Then, we have U 2 (t = K, F ) = α 2 ( z ) K 0 e r 2y df (y)+( α )( B (K ))e r 2(K ) K U 2 (t = K, F ) = α 2 ( z ) e r2y df (y)+e r 2K [ 0 2 (+α 2 α )p +( α )( B (K ))] 6

17 Therefore, U 2 (t = K, F ) U 2 (t = K, F ) > 0 for small values of because this difference is equal to o ( ) + o 2 ( ) + e r 2K 2 ( + α 2 α )p where o ( ) = α 2 ( z ) K K e r 2y df (y), o 2 ( ) = ( α )[( B (K )) ( B (K ))e r 2 ] and both o ( ) and o 2 ( ) approach zero as approaches zero. Thus we can conclude that if F has jump at time K, then player 2 prefers to wait for some time [K, K ) for > 0 small enough and concede at time K, contradicting with the fact that F 2 cannot be constant over [0, K ]. Recall that U i (t, F j ) = α j ( z j ) t 0 e r iy df j (y) + α i e r it [ ( z j )F j (t)] denote the expected payoff of rational player i who concedes at time t. Therefore, the utility functions are also continuous on [0, K ]. Then, it follows that D i := {t U i (t, F j ) = max s [0,K ] U i (s, F j )} is dense in [0, K ]. Hence, U i (t, F j ) is constant for all t [0, K ]. Consequently, D i = [0, K ]. Therefore, U i (t, F j ) is differentiable as a function of t. The differentiability of F and F 2 follows from the differentiability of the utility functions on [0, K ]. Differentiating the utility functions and applying the Leibnitz s rule, we get F i (t) = z i ( c i e λ it ) for all t K where c i = F i (0) and λ i = r j( α i ) α +α 2. Finally, since behavioral type player will leave at time K, player s type will be revealed at time K 2. Hence, rational player 2 will never concede after time K and rational player will accept α 2 and finish the game, if the game has not ended before. To prove the second part of Proposition 3, I suppose that ẑ 2 <. Since the equilibrium payoff F 2 is continuous on [0, K ] we must have that player 2 is indifferent between conceding at time K and not conceding. If rational player 2 concedes at time K his instantaneous payoff will be α. However, if he waits, then his expected payoff will be ( ẑ )α 2, since rational player will accept α 2 at time K 2 for sure. These two payoffs are equal if and only if ẑ = α +α 2 α 2 = z. Q.E.D. for the proof of Proposition 3. Proof of Theorem. As a result of all the propositions -6, in equilibrium the payoffs of the rational players in the game are as follows (assuming that α + α 2 > ) CASE : If K = 0, then. if z > z, then u = ( z 2 )α + z 2 ( α 2 ) and u 2 = α 7

18 2. if z < z, then u = α 2 and u 2 = ( z )α 2 3. if z = z, then u = ( z 2 )F 2 (0 )α + [( z 2 )( F 2 (0 )) + z 2 ]( α 2 ) and u 2 = α where F 2 (0 ) [0, ]. CASE 2: If 0 < K < T 0, then. if K < T, then 2. if T < K, then u = α 2 and u 2 = ( z z e λk )α 2 + z e λk ( α z ) u = ( z 2 e λ 2K )α + z 2 e λ 2K ( α ) and u 2 = α 3. if T = K, then CASE 3: If T 0 < K, then where c 2 [z 2 ( z z ) λ2 /λ, ]. if z > z λ /λ 2 2, then u = ( c 2 )α + c 2 ( α 2 ) and u 2 = α u = ( z 2 z λ 2/λ )α + z 2 z λ 2/λ ( α 2 ) and u 2 = α 2. if z λ /λ 2 2 > z, then u = α 2 and u 2 = ( z z λ /λ 2 2 )α 2 + z z λ /λ 2 2 ( α ) 3. if z = z λ /λ 2 2, then u = ( α 2 ) and u 2 = α First note that according to Proposition 3, reputation of player must reach z at time K. Hence, player needs T amount of time to build his reputation if z < z if he does not make any probabilistic concession at time zero. Clearly player can build his reputation to z much earlier than T if he makes a probabilistic concession at time zero. 8

19 However, in this case, player s expected payoff of the game will be the lowest payoff he can achieve, i.e. α 2. Therefore, in equilibrium player will choose K no less than T. To prove the first part of Theorem, suppose that z z. In this case player does not need any time to build his reputation. Also, it is easy to see from the above payoff functions that payoff of rational player is the highest when he chooses K = 0. Hence, in equilibrium, player will pick K = 0 and the equilibrium strategies in the concession game will be as given in Proposition. To prove the second part, suppose that z z λ /λ 2 2 < z < z. It immediately follows that K cannot be zero. Since z z λ /λ 2 2 < z holds, we have T < T 0. Thus, player will choose 0 < K < T 0, and so his equilibrium payoff is one of those given above in CASE 2. Clearly, player does not select K < T since he can achieve higher. However, for any K satisfying T < K, player can increase his payoff by choosing an exit time lower than K. Therefore, the optimal choice for player is K = T, and the equilibrium strategies are characterized by Proposition 6. However, in equilibrium, the value of c 2 is uniquely determined and it is equal to z 2 ( z z ) λ2 /λ. The proof of the last argument is easy. First note that player s payoff decreases as c 2 increases. Suppose for a contradiction that there is an equilibrium where ( ) player chooses T z λ2 /λ = K and c 2 > z. 2 z In this case, player could increase its expected payoff by deviating to ( K = T +ɛ for some sufficiently small ɛ > 0. This is true ( ) ) because payoff of player is z 2 e ɛλ 2 z λ2 /λ ( ) z α + z 2 e ɛλ 2 z λ2 /λ z ( α ) which is strictly higher than ( c 2 )α + c 2 ( α 2 ) for sufficiently small ɛ > 0. Finally to prove the third part of Theorem, suppose that z z z λ /λ 2 2. In this case, we have T 0 < T and z < z λ /λ 2 2. Therefore, expected payoff of player is as given under (2) in CASE 3. That says, expected payoff of player is α 2 and is independent of the value of K he chooses. Thus, any K [T 0, ) forms an equilibrium if F and F 2 are given in Proposition 2. Q.E.D. for the proof of Theorem. Proof of Proposition 7. Consider an equilibrium where players demands are α and α2. Suppose for a contradiction that z / D(α, α2). First, suppose that z is strictly greater than the upper end of the interval D(α, α2), i.e. z < z. According to Theorem, in equilibrium, player selects K = 0, which leads to expected payoff of ( z 2 )α + z 2 ( α2). However, if player demands α + ɛ where ɛ > 0 is small enough so that we still have z < z, then player s expected payoff increases to ( z 2 )(α +ɛ)+z 2 ( α2), contradicting the optimality of the equilibrium. 9

20 Now suppose that z is lower than or equal to the lower bound of the interval D(α, α 2). That is, z z z λ /λ 2 2. Then, according to Theorem, player selects any K > T 0 and his expected payoff in the game is α 2. However, if player demands α 2 + ɛ where ɛ > 0 is small enough so that ɛ = z α < z, then player s expected payoff increases to 2 ( z 2 )( α2 + ɛ) + z 2 ( α2), contradicting the optimality of the equilibrium. Next, I will show that we musty have α ᾱ. Suppose for a contradiction that α > ᾱ. That is, given the definition of ᾱ, there must exist some α 2 > α such that z z λ /λ 2 2 z. Note that α 2 is different from α 2 because, as we just proved, the equilibrium prices α and α 2 must satisfy z D(α, α 2). Finally, notice that in equilibrium player will choose K = ln(z /z ) λ, since z D(α, α2) holds, and so player 2 will be weak. That is, u 2 (α, α2) = α However, if player 2 deviates to α 2, then according to Theorem, player will choose K [T 0, ) and player 2 will become strong. In this case, player 2 s expected payoff in the game will be u 2 (α, α 2 ) = ( z z λ /λ 2 2 )α 2 + z z λ /λ 2 2 ( α ) which is larger than u 2 (α, α 2) because α 2 > α, contradicting with the optimality of equilibrium. Thus, in equilibrium, we must have α ᾱ. Furthermore, since the optimality of equilibrium implies α + α 2, we must have α 2 α ᾱ. Q.E.D. for the proof of Proposition 7. Proof of Theorem 2. The best response correspondences of the players are as follows: For any α (0, ), BR 2 (α ) = [ α, ), if α ᾱ { } arg max α 2 z (α + α 2 )z λ /λ 2 2, otherwise. α 2 [ α,) z z zλ /λ 2 2 For any α 2 (0, ), the best response correspondence of the first player is, BR (α 2 ) = { arg max α z 2 (α + α 2 )(z/z ) λ 2/λ } α [ α 2,ᾱ ] z D(α,α 2 ) Note that for any α 2 given, as α decreases down to α 2, then z decreases to zero. Hence, for any α 2 given, there always exists some α strictly higher than α 2 such that z D(α, α 2 ). By proposition 7 we know that the equilibrium demands α and α 2 must satisfy α 2 α ᾱ and z D(α, α 2). Given that we must have z D(α, α 2), Theorem implies that player is strong, and hence his expected payoff is ( z 2 ( z z ) λ2 /λ )α + z 2 ( z z ) λ2 /λ ( α 2 ) where the parameters z, λ and λ 2 are calculated for α and α 2. 20

21 Q.E.D. for the proof of Theorem 2. Sequential Demand Announcement: Player 2 is First Suppose now that player 2 makes his demand choice first, and then player chooses both his demand α and exit time K. In equilibrium, the optimal exit time of the first player will be determined according to Theorem. However, the equilibrium values of α and α 2 will be characterized as follows: First, start with defining the players best response correspondences. Given α 2 (0, ), the best response correspondence of the first player is ( where u (α, α 2 ) = α z 2 (α + α 2 ) BR (α 2 ) = arg max u (α, α 2 ) α [ α 2,) z D(α,α 2 ) α +α 2 z α 2 ) r( α2) r 2 ( α ). Hence, in equilibrium player 2 chooses α 2 (0, ) and player chooses a function α : (0, ) [ α 2, ) such that (i) α 2 arg min α 2 (0,) BR (α 2 ), and (ii) for any α 2 (0, ), α(α 2 ) arg max u (α, α 2 ). α [ α 2,) z D(α,α 2 ) Sequential Demand Announcement: Player is First Suppose now that player makes his demand choice first, and then player 2 chooses his demand. Finally, player chooses his exit time K. In equilibrium, the optimal exit time of the first player will be determined according to Theorem. values of α and α 2 will be characterized as follows. The equilibrium If player demands α > ᾱ, then player 2 will choose some α 2 : (0, ) [ α, ) such that for any α (0, ), α2(α ) arg max u 2 (α, α 2 ) α 2 [ α,) z z zλ /λ 2 where u 2 (α, α 2 ) = α 2 (α + α 2 )z z λ /λ 2 2. However, if player chooses α ᾱ, then player 2 will choose some mapping α 2 : (0, ) [ α, ). 2

22 Finally, player chooses his demand α such that α arg max Ū where α (0,) α 2α ( ), if α > ᾱ Ū = u (α, α2(α 2 )) if α ᾱ and α D(α, α2(α )) α 2α ( ), if α ᾱ and α / D(α, α2(α )). and u (α, α 2(α )) = α z 2 (α + α 2(α ) ) ( α +α 2 (α ) z α 2 (α ) ) r 2 ( α 2 (α )) r 2 ( α ). References [] Abreu, D., and F. Gul, (2000): Bargaining and Reputation, Econometrica, 68, [2] Abreu, D., and D. Pearce, (2007): Bargaining, Reputation and Equilibrium Selection in Repeated Games with Contracts, Econometrica, 75, [3] Abreu, D., D. Pearce, and E. Stacchetti, (202): One Sided Uncertainty and Delay in Reputational Bargaining, working paper, Princeton University. [4] Hendricks, K., A. Weiss, and R. Wilson (988): The War of Attrition in Continuous- Time with Complete Information, International Economic Review, 29, [5] Kambe, S. (999): Bargaining with Imperfect Commitment, Games and Economic Behavior, 28, [6] Kreps, D.M., and R. Wilson (982): Reputation and Imperfect Information, Journal of Economic Theory, 27, [7] Myerson, R. (99): Game Theory: Analysis of Conflict. Cambridge, MA: Harvard University Press. [8] Schelling, T. (960): The Strategy of Conflict. Harvard University Press. 22