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Optimal value commitment in bilateral bargaining
Games and Economic Behavior 77 (2013) 345–351
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Optimal value commitment in bilateral bargaining Volker Britz ∗ Center for Operations Research and Econometrics, Université Catholique de Louvain, Voie du Roman Pays 34, B-1348 Louvain-la-Neuve, Belgium
a r t i c l e
i n f o
Article history: Received 21 December 2010 Available online 5 November 2012 JEL classification: C72 C78 D74 Keywords: Strategic bargaining Commitment Subgame perfect equilibrium
a b s t r a c t We study the role of commitment as a source of strategic power in a non-cooperative bargaining game. Two impatient players bargain about the division of a shrinking surplus under a standard bargaining protocol in discrete time with constant recognition probabilities. Before bargaining, a player can commit to some part of the surplus. This commitment remains binding until the surplus has shrunk below the amount that the player is committed to. Intuitively, one cannot remain committed to something which has become impossible. The model offers insight on the relative importance of proposal power and commitment for the bargaining outcome. In a version of the game where both players may simultaneously choose their commitments, the equal split emerges from within a range of equilibrium divisions as a focal point which is robust to changes in the model parameters. © 2012 Elsevier Inc. All rights reserved.
1. Introduction We consider a standard surplus-division problem among two impatient players who play a non-cooperative bargaining game with complete information. In this context, we are interested in the ability to commit as a source of bargaining power. More in particular, our interest is in commitments which expire endogenously during the bargaining process. Endogenous expiration of commitments is of interest because of its influence on the players’ incentives to reach an agreement. In particular, as players anticipate that commitments are about to expire endogenously, they may want to come to an agreement as long as their own commitment is still binding, or may want to delay agreement until their opponent’s commitment has expired. In the paper at hand, our focus is on one particular kind of endogenously expiring commitment which leads to such incentives. The commitment device which we propose is based on two simple restrictions on the commitment’s credibility. We assume that any commitments are chosen by the players before the bargaining process starts. The first restriction on the commitment is that it must be expressed in present value terms. Since the players are impatient by assumption, this is different from a commitment to a share of the surplus. Loosely speaking, a player can credibly commit only to something that he cares about as by his utility function. This idea of value commitment has been introduced to the bargaining literature by Li (2007) who argues that “After all, what negotiating parties care about is the discounted value rather than the size of their share of the pie.” We impose a second restriction which says that a player cannot remain committed to something which has become impossible. To see what this means, suppose that a player is committed to a certain present value but disagreement persists long enough so that even the receipt of the entire surplus would no longer suffice to generate that present value. We assume that as of this moment of truth the commitment loses its credibility and thus its binding power.
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V. Britz / Games and Economic Behavior 77 (2013) 345–351
One crucial feature of such a commitment is that after each disagreement it becomes more demanding until it eventually collapses. It is a well-documented phenomenon in the management and psychology literatures that decision-makers tend to continue or even reinforce losing courses of action. This is often explained by appealing to a need for justification or to prospect theory1 . One frequently cited example of such behavior is a gambler who has lost money in one round of roulette and now wants to bet even higher in the next round so as to recover the previous losses. He avoids writing off his losses and admitting that the first bet was a mistake. If the gambler continues to make losses, the bets will keep increasing until they eventually collapse when the gambler’s budget constraint is reached. One interpretation of the commitment device introduced in this paper is that it reflects an analogous behavior in the context of non-cooperative bargaining with commitments. To be more precise, suppose that in some round of bargaining no agreement is reached. If a player is share-committed he automatically “writes off” the cost of delay since he now demands the same share of a smaller surplus. In contrast, a value-committed player refuses to realize the loss incurred due to the delay. As a result, his posture towards the opponent becomes ever more demanding. The opponent is now also supposed to bear the cost of delay alone. However, the refusal to account gradually for the cost of delay ultimately amplifies this cost. Just as the gambler experiences a rude awakening when he reaches the budget constraint, the value-committed player’s rude awakening is the moment of truth when the surplus has shrunk below the commitment. Our paper is most closely related to two strands of the bargaining literature. First, there are a number of papers in which commitments act as reference points in the utility function and typically evolve over time. In Fershtman and Seidmann (1993) and in Li (2007), a player’s commitment is given by the best offer which he has so far rejected. Fershtman and Seidmann evaluate the best rejected offer according to the share of the surplus whereas Li considers the present value. The main objective of these two papers is to explain equilibrium paths with gradual concessions. Since rejecting an offer creates a commitment, a proposer does not want to offer too much too early so as not to give the opponent the opportunity to be strongly committed. While the two aforementioned papers treat commitment as a function of previous decisions to accept or reject proposals, we consider commitment as an independent strategic choice which is made once and for all at the start of the bargaining process. In our analysis, there is no delay on the equilibrium path. Second, our paper relates to a number of studies on less-than-perfect commitments. In these studies, some extra features are added to the bargaining model which limit the binding power of the commitment. Two examples are a non-prohibitive cost of violating one’s commitment and the introduction of some noise in the commitment technology.2 In contrast to this literature, we stick to a simpler bargaining model in which the ability to commit is only limited by the two aforementioned restrictions, that is, commitments are in present value terms, and they expire. To be more precise, let us briefly introduce the model. We consider a game with two stages. In the initial commitment stage, either one or both players can choose a commitment of the type described above. Section 4 focuses on the case where only one player can commit at this stage, and Section 5 deals with the case where both players can commit. Once the commitment(s) have been determined, a bargaining stage with a potentially infinite number of rounds follows. In each such round, one of the two players is recognized as the proposer by a draw from a time-invariant probability distribution. The proposing player makes an offer and the game ends if this offer is accepted by the opponent. In case of a rejection the next round starts. However, any consumption in the next round will be discounted by a constant factor δ ∈ (0, 1). In line with our earlier discussion, the commitment device punishes the committed player if he accepts less than his commitment level while the pie’s value is still higher than that level. But once the “moment of truth“ where the pie’s value shrinks below the commitment level has passed, no punishment is given. The bargaining stage is analyzed in Section 3. Throughout the paper, we compare our results to those which one would expect if a share commitment was made before bargaining and were to remain effective forever. We refer to this kind of commitment as perfect commitment. When the discount factor is small, our results approximate those under perfect commitment. When the discount factor is large, however, we find quite different predictions. The key finding is that commitments tend to be moderate rather than extreme. For instance, even in the case where only one player can commit, the (unique) equilibrium payoff for that player may almost be as low as one half of the surplus, depending on the recognition probabilities. When both players are able to commit simultaneously, we find a range of equilibrium divisions which narrows down substantially as the discount factor approaches one. The equal split emerges as a uniquely robust focal point within the range of equilibrium divisions. 2. Game description Two players have a perfectly divisible pie of unit size at their disposal. They consume the pie once they have agreed on its division. Each player’s instantaneous utility is equal to his consumption of pie, but future consumption is discounted by
1 The classic paper by Staw (1976) as well as the later review papers by Staw (1981) and Brockner (1992) provide a good overview on this issue. Whyte (1986) emphasizes the link with prospect theory. 2 Commitments which can be revoked at a non-prohibitive cost have been studied by Muthoo (1992) and Muthoo (1996). Another approach proposed by Crawford (1982) and Ellingsen and Miettinen (2008) is to assume that a player’s attempt to commit only succeeds with some probability. A related approach frequently encountered in the literature is to introduce incomplete information; a player believes that with some probability his opponent is of a type which is exogenously committed. In such a model, an uncommitted type of player may have an incentive to mimic the committed type (Abreu and Gul, 2000; Kambe, 1999).
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a constant and common factor δ ∈ (0, 1). This implies that at any time t, the players can divide among themselves a surplus of value δt . In the sequel, we will mean by the surplus the time value, discounted to time t = 0, of the pie to be divided. The setup we consider consists of a commitment stage followed by a bargaining stage. In the commitment stage, a commitment pair c = (c 1 , c 2 ) ∈ [0, 1] × [0, 1] is determined. We consider one version of this setup where only one player has the ability to commit (the unilateral commitment game). That is, player 1 chooses c 1 ∈ [0, 1] but c 2 = 0 is exogenously fixed. We also consider a version (the bilateral commitment game) where both players simultaneously choose their commitment levels from [0, 1]. Once the commitments c have been determined, the bargaining stage begins. We denote this bargaining stage by G (c ). It is set in discrete time t = 0, 1, . . . . At the start of each such round t, one player is recognized as the proposer according to the probability distribution (β1 , β2 ), where β1 + β2 = 1 and βk > 0 for both k = 1, 2. This player then proposes a division of the surplus, i.e. a pair (x1 , x2 ) ∈ R2+ such that x1 + x2 δt . If the other player rejects the proposal, round t + 1 starts. If the other player accepts the proposal, it is implemented and the game ends with the following payoffs for the players:
u i ( xi , c i , t ) =
xi − λ if xi < c i δt xi otherwise
We assume λ is sufficiently large so that xi − λ < 0. We note that if c i = 0, then player i’s utility from the agreement x is simply xi . If players disagree forever, their payoffs are zero. If c i δt , we will say that the commitment c i is effective at time t. If c i > δt , we say that the commitment c i is void at time t. 3. Bargaining with commitments In this section, we focus on the bargaining stage G (c ) given the commitment(s). We begin by formalizing the idea of the “moment of truth” mentioned in the introduction. For some commitment level c i of player i, let
τ (c i ) =
min{t ∈ N|t > ln(c i )/ ln(δ)} 0
if c i ∈ (0, 1] if c i = 0
Notice that in round τ (c i ) and all later rounds, the commitment c i is void. We will say that the commitment c i expires at time τ (c i ). We will now write down a backward-induction algorithm which finds SPE payoffs in the bargaining stage G (c ). While spelling out this algorithm, we will assume without loss of generality that τ (c i ) τ (c j ).3 We will denote by v kt (c ) an SPE payoff of player k in a subgame of G (c ) which starts at round t. We use τ (c j ) as the starting point of the backward induction. A subgame starting at τ (c j ) is equivalent to a bargaining game without commitment. It is well known that in such a game, the unique SPE prediction is for the surplus to be divided in the proportion of the recognition probabilities; see for instance Binmore (1987). Indeed, let
v τ (c j ) (c ) = β1 δ τ (c j ) , β2 δ τ (c j ) For t < τ (c j ), define
(1)
αtj (c ) = max c j , v tj+1 (c ) and
α
t i (c )
=
(2)
v ti +1 (c )
if t τ (c i )
max{c i , v ti +1 (c )}
if t < τ (c i )
(3)
For player k = i , j, the aspiration αkt (c ) is the minimal amount of surplus which k would have to receive in round t in order to be at least as well-off as he would expect to be if the game continued to round t + 1. While k’s commitment is effective, his aspiration cannot be lower than his commitment level, reflecting the assumption that revoking the commitment is prohibitively costly. It can be shown that in any SPE of the unilateral and bilateral commitment games, a responding player accepts a proposal if and only it gives him at least his aspiration. If the sum of the aspirations exceeds the available surplus at t, then an agreement is not possible. Thus, round t + 1 will be reached and so v t (c ) = v t +1 (c ). If, however, the available surplus at t suffices to satisfy both aspirations, then agreement is reached. Under that agreement, each player receives his aspiration and the remainder of the surplus goes to the proposer as a premium. This is reflected in the following expression for v kt (c ). For t < τ (c j ), and k = i , j, and l = k, define
v kt (c )
=
αkt (c ) + βk (δt − αkt (c ) − αlt (c )) if αkt (c ) + αlt (c ) δt v kt +1 (c )
otherwise
By iteration, we find v 0 (c ) as the relevant SPE payoffs in the bargaining stage G (c ).
3
This means that either c i c j > 0 or c i = 0.
(4)
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4. The unilateral commitment game In this section, we are interested in the SPE of the unilateral commitment game and particularly in the choice of c 1 . In order to appreciate the main idea intuitively, suppose that player 1 has chosen some commitment level c 1 > 0. Player 2 has the option to delay agreement until τ (c 1 ). Then, a bargaining game without commitment will be played over the remaining surplus which is of size δ τ (c1 ) . But in the equilibrium of such a game, the surplus is split in the proportion of the recognition probabilities: Hence, player 2 can guarantee himself β2 δ τ (c1 ) by waiting for round τ (c 1 ). We refer to this course of action as the delay tactic in the sequel. We will see that, in equilibrium, player 1 keeps his commitment sufficiently low so that player 2 has no incentive to use the delay tactic. More formally, applying the backward-induction procedure from the previous section we can infer the following objective function which player 1 seeks to maximize by the appropriate choice of c 1 ∈ [0, 1].
⎧ if c 1 = 0 ⎨ β1 v 01 (c 1 , 0) = β1 + β2 c 1 − β1 β2 δ τ (c1 ) if c 1 + β2 δ τ (c1 ) 1 and c 1 > 0 ⎩ β1 δ τ (c1 ) if c 1 + β2 δ τ (c1 ) > 1
(5)
In the first two cases, we have v 02 (c 1 , 0) = 1 − v 01 (c 1 , 0), whereas in the third case, v 02 (c 1 , 0) = β2 δ τ (c1 ) . Intuitively, the first case corresponds to player 1 not using his commitment possibility and hence playing the bargaining game without commitment. In the second case, player 1 chooses a strictly positive commitment which is sufficiently low so as to allow an agreement at t = 0. In the third case, player 1 chooses a commitment which is so high that player 2 will respond with the aforementioned delay tactic. In the next theorem, we present the (unique) optimal commitment choice of player 1. Theorem 4.1. In any SPE of the unilateral commitment game, player 1 commits to
ψ1 =
δm˜
1 if δm˜ 2−δβ 1
1 − β2 δm˜
otherwise
˜ = min{m ∈ N0 | δm where m
ϕ1 =
v 01 (ψ1 , 0).
1 }. 1+β2 δ
Moreover, agreement is reached immediately on the division (ϕ1 , 1 −
ϕ1 ), where
The formal proof of this theorem is relegated to Appendix A. With perfect commitment, we would expect player 1 to receive the entire surplus. With the notion of commitment used here, the above theorem reveals that this is still approximately true if δ is close to zero. The intuition behind is that when the future is discounted very heavily, then the delay tactic is rather unattractive for player 2. Now consider the case with large δ . We have pointed out that in SPE an agreement is reached while player 1’s commitment is effective, therefore we have that ϕ1 ψ1 . In the case where c 1 = δm˜ , there is some “friction” between the commitment and the resulting payoff, which arises from the fact that δ τ (c1 ) changes in a “stepwise” fashion with c 1 . However, in the limit as δ → 1, these steps become ever smaller and the said friction vanishes as c 1 − δ τ (c1 ) → 0. It is not surprising, then, that ψ1 and ϕ1 do converge to the same limit. 1 and the SPE division of the surplus Corollary 4.2. In the limit as δ → 1, player 1’s optimal commitment level converges to ψ¯ 1 = 1+β 2 β2 1 converges to (ϕ¯ 1 , 1 − ϕ¯ 1 ) = ( 1+β , 1+β ). 2
2
The intuition behind the limit result is as follows. If δ is close enough to one, the surplus which remains at the moment of truth is nearly equal to the commitment. Thus, player 2 can obtain β2 times the committed amount through the delay tactic. Anticipating this, player 1 chooses the commitment just low enough to make player 2 willing to enter into an agreement immediately. Hence, the surplus is divided in the proportion 1 : β2 . We can use the above finding to address the following question. Suppose that one player has almost all the proposal power but the other player can commit. Is the commitment useful as a “weapon of the weak” to offset the lack of proposal power? Corollary 4.2 implies that if δ and β2 are both close to one, player 1 can obtain about one half of the pie. Hence, if δ is large, the power of one player to commit is just sufficient to compensate for the fact that proposal power is concentrated with the other player. 5. The bilateral commitment game In this section, we deal with the bilateral commitment game in which players 1 and 2 simultaneously choose their commitments before they start bargaining. Based on the analysis of Section 3, one can think of the bilateral commitment game as reducing to a one-shot simultaneous move game with the payoff function v 0 (c 1 , c 2 ). In an SPE of this game the two commitments are best responses to each other.
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The main insight needed to understand the analysis of the bilateral commitment game is the following. Suppose that player i commits to c i . One option which player j = i has at his disposal is a delay tactic similar to that discussed in the previous section: He can commit to ψ j δ τ (c i ) and delay agreement until τ (c i ). Then, the subgame which follows at τ (c i ) is analogous to a unilateral commitment game over a surplus of size δ τ (c i ) where player j has chosen his optimal commitment level. Hence, in the bilateral commitment game, a player can effectively choose to play a unilateral commitment game over the surplus which will be left once the opponent’s commitment has expired. In equilibrium, player i therefore wants to keep his commitment sufficiently low so that player j does not find this delay tactic attractive. This intuition makes it clear why players will not make “extreme” commitments in equilibrium. More formally, define the commitment level ηi = max{c i |c i + ϕ j δ τ (c i ) 1} for player i = 1, 2. If player i is committed to ηi , then the complement 1 − ηi of this commitment is no less than what player j could guarantee himself by using the delay tactic. The next theorem claims that there is a range of divisions of the surplus which can be supported by SPE of the bilateral commitment game. The endpoints of this range are determined by ηi for i = 1, 2. Theorem 5.1. A division (x1 , x2 ) ∈ R2+ of the surplus can be supported by an SPE of the bilateral commitment game if and only if x1 + x2 = 1 and xi 1 − η j for i = 1, 2 and j = i. The proof of Theorem 5.1 is relegated to Appendix A. If δ is small, then the range of SPE divisions specified in the above theorem will be broad. Intuitively, the delay tactic is unattractive when the future is discounted heavily. Hence, there is little incentive to moderate one’s commitment. As δ approaches one, however, the incentives to moderate commitments increase and thus the range of SPE divisions tends to shrink so that we obtain the following limit result. Corollary 5.2. If δ is sufficiently close to one, the surplus division (x1 , x2 ) ∈ R2+ can be supported by an SPE of the bilateral commitment 1 game if and only if x1 + x2 = 1 and xi 3−β for both i = 1, 2. i
When the bargaining friction vanishes, a player will not receive less than one third of the surplus even with arbitrarily small proposal power. The above corollary also implies that SPE leaves the allocation of at most one fifth of the surplus undetermined when δ is large enough. Under perfect commitments, the bilateral commitment game would resemble a Nash Demand Game without noise in which any efficient division is consistent with equilibrium. Making the commitment expire at the moment of truth and letting the bargaining friction vanish, we can reduce this multiplicity very substantially, namely to at most one fifth of the surplus. The next theorem goes one step further. It claims that the equal split is the only division which is consistent with equilibrium no matter how the recognition probabilities and the discount factor are chosen. The equal split therefore emerges as a uniquely robust focal point from within the range of equilibrium divisions. Theorem 5.3. A surplus division (x1 , x2 ) ∈ R2+ can be supported by an SPE of the bilateral commitment game for all δ and for all β if
and only if (x1 , x2 ) = ( 12 , 12 ).
The formal proof of Theorem 5.3 can be found in Appendix A. The main idea behind it is as follows. If each player is committed to one half of the surplus, they can equally split the surplus immediately. No player can gain anything from a delay tactic: When the commitment to one half of the surplus has expired, less than one half is left over. 6. Conclusion We have proposed a new model of commitment in bargaining where the ability to commit is subject to two natural and simple restrictions. In this game, players tend to make moderate commitments when the discount factor is sufficiently large. One question is what happens if a player can make a new commitment once his current commitment has expired. If a player is allowed to make a new commitment finitely many times, then we expect the analysis to remain similar qualitatively, although the leverage obtained from commitment would certainly increase. If a player can renew his commitment infinitely many times we expect the situation to be akin to that with perfect commitment. One may also wonder what happens if a player is not required to make his commitment choice before the first round of bargaining but at a later point. We conjecture that in equilibrium commitments would still be made before the first bargaining round and consequently the model predictions would not change. The intuition behind this conjecture is as follows. A commitment is effective for a certain limited time span. The amount of surplus that a player can capture by using the commitment increases with the size that the surplus has during that time span. It should therefore be optimal to commit in the beginning of the game when the surplus is largest.
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Acknowledgments The author would like to thank Jean-Jacques Herings, Arkadi Predtetchinski, two anonymous referees, and an associate editor for helpful comments and suggestions. Financial support by the Netherlands Organization for Scientific Research (NWO) is gratefully acknowledged. Appendix A Proof of Theorem 4.1. Step 1. We show first that c 1 = 0 in SPE of the unilateral commitment game. Since limc1 ↓0 [c 1 + β2 δ τ (c1 ) ] = 0, we can find c 1 > 0 sufficiently small such that c 1 + β2 δ τ (c1 ) 1. Then, v 01 (c 1 , 0) = β1 + β2 c 1 − β1 β2 δ τ (c1 ) . By definition of τ (c 1 ), it holds
that c 1 > δ τ (c1 ) for any c 1 > 0. If β1 < 1, this implies that v 01 (c 1 , 0) > β1 + β22 δ τ (c1 ) , which readily implies v 01 (c 1 , 0) > β1 . But β1 is the payoff to player 1 from the choice of c 1 = 0. We have shown that player 1 chooses c 1 > 0 in SPE. Step 2. The next step is to show that c 1 + β2 δ τ (c 1 ) 1 in SPE. Suppose not. Then, player 1’s SPE payoff in the subgame G (c 1 , 0) equals β1 δ τ (c1 ) β1 δ . Player 1 may deviate from his choice of c 1 to a commitment level of zero. In that case, a payoff of β1 > 0 will result. Since β1 > β1 δ , this deviation is profitable, a contradiction. Step 3. We have shown so far that in SPE, agreement is reached immediately under an effective commitment c 1 > 0 such that c 1 + β2 δ τ (c1 ) 1. Using the expression for v 0 (c ), the following statement is easily verified: For some c 1 > 0 such that c 1 + β2 δ τ (c1 ) < 1, suppose that there exists ε > 0 sufficiently small so that τ (c 1 + ε ) = τ (c 1 ) and c 1 + ε + β2 δ τ (c1 ) 1. Then, v 01 (c 1 + ε , 0) > v 01 (c 1 , 0). Hence, c 1 cannot be the optimal choice of commitment. Conversely, we have shown that if c 1 is optimal, then either it holds that c 1 = δm for some m ∈ N0 , or that c 1 + β2 δ τ (c1 ) = 1. ˜ any c 1 δm˜ −1 would violate the condition c 1 + β2 δ τ (c1 ) 1, and can thus not be optimal. Step 4. By construction of m, We have shown that the optimal commitment level satisfies c 1 < δm˜ −1 . Step 5. We show next that c 1 δm˜ in SPE. To see this, suppose by way of contradiction that some c 1 < δm˜ is optimal. By ˜ we have c 1 + β2 δ τ (c1 ) < 1. Since the inequality is strict, the argument in Step 1 above implies that c 1 = δm definition of m, ˜ ˜ 0 0 m m m ˜ because c 1 < δ . Plugging into the payoff function, we see that v 1 (δ , 0) > v 1 (δ , 0). for some m ∈ N0 . But then m > m
Thus, player 1 could profitably deviate from c 1 = δm to a commitment of δm˜ , a contradiction. We have now established that c 1 ∈ [δm˜ , δm˜ −1 ) in SPE. ˜ + 1, whereas τ (c 1 ) = m ˜ for any c 1 ∈ (δm˜ , δm˜ −1 ). For clarification, we remark that τ (δm˜ ) = m Step 6. In this step, we derive a condition under which there exists some h > 0 so that δm˜ + h + β2 δm˜ 1 and v 01 (δm˜ + h, 0) > v 01 (δm˜ , 0). Rewriting the first condition, we find h 1 − (1 + β2 )δm˜ . Since we are interested in h > 0 which
satisfy the first condition, we know that v 01 (δm˜ + h, 0) = β1 + β2 (δm˜ + h) − β1 β2 δm˜ . The condition v 01 (δm˜ + h, 0) > v 01 (δm˜ , 0)
can then be written as (β1 + β2 (δm˜ + h) − β1 β2 δm˜ ) − (β1 + β2 δm˜ − β1 β2 δm˜ +1 ) > 0. Suitably rearranging the terms, this can be reduced to h > β1 δm˜ (1 − δ). We are now looking for some h such that 1 − (1 + β2 )δm˜ h > β1 δm˜ (1 − δ). Such h exists if 1 1 ˜ m . If δm˜ 2−β and only if δm˜ < 2−δβ δ , then a commitment of δ is indeed optimal, as claimed in the theorem. 1
1
1 Step 7. Now turn to the case where δm˜ < 2−β . In this case, we have shown in Step 6 that there does exist h > 0 such 1δ ˜ ˜ ˜ ˜ 0 m 0 m m m that δ + h + β2 δ 1 and v 1 (δ + h, 0) > v 1 (δ , 0). Thus, δm˜ is not optimal, and so it follows from the conclusion of Step 5
above that the optimal c 1 belongs to the open interval (δm˜ , δm˜ −1 ). Since this interval is open, the optimal commitment level cannot satisfy c 1 = δm for any m ∈ N0 . But then, by the argument in Step 3, the optimal commitment level must satisfy ˜ for any c 1 ∈ (δm˜ , δm˜ −1 ), we can conclude that the optimal commitment level is equal to c 1 + β2 δ τ (c1 ) = 1. Since τ (c 1 ) = m ˜ m 1 − β2 δ , as claimed in the lemma. Step 8. We have shown that in SPE, player 1 commits to ψ1 , as defined in the statement of the lemma. Agreement is immediate and efficient. Moreover, the payoff to player 1 is v 01 (ψ1 , 0), as desired. 2 The proof of Theorem 5.1 requires the following auxiliary. Lemma A.1. If c i + c j 1 and c i c j , then v 0j (c i , c j ) c j . Proof. Suppose by way of contradiction that c i + c j 1 and c i c j but v 0j (c ) < c j . Since the cost of violating an effective commitment is prohibitive, this implies that no agreement is reached before τ (c i ). At time τ (c j ) − 1, player j’s aspiration is max{c j , β j δ τ (c j ) } = c j . Let b j (c j ) = max{t ∈ N0 |δt c j + βi δ τ (c j )}. Suppose that b(c j ) τ (c i ). Then, agreement is possible at b(c j ) < τ (c j ), a contradiction. Now suppose that b(c j ) τ (c j ). Then consider round τ (c i ) − 1. The aspirations in that round are c i and c j . Since c i + c j 1, we can find b˜ (c ) = max{t ∈ N0 |δt c i + c j }. Agreement at b˜ (c ) is possible, a contradiction.
2
Proof of Theorem 5.1. If: Take a pair of commitments c = (c 1 , c 2 ) such that c 1 + c 2 = 1 and c i 1 − η j for i = 1, 2 and j = i. Then, v 0 (c ) = c. Suppose player i deviates by choosing a different commitment than c i . If this deviation results in some agreement x before τ (c j ), then x j c j and so xi 1 − c j = c i , the deviation is not profitable. If the deviation results
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in an agreement x after τ (c j ), then xi ϕi δ τ (c j ) . The deviation is not profitable. Finally, if the deviation results in perpetual disagreement, it is clearly not profitable either. Only if: Suppose by way of contradiction that in some SPE an agreement x is reached and xi < 1 − η j . If c j η j , it is a profitable deviation for player i to commit to ψi δ τ (c j ) and delay agreement until τ (c j ). The resulting payoff will be ϕi δ τ (c j ) 1 − η j . Hence, it must hold that c j < η j . We show that choosing c i = 1 − η j is a profitable deviation for player i. Suppose that c j c i . Then, v 0 (c i , c j ) c i = 1 − η j by Lemma A.1, as desired. Now consider the case where c i > c j . We want to show that i receives at least c i . Suppose not. Then t¯ = τ (c i ) is reached. We have t¯−1
t¯−1
t¯
t¯−1
αit¯−1 (c i , c j ) = c i = 1 − η j and
α j (c i , c j ) = max{c j , v j } < 1 − η j . Thus, αi (c i , c j ) + α j (c i , c j ) < 2(1 − η j ) 1. (The last inequality follows from the fact that η j 12 which can be easily verified from the definition of η j .) Consequently, there is t < τ (c i ) such that δt
exceeds the sum of the aspirations at t. Agreement can be reached, a contradiction. It remains to show that SPE is efficient. Perpetual disagreement is not possible in SPE since both commitments expire in finite time, and in the bargaining game with no effective commitments an agreement is reached. Furthermore, Eq. (4) shows that if an agreement x is reached at t, then x1 + x2 = δt . Hence, it will suffice to show that agreement is reached immediately in SPE. Suppose to the contrary that no agreement is reached at t = 0. Then, delay will last until τ (c i ). But then SPE requires that c j = ψ j δ τ (c i ) , where we use the convention that c i c j . In that case, we know from the analysis of the unilateral bargaining game that payoffs will be ϕ j δ τ (c i ) for player j and (1 − ϕ j )δ τ (c i ) for player i. We claim that it is a profitable deviation for player i to choose a zero commitment rather than c i . Indeed, v 0i (0, c j ) = (1 − ϕ j )δ τ (c i ) + βi (1 − δ τ (c i ) ). The claim follows since we have assumed βi > 0. 2 Proof of Theorem 5.4. If: Since
ηi
1 2
for i = 1, 2, we also have 1 − ηi 12 . The claim follows from Theorem 5.1.
Only if: Consider a pie division in which player k = 1, 2 obtains a payoff of sufficiently large, SPE requires 1 2 −3ε 1 2 −ε
1 2
−ε
1
3−βk
. This can be rewritten as βk
< 1. Consequently, choosing δ sufficiently large and βk ∈ (
is not supported by any SPE.
2
1 2 −3ε 1 2 −ε
1 2
− ε , where ε > 0. By Theorem 5.2, if δ is
1 2 −3ε 1 −ε 2
. But we have assumed that
ε > 0, thus
, 1) ensures that the pie division under consideration
References Abreu, D., Gul, F., 2000. Bargaining and reputation. Econometrica 68, 85–117. Binmore, K.G., 1987. Perfect equilibria in bargaining models. In: Binmore, K., Dasgupta, P. (Eds.), The Economics of Bargaining. Blackwell, Oxford. Brockner, J., 1992. The escalation of commitment to a failing course of action: Toward theoretical progress. Acad. Manage. Rev. 17, 39–61. Crawford, V.P., 1982. A theory of disagreement in bargaining. Econometrica 50, 607–637. Ellingsen, T., Miettinen, T., 2008. Commitment and conflict in bilateral bargaining. Amer. Econ. Rev. 98, 1629–1635. Fershtman, C., Seidmann, D., 1993. Deadline effects and inefficient delay in bargaining with endogenous commitment. J. Econ. Theory 60, 306–321. Kambe, S., 1999. Bargaining with imperfect commitment. Games Econ. Behav. 28, 217–237. Li, D., 2007. Bargaining with history-dependent preferences. J. Econ. Theory 136, 695–708. Muthoo, A., 1992. Revocable commitment and sequential bargaining. Econ. J. 102, 378–387. Muthoo, A., 1996. A bargaining model based on the commitment tactic. J. Econ. Theory 69, 134–152. Staw, B.M., 1976. Knee-deep in the big muddy: A study of escalating commitment to a chosen course of action. Organ. Behav. Hum. Perform. 16, 27–44. Staw, B.M., 1981. The escalation of commitment to a course of action. Acad. Manage. Rev. 6, 577–587. Whyte, G., 1986. Escalating commitment to a course of action: A reinterpretation. Acad. Manage. Rev. 11, 311–321.
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Note
Optimal value commitment in bilateral bargaining Volker Britz ∗ Center for Operations Research and Econometrics, Université Catholique de Louvain, Voie du Roman Pays 34, B-1348 Louvain-la-Neuve, Belgium
a r t i c l e
i n f o
Article history: Received 21 December 2010 Available online 5 November 2012 JEL classification: C72 C78 D74 Keywords: Strategic bargaining Commitment Subgame perfect equilibrium
a b s t r a c t We study the role of commitment as a source of strategic power in a non-cooperative bargaining game. Two impatient players bargain about the division of a shrinking surplus under a standard bargaining protocol in discrete time with constant recognition probabilities. Before bargaining, a player can commit to some part of the surplus. This commitment remains binding until the surplus has shrunk below the amount that the player is committed to. Intuitively, one cannot remain committed to something which has become impossible. The model offers insight on the relative importance of proposal power and commitment for the bargaining outcome. In a version of the game where both players may simultaneously choose their commitments, the equal split emerges from within a range of equilibrium divisions as a focal point which is robust to changes in the model parameters. © 2012 Elsevier Inc. All rights reserved.
1. Introduction We consider a standard surplus-division problem among two impatient players who play a non-cooperative bargaining game with complete information. In this context, we are interested in the ability to commit as a source of bargaining power. More in particular, our interest is in commitments which expire endogenously during the bargaining process. Endogenous expiration of commitments is of interest because of its influence on the players’ incentives to reach an agreement. In particular, as players anticipate that commitments are about to expire endogenously, they may want to come to an agreement as long as their own commitment is still binding, or may want to delay agreement until their opponent’s commitment has expired. In the paper at hand, our focus is on one particular kind of endogenously expiring commitment which leads to such incentives. The commitment device which we propose is based on two simple restrictions on the commitment’s credibility. We assume that any commitments are chosen by the players before the bargaining process starts. The first restriction on the commitment is that it must be expressed in present value terms. Since the players are impatient by assumption, this is different from a commitment to a share of the surplus. Loosely speaking, a player can credibly commit only to something that he cares about as by his utility function. This idea of value commitment has been introduced to the bargaining literature by Li (2007) who argues that “After all, what negotiating parties care about is the discounted value rather than the size of their share of the pie.” We impose a second restriction which says that a player cannot remain committed to something which has become impossible. To see what this means, suppose that a player is committed to a certain present value but disagreement persists long enough so that even the receipt of the entire surplus would no longer suffice to generate that present value. We assume that as of this moment of truth the commitment loses its credibility and thus its binding power.
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V. Britz / Games and Economic Behavior 77 (2013) 345–351
One crucial feature of such a commitment is that after each disagreement it becomes more demanding until it eventually collapses. It is a well-documented phenomenon in the management and psychology literatures that decision-makers tend to continue or even reinforce losing courses of action. This is often explained by appealing to a need for justification or to prospect theory1 . One frequently cited example of such behavior is a gambler who has lost money in one round of roulette and now wants to bet even higher in the next round so as to recover the previous losses. He avoids writing off his losses and admitting that the first bet was a mistake. If the gambler continues to make losses, the bets will keep increasing until they eventually collapse when the gambler’s budget constraint is reached. One interpretation of the commitment device introduced in this paper is that it reflects an analogous behavior in the context of non-cooperative bargaining with commitments. To be more precise, suppose that in some round of bargaining no agreement is reached. If a player is share-committed he automatically “writes off” the cost of delay since he now demands the same share of a smaller surplus. In contrast, a value-committed player refuses to realize the loss incurred due to the delay. As a result, his posture towards the opponent becomes ever more demanding. The opponent is now also supposed to bear the cost of delay alone. However, the refusal to account gradually for the cost of delay ultimately amplifies this cost. Just as the gambler experiences a rude awakening when he reaches the budget constraint, the value-committed player’s rude awakening is the moment of truth when the surplus has shrunk below the commitment. Our paper is most closely related to two strands of the bargaining literature. First, there are a number of papers in which commitments act as reference points in the utility function and typically evolve over time. In Fershtman and Seidmann (1993) and in Li (2007), a player’s commitment is given by the best offer which he has so far rejected. Fershtman and Seidmann evaluate the best rejected offer according to the share of the surplus whereas Li considers the present value. The main objective of these two papers is to explain equilibrium paths with gradual concessions. Since rejecting an offer creates a commitment, a proposer does not want to offer too much too early so as not to give the opponent the opportunity to be strongly committed. While the two aforementioned papers treat commitment as a function of previous decisions to accept or reject proposals, we consider commitment as an independent strategic choice which is made once and for all at the start of the bargaining process. In our analysis, there is no delay on the equilibrium path. Second, our paper relates to a number of studies on less-than-perfect commitments. In these studies, some extra features are added to the bargaining model which limit the binding power of the commitment. Two examples are a non-prohibitive cost of violating one’s commitment and the introduction of some noise in the commitment technology.2 In contrast to this literature, we stick to a simpler bargaining model in which the ability to commit is only limited by the two aforementioned restrictions, that is, commitments are in present value terms, and they expire. To be more precise, let us briefly introduce the model. We consider a game with two stages. In the initial commitment stage, either one or both players can choose a commitment of the type described above. Section 4 focuses on the case where only one player can commit at this stage, and Section 5 deals with the case where both players can commit. Once the commitment(s) have been determined, a bargaining stage with a potentially infinite number of rounds follows. In each such round, one of the two players is recognized as the proposer by a draw from a time-invariant probability distribution. The proposing player makes an offer and the game ends if this offer is accepted by the opponent. In case of a rejection the next round starts. However, any consumption in the next round will be discounted by a constant factor δ ∈ (0, 1). In line with our earlier discussion, the commitment device punishes the committed player if he accepts less than his commitment level while the pie’s value is still higher than that level. But once the “moment of truth“ where the pie’s value shrinks below the commitment level has passed, no punishment is given. The bargaining stage is analyzed in Section 3. Throughout the paper, we compare our results to those which one would expect if a share commitment was made before bargaining and were to remain effective forever. We refer to this kind of commitment as perfect commitment. When the discount factor is small, our results approximate those under perfect commitment. When the discount factor is large, however, we find quite different predictions. The key finding is that commitments tend to be moderate rather than extreme. For instance, even in the case where only one player can commit, the (unique) equilibrium payoff for that player may almost be as low as one half of the surplus, depending on the recognition probabilities. When both players are able to commit simultaneously, we find a range of equilibrium divisions which narrows down substantially as the discount factor approaches one. The equal split emerges as a uniquely robust focal point within the range of equilibrium divisions. 2. Game description Two players have a perfectly divisible pie of unit size at their disposal. They consume the pie once they have agreed on its division. Each player’s instantaneous utility is equal to his consumption of pie, but future consumption is discounted by
1 The classic paper by Staw (1976) as well as the later review papers by Staw (1981) and Brockner (1992) provide a good overview on this issue. Whyte (1986) emphasizes the link with prospect theory. 2 Commitments which can be revoked at a non-prohibitive cost have been studied by Muthoo (1992) and Muthoo (1996). Another approach proposed by Crawford (1982) and Ellingsen and Miettinen (2008) is to assume that a player’s attempt to commit only succeeds with some probability. A related approach frequently encountered in the literature is to introduce incomplete information; a player believes that with some probability his opponent is of a type which is exogenously committed. In such a model, an uncommitted type of player may have an incentive to mimic the committed type (Abreu and Gul, 2000; Kambe, 1999).
V. Britz / Games and Economic Behavior 77 (2013) 345–351
347
a constant and common factor δ ∈ (0, 1). This implies that at any time t, the players can divide among themselves a surplus of value δt . In the sequel, we will mean by the surplus the time value, discounted to time t = 0, of the pie to be divided. The setup we consider consists of a commitment stage followed by a bargaining stage. In the commitment stage, a commitment pair c = (c 1 , c 2 ) ∈ [0, 1] × [0, 1] is determined. We consider one version of this setup where only one player has the ability to commit (the unilateral commitment game). That is, player 1 chooses c 1 ∈ [0, 1] but c 2 = 0 is exogenously fixed. We also consider a version (the bilateral commitment game) where both players simultaneously choose their commitment levels from [0, 1]. Once the commitments c have been determined, the bargaining stage begins. We denote this bargaining stage by G (c ). It is set in discrete time t = 0, 1, . . . . At the start of each such round t, one player is recognized as the proposer according to the probability distribution (β1 , β2 ), where β1 + β2 = 1 and βk > 0 for both k = 1, 2. This player then proposes a division of the surplus, i.e. a pair (x1 , x2 ) ∈ R2+ such that x1 + x2 δt . If the other player rejects the proposal, round t + 1 starts. If the other player accepts the proposal, it is implemented and the game ends with the following payoffs for the players:
u i ( xi , c i , t ) =
xi − λ if xi < c i δt xi otherwise
We assume λ is sufficiently large so that xi − λ < 0. We note that if c i = 0, then player i’s utility from the agreement x is simply xi . If players disagree forever, their payoffs are zero. If c i δt , we will say that the commitment c i is effective at time t. If c i > δt , we say that the commitment c i is void at time t. 3. Bargaining with commitments In this section, we focus on the bargaining stage G (c ) given the commitment(s). We begin by formalizing the idea of the “moment of truth” mentioned in the introduction. For some commitment level c i of player i, let
τ (c i ) =
min{t ∈ N|t > ln(c i )/ ln(δ)} 0
if c i ∈ (0, 1] if c i = 0
Notice that in round τ (c i ) and all later rounds, the commitment c i is void. We will say that the commitment c i expires at time τ (c i ). We will now write down a backward-induction algorithm which finds SPE payoffs in the bargaining stage G (c ). While spelling out this algorithm, we will assume without loss of generality that τ (c i ) τ (c j ).3 We will denote by v kt (c ) an SPE payoff of player k in a subgame of G (c ) which starts at round t. We use τ (c j ) as the starting point of the backward induction. A subgame starting at τ (c j ) is equivalent to a bargaining game without commitment. It is well known that in such a game, the unique SPE prediction is for the surplus to be divided in the proportion of the recognition probabilities; see for instance Binmore (1987). Indeed, let
v τ (c j ) (c ) = β1 δ τ (c j ) , β2 δ τ (c j ) For t < τ (c j ), define
(1)
αtj (c ) = max c j , v tj+1 (c ) and
α
t i (c )
=
(2)
v ti +1 (c )
if t τ (c i )
max{c i , v ti +1 (c )}
if t < τ (c i )
(3)
For player k = i , j, the aspiration αkt (c ) is the minimal amount of surplus which k would have to receive in round t in order to be at least as well-off as he would expect to be if the game continued to round t + 1. While k’s commitment is effective, his aspiration cannot be lower than his commitment level, reflecting the assumption that revoking the commitment is prohibitively costly. It can be shown that in any SPE of the unilateral and bilateral commitment games, a responding player accepts a proposal if and only it gives him at least his aspiration. If the sum of the aspirations exceeds the available surplus at t, then an agreement is not possible. Thus, round t + 1 will be reached and so v t (c ) = v t +1 (c ). If, however, the available surplus at t suffices to satisfy both aspirations, then agreement is reached. Under that agreement, each player receives his aspiration and the remainder of the surplus goes to the proposer as a premium. This is reflected in the following expression for v kt (c ). For t < τ (c j ), and k = i , j, and l = k, define
v kt (c )
=
αkt (c ) + βk (δt − αkt (c ) − αlt (c )) if αkt (c ) + αlt (c ) δt v kt +1 (c )
otherwise
By iteration, we find v 0 (c ) as the relevant SPE payoffs in the bargaining stage G (c ).
3
This means that either c i c j > 0 or c i = 0.
(4)
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4. The unilateral commitment game In this section, we are interested in the SPE of the unilateral commitment game and particularly in the choice of c 1 . In order to appreciate the main idea intuitively, suppose that player 1 has chosen some commitment level c 1 > 0. Player 2 has the option to delay agreement until τ (c 1 ). Then, a bargaining game without commitment will be played over the remaining surplus which is of size δ τ (c1 ) . But in the equilibrium of such a game, the surplus is split in the proportion of the recognition probabilities: Hence, player 2 can guarantee himself β2 δ τ (c1 ) by waiting for round τ (c 1 ). We refer to this course of action as the delay tactic in the sequel. We will see that, in equilibrium, player 1 keeps his commitment sufficiently low so that player 2 has no incentive to use the delay tactic. More formally, applying the backward-induction procedure from the previous section we can infer the following objective function which player 1 seeks to maximize by the appropriate choice of c 1 ∈ [0, 1].
⎧ if c 1 = 0 ⎨ β1 v 01 (c 1 , 0) = β1 + β2 c 1 − β1 β2 δ τ (c1 ) if c 1 + β2 δ τ (c1 ) 1 and c 1 > 0 ⎩ β1 δ τ (c1 ) if c 1 + β2 δ τ (c1 ) > 1
(5)
In the first two cases, we have v 02 (c 1 , 0) = 1 − v 01 (c 1 , 0), whereas in the third case, v 02 (c 1 , 0) = β2 δ τ (c1 ) . Intuitively, the first case corresponds to player 1 not using his commitment possibility and hence playing the bargaining game without commitment. In the second case, player 1 chooses a strictly positive commitment which is sufficiently low so as to allow an agreement at t = 0. In the third case, player 1 chooses a commitment which is so high that player 2 will respond with the aforementioned delay tactic. In the next theorem, we present the (unique) optimal commitment choice of player 1. Theorem 4.1. In any SPE of the unilateral commitment game, player 1 commits to
ψ1 =
δm˜
1 if δm˜ 2−δβ 1
1 − β2 δm˜
otherwise
˜ = min{m ∈ N0 | δm where m
ϕ1 =
v 01 (ψ1 , 0).
1 }. 1+β2 δ
Moreover, agreement is reached immediately on the division (ϕ1 , 1 −
ϕ1 ), where
The formal proof of this theorem is relegated to Appendix A. With perfect commitment, we would expect player 1 to receive the entire surplus. With the notion of commitment used here, the above theorem reveals that this is still approximately true if δ is close to zero. The intuition behind is that when the future is discounted very heavily, then the delay tactic is rather unattractive for player 2. Now consider the case with large δ . We have pointed out that in SPE an agreement is reached while player 1’s commitment is effective, therefore we have that ϕ1 ψ1 . In the case where c 1 = δm˜ , there is some “friction” between the commitment and the resulting payoff, which arises from the fact that δ τ (c1 ) changes in a “stepwise” fashion with c 1 . However, in the limit as δ → 1, these steps become ever smaller and the said friction vanishes as c 1 − δ τ (c1 ) → 0. It is not surprising, then, that ψ1 and ϕ1 do converge to the same limit. 1 and the SPE division of the surplus Corollary 4.2. In the limit as δ → 1, player 1’s optimal commitment level converges to ψ¯ 1 = 1+β 2 β2 1 converges to (ϕ¯ 1 , 1 − ϕ¯ 1 ) = ( 1+β , 1+β ). 2
2
The intuition behind the limit result is as follows. If δ is close enough to one, the surplus which remains at the moment of truth is nearly equal to the commitment. Thus, player 2 can obtain β2 times the committed amount through the delay tactic. Anticipating this, player 1 chooses the commitment just low enough to make player 2 willing to enter into an agreement immediately. Hence, the surplus is divided in the proportion 1 : β2 . We can use the above finding to address the following question. Suppose that one player has almost all the proposal power but the other player can commit. Is the commitment useful as a “weapon of the weak” to offset the lack of proposal power? Corollary 4.2 implies that if δ and β2 are both close to one, player 1 can obtain about one half of the pie. Hence, if δ is large, the power of one player to commit is just sufficient to compensate for the fact that proposal power is concentrated with the other player. 5. The bilateral commitment game In this section, we deal with the bilateral commitment game in which players 1 and 2 simultaneously choose their commitments before they start bargaining. Based on the analysis of Section 3, one can think of the bilateral commitment game as reducing to a one-shot simultaneous move game with the payoff function v 0 (c 1 , c 2 ). In an SPE of this game the two commitments are best responses to each other.
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The main insight needed to understand the analysis of the bilateral commitment game is the following. Suppose that player i commits to c i . One option which player j = i has at his disposal is a delay tactic similar to that discussed in the previous section: He can commit to ψ j δ τ (c i ) and delay agreement until τ (c i ). Then, the subgame which follows at τ (c i ) is analogous to a unilateral commitment game over a surplus of size δ τ (c i ) where player j has chosen his optimal commitment level. Hence, in the bilateral commitment game, a player can effectively choose to play a unilateral commitment game over the surplus which will be left once the opponent’s commitment has expired. In equilibrium, player i therefore wants to keep his commitment sufficiently low so that player j does not find this delay tactic attractive. This intuition makes it clear why players will not make “extreme” commitments in equilibrium. More formally, define the commitment level ηi = max{c i |c i + ϕ j δ τ (c i ) 1} for player i = 1, 2. If player i is committed to ηi , then the complement 1 − ηi of this commitment is no less than what player j could guarantee himself by using the delay tactic. The next theorem claims that there is a range of divisions of the surplus which can be supported by SPE of the bilateral commitment game. The endpoints of this range are determined by ηi for i = 1, 2. Theorem 5.1. A division (x1 , x2 ) ∈ R2+ of the surplus can be supported by an SPE of the bilateral commitment game if and only if x1 + x2 = 1 and xi 1 − η j for i = 1, 2 and j = i. The proof of Theorem 5.1 is relegated to Appendix A. If δ is small, then the range of SPE divisions specified in the above theorem will be broad. Intuitively, the delay tactic is unattractive when the future is discounted heavily. Hence, there is little incentive to moderate one’s commitment. As δ approaches one, however, the incentives to moderate commitments increase and thus the range of SPE divisions tends to shrink so that we obtain the following limit result. Corollary 5.2. If δ is sufficiently close to one, the surplus division (x1 , x2 ) ∈ R2+ can be supported by an SPE of the bilateral commitment 1 game if and only if x1 + x2 = 1 and xi 3−β for both i = 1, 2. i
When the bargaining friction vanishes, a player will not receive less than one third of the surplus even with arbitrarily small proposal power. The above corollary also implies that SPE leaves the allocation of at most one fifth of the surplus undetermined when δ is large enough. Under perfect commitments, the bilateral commitment game would resemble a Nash Demand Game without noise in which any efficient division is consistent with equilibrium. Making the commitment expire at the moment of truth and letting the bargaining friction vanish, we can reduce this multiplicity very substantially, namely to at most one fifth of the surplus. The next theorem goes one step further. It claims that the equal split is the only division which is consistent with equilibrium no matter how the recognition probabilities and the discount factor are chosen. The equal split therefore emerges as a uniquely robust focal point from within the range of equilibrium divisions. Theorem 5.3. A surplus division (x1 , x2 ) ∈ R2+ can be supported by an SPE of the bilateral commitment game for all δ and for all β if
and only if (x1 , x2 ) = ( 12 , 12 ).
The formal proof of Theorem 5.3 can be found in Appendix A. The main idea behind it is as follows. If each player is committed to one half of the surplus, they can equally split the surplus immediately. No player can gain anything from a delay tactic: When the commitment to one half of the surplus has expired, less than one half is left over. 6. Conclusion We have proposed a new model of commitment in bargaining where the ability to commit is subject to two natural and simple restrictions. In this game, players tend to make moderate commitments when the discount factor is sufficiently large. One question is what happens if a player can make a new commitment once his current commitment has expired. If a player is allowed to make a new commitment finitely many times, then we expect the analysis to remain similar qualitatively, although the leverage obtained from commitment would certainly increase. If a player can renew his commitment infinitely many times we expect the situation to be akin to that with perfect commitment. One may also wonder what happens if a player is not required to make his commitment choice before the first round of bargaining but at a later point. We conjecture that in equilibrium commitments would still be made before the first bargaining round and consequently the model predictions would not change. The intuition behind this conjecture is as follows. A commitment is effective for a certain limited time span. The amount of surplus that a player can capture by using the commitment increases with the size that the surplus has during that time span. It should therefore be optimal to commit in the beginning of the game when the surplus is largest.
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V. Britz / Games and Economic Behavior 77 (2013) 345–351
Acknowledgments The author would like to thank Jean-Jacques Herings, Arkadi Predtetchinski, two anonymous referees, and an associate editor for helpful comments and suggestions. Financial support by the Netherlands Organization for Scientific Research (NWO) is gratefully acknowledged. Appendix A Proof of Theorem 4.1. Step 1. We show first that c 1 = 0 in SPE of the unilateral commitment game. Since limc1 ↓0 [c 1 + β2 δ τ (c1 ) ] = 0, we can find c 1 > 0 sufficiently small such that c 1 + β2 δ τ (c1 ) 1. Then, v 01 (c 1 , 0) = β1 + β2 c 1 − β1 β2 δ τ (c1 ) . By definition of τ (c 1 ), it holds
that c 1 > δ τ (c1 ) for any c 1 > 0. If β1 < 1, this implies that v 01 (c 1 , 0) > β1 + β22 δ τ (c1 ) , which readily implies v 01 (c 1 , 0) > β1 . But β1 is the payoff to player 1 from the choice of c 1 = 0. We have shown that player 1 chooses c 1 > 0 in SPE. Step 2. The next step is to show that c 1 + β2 δ τ (c 1 ) 1 in SPE. Suppose not. Then, player 1’s SPE payoff in the subgame G (c 1 , 0) equals β1 δ τ (c1 ) β1 δ . Player 1 may deviate from his choice of c 1 to a commitment level of zero. In that case, a payoff of β1 > 0 will result. Since β1 > β1 δ , this deviation is profitable, a contradiction. Step 3. We have shown so far that in SPE, agreement is reached immediately under an effective commitment c 1 > 0 such that c 1 + β2 δ τ (c1 ) 1. Using the expression for v 0 (c ), the following statement is easily verified: For some c 1 > 0 such that c 1 + β2 δ τ (c1 ) < 1, suppose that there exists ε > 0 sufficiently small so that τ (c 1 + ε ) = τ (c 1 ) and c 1 + ε + β2 δ τ (c1 ) 1. Then, v 01 (c 1 + ε , 0) > v 01 (c 1 , 0). Hence, c 1 cannot be the optimal choice of commitment. Conversely, we have shown that if c 1 is optimal, then either it holds that c 1 = δm for some m ∈ N0 , or that c 1 + β2 δ τ (c1 ) = 1. ˜ any c 1 δm˜ −1 would violate the condition c 1 + β2 δ τ (c1 ) 1, and can thus not be optimal. Step 4. By construction of m, We have shown that the optimal commitment level satisfies c 1 < δm˜ −1 . Step 5. We show next that c 1 δm˜ in SPE. To see this, suppose by way of contradiction that some c 1 < δm˜ is optimal. By ˜ we have c 1 + β2 δ τ (c1 ) < 1. Since the inequality is strict, the argument in Step 1 above implies that c 1 = δm definition of m, ˜ ˜ 0 0 m m m ˜ because c 1 < δ . Plugging into the payoff function, we see that v 1 (δ , 0) > v 1 (δ , 0). for some m ∈ N0 . But then m > m
Thus, player 1 could profitably deviate from c 1 = δm to a commitment of δm˜ , a contradiction. We have now established that c 1 ∈ [δm˜ , δm˜ −1 ) in SPE. ˜ + 1, whereas τ (c 1 ) = m ˜ for any c 1 ∈ (δm˜ , δm˜ −1 ). For clarification, we remark that τ (δm˜ ) = m Step 6. In this step, we derive a condition under which there exists some h > 0 so that δm˜ + h + β2 δm˜ 1 and v 01 (δm˜ + h, 0) > v 01 (δm˜ , 0). Rewriting the first condition, we find h 1 − (1 + β2 )δm˜ . Since we are interested in h > 0 which
satisfy the first condition, we know that v 01 (δm˜ + h, 0) = β1 + β2 (δm˜ + h) − β1 β2 δm˜ . The condition v 01 (δm˜ + h, 0) > v 01 (δm˜ , 0)
can then be written as (β1 + β2 (δm˜ + h) − β1 β2 δm˜ ) − (β1 + β2 δm˜ − β1 β2 δm˜ +1 ) > 0. Suitably rearranging the terms, this can be reduced to h > β1 δm˜ (1 − δ). We are now looking for some h such that 1 − (1 + β2 )δm˜ h > β1 δm˜ (1 − δ). Such h exists if 1 1 ˜ m . If δm˜ 2−β and only if δm˜ < 2−δβ δ , then a commitment of δ is indeed optimal, as claimed in the theorem. 1
1
1 Step 7. Now turn to the case where δm˜ < 2−β . In this case, we have shown in Step 6 that there does exist h > 0 such 1δ ˜ ˜ ˜ ˜ 0 m 0 m m m that δ + h + β2 δ 1 and v 1 (δ + h, 0) > v 1 (δ , 0). Thus, δm˜ is not optimal, and so it follows from the conclusion of Step 5
above that the optimal c 1 belongs to the open interval (δm˜ , δm˜ −1 ). Since this interval is open, the optimal commitment level cannot satisfy c 1 = δm for any m ∈ N0 . But then, by the argument in Step 3, the optimal commitment level must satisfy ˜ for any c 1 ∈ (δm˜ , δm˜ −1 ), we can conclude that the optimal commitment level is equal to c 1 + β2 δ τ (c1 ) = 1. Since τ (c 1 ) = m ˜ m 1 − β2 δ , as claimed in the lemma. Step 8. We have shown that in SPE, player 1 commits to ψ1 , as defined in the statement of the lemma. Agreement is immediate and efficient. Moreover, the payoff to player 1 is v 01 (ψ1 , 0), as desired. 2 The proof of Theorem 5.1 requires the following auxiliary. Lemma A.1. If c i + c j 1 and c i c j , then v 0j (c i , c j ) c j . Proof. Suppose by way of contradiction that c i + c j 1 and c i c j but v 0j (c ) < c j . Since the cost of violating an effective commitment is prohibitive, this implies that no agreement is reached before τ (c i ). At time τ (c j ) − 1, player j’s aspiration is max{c j , β j δ τ (c j ) } = c j . Let b j (c j ) = max{t ∈ N0 |δt c j + βi δ τ (c j )}. Suppose that b(c j ) τ (c i ). Then, agreement is possible at b(c j ) < τ (c j ), a contradiction. Now suppose that b(c j ) τ (c j ). Then consider round τ (c i ) − 1. The aspirations in that round are c i and c j . Since c i + c j 1, we can find b˜ (c ) = max{t ∈ N0 |δt c i + c j }. Agreement at b˜ (c ) is possible, a contradiction.
2
Proof of Theorem 5.1. If: Take a pair of commitments c = (c 1 , c 2 ) such that c 1 + c 2 = 1 and c i 1 − η j for i = 1, 2 and j = i. Then, v 0 (c ) = c. Suppose player i deviates by choosing a different commitment than c i . If this deviation results in some agreement x before τ (c j ), then x j c j and so xi 1 − c j = c i , the deviation is not profitable. If the deviation results
V. Britz / Games and Economic Behavior 77 (2013) 345–351
351
in an agreement x after τ (c j ), then xi ϕi δ τ (c j ) . The deviation is not profitable. Finally, if the deviation results in perpetual disagreement, it is clearly not profitable either. Only if: Suppose by way of contradiction that in some SPE an agreement x is reached and xi < 1 − η j . If c j η j , it is a profitable deviation for player i to commit to ψi δ τ (c j ) and delay agreement until τ (c j ). The resulting payoff will be ϕi δ τ (c j ) 1 − η j . Hence, it must hold that c j < η j . We show that choosing c i = 1 − η j is a profitable deviation for player i. Suppose that c j c i . Then, v 0 (c i , c j ) c i = 1 − η j by Lemma A.1, as desired. Now consider the case where c i > c j . We want to show that i receives at least c i . Suppose not. Then t¯ = τ (c i ) is reached. We have t¯−1
t¯−1
t¯
t¯−1
αit¯−1 (c i , c j ) = c i = 1 − η j and
α j (c i , c j ) = max{c j , v j } < 1 − η j . Thus, αi (c i , c j ) + α j (c i , c j ) < 2(1 − η j ) 1. (The last inequality follows from the fact that η j 12 which can be easily verified from the definition of η j .) Consequently, there is t < τ (c i ) such that δt
exceeds the sum of the aspirations at t. Agreement can be reached, a contradiction. It remains to show that SPE is efficient. Perpetual disagreement is not possible in SPE since both commitments expire in finite time, and in the bargaining game with no effective commitments an agreement is reached. Furthermore, Eq. (4) shows that if an agreement x is reached at t, then x1 + x2 = δt . Hence, it will suffice to show that agreement is reached immediately in SPE. Suppose to the contrary that no agreement is reached at t = 0. Then, delay will last until τ (c i ). But then SPE requires that c j = ψ j δ τ (c i ) , where we use the convention that c i c j . In that case, we know from the analysis of the unilateral bargaining game that payoffs will be ϕ j δ τ (c i ) for player j and (1 − ϕ j )δ τ (c i ) for player i. We claim that it is a profitable deviation for player i to choose a zero commitment rather than c i . Indeed, v 0i (0, c j ) = (1 − ϕ j )δ τ (c i ) + βi (1 − δ τ (c i ) ). The claim follows since we have assumed βi > 0. 2 Proof of Theorem 5.4. If: Since
ηi
1 2
for i = 1, 2, we also have 1 − ηi 12 . The claim follows from Theorem 5.1.
Only if: Consider a pie division in which player k = 1, 2 obtains a payoff of sufficiently large, SPE requires 1 2 −3ε 1 2 −ε
1 2
−ε
1
3−βk
. This can be rewritten as βk
< 1. Consequently, choosing δ sufficiently large and βk ∈ (
is not supported by any SPE.
2
1 2 −3ε 1 2 −ε
1 2
− ε , where ε > 0. By Theorem 5.2, if δ is
1 2 −3ε 1 −ε 2
. But we have assumed that
ε > 0, thus
, 1) ensures that the pie division under consideration
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