- Email: [email protected]
The power to delay
Economics Letters 112 (2011) 155–157
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The power to delay Duozhe Li ∗ Department of Economics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
article
info
Article history: Received 15 March 2010 Received in revised form 29 March 2011 Accepted 20 April 2011 Available online 29 April 2011
abstract This paper studies a bilateral bargaining game in which one party can suspend the game indefinitely. If suspension is costless, multiple equilibria arise; if suspension is costly and subject to a budget constraint, a unique equilibrium with immediate agreement is obtained. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C78 Keywords: Bargaining Suspension Sanction Budget constraint
1. Introduction In international peace talks, there are often occasions when one party unilaterally and indefinitely suspends the ongoing negotiations, which often incurs economic and political sanctions by third parties, e.g. the United Nations.1 Similar scenarios occur in other bargaining situations such as land acquisitions, merger negotiations, etc. Suspension of negotiations is somewhat different from bargaining delay in the usual sense because no proposal is heard during a suspension. Meanwhile, as suspension is a reversible decision, it is also different from taking up an outside option. This paper studies the role of ‘‘the power to delay’’ in bilateral bargaining. It analyzes the bargaining game between a committed player and an uncommitted player, in which the former is committed to bargaining all of the time and the latter is endowed with the power to delay. More specifically, after each bargaining period, the uncommitted player has the option of suspending the game. During a suspension, no offers and responses can be exchanged. The uncommitted player can resume the bargaining game at any time, and he has to pay an extra cost for each period of suspension. The cost can be naturally interpreted as a sanction imposed by a third party.
∗
Tel.: +852 2609 8183; fax: +852 2603 5805. E-mail address: [email protected].
1 In April 2009, North Korea pulled out of the six-party talks on its nuclear program. Sanctions were imposed, and not until recently was there any hope that the negotiations would be resumed. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.04.008
There are two main results. First, the uncommitted player can exploit the power to delay to create an endogenous disparity of bargaining power sustaining highly asymmetric equilibrium divisions. More interestingly, if the uncommitted player is subject to a budget constraint or the suspension cost increases over time and will eventually exceed a certain threshold, the bargaining game has a unique efficient equilibrium with a symmetric agreement and suspension never occurs. This result, clearly at odds with our daily observations, resembles the familiar chainstore paradox. Since Rubinstein (1982) showed that two impatient bargainers will always reach an immediate agreement in a unique equilibrium, a large body of literature has evolved to explain the inefficient bargaining delay often observed in real life. Early research mainly focused on the case with information asymmetry between players, where delay serves as a signaling device. It is also well known that some models of complete information admit multiple equilibria with immediate agreement, from which one can construct equilibria with delay.2 Avery and Zemsky (1994) synthesize the multiplicity results from complete information bargaining models by identifying a general principle: ‘‘money burning’’. In general, multiple equilibria arise when at least one player can destroy some bargaining value after his own proposal is rejected. The first result of the current study is in line with this principle.
2 Haller and Holden (1990) and Fernandez and Glazer (1991) analyze a wage bargaining game in which whether or not to strike is a strategic choice of the union. Shaked (1994) and Ponsatí and Sákovics (1998) study bargaining with outside options. Busch and Wen (1995) study a bargaining model in which players’ interim disagreement payoffs are determined by a normal form game. They all obtain multiple equilibria under complete information.
156
D. Li / Economics Letters 112 (2011) 155–157
However, existing studies do not recognize that in reality money-burning actions often incur sanctions. As it is a realistic description that negotiating parties may face budget constraints and increasingly intensive sanctions for refusing to negotiate, the second result of the current study, in some sense, casts doubt on the explanatory power of money-burning actions on inefficient bargaining delay. 2. Model Two players, 1 and 2, bargain over a surplus of size one. Time is discrete, indexed by t ∈ {1, 2, . . .}. In each bargaining period, one player is randomly selected to make a proposal, and the other player decides whether to accept or reject it. Assume that both players are always equally likely to become the proposer. Bargaining ends with agreement at the first time that a proposal is accepted. A new feature of the bargaining game here is that one of the players, say player 1, is endowed with the power to suspend the game.3 More specifically, following a rejection in period t, player 1 chooses between two options: one is to continue to bargain in the following period and the other is to suspend the game indefinitely. During a suspension, player 1 decides at the beginning of every period whether or not to resume bargaining. The structure of the game remains the same when it is resumed from suspension. Finally, player 1 has to pay an extra cost of c ≥ 0 for each period the suspension lasts. A bargaining outcome is denoted as x, hT , where x ∈ [0, 1] is the agreement reached in period T , by which player 1 (2 resp.)
T
receives a share of x (1 − x resp.), and the sequence hTs s=1 summarizes the suspension decisions during the bargaining process. Let hTs = 1 if the sth period of bargaining is suspended and hTs = 0 if it is not suspended. Clearly, it must be that hTT = 0. The players discount future payoffs by a common factor δ ∈ (0, 1). Hence, given x, hT , player 1’s payoff evaluated in period t ≤ T is u1 x , h T ; t = δ T − t x −
T −
hTs δ s−t c ,
s=t
and player 2’s payoff in period t is simply u2 x, hT ; t = δ T −t (1 − x) .
Note that it is crucial that player 1 can start a suspension after his own proposal is rejected. In this regard, it is similar to the study of the outside option effect in bargaining. If only the responder can opt out, the bargaining game has a unique equilibrium.4 If, instead, the proposer can also opt out after his own proposal is rejected, multiple equilibria arise. Shaked (1994) and Ponsatí and Sákovics (1998) elaborate on this insight.
Lemma 1. The following strategy profile is a subgame perfect equilibrium: (i) the proposer always offers the responder δ/2; (ii) the responder always accepts any offer no less than δ/2; (iii) player 1 never suspends the game. Player 1 can indeed benefit from the power to delay. The key insight here is that the power to delay can create an endogenous disparity of bargaining power measured by the cost of rejection. More precisely, suppose that player 1 suspends the game for k ≥ 1 periods after every rejection of player 2, and does not suspend after his own rejection. Then, the rejection cost of player 2 is (k + 1) times that of player 1, and this gives player 1 greater bargaining power than player 2. If there is an equilibrium in which player 1 bargains in this way, the ratio of equilibrium shares shall tend to (k + 1) : 1 as the discount factor δ tends to one. It remains to be shown that the threat to suspend the game for k periods is credible. Note that suspension is imposed as a punishment on player 2. If player 1 carries it out completely, he is rewarded by receiving the large share sustained by this punishment; otherwise, the Rubinstein outcome awaits. Hence, when the suspension cost is sufficiently low and player 1 is sufficiently patient, suspension is indeed a credible threat. Proposition 1. For any integer k ≥ 1, when δ is sufficiently close to 1 and c is sufficiently small, there exists a subgame perfect equilibrium with an immediate agreement in which player 1’s share tends to (k + 1) / (k + 2) as δ tends to 1. Proof. The equilibrium strategy profile consists of three phases. The play begins in phase 1, in which player 1 (2 resp.) proposes (x, 1 − x) ((y, 1 − y) resp.) and accepts any proposal that gives him no less than y (1 − x resp.). The play remains in phase 1 if player 1 deviates. Following player 2’s deviation, the play enters phase 2, in which player 1 suspends the game for k periods. Move back to phase 1 after phase 2 is completed. If player 1 ends phase 2 early, then phase 3 begins. This is an absorbing phase, in which the Rubinstein outcome is played. The equilibrium proposals (in phase 1) must satisfy the following indifference conditions:
[
] 1 (1 − x) + (1 − y) 2 2 1 1 y=δ x+ y , 1 − x = δ k+1
2
1
2
by which we can obtain x = 1 − δ k+1 η−1
y = δ 1 − η −1 ,
and
where 3. Results This section characterizes the subgame perfect equilibria of the bargaining game, focusing on the equilibria with immediate agreements. Once the multiplicity of the efficient equilibrium outcomes is established, it is straightforward to construct equilibria with lengthy delays. If player 1 does not have the power to delay, the bargaining game becomes a straightforward variant of the Rubinstein game, in which there is a unique subgame perfect equilibrium with immediate agreement, i.e., the Rubinstein outcome. In this equilibrium, whoever is the proposer offers the other a share of δ/2, which is accepted. In the current model, this remains an equilibrium outcome. The only amendment of the strategy profile is that player 1 never suspends the game.
3 The results can easily be carried over to the case with two-sided suspension. 4 See Shaked and Sutton (1984), Binmore (1985), and Sutton (1986) for details.
η = 1+
k −
δ
.
s
s =0
For this to be an equilibrium, it has to be optimal for player 1 to carry out phase 2 completely. Clearly, if it is not profitable for player 1 to deviate in the first period of phase 2, it cannot be profitable for him to deviate in any other period. Hence, it suffices to have
δk
1 2
x+
1 2
y
−
k−1 −
δs c ≥
s=0
1 2
,
(1)
where the left-hand side is player 1’s expected payoff from carrying out phase 2 completely evaluated in the first period of phase 2, and the right-hand side is his expected payoff from deviating in the first period of phase 2. It is easy to check that lim x = lim y =
δ→1
δ→1
k+1 k+2
.
D. Li / Economics Letters 112 (2011) 155–157
Thus, as δ → 1, the left-hand side of condition (1) goes to (k + 1) / (k + 2) − kc, which is greater than 1/2 for any k ≥ 1 when c is sufficiently small. This completes the proof. For any given δ and c, let K be the greatest integer such that condition (1) is satisfied. Denote by s(K ) player 1’s expected equilibrium share, i.e., s(K ) = 12 [x(K ) + y(K )]. It is easy to see that if c ≥ 1/6, there does not exist any k such that condition (1) is satisfied. In this case, K = 0, the Rubinstein outcome is the unique equilibrium outcome. If 0 ≤ c < 1/6, K is well defined when δ is sufficiently close to 1. Finally, if c = 0, then as δ tends to 1, K tends to infinity and s(K ) tends to 1. In other words, if suspension is costless, player 1 can exploit the power to delay to obtain almost the entire surplus when he is extremely patient. Next, consider the situation in which suspension is subject to a budget constraint. Player 1 is endowed with a budget of B0 > 0, which can be used to cover the suspension cost. After B0 is exhausted, suspension is no longer an option for player 1. Proposition 2 shows that when there is a budget constraint, the threat of suspension loses its credibility completely and the bargaining game has the Rubinstein outcome as its unique equilibrium outcome. Proposition 2. For any B0 > 0 and c > 0, the bargaining game has a unique equilibrium, in which whoever is the proposer offers the other a share of δ/2, and the offer is accepted. Proof. Suppose that Tc ≤ B0 < (T + 1) c: player 1 has the budget to afford at most T periods of suspension. Denote the subgame starting from period t as Gt (Bt ), where Bt is the remaining budget at the beginning of period t. Clearly, if Bt < c, player 1 cannot afford another suspension, then, Gt (Bt ) is equivalent to the game in which the suspension option is not available, and the Rubinstein outcome is its unique equilibrium outcome. Using backward induction, it is easy to establish that in G0 (B0 ) with B0 < ∞, player 1 never chooses the suspension option, on or off the equilibrium path. Hence, the Rubinstein outcome is also the unique equilibrium outcome of G0 (B0 ). A similar situation is that player 1 has increasing suspension costs, i.e., ct +1 ≥ ct > 0. If the suspension cost will eventually exceed 1/6, by backward induction the bargaining game has the same unique equilibrium as above, even without a budget constraint. Meanwhile, if there exists an upper bound c¯ < 1/6 on
157
the suspension cost, then the set of equilibria is the same as in the game with a constant cost c¯ . Corollary 1. (i) If there exists T such that ct ≥ 1/6 for any t ≥ T , then the bargaining game has the same unique equilibrium as in Proposition 2. (ii) If ct → c¯ < 1/6 as t → ∞, then for any δ < 1, the set of equilibria is the same as in the game with a constant cost c¯ . The uniqueness result in Proposition 2 and Corollary 1(i) is clearly at odds with our daily observation that bargainers often exploit their power to delay to a certain extent. One may consider it reminiscent of the familiar chainstore paradox. Along this line of thought, one way to reconcile the tension between the game theoretic prediction and intuition based on daily observations is to relax the assumption of common knowledge of rationality; then, even a rational bargainer has the incentive to build up a reputation of being obstinate. Alternatively, one may argue that many real-life bargaining situations might not meet the assumptions sustaining the uniqueness result: first of all, negotiating parties (e.g. countries or corporations) often have steady streams of revenue to replenish their reserves so that repeated sanctions can never bankrupt them; secondly, a suspension cost greater than 1/6 of the entire surplus seems unrealistically high. In the light of this argument, our study suggests that, in order to have an appropriate model of bargaining, one has to take into consideration several new parameters. References Avery, C., Zemsky, P.B., 1994. Money burning and multiple equilibria in bargaining. Games Econom. Behav. 7, 154–168. Binmore, K., 1985. Bargaining and coalitions. In: Roth, A.E. (Ed.), Game Theoretic Models of Bargaining. Cambridge University Press, Cambridge, UK. Busch, L.-A., Wen, Q., 1995. Perfect equilibria in a negotiation model. Econometrica 63, 545–565. Fernandez, R., Glazer, J., 1991. Striking for a bargain between two completely informed agents. Am. Econ. Rev. 81, 240–252. Haller, H., Holden, S., 1990. A letter to editor on wage bargaining. J. Econom. Theory 52, 232–236. Ponsatí, C., Sákovics, J., 1998. Rubinstein bargaining with two-sided outside options. J. Econom. Theory 11, 667–672. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–110. Shaked, A., 1994. Opting out: bazaars versus ‘‘hi tech’’ markets. Invest. Econ. 18, 421–432. Shaked, A., Sutton, J., 1984. Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52, 1351–1364. Sutton, J., 1986. Non-cooperative bargaining theory: an introduction. Rev. Econom. Stud. 53, 709–724.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
The power to delay Duozhe Li ∗ Department of Economics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
article
info
Article history: Received 15 March 2010 Received in revised form 29 March 2011 Accepted 20 April 2011 Available online 29 April 2011
abstract This paper studies a bilateral bargaining game in which one party can suspend the game indefinitely. If suspension is costless, multiple equilibria arise; if suspension is costly and subject to a budget constraint, a unique equilibrium with immediate agreement is obtained. © 2011 Elsevier B.V. All rights reserved.
JEL classification: C78 Keywords: Bargaining Suspension Sanction Budget constraint
1. Introduction In international peace talks, there are often occasions when one party unilaterally and indefinitely suspends the ongoing negotiations, which often incurs economic and political sanctions by third parties, e.g. the United Nations.1 Similar scenarios occur in other bargaining situations such as land acquisitions, merger negotiations, etc. Suspension of negotiations is somewhat different from bargaining delay in the usual sense because no proposal is heard during a suspension. Meanwhile, as suspension is a reversible decision, it is also different from taking up an outside option. This paper studies the role of ‘‘the power to delay’’ in bilateral bargaining. It analyzes the bargaining game between a committed player and an uncommitted player, in which the former is committed to bargaining all of the time and the latter is endowed with the power to delay. More specifically, after each bargaining period, the uncommitted player has the option of suspending the game. During a suspension, no offers and responses can be exchanged. The uncommitted player can resume the bargaining game at any time, and he has to pay an extra cost for each period of suspension. The cost can be naturally interpreted as a sanction imposed by a third party.
∗
Tel.: +852 2609 8183; fax: +852 2603 5805. E-mail address: [email protected].
1 In April 2009, North Korea pulled out of the six-party talks on its nuclear program. Sanctions were imposed, and not until recently was there any hope that the negotiations would be resumed. 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.04.008
There are two main results. First, the uncommitted player can exploit the power to delay to create an endogenous disparity of bargaining power sustaining highly asymmetric equilibrium divisions. More interestingly, if the uncommitted player is subject to a budget constraint or the suspension cost increases over time and will eventually exceed a certain threshold, the bargaining game has a unique efficient equilibrium with a symmetric agreement and suspension never occurs. This result, clearly at odds with our daily observations, resembles the familiar chainstore paradox. Since Rubinstein (1982) showed that two impatient bargainers will always reach an immediate agreement in a unique equilibrium, a large body of literature has evolved to explain the inefficient bargaining delay often observed in real life. Early research mainly focused on the case with information asymmetry between players, where delay serves as a signaling device. It is also well known that some models of complete information admit multiple equilibria with immediate agreement, from which one can construct equilibria with delay.2 Avery and Zemsky (1994) synthesize the multiplicity results from complete information bargaining models by identifying a general principle: ‘‘money burning’’. In general, multiple equilibria arise when at least one player can destroy some bargaining value after his own proposal is rejected. The first result of the current study is in line with this principle.
2 Haller and Holden (1990) and Fernandez and Glazer (1991) analyze a wage bargaining game in which whether or not to strike is a strategic choice of the union. Shaked (1994) and Ponsatí and Sákovics (1998) study bargaining with outside options. Busch and Wen (1995) study a bargaining model in which players’ interim disagreement payoffs are determined by a normal form game. They all obtain multiple equilibria under complete information.
156
D. Li / Economics Letters 112 (2011) 155–157
However, existing studies do not recognize that in reality money-burning actions often incur sanctions. As it is a realistic description that negotiating parties may face budget constraints and increasingly intensive sanctions for refusing to negotiate, the second result of the current study, in some sense, casts doubt on the explanatory power of money-burning actions on inefficient bargaining delay. 2. Model Two players, 1 and 2, bargain over a surplus of size one. Time is discrete, indexed by t ∈ {1, 2, . . .}. In each bargaining period, one player is randomly selected to make a proposal, and the other player decides whether to accept or reject it. Assume that both players are always equally likely to become the proposer. Bargaining ends with agreement at the first time that a proposal is accepted. A new feature of the bargaining game here is that one of the players, say player 1, is endowed with the power to suspend the game.3 More specifically, following a rejection in period t, player 1 chooses between two options: one is to continue to bargain in the following period and the other is to suspend the game indefinitely. During a suspension, player 1 decides at the beginning of every period whether or not to resume bargaining. The structure of the game remains the same when it is resumed from suspension. Finally, player 1 has to pay an extra cost of c ≥ 0 for each period the suspension lasts. A bargaining outcome is denoted as x, hT , where x ∈ [0, 1] is the agreement reached in period T , by which player 1 (2 resp.)
T
receives a share of x (1 − x resp.), and the sequence hTs s=1 summarizes the suspension decisions during the bargaining process. Let hTs = 1 if the sth period of bargaining is suspended and hTs = 0 if it is not suspended. Clearly, it must be that hTT = 0. The players discount future payoffs by a common factor δ ∈ (0, 1). Hence, given x, hT , player 1’s payoff evaluated in period t ≤ T is u1 x , h T ; t = δ T − t x −
T −
hTs δ s−t c ,
s=t
and player 2’s payoff in period t is simply u2 x, hT ; t = δ T −t (1 − x) .
Note that it is crucial that player 1 can start a suspension after his own proposal is rejected. In this regard, it is similar to the study of the outside option effect in bargaining. If only the responder can opt out, the bargaining game has a unique equilibrium.4 If, instead, the proposer can also opt out after his own proposal is rejected, multiple equilibria arise. Shaked (1994) and Ponsatí and Sákovics (1998) elaborate on this insight.
Lemma 1. The following strategy profile is a subgame perfect equilibrium: (i) the proposer always offers the responder δ/2; (ii) the responder always accepts any offer no less than δ/2; (iii) player 1 never suspends the game. Player 1 can indeed benefit from the power to delay. The key insight here is that the power to delay can create an endogenous disparity of bargaining power measured by the cost of rejection. More precisely, suppose that player 1 suspends the game for k ≥ 1 periods after every rejection of player 2, and does not suspend after his own rejection. Then, the rejection cost of player 2 is (k + 1) times that of player 1, and this gives player 1 greater bargaining power than player 2. If there is an equilibrium in which player 1 bargains in this way, the ratio of equilibrium shares shall tend to (k + 1) : 1 as the discount factor δ tends to one. It remains to be shown that the threat to suspend the game for k periods is credible. Note that suspension is imposed as a punishment on player 2. If player 1 carries it out completely, he is rewarded by receiving the large share sustained by this punishment; otherwise, the Rubinstein outcome awaits. Hence, when the suspension cost is sufficiently low and player 1 is sufficiently patient, suspension is indeed a credible threat. Proposition 1. For any integer k ≥ 1, when δ is sufficiently close to 1 and c is sufficiently small, there exists a subgame perfect equilibrium with an immediate agreement in which player 1’s share tends to (k + 1) / (k + 2) as δ tends to 1. Proof. The equilibrium strategy profile consists of three phases. The play begins in phase 1, in which player 1 (2 resp.) proposes (x, 1 − x) ((y, 1 − y) resp.) and accepts any proposal that gives him no less than y (1 − x resp.). The play remains in phase 1 if player 1 deviates. Following player 2’s deviation, the play enters phase 2, in which player 1 suspends the game for k periods. Move back to phase 1 after phase 2 is completed. If player 1 ends phase 2 early, then phase 3 begins. This is an absorbing phase, in which the Rubinstein outcome is played. The equilibrium proposals (in phase 1) must satisfy the following indifference conditions:
[
] 1 (1 − x) + (1 − y) 2 2 1 1 y=δ x+ y , 1 − x = δ k+1
2
1
2
by which we can obtain x = 1 − δ k+1 η−1
y = δ 1 − η −1 ,
and
where 3. Results This section characterizes the subgame perfect equilibria of the bargaining game, focusing on the equilibria with immediate agreements. Once the multiplicity of the efficient equilibrium outcomes is established, it is straightforward to construct equilibria with lengthy delays. If player 1 does not have the power to delay, the bargaining game becomes a straightforward variant of the Rubinstein game, in which there is a unique subgame perfect equilibrium with immediate agreement, i.e., the Rubinstein outcome. In this equilibrium, whoever is the proposer offers the other a share of δ/2, which is accepted. In the current model, this remains an equilibrium outcome. The only amendment of the strategy profile is that player 1 never suspends the game.
3 The results can easily be carried over to the case with two-sided suspension. 4 See Shaked and Sutton (1984), Binmore (1985), and Sutton (1986) for details.
η = 1+
k −
δ
.
s
s =0
For this to be an equilibrium, it has to be optimal for player 1 to carry out phase 2 completely. Clearly, if it is not profitable for player 1 to deviate in the first period of phase 2, it cannot be profitable for him to deviate in any other period. Hence, it suffices to have
δk
1 2
x+
1 2
y
−
k−1 −
δs c ≥
s=0
1 2
,
(1)
where the left-hand side is player 1’s expected payoff from carrying out phase 2 completely evaluated in the first period of phase 2, and the right-hand side is his expected payoff from deviating in the first period of phase 2. It is easy to check that lim x = lim y =
δ→1
δ→1
k+1 k+2
.
D. Li / Economics Letters 112 (2011) 155–157
Thus, as δ → 1, the left-hand side of condition (1) goes to (k + 1) / (k + 2) − kc, which is greater than 1/2 for any k ≥ 1 when c is sufficiently small. This completes the proof. For any given δ and c, let K be the greatest integer such that condition (1) is satisfied. Denote by s(K ) player 1’s expected equilibrium share, i.e., s(K ) = 12 [x(K ) + y(K )]. It is easy to see that if c ≥ 1/6, there does not exist any k such that condition (1) is satisfied. In this case, K = 0, the Rubinstein outcome is the unique equilibrium outcome. If 0 ≤ c < 1/6, K is well defined when δ is sufficiently close to 1. Finally, if c = 0, then as δ tends to 1, K tends to infinity and s(K ) tends to 1. In other words, if suspension is costless, player 1 can exploit the power to delay to obtain almost the entire surplus when he is extremely patient. Next, consider the situation in which suspension is subject to a budget constraint. Player 1 is endowed with a budget of B0 > 0, which can be used to cover the suspension cost. After B0 is exhausted, suspension is no longer an option for player 1. Proposition 2 shows that when there is a budget constraint, the threat of suspension loses its credibility completely and the bargaining game has the Rubinstein outcome as its unique equilibrium outcome. Proposition 2. For any B0 > 0 and c > 0, the bargaining game has a unique equilibrium, in which whoever is the proposer offers the other a share of δ/2, and the offer is accepted. Proof. Suppose that Tc ≤ B0 < (T + 1) c: player 1 has the budget to afford at most T periods of suspension. Denote the subgame starting from period t as Gt (Bt ), where Bt is the remaining budget at the beginning of period t. Clearly, if Bt < c, player 1 cannot afford another suspension, then, Gt (Bt ) is equivalent to the game in which the suspension option is not available, and the Rubinstein outcome is its unique equilibrium outcome. Using backward induction, it is easy to establish that in G0 (B0 ) with B0 < ∞, player 1 never chooses the suspension option, on or off the equilibrium path. Hence, the Rubinstein outcome is also the unique equilibrium outcome of G0 (B0 ). A similar situation is that player 1 has increasing suspension costs, i.e., ct +1 ≥ ct > 0. If the suspension cost will eventually exceed 1/6, by backward induction the bargaining game has the same unique equilibrium as above, even without a budget constraint. Meanwhile, if there exists an upper bound c¯ < 1/6 on
157
the suspension cost, then the set of equilibria is the same as in the game with a constant cost c¯ . Corollary 1. (i) If there exists T such that ct ≥ 1/6 for any t ≥ T , then the bargaining game has the same unique equilibrium as in Proposition 2. (ii) If ct → c¯ < 1/6 as t → ∞, then for any δ < 1, the set of equilibria is the same as in the game with a constant cost c¯ . The uniqueness result in Proposition 2 and Corollary 1(i) is clearly at odds with our daily observation that bargainers often exploit their power to delay to a certain extent. One may consider it reminiscent of the familiar chainstore paradox. Along this line of thought, one way to reconcile the tension between the game theoretic prediction and intuition based on daily observations is to relax the assumption of common knowledge of rationality; then, even a rational bargainer has the incentive to build up a reputation of being obstinate. Alternatively, one may argue that many real-life bargaining situations might not meet the assumptions sustaining the uniqueness result: first of all, negotiating parties (e.g. countries or corporations) often have steady streams of revenue to replenish their reserves so that repeated sanctions can never bankrupt them; secondly, a suspension cost greater than 1/6 of the entire surplus seems unrealistically high. In the light of this argument, our study suggests that, in order to have an appropriate model of bargaining, one has to take into consideration several new parameters. References Avery, C., Zemsky, P.B., 1994. Money burning and multiple equilibria in bargaining. Games Econom. Behav. 7, 154–168. Binmore, K., 1985. Bargaining and coalitions. In: Roth, A.E. (Ed.), Game Theoretic Models of Bargaining. Cambridge University Press, Cambridge, UK. Busch, L.-A., Wen, Q., 1995. Perfect equilibria in a negotiation model. Econometrica 63, 545–565. Fernandez, R., Glazer, J., 1991. Striking for a bargain between two completely informed agents. Am. Econ. Rev. 81, 240–252. Haller, H., Holden, S., 1990. A letter to editor on wage bargaining. J. Econom. Theory 52, 232–236. Ponsatí, C., Sákovics, J., 1998. Rubinstein bargaining with two-sided outside options. J. Econom. Theory 11, 667–672. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–110. Shaked, A., 1994. Opting out: bazaars versus ‘‘hi tech’’ markets. Invest. Econ. 18, 421–432. Shaked, A., Sutton, J., 1984. Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52, 1351–1364. Sutton, J., 1986. Non-cooperative bargaining theory: an introduction. Rev. Econom. Stud. 53, 709–724.