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A note on three-player noncooperative bargaining with restricted pairwise meetings
Economics Letters 65 (1999) 47–54
A note on three-player noncooperative bargaining with restricted pairwise meetings a,b , * ´ Antoni Calvo-Armengol a
Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25 – 27, 08005 Barcelona, Spain b ` , 75007 Paris, France CERAS–ENPC, 28 rue des Saints-Peres Received 23 February 1999; accepted 26 May 1999
Abstract A three-player bargaining game where only the central player can negotiate with two different partners has a unique perfect equilibrium outcome. We discuss when the a priori favored position of the central player is really advantageous in terms of equilibrium payoffs. 1999 Elsevier Science S.A. All rights reserved. Keywords: Bargaining; Central player; Peripheral players; Impatience rates JEL classification: C78; D20
1. Introduction In many social and economic situations, agents are in direct contact with only a limited subset of other agents.1 When this is the case, bilateral volunteer meetings are limited to those pairs of agents that can communicate with each other directly (or that know each other directly). The aim of this paper is to explore the implications of the restrictions on feasible pairwise meetings in a simple bargaining context. The bargaining game we consider is a particular version of the ‘three-player / onecake’ game where players hold asymmetric positions. More explicitly, we model a negotiation process where a central player bargains for a given number of rounds with the same partner, and then switches over to another available partner for another negotiation phase with fixed length. If no bilateral agreement is still reached, the central player moves back to the first partner. And so on. Therefore, any (final) agreement between two partners must necessarily involve the central player. *Tel.: 134-93-542-2688; fax: 134-93-542-1746. ´ E-mail address: [email protected] (A. Calvo-Armengol) 1 Jackson and Wolinsky (1996) provide examples illustrating the importance of social and economic networks, and a survey of the related literature. 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00136-6
1999 Elsevier Science S.A. All rights reserved.
48
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
This situation arises, for instance, when a firm requiring the services of one employer to produce one unit of profit faces two potential groups of workers, and a negotiation over the wages is engaged. The final agreement necessarily involves the firm (central player). Shaked and Sutton (1984) consider a similar employer–employee bargaining situation (where a firm bargains for wages with the operating workforce for a certain number of periods after which the firm is free to make an offer to an outsider) to study unemployment. Horn and Wolinsky (1988) and Jun (1989) investigate unionization in a bargaining model where the three players are the firm and two groups of workers of different types. An alternative situation fitting to the problem considered is when a seller owns an indivisible object to be sold to one of two (or more) potential buyers through a bargaining procedure. Related models are discussed by Fudenberg et al. (1987) in an incomplete information setting, by Hendon and Tranæs (1991) where the buyers have different valuations for the good, and by Jehiel and Moldovanu (1995a,b) who investigate the effect of externalities between buyers. Section 2 presents the game and states the main result of the paper: the noncooperative bargaining game with restricted pairwise meetings has a unique perfect equilibrium outcome. The corresponding equilibrium shares are fully characterized. The central player is the only one that can negotiate with two different partners, and is involved in any bilateral agreement of the game. We determine in Section 3 when this a priori favored position is really advantageous in terms of equilibrium payoffs, and provide the bargaining procedure ensuring the central player a higher outcome. Section 4 contains all the proofs.
2. The model
2.1. The game ˚ alternating offers Our bargaining game is simple in that it is an adaptation of the Rubinstein-Stahl game. In particular, no trade takes place and bargaining is bilateral: one player can only bargain with one partner at a time. We assume that the central player initiates the negotiations. Players are consequently labeled 0 (central player) and 61 (peripheral players). Without loss of generality, the game begins with a contest between the central player and the player on her right-hand side, player 1 1. Player 0 and player 1 1 then bargain for r $ 1 rounds, each round being composed of two different stages consisting of an offer from player 0 to player 1 1 and, in case of rejection, of a counter-offer from player 1 1 to player 0. Players discount stages with a discount rate di [s0, 1d, i [h 2 1, 0, 1 1j. If these two players haven’t still reached an agreement after r rounds of negotiations, player 0 then bargains for l $ 1 rounds with the player on her left-hand side, player 2 1. And so on. The game ends when, and if, an agreement is reached between the central player and one of the peripheral partners. In this game, pairwise meetings of players are restricted: the peripheral players can never make proposals one to the other, and the final agreement always involves the central player. In that sense, players hold asymmetric positions as it is the case, for instance, in a wage bargaining context where an employer faces two isolated potential workers. We can represent these restrictions on the feasible pairwise meetings (and, consequently, on the possible bilateral contests) by a graph where the nodes
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
49
are identified with the players and a link between two nodes means that the corresponding players can bargain together.2 More formally, denote by ( pt , qt ) the pair of proposer and respondent at round t 5 0, 1, 2, . . . At t, player pt makes an offer to qt and the game terminates if the offer is accepted. If not, we go to the next round t 1 1 and qt becomes the new proposer, that is pt11 5 qt . Now it is pt11 ’s turn to make an offer to qt 11 5 pt 12 . And so on. The active bargaining pair of players at round t is then ( pt , pt11 ), with p0 5 0, p1 5 1 1, p2 5 0, . . . , p2r21 5 1 1, p2r 5 0, p2r11 5 2 1, p2r12 5 0, . . . , p2(r1l ) 5 0, p2(r1l )11 5 1 1, . . .
2.2. The main result This bargaining procedure defines a noncooperative sequential bargaining game that we analyze. We restrict attention to subgame perfect equilibria. We first introduce some useful notations. For any two periods of time t and s (t $ 0, s . 0), let v (t, t 1 s) be the product of the discount factors of the players that respond to offers from round t to round t 1 s 2 1 (or, equivalently, that make offers from round t 1 1 to round t 1 s):
Pd s
v (t, t 1 s) 5 dpt 11 3 ? ? ? 3 dpt 1s 5
i 51
p t 1i 2(r1l )21
i
Let also a t 5 [1 / 1 2 v (0, 2(r 1 l))]f1 1 o i51 (21) v (t, t 1 i)g, for all t 5 0, 1, . . . Finally, denote by D3 5hsx 1 , x 2 , x 3dux 1 1 x 2 1 x 3 5 1; x 1 , . . . , x 3 $ 0j the unit simplex of R 31 . The main result is: Proposition 1. The above game has a unique perfect equilibrium outcome supported by the following set of stationary strategies, where offers made at any stage correspond to the perfect equilibrium outcome and players agree to any offer of at least this amount: at round t, player pt proposes the share 1 2 a t to player pt 11 and accepts any offer x [ D3 from player pt 21 as long as x pt $ dpt a t .
˚ (1972) analyze a two-player sequential bargaining game Remark 1. Rubinstein (1982) and Stahl where players alternate in making proposals that has a unique subgame perfect equilibrium. Shaked (see Sutton, 1986) gives an example of a three-player bargaining game of which the perfect equilibrium is not unique. Herrero (1985) and Haller (1986) observe that games involving more than two bargainers display multiple equilibria. Jun (1987) recovers uniqueness with three players by allowing them to sign contracts. This result is lately extended to the n-player case by Chae and Yang ˚ (1994). Uniqueness also holds in our three-player Rubinstein-Stahl-type bargaining model where bilateral agreements always involve the central player. Remark 2. We can easily check that 1 2 a t 5 dpt 11 a t 11 , ;t 5 0, 1, . . . These equations correspond to ˚ solution where, at equilibrium, players are indifferent between their the standard Rubinstein-Stahl share as a respondent and their share as a delayed proposer. Proposition 1 means in particular that 2 We refer the reader to Myerson (1977) for a first representation of the restricted communication possibilities by an undirected graph.
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´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
the bargaining outcome corresponding to the above game is predictable: at t 5 0 the central player proposes a share 1 2 a 0 to player 1 1, that immediately accepts this offer in which the left-hand side player 2 1 ends up with no share of the cake. Remark 3. It is straightforward to show that the uniqueness of the perfect equilibrium outcome holds in a more general setting where a central player bargains successively with N players labeled from 1 to n with discount factors d1 , . . . ,dn [s0, 1d (n $ 2) for, respectively, r 1 , . . . ,r n $ 1 rounds, each round consisting of an offer and a counter-offer stage. The central player’s share of the cake at the unique perfect equilibrium of this game is a 0 5 [1 / 1 2 v (0, 2R)]f1 1 o 2R21 (21)i v (t, t 1 i)g, where i 51 n R 5 o i51 r i . Similarly, in the three-player case, we can allow for a sequential bargaining procedure consisting of (successively) r 1 rounds with player 1 1, l 1 rounds with player 2 1, . . . ,r p rounds with player 1 1, l p rounds with player 2 1, followed by a similar phase of negotiations, and so on. The central player’s share of the cake at the unique perfect equilibrium of this game is a 0 5 [1 / 1 2 v (0, )21 p 2(R 1 L))]f1 1 o 2(R1L (21)i v (t, t 1 i)g, where R 5 o ip51 r i and L 5 o i51 li . i 51
3. Comparative statics The central player is the only one that can negotiate with two different partners. Moreover, our game requires that any bilateral agreement of the game involves player 0. We shall then ask whether the central player derives any advantage from her a priori favored position, and if so, what bargaining procedure (i.e. what length r and l of the negotiation rounds) ensures her a higher payoff. Corollaries 1–3 answer these questions. Corollary 1. If the two peripheral players have a common discount factor d, at the unique perfect equilibrium of the above game the central player’s share is a 0 5 (1 2 d / 1 2 d0d ). This share coincides with the standard agreement in the two-player case. Hence, when the peripheral players have identical time preferences, the central player does not derive any advantage from his particular location. In fact, abandoning the current partner after a fixed round of negotiations has no effect. Indeed, the new partner has exactly the same time preferences as the previous one. As for the forces that drive the negotiation agreement (time preferences and credible outside options), everything is as if the central player hadn’t changed her partner. In other words, the central player faces an identical situation irrespective of the peripheral player she is bargaining with. Denote by a 0 (r, l) the central player’s share of the cake at the unique perfect equilibrium outcome where player 0 bargains successively with player 1 1 and with player 2 1, respectively for r rounds and l rounds. Corollary 2. At the unique perfect equilibrium of the above game, the central player’s share of the cake increases with r and decreases with l if and only if d11 , d21 . In words, the longer the negotiation phase with the relatively more impatient partner, the higher the outcome of the bargaining game for the central player. Symmetrically, the longer the negotiation phase with the relatively more patient neighbor, the lower the central player’s equilibrium payoff. In
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
51
fact, when bargaining with the relatively more patient neighbor, the central player can effectively threaten him to abandon the current negotiations and turn to the outside partner, that is relatively more impatient and thus more eager to accept tougher proposals. Corollary 2 just states that the sooner the relatively more impatient neighbor is effectively introduced into the bargaining process (the sooner the central player ‘opts out’), the better it is for the central player (the higher the central player’s share of the cake at the unique perfect equilibrium outcome). In particular, it is harmful for the central to delay the effective 3-sided bargaining game when her current partner is the relatively more patient peripheral player. Corollary 3. Assume that d11 , d21 . The highest payoff the central player can obtain is lim r→` a 0 (r, l) 5 (1 2 d11 / 1 2 d0d11 ), ;l $ 1. This share coincides with the unique perfect equilibrium outcome of a standard two-player infinite-horizon alternating-offers game between the central player and the relatively more impatient partner, player 1 1. Hence, if we allow the central player to choose the type of the game that is played, the best thing to choose (that is, the game ensuring her the highest equilibrium outcome) is to bargain bilaterally with the relatively more impatient peripheral player, and ignore the other potential partner. In that case, negotiating with many partners is harmful. Consider the standard two-player alternating offers game between two players, labeled 1 and 2. If player 1 is the initiator of the bargaining game, at the unique perfect equilibrium of the game she gets a share a 1 5 (1 2 d2 / 1 2 d1d2 ) of the cake. This payoff satisfies (≠a 1 / ≠d2 ) 5 2 [1 2 d2 /s1 2 d1d2d 2 ] , 0. Hence, in the two-player game, the more one’s rival is impatient (the smaller d2 in this case), the bigger the share obtained. Therefore, if player 1 can choose her bargaining partner among a set of potential partners, she selects the relatively more impatient one. Corollary 3 recovers this result for the three-player game with restricted pairwise meetings.
4. Proofs Proof of Proposition 1. It is easy to check that the strategies described above are a subgame perfect equilibrium. We follow Shaked and Sutton’s (1984) method to show uniqueness, i.e. we show that the infimum and the supremum of the set of equilibria payoffs coincide. For all t 5 0, 1, . . . , let S t be the set of perfect equilibrium payoffs of player pt at period t for the subgame where pt is the initiator of the bargaining procedure (subgame beginning at round t). Let St 5 sup S t and s t 5 inf S t , ;t 5 0, 1, . . . We establish the proof in four steps. Step 1. St # 1 2 dpt 11 s t11 , ;t 5 0, 1, . . . Proof. At round t 1 1, the proposer pt 11 can get at least s t 11 which corresponds to a discounted payoff of dpt 11 s t 11 at round t. Hence, at round t the proposer pt can not offer less than dpt 11 s t11 to her bargaining partner (respondent) pt 11 for the offer to be accepted. Therefore, at round t the proposer pt can not get more than a share of 1 2 dpt 11 s t 11 , that is St # 1 2 dpt 11 s t11 . Q.E.D. Step 2. s t $ 1 2 dpt 11 St11 , ;t 5 0, 1, . . .
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
52
Proof. Similar argument than in the previous step. Q.E.D. 21)21 Step 3. St # [1 / 1 2 v (0, 2(n 2 1))]f1 1 o 2(n (21)i v (t, t 1 i)g, ;t 5 0, 1, . . . i51
Proof. Let t 5 0, 1, . . . . We show by induction on k that:
O (21) Pd 1Pd
2k21
St # 1 1
i
2k
i
s51
i 51
p t 1s
p t 1s
s 51
3 St12k , ;k 5 1, 2, . . .
(4.1)
Combining the results of the two previous steps we get: St # 1 2 dpt 11s1 2 dpt 12 St12d⇔St # 1 2 dpt 11 1 dpt 11dpt 12 St 12 Hence, the result is true for k 5 1. Suppose then that it is true for some k $ 1. We show that it still holds at k 1 1. Indeed, combining again the results of Steps 1 and 2 we have: St12k # 1 2 dpt 12k 11 1 dpt 12k 11dpt 12(k 11 ) St12(k 11) We then obtain a new upper bound for St :
O (21) Pd 1Pd s1 2 d # 1 1 O (21) P d 1 P d 3 S 2k21
i
2k
i
St # 1 1
s51
i 51
2k11
p t 1s
p t 1s
s 51
p t 12k 11
1 dpt 12k 11dpt 12(k 11 ) St12(k 11)d⇔St
2(k 11 )
i
i
s51
i 51
p t 1s
s51
t12(k 11 )
p t 1s
Hence, if the inequality is true at k it still holds at k 1 1. Therefore, by induction, the inequality (4.1) holds for all integer k $ 1, and for all integer t $ 0. We know from the description of the bargaining procedure that for all t 5 0, 1, . . . , pt 12(r1l ) 5 pt . Indeed, after 2(l 1 r) stages, the r rounds of negotiations between player 0 and player 1 1 and the l rounds of negotiations between player 0 and player 2 1 have been completed, and a new tour of bilateral bargaining contests begins. Therefore, St12(r1l ) 5 St , ;t 5 0, 1, . . . Then, if we let k 5 r 1 l in (4.1) we obtain:
S P D
2(r 1l )21
2(r1l )
St 1 2
s51
dpt 1s # 1 1
O
Pd i
(21)
i
s51
i51
p t 1s
And with the notation introduced at the beginning of this section:
F O
2(r1l )21
St #
1 ]]]] 1 2 v (0, 2(r 1 l ))
11
i
G
(21) v (t, t 1 i) , ;t 5 0, 1, . . .
i 51
Q.E.D.
)21 Step 4. s t 5 St 5 [1 / 1 2 v (0, 2(r 1 l))]f1 1 o i2(r1l (21)i v (t, t 1 i)g, ;t 5 0, 1, . . . 51
Proof. With a similar argument as in the previous step we show that:
F O
2(r1l )21
st $
1 ]]]] 1 2 v (0,2(r 1 l ))
11
i51
i
G
(21) v (t, t 1 i) , ;t 5 0, 1, . . .
Q.E.D.
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
53
And the result follows immediately as, by definition, s t # St , ;t 5 0, 1, . . . .
h
Proof of Corollary 1. Let D11 5 d0d11 and D21 5 d0d21 . From Proposition 1 and after some algebra we get for r, l $ 1:
F
l
21
5
11
1 ]]] 1 2 Dl Dr 21
r
11
F
OD
OD
r21
1 a 0 (r,l) 5 ]]] S1 1 D 21 D 11D (1 2 d11 ) 1 2 D 2l D 2r
k 50
r
1 (1 2 d21 )D 11 11
k
k50
21
0
OD G
l 21
r21
(1 2 d11 )
O D G⇔a (r,l)
l21 k
k
k50
11
r
1 (1 2 d21 )D 11
k
21
k 50
Suppose that the peripheral players have the same time preferences. Let d 5 d11 5 d21 and D 5 D11 5 D21 5 d0d. Then,
SO
r21
a 0 (r, l) 5
12d ]] 1 2 D r 1l
O D D5
l21 k
D 1D
r
k 50
k
k50
O
r1l21 12d ]] 1 2 D r 1l
12d D k 5 ]] 12d d .
h
0
k 50
Proof of Corollary 2. The central player’s share a 0 (r, l) increases with r if and only if for all r, l $ 1, a 0 (r 1 1, l) . a 0 (r, l)
OD
l 21
⇔(1 2 d11 )D 11 2 (1 2 d21 )s1 2 D11dD 11 r
r
k 50
k 21
r
l
2 (1 2 d11 )D 11 D 21 . 0
⇔(1 2 d11 )s1 2 D21d 2 (1 2 d21 )s1 2 D11dS1 2 D l21D 2 (1 2 d11 )(1 2 d21 )D l21 . 0 ⇔S1 2 D l21Df(1 2 d11 )s1 2 D21d 2 (1 2 d21 )s1 2 D11dg . 0⇔d21 . d11 . Similarly, we prove that a 0 (r, l) decreases with l if and only if d21 . d11 .
h
Proof of Corollary 3. Assume that d11 , d21 . Corollary 2 implies that the highest payoff the central player can obtain is lim r →` a 0 (r, l), ;l $ 1. We have
21
11
F
OD
r 21
1 a 0 (r, l) 5 ]]] (1 2 d11 ) 1 2 Dl Dr
F
21
11
r
k 21
1 (1 2 d21 )D 11
k 50
1 ⇔a 0 (r, l) 5 ]]] (1 2 d11 ) 1 2 Dl Dr
OD G )D O D G l21
k 11
k50
l21
1 2 D r11 ]] 1 2 D 11
1 (1 2 d21
r 11
k 21
k50
From lim r→` D r11 5 0 we deduce that lim r→` a 0 (r, l) 5 (1 2 d11 / 1 2 d0d11 ), ;l $ 1.
h
Acknowledgements I am especially indebted to Andreu Mas-Colell. I thank Antonio Cabrales, Olivier Compte, Margarida Corominas, Nir Dagan, Sjaak Hurkens, Philippe Jehiel and the seminar participants at Universitat Pompeu Fabra (Barcelona) and LEI-CERAS (Paris). Financial support from the Spanish
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´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
Ministry of Education through research grant DGESIC PB96-0302, and from the Ecole Nationale des ´ Ponts et Chaussees, Paris is gratefully acknowledged. All errors are of course mine.
References Chae, S., Yang, J.A., 1994. An n-person pure bargaining game. Journal of Economic Theory 62, 86–102. Fudenberg, D., Levine, D.K., Tirole, J., 1987. Incomplete information bargaining with outside opportunities. Quarterly Journal of Economics 102, 37–51. Haller, H., 1986. Non-cooperative bargaining of N $ 3 players. Economics Letters 22, 11–13. Hendon, E., Tranæs, T., 1991. Sequential bargaining in a market with one seller and two different buyers. Games and Economic Behavior 3, 453–466. Herrero, M., 1985. A strategic bargaining approach to market institutions, University of London, Ph.D. thesis. Horn, H., Wolinsky, A., 1988. Worker substitutability and patterns of unionization. Economic Journal 98, 484–497. Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks. Journal of Economic Theory 71, 44–74. Jehiel, P., Moldovanu, B., 1995a. Cyclical delay in bargaining with externalities. Review of Economic Studies 62, 619–638. Jehiel, P., Moldovanu, B., 1995b. Negative externalities may cause delay in negotiation. Econometrica 63, 1321–1336. Jun, B.H., 1987. A structural consideration on a 3-person bargaining game, University of Pennsylvania, Ph.D. thesis. Jun, B.H., 1989. Non-cooperative bargaining and union formation. Review of Economic Studies 56, 59–76. Myerson, R., 1977. Graphs and cooperation in games. Mathematics of Operation Research 2, 225–229. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–110. Shaked, A., Sutton, J., 1984. Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52, 1351–1364. ˚ Stahl, I., 1972. Bargaining Theory, Economics Research Institute, Stockholm. Sutton, J., 1986. Non-cooperative bargaining theory: an introduction. Review of Economic Studies 53, 709–724.
A note on three-player noncooperative bargaining with restricted pairwise meetings a,b , * ´ Antoni Calvo-Armengol a
Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25 – 27, 08005 Barcelona, Spain b ` , 75007 Paris, France CERAS–ENPC, 28 rue des Saints-Peres Received 23 February 1999; accepted 26 May 1999
Abstract A three-player bargaining game where only the central player can negotiate with two different partners has a unique perfect equilibrium outcome. We discuss when the a priori favored position of the central player is really advantageous in terms of equilibrium payoffs. 1999 Elsevier Science S.A. All rights reserved. Keywords: Bargaining; Central player; Peripheral players; Impatience rates JEL classification: C78; D20
1. Introduction In many social and economic situations, agents are in direct contact with only a limited subset of other agents.1 When this is the case, bilateral volunteer meetings are limited to those pairs of agents that can communicate with each other directly (or that know each other directly). The aim of this paper is to explore the implications of the restrictions on feasible pairwise meetings in a simple bargaining context. The bargaining game we consider is a particular version of the ‘three-player / onecake’ game where players hold asymmetric positions. More explicitly, we model a negotiation process where a central player bargains for a given number of rounds with the same partner, and then switches over to another available partner for another negotiation phase with fixed length. If no bilateral agreement is still reached, the central player moves back to the first partner. And so on. Therefore, any (final) agreement between two partners must necessarily involve the central player. *Tel.: 134-93-542-2688; fax: 134-93-542-1746. ´ E-mail address: [email protected] (A. Calvo-Armengol) 1 Jackson and Wolinsky (1996) provide examples illustrating the importance of social and economic networks, and a survey of the related literature. 0165-1765 / 99 / $ – see front matter PII: S0165-1765( 99 )00136-6
1999 Elsevier Science S.A. All rights reserved.
48
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
This situation arises, for instance, when a firm requiring the services of one employer to produce one unit of profit faces two potential groups of workers, and a negotiation over the wages is engaged. The final agreement necessarily involves the firm (central player). Shaked and Sutton (1984) consider a similar employer–employee bargaining situation (where a firm bargains for wages with the operating workforce for a certain number of periods after which the firm is free to make an offer to an outsider) to study unemployment. Horn and Wolinsky (1988) and Jun (1989) investigate unionization in a bargaining model where the three players are the firm and two groups of workers of different types. An alternative situation fitting to the problem considered is when a seller owns an indivisible object to be sold to one of two (or more) potential buyers through a bargaining procedure. Related models are discussed by Fudenberg et al. (1987) in an incomplete information setting, by Hendon and Tranæs (1991) where the buyers have different valuations for the good, and by Jehiel and Moldovanu (1995a,b) who investigate the effect of externalities between buyers. Section 2 presents the game and states the main result of the paper: the noncooperative bargaining game with restricted pairwise meetings has a unique perfect equilibrium outcome. The corresponding equilibrium shares are fully characterized. The central player is the only one that can negotiate with two different partners, and is involved in any bilateral agreement of the game. We determine in Section 3 when this a priori favored position is really advantageous in terms of equilibrium payoffs, and provide the bargaining procedure ensuring the central player a higher outcome. Section 4 contains all the proofs.
2. The model
2.1. The game ˚ alternating offers Our bargaining game is simple in that it is an adaptation of the Rubinstein-Stahl game. In particular, no trade takes place and bargaining is bilateral: one player can only bargain with one partner at a time. We assume that the central player initiates the negotiations. Players are consequently labeled 0 (central player) and 61 (peripheral players). Without loss of generality, the game begins with a contest between the central player and the player on her right-hand side, player 1 1. Player 0 and player 1 1 then bargain for r $ 1 rounds, each round being composed of two different stages consisting of an offer from player 0 to player 1 1 and, in case of rejection, of a counter-offer from player 1 1 to player 0. Players discount stages with a discount rate di [s0, 1d, i [h 2 1, 0, 1 1j. If these two players haven’t still reached an agreement after r rounds of negotiations, player 0 then bargains for l $ 1 rounds with the player on her left-hand side, player 2 1. And so on. The game ends when, and if, an agreement is reached between the central player and one of the peripheral partners. In this game, pairwise meetings of players are restricted: the peripheral players can never make proposals one to the other, and the final agreement always involves the central player. In that sense, players hold asymmetric positions as it is the case, for instance, in a wage bargaining context where an employer faces two isolated potential workers. We can represent these restrictions on the feasible pairwise meetings (and, consequently, on the possible bilateral contests) by a graph where the nodes
´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
49
are identified with the players and a link between two nodes means that the corresponding players can bargain together.2 More formally, denote by ( pt , qt ) the pair of proposer and respondent at round t 5 0, 1, 2, . . . At t, player pt makes an offer to qt and the game terminates if the offer is accepted. If not, we go to the next round t 1 1 and qt becomes the new proposer, that is pt11 5 qt . Now it is pt11 ’s turn to make an offer to qt 11 5 pt 12 . And so on. The active bargaining pair of players at round t is then ( pt , pt11 ), with p0 5 0, p1 5 1 1, p2 5 0, . . . , p2r21 5 1 1, p2r 5 0, p2r11 5 2 1, p2r12 5 0, . . . , p2(r1l ) 5 0, p2(r1l )11 5 1 1, . . .
2.2. The main result This bargaining procedure defines a noncooperative sequential bargaining game that we analyze. We restrict attention to subgame perfect equilibria. We first introduce some useful notations. For any two periods of time t and s (t $ 0, s . 0), let v (t, t 1 s) be the product of the discount factors of the players that respond to offers from round t to round t 1 s 2 1 (or, equivalently, that make offers from round t 1 1 to round t 1 s):
Pd s
v (t, t 1 s) 5 dpt 11 3 ? ? ? 3 dpt 1s 5
i 51
p t 1i 2(r1l )21
i
Let also a t 5 [1 / 1 2 v (0, 2(r 1 l))]f1 1 o i51 (21) v (t, t 1 i)g, for all t 5 0, 1, . . . Finally, denote by D3 5hsx 1 , x 2 , x 3dux 1 1 x 2 1 x 3 5 1; x 1 , . . . , x 3 $ 0j the unit simplex of R 31 . The main result is: Proposition 1. The above game has a unique perfect equilibrium outcome supported by the following set of stationary strategies, where offers made at any stage correspond to the perfect equilibrium outcome and players agree to any offer of at least this amount: at round t, player pt proposes the share 1 2 a t to player pt 11 and accepts any offer x [ D3 from player pt 21 as long as x pt $ dpt a t .
˚ (1972) analyze a two-player sequential bargaining game Remark 1. Rubinstein (1982) and Stahl where players alternate in making proposals that has a unique subgame perfect equilibrium. Shaked (see Sutton, 1986) gives an example of a three-player bargaining game of which the perfect equilibrium is not unique. Herrero (1985) and Haller (1986) observe that games involving more than two bargainers display multiple equilibria. Jun (1987) recovers uniqueness with three players by allowing them to sign contracts. This result is lately extended to the n-player case by Chae and Yang ˚ (1994). Uniqueness also holds in our three-player Rubinstein-Stahl-type bargaining model where bilateral agreements always involve the central player. Remark 2. We can easily check that 1 2 a t 5 dpt 11 a t 11 , ;t 5 0, 1, . . . These equations correspond to ˚ solution where, at equilibrium, players are indifferent between their the standard Rubinstein-Stahl share as a respondent and their share as a delayed proposer. Proposition 1 means in particular that 2 We refer the reader to Myerson (1977) for a first representation of the restricted communication possibilities by an undirected graph.
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´ / Economics Letters 65 (1999) 47 – 54 A. Calvo-Armengol
the bargaining outcome corresponding to the above game is predictable: at t 5 0 the central player proposes a share 1 2 a 0 to player 1 1, that immediately accepts this offer in which the left-hand side player 2 1 ends up with no share of the cake. Remark 3. It is straightforward to show that the uniqueness of the perfect equilibrium outcome holds in a more general setting where a central player bargains successively with N players labeled from 1 to n with discount factors d1 , . . . ,dn [s0, 1d (n $ 2) for, respectively, r 1 , . . . ,r n $ 1 rounds, each round consisting of an offer and a counter-offer stage. The central player’s share of the cake at the unique perfect equilibrium of this game is a 0 5 [1 / 1 2 v (0, 2R)]f1 1 o 2R21 (21)i v (t, t 1 i)g, where i 51 n R 5 o i51 r i . Similarly, in the three-player case, we can allow for a sequential bargaining procedure consisting of (successively) r 1 rounds with player 1 1, l 1 rounds with player 2 1, . . . ,r p rounds with player 1 1, l p rounds with player 2 1, followed by a similar phase of negotiations, and so on. The central player’s share of the cake at the unique perfect equilibrium of this game is a 0 5 [1 / 1 2 v (0, )21 p 2(R 1 L))]f1 1 o 2(R1L (21)i v (t, t 1 i)g, where R 5 o ip51 r i and L 5 o i51 li . i 51
3. Comparative statics The central player is the only one that can negotiate with two different partners. Moreover, our game requires that any bilateral agreement of the game involves player 0. We shall then ask whether the central player derives any advantage from her a priori favored position, and if so, what bargaining procedure (i.e. what length r and l of the negotiation rounds) ensures her a higher payoff. Corollaries 1–3 answer these questions. Corollary 1. If the two peripheral players have a common discount factor d, at the unique perfect equilibrium of the above game the central player’s share is a 0 5 (1 2 d / 1 2 d0d ). This share coincides with the standard agreement in the two-player case. Hence, when the peripheral players have identical time preferences, the central player does not derive any advantage from his particular location. In fact, abandoning the current partner after a fixed round of negotiations has no effect. Indeed, the new partner has exactly the same time preferences as the previous one. As for the forces that drive the negotiation agreement (time preferences and credible outside options), everything is as if the central player hadn’t changed her partner. In other words, the central player faces an identical situation irrespective of the peripheral player she is bargaining with. Denote by a 0 (r, l) the central player’s share of the cake at the unique perfect equilibrium outcome where player 0 bargains successively with player 1 1 and with player 2 1, respectively for r rounds and l rounds. Corollary 2. At the unique perfect equilibrium of the above game, the central player’s share of the cake increases with r and decreases with l if and only if d11 , d21 . In words, the longer the negotiation phase with the relatively more impatient partner, the higher the outcome of the bargaining game for the central player. Symmetrically, the longer the negotiation phase with the relatively more patient neighbor, the lower the central player’s equilibrium payoff. In
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fact, when bargaining with the relatively more patient neighbor, the central player can effectively threaten him to abandon the current negotiations and turn to the outside partner, that is relatively more impatient and thus more eager to accept tougher proposals. Corollary 2 just states that the sooner the relatively more impatient neighbor is effectively introduced into the bargaining process (the sooner the central player ‘opts out’), the better it is for the central player (the higher the central player’s share of the cake at the unique perfect equilibrium outcome). In particular, it is harmful for the central to delay the effective 3-sided bargaining game when her current partner is the relatively more patient peripheral player. Corollary 3. Assume that d11 , d21 . The highest payoff the central player can obtain is lim r→` a 0 (r, l) 5 (1 2 d11 / 1 2 d0d11 ), ;l $ 1. This share coincides with the unique perfect equilibrium outcome of a standard two-player infinite-horizon alternating-offers game between the central player and the relatively more impatient partner, player 1 1. Hence, if we allow the central player to choose the type of the game that is played, the best thing to choose (that is, the game ensuring her the highest equilibrium outcome) is to bargain bilaterally with the relatively more impatient peripheral player, and ignore the other potential partner. In that case, negotiating with many partners is harmful. Consider the standard two-player alternating offers game between two players, labeled 1 and 2. If player 1 is the initiator of the bargaining game, at the unique perfect equilibrium of the game she gets a share a 1 5 (1 2 d2 / 1 2 d1d2 ) of the cake. This payoff satisfies (≠a 1 / ≠d2 ) 5 2 [1 2 d2 /s1 2 d1d2d 2 ] , 0. Hence, in the two-player game, the more one’s rival is impatient (the smaller d2 in this case), the bigger the share obtained. Therefore, if player 1 can choose her bargaining partner among a set of potential partners, she selects the relatively more impatient one. Corollary 3 recovers this result for the three-player game with restricted pairwise meetings.
4. Proofs Proof of Proposition 1. It is easy to check that the strategies described above are a subgame perfect equilibrium. We follow Shaked and Sutton’s (1984) method to show uniqueness, i.e. we show that the infimum and the supremum of the set of equilibria payoffs coincide. For all t 5 0, 1, . . . , let S t be the set of perfect equilibrium payoffs of player pt at period t for the subgame where pt is the initiator of the bargaining procedure (subgame beginning at round t). Let St 5 sup S t and s t 5 inf S t , ;t 5 0, 1, . . . We establish the proof in four steps. Step 1. St # 1 2 dpt 11 s t11 , ;t 5 0, 1, . . . Proof. At round t 1 1, the proposer pt 11 can get at least s t 11 which corresponds to a discounted payoff of dpt 11 s t 11 at round t. Hence, at round t the proposer pt can not offer less than dpt 11 s t11 to her bargaining partner (respondent) pt 11 for the offer to be accepted. Therefore, at round t the proposer pt can not get more than a share of 1 2 dpt 11 s t 11 , that is St # 1 2 dpt 11 s t11 . Q.E.D. Step 2. s t $ 1 2 dpt 11 St11 , ;t 5 0, 1, . . .
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Proof. Similar argument than in the previous step. Q.E.D. 21)21 Step 3. St # [1 / 1 2 v (0, 2(n 2 1))]f1 1 o 2(n (21)i v (t, t 1 i)g, ;t 5 0, 1, . . . i51
Proof. Let t 5 0, 1, . . . . We show by induction on k that:
O (21) Pd 1Pd
2k21
St # 1 1
i
2k
i
s51
i 51
p t 1s
p t 1s
s 51
3 St12k , ;k 5 1, 2, . . .
(4.1)
Combining the results of the two previous steps we get: St # 1 2 dpt 11s1 2 dpt 12 St12d⇔St # 1 2 dpt 11 1 dpt 11dpt 12 St 12 Hence, the result is true for k 5 1. Suppose then that it is true for some k $ 1. We show that it still holds at k 1 1. Indeed, combining again the results of Steps 1 and 2 we have: St12k # 1 2 dpt 12k 11 1 dpt 12k 11dpt 12(k 11 ) St12(k 11) We then obtain a new upper bound for St :
O (21) Pd 1Pd s1 2 d # 1 1 O (21) P d 1 P d 3 S 2k21
i
2k
i
St # 1 1
s51
i 51
2k11
p t 1s
p t 1s
s 51
p t 12k 11
1 dpt 12k 11dpt 12(k 11 ) St12(k 11)d⇔St
2(k 11 )
i
i
s51
i 51
p t 1s
s51
t12(k 11 )
p t 1s
Hence, if the inequality is true at k it still holds at k 1 1. Therefore, by induction, the inequality (4.1) holds for all integer k $ 1, and for all integer t $ 0. We know from the description of the bargaining procedure that for all t 5 0, 1, . . . , pt 12(r1l ) 5 pt . Indeed, after 2(l 1 r) stages, the r rounds of negotiations between player 0 and player 1 1 and the l rounds of negotiations between player 0 and player 2 1 have been completed, and a new tour of bilateral bargaining contests begins. Therefore, St12(r1l ) 5 St , ;t 5 0, 1, . . . Then, if we let k 5 r 1 l in (4.1) we obtain:
S P D
2(r 1l )21
2(r1l )
St 1 2
s51
dpt 1s # 1 1
O
Pd i
(21)
i
s51
i51
p t 1s
And with the notation introduced at the beginning of this section:
F O
2(r1l )21
St #
1 ]]]] 1 2 v (0, 2(r 1 l ))
11
i
G
(21) v (t, t 1 i) , ;t 5 0, 1, . . .
i 51
Q.E.D.
)21 Step 4. s t 5 St 5 [1 / 1 2 v (0, 2(r 1 l))]f1 1 o i2(r1l (21)i v (t, t 1 i)g, ;t 5 0, 1, . . . 51
Proof. With a similar argument as in the previous step we show that:
F O
2(r1l )21
st $
1 ]]]] 1 2 v (0,2(r 1 l ))
11
i51
i
G
(21) v (t, t 1 i) , ;t 5 0, 1, . . .
Q.E.D.
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And the result follows immediately as, by definition, s t # St , ;t 5 0, 1, . . . .
h
Proof of Corollary 1. Let D11 5 d0d11 and D21 5 d0d21 . From Proposition 1 and after some algebra we get for r, l $ 1:
F
l
21
5
11
1 ]]] 1 2 Dl Dr 21
r
11
F
OD
OD
r21
1 a 0 (r,l) 5 ]]] S1 1 D 21 D 11D (1 2 d11 ) 1 2 D 2l D 2r
k 50
r
1 (1 2 d21 )D 11 11
k
k50
21
0
OD G
l 21
r21
(1 2 d11 )
O D G⇔a (r,l)
l21 k
k
k50
11
r
1 (1 2 d21 )D 11
k
21
k 50
Suppose that the peripheral players have the same time preferences. Let d 5 d11 5 d21 and D 5 D11 5 D21 5 d0d. Then,
SO
r21
a 0 (r, l) 5
12d ]] 1 2 D r 1l
O D D5
l21 k
D 1D
r
k 50
k
k50
O
r1l21 12d ]] 1 2 D r 1l
12d D k 5 ]] 12d d .
h
0
k 50
Proof of Corollary 2. The central player’s share a 0 (r, l) increases with r if and only if for all r, l $ 1, a 0 (r 1 1, l) . a 0 (r, l)
OD
l 21
⇔(1 2 d11 )D 11 2 (1 2 d21 )s1 2 D11dD 11 r
r
k 50
k 21
r
l
2 (1 2 d11 )D 11 D 21 . 0
⇔(1 2 d11 )s1 2 D21d 2 (1 2 d21 )s1 2 D11dS1 2 D l21D 2 (1 2 d11 )(1 2 d21 )D l21 . 0 ⇔S1 2 D l21Df(1 2 d11 )s1 2 D21d 2 (1 2 d21 )s1 2 D11dg . 0⇔d21 . d11 . Similarly, we prove that a 0 (r, l) decreases with l if and only if d21 . d11 .
h
Proof of Corollary 3. Assume that d11 , d21 . Corollary 2 implies that the highest payoff the central player can obtain is lim r →` a 0 (r, l), ;l $ 1. We have
21
11
F
OD
r 21
1 a 0 (r, l) 5 ]]] (1 2 d11 ) 1 2 Dl Dr
F
21
11
r
k 21
1 (1 2 d21 )D 11
k 50
1 ⇔a 0 (r, l) 5 ]]] (1 2 d11 ) 1 2 Dl Dr
OD G )D O D G l21
k 11
k50
l21
1 2 D r11 ]] 1 2 D 11
1 (1 2 d21
r 11
k 21
k50
From lim r→` D r11 5 0 we deduce that lim r→` a 0 (r, l) 5 (1 2 d11 / 1 2 d0d11 ), ;l $ 1.
h
Acknowledgements I am especially indebted to Andreu Mas-Colell. I thank Antonio Cabrales, Olivier Compte, Margarida Corominas, Nir Dagan, Sjaak Hurkens, Philippe Jehiel and the seminar participants at Universitat Pompeu Fabra (Barcelona) and LEI-CERAS (Paris). Financial support from the Spanish
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Ministry of Education through research grant DGESIC PB96-0302, and from the Ecole Nationale des ´ Ponts et Chaussees, Paris is gratefully acknowledged. All errors are of course mine.
References Chae, S., Yang, J.A., 1994. An n-person pure bargaining game. Journal of Economic Theory 62, 86–102. Fudenberg, D., Levine, D.K., Tirole, J., 1987. Incomplete information bargaining with outside opportunities. Quarterly Journal of Economics 102, 37–51. Haller, H., 1986. Non-cooperative bargaining of N $ 3 players. Economics Letters 22, 11–13. Hendon, E., Tranæs, T., 1991. Sequential bargaining in a market with one seller and two different buyers. Games and Economic Behavior 3, 453–466. Herrero, M., 1985. A strategic bargaining approach to market institutions, University of London, Ph.D. thesis. Horn, H., Wolinsky, A., 1988. Worker substitutability and patterns of unionization. Economic Journal 98, 484–497. Jackson, M.O., Wolinsky, A., 1996. A strategic model of social and economic networks. Journal of Economic Theory 71, 44–74. Jehiel, P., Moldovanu, B., 1995a. Cyclical delay in bargaining with externalities. Review of Economic Studies 62, 619–638. Jehiel, P., Moldovanu, B., 1995b. Negative externalities may cause delay in negotiation. Econometrica 63, 1321–1336. Jun, B.H., 1987. A structural consideration on a 3-person bargaining game, University of Pennsylvania, Ph.D. thesis. Jun, B.H., 1989. Non-cooperative bargaining and union formation. Review of Economic Studies 56, 59–76. Myerson, R., 1977. Graphs and cooperation in games. Mathematics of Operation Research 2, 225–229. Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–110. Shaked, A., Sutton, J., 1984. Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52, 1351–1364. ˚ Stahl, I., 1972. Bargaining Theory, Economics Research Institute, Stockholm. Sutton, J., 1986. Non-cooperative bargaining theory: an introduction. Review of Economic Studies 53, 709–724.