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Bargaining with asymmetric threat points
*
Margarida Corominas-Bosch
Department of Economics, Universitat Pompeu Fabra, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain Received 12 July 1999; accepted 25 April 2000
Abstract
The 2-person version of Hart and Mas-Colell [Econometrica 64 (1996) 357] is studied, reinterpreted as a bargaining game with asymmetric threat points. Multiplicity of stationary subgame perfect equilibria is necessary for multiplicity of subgame perfect equilibria, and depends on risk aversion and the point constructed with the payoff that responders get if breakdown occurs. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Bargaining; Threat points; Delay in reaching agreement
JEL classification: C78
1. Introduction
In Binmore et al. (1986) the bargaining model of Rubinstein (1982) is modified by incorporating players that do not discount payoffs but face a probability of breakdown after every rejection. If breakdown happens, both players get a payoff of zero. In the present paper a game in which threat points are different across agents is constructed. In the event of breakdown, agents do not get both the worst possible outcome (that is, zero) but some fixed exogenous amount that might depend upon which of the two players rejected. We regard these exogenously fixed threat points as some sort of outside opportunities available to the agents. With this view there is no particular reason for the threat point to be independent of the proposer or rejector.
The set of subgame perfect equilibria payoffs of the game is characterized and it is shown that uniqueness of stationary subgame perfect equilibria implies uniqueness of subgame perfect equilibria. We also show that we need risk averse agents and develop on the relative position of the two threat points in order to support multiplicity.
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This paper was first presented in January 1997 in the Micro Workshop at UPF. *Tel.:134-93-542-2670; fax: 134-93-542-1746.
E-mail address: [email protected] (M. Corominas-Bosch).
Although this model shares common features with Merlo and Wilson (1995) and, with adequate reformulation, Houba (1997, section 4.1), our simpler set-up allows a very transparent analysis of the role that risk aversion and asymmetric threats play in generating multiplicity in 2 person perfect information models.
2. The model
1
]
At the beginning of the game, a proposer is chosen among players 1 and 2 with probability 2. The chosen proposer makes a proposal [S (the set of feasible payoffs), that the other player accepts (in
which case the game finishes) or rejects. In the latter case, with probability r the game starts again by choosing a random proposer and with probability 12r the game ends with the following payoffs: a if player 2 was the responder and b if player 1 was the responder, where both a5(a , a ) and b1 2 5(b ,1
b ) belong to S.2
Unlike Binmore et al. (1986), these exogenously fixed threat points may depend on who rejected the offer (a±b).
2
The set S,R1 is assumed to be closed, convex and comprehensive. The concave, strictly decreasing function defining the Pareto frontier of S is called f.
We now turn to the study of subgame perfect equilibrium (subsequently, PE) and subgame perfect equilibrium payoffs (PEP), starting from the stationary ones (referred as SPE and SPEP).
3. Stationary equilibria
The proposition below characterizes the SPE, i.e., the PE where players always use the same rule for making and accepting proposals. Call x5(x , x ) player 1’s proposal whenever he is the proposer1 2 and y5( y , y ) player 2’s. Observe that the proposals x and y are efficient and leave the responder1 2
indifferent between accepting or rejecting. The proof is straightforward and is therefore omitted (see for instance Osborne and Rubinstein (1990), chapter 3, section 8).
Proposition 1. The SPE are given by:
player 1: offers (x , x ) and rejects any offer in which player 1’s payoff is smaller than y1 2 1 player 2: offers ( y , y ) and rejects any offer in which player 2’s payoff is smaller than x , where:1 2 2
1 1
S
]D
S
]D
x25r (x21y )2 1(12r)a ,2 y15r (x11y )1 1(12r)b ,1 x25f(x ),1
2 2
y25f( y )1 (1)
~
It follows from (1) that only the amount that responders will get (a and b ) is relevant. Let us for2 1
convenience define the point r5(r , r )1 2 5(b , a ), which can be feasible (when r and r are small,1 2 1 2 that is, the costs of disagreement are quite severe) or can lie outside of the feasible set (at least one of the r is large). The location of r is going to be crucial for the whole analysis.i
( y12r )( y1 22r )2 5(x12r )(x1 22r )2 r
]]
y12r15 (x12r ),1 x25f(x ),1 y25f( y )1
22r
That is, x and y are efficient, they lie in the same hyperbola centered in r, and the points x12r and1 y12r are in the proportion 1 to1 r/ 22r. Using this geometrical characterization it can easily be shown that when r is feasible there is uniqueness of stationary PE (see Houba (1993) for details).
Corollary 1. If r is feasible, the SPE is unique.
If r is not feasible, multiplicity of SPEP may arise provided that the Pareto frontier of the set S is not linear, that is, when agents are risk averse. Note that if r is not feasible proposers offer more to their opponent than what the responders would demand for themselves (x1,y , y1 2,x ), since when2
rejecting, a responder may get with probability 12r the amount r which is relatively high. We nowi give an example in which there exists multiplicity of SPEP. Consider a hyperbola h(x) centered in r,
] ]
Œ Œ
h(x)5k /x2r11r with k2 .0 being such that r1. k, r2. k. Now, choose four points a5(a ,1
a ), b2 5(b , b ), c1 2 5(c , c ) and d1 2 5(d , d ), all in the positive orthant and lying in h(x), such that:1 2
a1,r ,1 b1,r ,1 c1,r ,1 d1,r1
r r r
]](a12r )1 5b12r ,1 ]](b12r )1 5c12r ,1 ]](c12r )1 5d12r1
22r 22r 22r
Call s the line that goes through a and b and s the line that goes through c and d. Define the frontier1 2 of the feasible set S as being equal to s until the point in which it intersects with s and let it coincide1 2
2
with s from thereafter. We then define S as the points of R2 1 under this Pareto frontier. It is easy to check that S satisfies our assumptions and that there are 3 SPEP given by a1b / 2 (with player 1
proposing a and player 2 proposing b), b1c / 2 (with player 1 proposing b and player 2 proposing c),
and c1d / 2 (with player 1 proposing c and player 2 proposing d ). Indeed, all these proposals are
efficient, lie in a hyperbola centered in r and are in the right proportions (i.e., fulfill Eq. (1)). The set can be smoothed by chosing a polynomial between c and d that would still fulfill our assumptions.
4. Non stationary equilibria
We now proceed to characterize the set of non stationary equilibria. We will first define E, a set containing all PEP of the game. Next we will verify that the extreme points of the Pareto frontier of this set are supportable as SPEP. Then our main result will follow: multiplicity of SPEP is necessary for multiplicity of PEP. Finally, we will show that the set E contains all PEP and only PEP.
4.1. Characterization of the set of PEP
9
subgame starting at time t11 (in expected terms from period t’s view) satisfies m #z #M
2 2 2
12r players get (0, r ) whereas with probability2 r the game starts again. Therefore, the set denoted
1 1
We can then recursively construct, by taking into account that all PEP at period n are contained in
S:
Note that x and x
s
y and yd
are the upper and lower extremes respectively of the Pareton n
The limits will exist since A and A are compact sets. We are now ready to characterize the set of1 2
PEP.
1 2 2
n
Now, denote by z and z the two extreme points of the Pareto frontier of E . Similarly, denote
n
9
n n n9
n9
nlimn→` z 5z, limn→` z 5z9, limn→` x 5x, limn→` y 5y, limn→` x 5x9, limn→` y 5y9, so that z and z9, x and x9, y and y9, are the two extreme points of the Pareto frontier of E, A and A1 2 respectively.
We now show that z and z9 can be supported as SPEP. This implies the main result of this paper, namely, that multiplicity of SPEP is necessary for multiplicity of PEP to exist.
Theorem 1. If there exists a unique stationary PEP, then we can assert that:
(a) There exists a unique PEP.
2
(b) Furthermore, if S is not a singleton , there exists a unique PE (i.e., the unique PEP can be
supported with uniquely determined strategies).
Proof. (a) Suppose there existed two PEP. This implies that z±z9. Taking limits to (2), we know that
1 1
Since the equations above coincide with (1), z and z9 can be supported as SPEP.
(b) Denote by z the unique PEP. First we argue that in any PE strategy profile supporting z, offers are immediately accepted. Suppose w.l.o.g. that we are in period 0 and in G and assume to the1 contrary that player 2 rejects the offer. If there is not breakdown, the payoff players expect to get in period 1 after player 2’s rejection is rz1(12r)(0, r )2 5(rz , x ). Since the PEP at period 0 equals1 2
Since PE proposals are always accepted, it follows that the PE strategies must be the stationary ones. h
We have seen in the previous proof that the two extreme points of the Pareto frontier of E are SPEP. This allows us to conclude directly that in the case in which the disagreement point r is feasible, there exists only one PE of the game, since the stationary PE is unique.
2
Corollary 2. If r is feasible, there exists only one PE of the game.
In the case in which r is not feasible, the set of PEP may not be a singleton. We will now use the
3
extreme SPEP to support all the other points of E as PEP.
Proposition 2. A point in S is a PEP ⇔ it belongs to E.
Proof. ⇒) see Lemma 2.
⇐) The two points z and z9 can be supported through the stationary strategies. Now, consider the
] ] ] ] ] ]1] ]
points z [E with z[⁄ hz, z9j. There exist x[A and y1 [A such that z2 52sx1y . Consider thed
following strategies:
]
At period 0, player 1 proposes x and accepts a proposal by player 2 iff his payoff is greater or equal
] ]
than y . Similarly, at period 0, player 2 proposes y and accepts a proposal by player 2 iff his payoff is1
]
greater or equal than x .2
From period 1 onwards, following deviations that happen after 1 (2) has proposed at period 0, strategies prescribe to play the SPEP z (z9) from there after if 2 (1) has not deviated in his response, and z9 (z) if 2 (1) has deviated in his response.
Observe that z (z9) is the worst (best) PEP payoff 1 can expect. In our strategies z is used as a punishment for 1 (it is implemented if player 1 deviates as a proposer trying to get a higher payoff) and z9 is used as a reward. The roles get reversed for player 2. h
4.2. Properties of the set of PEP
We now remark on some of the properties of the set E of PEP. Note first of all that from its construction E is a convex and compact set limited by a vertical segment on the left and a horizontal segment on the bottom that intersect forming an orthogonal angle, and a Pareto frontier given by a strictly decreasing and concave function. Note also that in order to find the set E it is not necessary to compute the limit of the sequence of E . Indeed, by solving (1) we can find all the SPEP and throughn them determine x, y, x9, y9 which directly define E.
Interestingly, from Proposition 2 it follows that if there is multiplicity some of the PEP are inefficient because proposals that are not in the Pareto frontier are accepted. Moreover, even points which Pareto dominate z and z9 (the extremes of E ) can not be supported in equilibrium. Note that also the stationary PEP are inefficient since they are the average of two points lying on the frontier of a strictly convex set.
Finally, note that asr approaches 1, if r is feasible the unique PEP approaches the Nash solution of
S with disagreement point equal to r. This is easy to see in our case: asr tends to 1, the two points x and y defining z get closer and keep lying in a hyperbola centered in r, and in the limit they coincide
*
*
with the efficient point z* that maximizes the product (z1 2r )(z1 2 2r ). If r is not feasible, in the2
limit the (possibly multiple) SPEP coincide with the efficient points z* that maximize the product
*
*
(r12z )(r1 22z ), which constitute the ‘reinterpreted kernel’ or ‘Nash set’ studied in Serrano (1997).2
3
5. Final remarks
We would like to remark that SPEP are always achieved through efficient proposals, but many of the non stationary PEP are achieved through non efficient proposals. Agents accept those proposals immediately even if they are dominated by other payoffs in the feasible set. Therefore, this model explains inefficiency without using any sort of imperfect information or bounded rationality. The introduction of large payoffs received by the responder in the event of breakdown together with risk aversion can lead to accepting non efficient outcomes.
Hart and Mas-Colell (1996) study the stationary PEP for a bargaining game in which feasible payoffs at each stage are given by a cooperative game (N, V ) depending on what is the current set of proposers. The analysis carried out here studies the set of all PEP for the case in which the cooperative game (N, V ) is NTU, which in this context means that the feasible set is convex and that r may lie above the Pareto frontier. Krishna and Serrano (1995) study the set of all PEP associated with the TU case. The property of multiplicity of SPEP needed for there to exist multiplicity of PEP, that we have shown to hold for the general NTU case with 2 players, does not hold any longer for N.2.
Acknowledgements
I wish to thank my thesis advisor Andreu Mas-Colell and Clara Ponsati for their constant guidance and support. I also benefited from very helpful conversations with Sergiu Hart.
References
Binmore, K.G., 1987. The Economics of Bargaining, Binmore, K.G., Dasgupta, P., (Eds.). Blackwell, Oxford, Cambridge. Binmore, K.G., Rubinstein, A., Wolinsky, A., 1986. The Nash bargaining solution in economic modelling. Rand Journal of
Economics 17, 176–188.
Hart, S., Mas-Colell, A., 1996. Bargaining and value. Econometrica 64, 357–380.
Houba, H., 1997. The policy bargaining model. Journal of Mathematical Economics 28, 1–27.
Houba, H., 1993. An alternative proof of uniqueness in non-cooperative bargaining. Economic Letters 41, 253–256. Krishna, V., Serrano, R., 1995. Perfect equilibria of a model of N-person noncooperative bargaining. International Journal of
Game Theory 24, 259–272.
Merlo, A., Wilson, C., 1995. A stochastic model of sequential bargaining with complete information. Econometrica 63, 371–399.
