Abstract
Two players bargain to select a utility allocation in some set \(X\subset {\mathbb {R}}_+^2\). Bargaining takes place in infinite discrete time, where each period t is divided into two sub-periods. In the first sub-period, the players play a simultaneous-move game to determine that period’s proposer, and bargaining takes place in the second sub-period. Rejection triggers a one-period delay and move to \(t+1\). For every \(x\in X\cap {\mathbb {R}}^2_{++}\), there exists a cutoff \(\delta (x)<1\), such that if at least one player has a discount factor above \(\delta (x)\), then for every \(y\in X\) that satisfies \(y\ge x\) there exists a subgame perfect equilibrium with immediate agreement on y. The equilibrium is supported by “dictatorial threats.” These threats can be dispensed with if X is the unit simplex and the target-vector is Pareto efficient. The results can be modified in a way that allows for arbitrarily long delays in equilibrium.
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Notes
If no proposal is ever accepted, the game results in perpetual disagreement.
A player is allowed to mix between these messages.
In the terminology of Abreu (1988), the dictatorial equilibria are penal codes.
The idea that rejection of an offer leads to unfavorable continuation play, and this what sustains equilibrium-multiplicity, is at the heart of the literature on bargaining with “money burning.” “Money burning” is a threat by a player to destroy surplus in case his offer is rejected, which increases this player’s bargaining power if it is credible (see Avery and Zemsky 1994; Fernandez and Glazer 1991; Haller and Holden 1990). The rationale of my folk theorem is similar to the money burning logic, but not identical to it, because the threat of a dictatorial equilibrium is different from that of surplus destruction: the former favors one player at the expense of the other, the latter hurts both players.
If no one has made a proposal yet (as is the case, in particular, in the beginning of the game), then player 1 is defined to be the last responder.
The derivation of alternating-offers is also the subject of a work by McKelvey and Palfrey (1997), who study a war of attrition model in which an “alternating-offer type” equilibrium is derived.
Vector inequalities: uRv if and only if \(u_iRv_i\) for both i, for \(R\in \{>,\ge \}\); \(u\gneqq v\) if and only if \(u\ge v\) and \(u\ne v\).
The existence of these f and g does not follows from the assumption of a strictly decreasing frontier, hence it needs to be assumed. For example, consider the set \(X=\{(1,0),(0,1),(\frac{1}{2},\frac{1}{2}),(\frac{1}{4},\frac{1}{4})\}\). The payoff \(\frac{1}{4}\) is a feasible payoff for player 1, but there is no \(f(\frac{1}{4})\) such that \((\frac{1}{4},f(\frac{1}{4}))\in \partial X\). I owe this observation to a helpful referee.
Whenever players i and j are mentioned in the same sentence, it is assumed that \(i\ne j\).
Trusting that no confusion will arise, I skip a fully-specified formalism for \(\sigma _i^{B}\). This part of the strategy, like in Rubinstein’s game, describes a player’s proposals when he is the proposer, and accept/reject decision when he is the responder. Both the proposals and the accept/reject decisions are pure.
More formally, let \(A=\{(a_1,a_2)\in X: a_i\ge \delta _ix_i^{*i}\}\). Player j selects the point of A that maximizes his own payoff. This point is unique, because X’s Pareto frontier is strictly decreasing. To see this, assume by contradiction, w.l.o.g, that player 2 is the non-dictator and that there are two distinct points of A, \(a=(a_1,a_2)\) and \(b=(b_1,b_2)\), that maximize his payoff. In particular, \(a_2=b_2\). Since \(a\ne b\), assume w.l.o.g that \(a_1<b_1\). Since \(a\lneqq b\) and the frontier is strictly decreasing, it follows that there exists a \(c\in X\) with \(c>a\), in contradiction to the optimality of a.
In the dictatorial equilibrium D(i) player i may find himself in the position of the proposer also off the path; this can happen if \((m_1,m_2)=(you,you)\).
Namely the treatment here is the same as in case 1.
In short, case 4 is redundant, in the sense that no matter what scenario materializes given case 4, the specification of continuation play is as described in one of cases 1–3.
In case (II) one can take the dictator to be either the proposer or the responder, or, alternatively, the dictator can be a pre-determined player i; this choice does not matter, as any dictator-selection can be made part of an equilibrium.
Rubinstein’s model corresponds to \(\omega =(1,2,1,2,1,\cdots )\).
Mao allows \(\omega \) to be either finite or infinite. There is no loss of generality in assuming that it is infinite, since every finite procedure of length T in which i is the last proposer can be replaced by an infinite one that coincides with the finite procedure in the first T elements, and in which i is the proposer in all periods from T onwards; practically, there is no difference between the two.
There are three possibilities given which the above applies: it may be that player 2 is the sole deviator (he rejected the equilibrium offer), it may be that player 1 is the sole deviator (he made a non-equilibrium offer, which player 2 justifiably rejected), and it may also be that both players deviated.
Note that the entire proof focuses on an arbitrary period k. The proof’s logic lies in the fact that the equilibrium e is mimicked in the game G, and all the arguments involved in this mimicking do not depend on whether the stage game in question (i.e., the period-k game) is on or off the path; more broadly, it does not matter what happened in periods before k.
If both players deviated simultaneously, let \(\epsilon \) be the payoff of player 1.
For example, Britz et al. (2010) study a general bargaining model and show that all stationary subgame perfect equilibrium payoffs converge to those of a certain asymmetric Nash solution as the probability of negotiation breakdown vanishes.
One can show that in the present model, if the feasible set is convex and comprehensive, then the set of stationary equilibria is degenerate. Roughly, not much more can be achieved beyond the dictatorial equilibria and disagreement equilibria (comprehensiveness of X means that it contains any \(y\in {\mathbb {R}}_+^2\) that satisfies \(y\le x\) for some \(x\in X\)).
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Acknowledgements
I am thankful for Emin Karagözoğlu’s helpful comments, and for two exceptionally effective referee reports.
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Rachmilevitch, S. Folk theorems in a bargaining game with endogenous protocol. Theory Decis 86, 389–399 (2019). https://doi.org/10.1007/s11238-019-09688-6
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DOI: https://doi.org/10.1007/s11238-019-09688-6