A Non­cooperative Solution to the Bargaining Problem

2021

In this thesis we discuss some novel concepts of stability in bargaining games, over a network setting. So far, the studies on bargaining games were done as profit sharing problems, whose underlying combinatorial optimization problems are of packing type. In our work, we study bargaining games from a cost sharing perspective, where the underlying combinatorial optimization problems are covering type problems. Unlike previous studies, where bargaining processes are restricted to only two players, we study bargaining games over a more generic hypergraph setting, which allows any bargaining process to be formed among any number of players. In previous studies of bargaining games, the objects that are being negotiated are assumed to be uniform and only the outcomes of the negotiations are allowed to be different. However, in our study, we accommodate possibilities of non-uniform weights of the objects that are being negotiated, which is closer to any real life scenario. Finally we exten...

International Journal of Game Theory, 2007

We consider an n-player bargaining problem where the utility possibility set is compact, convex, and stricly comprehensive. We show that a stationary subgame perfect Nash equilibrium exists, and that, if the Pareto surface is differentiable, all such equilibria converge to the Nash bargaining solution as the length of a time period between offers goes to zero. Without the differentiability assumption, convergence need not hold.

Annals of Operations Research, 2012

We extend well-known results for convex games (Maschler et al., 1972) to a more general class of games named almost-convex games, that is, games where all proper subgames are convex. We prove that for any balanced almost-convex game the bargaining set coincides with the core. We also prove that the kernel of any zero-monotonic almost-convex game reduces to the nucleolus. In contrast to the above results, the core of a balanced almostconvex game is stable in the sense of von Neumann-Morgenstern if and only if the game is convex.

2012

International Journal of Game Theory

Within the class of zero-monotonic and grand coalition superadditive 4 cooperative games with transferable utility, the convexity of a game is character-5 ized by the coincidence of its core and the steady bargaining set. As a consequence 6 it is proved that convexity can also be characterized by the coincidence of the core 7 of a game and the modified Zhou bargaining setà la Shimomura. 8 Keywords cooperative game • convex games • core • bargaining set 9 The authors acknowledge the support from research grants ECO2014-52340-P (Ministerio de Economía y Competitividad) and 2014SGR40 (Generalitat de Catalunya).

2018

We consider two-player bargaining problems with compact star-shaped choice sets arising from a class of economic environments. We characterize single-valued solutions satisfying the Nash axioms on this class of bargaining problems. Our results show that there are exactly two Nash solutions with each being a dictatorial (in favor of one player) selection of Nash product maximizers. We also provide an extensive form for implementing these two Nash solutions.

SSRN Electronic Journal, 2015

We consider a standard coalitional bargaining game where once a coalition forms it exits as in Okada (2011), however, instead of alternating o¤ers, we have simultaneous payo¤ demands. We focus in the producer game he studies. Each player is chosen with equal probability. If that is the case, she can choose any coalition she belongs to. However, a coalition can form if an only if payo¤ demands are feasible as in the Nash (1953) demand game. After smoothing the game (as in Van Damme (1991)), when the noise vanishes, when the discount factor is close to 1, and as in Okada´s ( ), the coalitional Nash bargaining solution is the unique stationary subgameperfect equilibrium.

Computing Research Repository - CORR, 2008

We consider bargaining problems which involve two participants, with a nonempty closed, bounded convex bargaining set of points in the real plane representing all realizable bargains. We also assume that there is no definite threat or disagreement point which will provide the default bargain if the players cannot agree on some point in the bargaining set. However, there is a nondeterministic threat: if the players fail to agree on a bargain, one of them will be chosen at random with equal probability, and that chosen player will select any realizable bargain as the solution, subject to a reasonable restriction.

1989

*Parts of this chapter use material from Rubinstein [1987], a survey of sequential bargaining models. Parts of Sections 5, 6, 7 and 8 are based on a draft of parts of Osborne and Rubinstein [1990]. **The first author wishes to thank Avner Shaked and John Sutton and the third author wishes to thank Asher Wolinsky for long and fruitful collaborations. Result 1 (Rubinstein, 1982) Under assumptions (TPO)-(TP5) the bargaining game has a unique sub-game-perfect equilibrium. In this equilibrium, agreement is reached immediately, and the players' utilities satisfy (2). Alternative versions of Rubinstein's proof appear in Binmore [1987b) and Shaked/Sutton [1984]. The following proof of Shaked and Sutton is especially useful for extensions and modifications of the theorem.

International Journal of Game Theory, 1998

Maschler, Owen and Peleg (1988) constructed a dynamic system for modelling a possible negotiation process for players facing a smooth n-person pure bargaining game, and showed that all paths of this system lead to the Nash point. They also considered the non-convex case, and found in this case that the limiting points of solutions of the dynamic system belong to the Nash set. Here we extend the model to i) general convex pure bargaining games, and to ii) games generated by``divide the cake'' problems. In each of these cases we construct a dynamic system consisting of a di¨erential inclusion (generalizing the Maschler-Owen-Peleg system of di¨erential equations), prove existence of solutions, and show that the solutions converge to the Nash point (or Nash set). The main technical point is proving existence, as the system is neither convex valued nor continuous. The intuition underlying the dynamics is the same as (in the convex case) or analogous to (in the division game) that of Maschler, Owen, and Peleg.