On the implementation of the L-Nash bargaining solution in two-person bargaining games

The “Nash program” initiated by Nash (Econometrica 21:128–140, 1953) is a research agenda aiming at representing every axiomatically determined cooperative solution to a game as a Nash outcome of a reasonable noncooperative bargaining game. The L-Nash solution first defined by Forgó (Interactive Decisions. Lecture Notes in Economics and Mathematical Systems, vol 229. Springer, Berlin, pp 1–15, 1983) is obtained as the limiting point of the Nash bargaining solution when the disagreement point goes to negative infinity in a fixed direction. In Forgó and Szidarovszky (Eur J Oper Res 147:108–116, 2003), the L-Nash solution was related to the solution of multiciteria decision making and two different axiomatizations of the L-Nash solution were also given in this context. In this paper, finite bounds are established for the penalty of disagreement in certain special two-person bargaining problems, making it possible to apply all the implementation models designed for Nash bargaining problems with a finite disagreement point to obtain the L-Nash solution as well. For another set of problems where this method does not work, a version of Rubinstein’s alternative offer game (Econometrica 50:97–109, 1982) is shown to asymptotically implement the L-Nash solution. If penalty is internalized as a decision variable of one of the players, then a modification of Howard’s game (J Econ Theory 56:142–159, 1992) also implements the L-Nash solution.

Finite horizon bargaining and the consistent field

This paper explores the relationships between noncooperative bargaining games and the consistent value for non-transferable utility (NTU) cooperative games. A dynamic approach to the consistent value for NTU games is introduced: the consistent vector field. The main contribution of the paper is to show that the consistent field is intimately related to the concept of subgame perfection for finite horizon noncooperative bargaining games, as the horizon goes to infinity and the cost of delay goes to zero. The solutions of the dynamic system associated to the consistent field characterize the subgame perfect equilibrium payoffs of the noncooperative bargaining games. We show that for transferable utility, hyperplane and pure bargaining games, the dynamics of the consistent fields converge globally to the unique consistent value. However, in the general NTU case, the dynamics of the consistent field can be complex. An example is constructed where the consistent field has cyclic solutions; moreover, the finite horizon subgame perfect equilibria do not approach the consistent value.

Um modelo fuzzy-dea-game para estratégias de produção sob incerteza

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bargaining over strategies of non-cooperative games

We propose a bargaining process supergame over the strategies to play in a non-cooperative game. The agreement reached by players at the end of the bargaining process is the strategy profile that they will play in the original non-cooperative game. We analyze the subgame perfect equilibria of this supergame, and its implications on the original game. We discuss existence, uniqueness, and efficiency of the agreement reachable through this bargaining process. We illustrate the consequences of applying such a process to several common two-player non-cooperative games: the Prisoner’s Dilemma, the Hawk-Dove Game, the Trust Game, and the Ultimatum Game. In each of them, the proposed bargaining process gives rise to Pareto-efficient agreements that are typically different from the Nash equilibrium of the original games.

Three essays on game theory and bargaining

A PhD Dissertation, presented as part of the requirements for the Degree of Doctor of Philosophy from the NOVA - School of Business and Economics

Simultaneous Nash bargaining with consistent beliefs

We propose and analyze a new solution concept, the R solution, for three-person, transferable utility, cooperative games. In the spirit of the Nash Bargaining Solution, our concept is founded on the predicted outcomes of simultaneous, two-party negotiations that would be the alternative to the grand coalition. These possibly probabilistic predictions are based on consistent beliefs. We analyze the properties of the R solution and compare it with the Shapley value and other concepts. The R solution exists and is unique. It belongs to the bargaining set and to the core whenever the latter is not empty. In fact, when the grand coalition can simply execute one of the three possible bilateral trades, the R solution is the most egalitarian selection of the bargaining set. Finally, we discuss how the R solution changes important conclusions of several well known Industrial Organization models.

Cooperative games in strategic form

In this paper we view bargaining and cooperation as an interaction superimposed on a strategic form game. A multistage bargaining procedure for N players, the proposer commitment procedure, is presented. It is inspired by Nash s two-player variable-threat model; a key feature is the commitment to threats. We establish links to classical cooperative game theory solutions, such as the Shapley value in the transferable utility case. However, we show that even in standard pure exchange economies the traditional coalitional function may not be adequate when utilities are not transferable.

An axiomatization of the prekernel of nontransferable utility games

We characterize the prekernel of NTU games by means of consistency,converse consistency, and five axioms of the Nash type on bilateral problems.The intersection of the prekernel and the core is also characterized with thesame axioms over the class of games where the core is nonempty.

Coalitionally monotonic set-solutions for cooperative TU games

A static comparative study on set-solutions for cooperative TU games is carried out. The analysis focuses on studying the compatibility between two classical and reasonable properties introduced by Young (1985) in the context of single valued solutions, namely core-selection and coalitional monotonicity. As the main result, it is showed that coalitional monotonicity is not only incompatible with the core-selection property but also with the bargaining-selection property. This new impossibility result reinforces the tradeoff between these kinds of interesting and intuitive economic properties. Positive results about compatibility between desirable economic properties are given replacing the core selection requirement by the core-extension property.

Coalitionally monotonic set-solutions for cooperative TU games

A static comparative study on set-solutions for cooperative TU games is carried out. The analysis focuses on studying the compatibility between two classical and reasonable properties introduced by Young (1985) in the context of single valued solutions, namely core-selection and coalitional monotonicity. As the main result, it is showed that coalitional monotonicity is not only incompatible with the core-selection property but also with the bargaining-selection property. This new impossibility result reinforces the tradeoff between these kinds of interesting and intuitive economic properties. Positive results about compatibility between desirable economic properties are given replacing the core selection requirement by the core-extension property.

On the coincidence of the Mas-Colell bargaining set and the core

In this paper we prove that the Mas-Colell bargaining set coincides with the core for three-player balanced and superadditive cooperative games. This is no longer true without the superadditivity condition or for games with more than three-players. Furthermore, under the same assumptions, the coincidence between the Mas-Collel and the individual rational bargaining set (Vohra (1991)) is revealed. Keywords: Cooperative game, Mas-Colell bargaining set, balancedness, individual rational bargaining set. JEL classi fication: C71, D63, D71.

Axiomatization of the nucl eolus of assignment games

On the domain of general assignment games (with possible reservation prices) the core is axiomatized as the unique solution satisfying two consistency principles: projection consistency and derived consistency. Also, an axiomatic characterization of the nucleolus is given as the unique solution that satisfies derived consistency and equal maximum complaint between groups. As a consequence, we obtain a geometric characterization of the nucleolus. Maschler et al. (1979) provide a geometrical characterization for the intersection of the kernel and the core of a coalitional game, showing that those allocations that lie in both sets are always the midpoint of certain bargaining range between each pair of players. In the case of the assignment game, this means that the kernel can be determined as those core allocations where the maximum amount, that can be transferred without getting outside the core, from one agent to his / her optimally matched partner equals the maximum amount that he / she can receive from this partner, also remaining inside the core. We now prove that the nucleolus of the assignment game can be characterized by requiring this bisection property be satisfied not only for optimally matched pairs but also for optimally matched coalitions. Key words: cooperative games, assignment game, core, nucleolus

Efficiency in Uncertain Cooperative Games

A contingent contract in a transferable utility game under uncertainty specifies an outcome for each possible state. It is assumed that coalitions evaluate these contracts by considering the minimal possible excesses. A main question of the paper concerns the existence and characterization of efficient contracts. It is shown that they exist if and only if the set of possible coalitions contains a balanced subset. Moreover, a characterization of values that result in efficient contracts in the case of minimally balanced collections is provided.

On Limits to the Use of Linear Markov Strategies in Common Property Natural Resource Games

We derive conditions that must be satisfied by the primitives of the problem in order for an equilibrium in linear Markov strategies to exist in some common property natural resource differential games. These conditions impose restrictions on the admissible form of the natural growth function, given a benefit function, or on the admissible form of the benefit function, given a natural growth function.

Gender differences in ultimatum games: despite rather than due to risk attitudes

We analyze experimental data obtained from an ultimatum game framed as a situation of employee-employer negotiation over salaries. Parallel to this, we elicit subjects' risk attitudes. In the existing literature, it has often been conjectured that gender differences in strategic environments are partly due to differences in risky decision making. Our evidence suggests that both gender and risk-related effects co-exist in ultimatum bargaining. However, differences in risk attitudes cannot explain gender effects in ultimatum bargaining.