Game Theory: The Language of Social Science?

The present paper tries in a largely non-technical way to discuss the aim, the basic notions and methods as well as the limits of game theory under the aspect of providing a general modelling method or language for social sciences.

An exact non-cooperative support for the sequential Raiffa solution

This article provides an exact non-cooperative foundation of the sequential Raiffa solution for two person bargaining games. Based on an approximate foundation due to Myerson (1997) for any two-person bargaining game (S, d) an extensive form game G^{S,d} is defined that has an infinity of weakly subgame perfect equilibria whose payoff vectors coincide with that of the sequential Raiffa solution of (S, d). Moreover all those equilibria share the same equilibrium path consisting of proposing the Raiffa solution and accepting it in the first stage of the game. By a modification of G^{S,d} the analogous result is provided for subgame perfect equilibria. Finally, it is indicated how these results can be extended to implementation of a sequential Raiffa (solution based) social choice rule in subgame perfect equilibrium.Raiffa solution, non-cooperative foundation Nash program, implementation, solution based social choice rule

Core-Equivalence for the Nash Bargaining Solution.

Core equivalence and shrinking of the core results are well known for economies. The present paper establishes counterparts for bargaining economies, a specific class of production economies (finite and infinite) representing standard two-person bargaining games and their continuum counterparts as coalition production economies. Thereby we get core equivalence of the Nash solution. The results reconfirm the Walrasian approach to Nash bargaining of Trockel (1996). Moreover we establish the same speed of convergence as is known from Debreu (1975) and Grodal (1975) for replicated pure exchange economies and for regular purely competitive sequences of economies, respectively.

The Chain-Store Paradox Revisited

Trockel W. The Chain-Store Paradox Revisited. Theory and Decision. 1986;21(2):163-179

On Maskin monotonicity of solution based social choice rules

Howard (1992) argues that the Nash bargaining solution is not Nash implementable, as it does not satisfy Maskin monotonicity. His arguments can be extended to other bargaining solutions as well. However, by de.ning a social choice correspondence that is based on the solution rather than on its realizations, one can overcome this shortcoming. We even show that such correspondences satisfy a stronger version of monotonicity that is even su.cient for Nash implementability.Maskin monotonicity, social choice rule, bargaining games, Nash program, mechanism, implementation

Nash Smoothing on the Test Bench: HαH_{\alpha} -Essential Equilibria

Duman P, Trockel W. Nash Smoothing on the Test Bench: $H_{\alpha}$ -Essential Equilibria. Center for Mathematical Economics Working Papers. Vol 632. Bielefeld: Center for Mathematical Economics; 2020.We extend the analysis of van Damme (1987, Section 7.5) of the famous smoothing demand in Nash (1953) as an argument for the singular stability of the symmetric Nash bargaining solution among all Pareto efficient equilibria of the Nash demand game. Van Damme's analysis provides a clean mathematical framework where he substantiates Nash's conjecture by two fundamental theorems in which he proves that the Nash solution is among all Nash equilibria of the Nash demand game the only one that is *H*-essential. We show by generalizing this analysis that for any asymmetric Nash bargaining solution a similar stability property can be established that we call $H_{\alpha}$-essentiality. A special case of our result for α = 1/2 is $H_{1/2}$-essentiality that coincides with van Damme's *H*-essentiality. Our analysis deprives the symmetric Nash solution equilibrium of Nash's demand game of its exposed position and fortifies our conviction that, in contrast to the predominant view in the related literature, the only structural diffeerence between the asymmetric Nash solutions and the symmetric one is that the latter one is symmetric. While our proofs are mathematically straightforward given the analysis of van Damme (1987), our results change drastically the prevalent interpretation of Nash's smoothing of his demand game and dilute its conceptual importance

On approximate cores of non-convex economies

Grodal B, Trockel W, Weber S. On approximate cores of non-convex economies. Economics letters. 1984;15(3-4):197-202.In this note we investigate, for non-convex finite economies, the relationship between the existence of approximate core allocations and the size of an economy

An Exact Implementation of the Nash Bargaining Solution in Dominant Strategies

For any abstract bargaining problem a non-cooperative one stage strategic game is constructed whose unique dominant strategies Nash equilibrium implements the Nash solution of the bargaining problem.Nash Programm, Implementation, Nash Bargaining Solution