International Game Theory Review

The purpose of this work is to offer for each player and any Nash equilibrium (NE), a measure for the potential risk in deviating from the NE strategy in any two person matrix game. We present two approaches regarding the nature of deviations: Strategic and Accidental. Accordingly, we define two models: S-model and T-model. The S-model defines a new game in which players deviate in the least dangerous direction. The risk defined in the T-model can serve as a refinement for the notion of “trembling hand perfect equilibrium” introduced by R. Selten. The risk measures enable testing and evaluating predictions on the behavior of players. For example: do players deviate more from a NE that is less risky? This may be relevant to the design of experiments. We present an Integer programming problem that computes the risk for any given player and NE. In the special case of zero-sum games with a unique strictly mixed NE, we prove that the risks of the players always coincide, even if the game is far from symmetry. This result holds for any norm we use for the size of deviations. We compare our risk measures to the risk measure defined by Harsanyi and Selten which is based on criteria of stability rather than on potential damage. We show that the measures may contradict.

Recently models and solution concepts of games with unawareness have been developed. This paper discusses the issue of reducing static games with unawareness to standard Bayesian games and shows that equilibrium concepts for the two formulations, i.e., generalized Nash equilibrium and Bayesian equilibrium, are equivalent. We discuss implications of the result.

In this paper, we derive a cost-minimizing approach for governments in the eurozone that have to define their debt structure in terms of currency of issuance and maturity. We introduce and characterize the concept of weighted coalitional $\alpha$-equilibrium to assure stability of two tiers of the eurozone, the stronger (White) countries and the weaker (Yolk) countries. The equilibrium presents both the Pareto optimality of the countries in each tier and the stability against any deviation of each subcoalition. In a practical way, it would require either strong political integration within each tier or considerable market pressure.

We propose a new noncooperative approach to implement a cooperative bargaining solution. This approach relies on a mechanism which specifies what happens when, at the end of the bargaining phase, players’ propositions are still not compatible. The mechanism uses a random lottery and the history of proposals. At equilibrium, under the threat of this mechanism, provided that the use of it carries a small cost, players reach consensus after a finite number of proposals. No discounting is needed. The equilibrium strategies implement the full Raiffa bargaining process and players agree on a solution arbitrarily close to the Raiffa solution.

The methodology of relative skill, by means of a detailed mathematical analysis of realistic approximations of Texas Hold’em cash games and tournaments as provided in van der Genugten and Borm [2012], and Borm and van der Genugten [2009] respectively, reconfirms the findings of the court in The Hague (cf. [Rechtbank Gravenhage [2011]) that Texas Hold’em in all its practical variants is a game of skill.

The seminal work by Green and Laffont [(1977) characterization of satisfactory mechanisms for the revelation of preferences for public goods, Econometrica 45, 427–438] shows that efficient mechanisms with Vickrey–Clarke–Groves prices satisfy the properties of dominant-strategy incentive compatible (DSIC) and individually rational in the quasilinear utilities model. Nevertheless in many real-world situations some players have a gap between their willingness to pay and their ability to pay, i.e., a budget. We show that once budgets are integrated into the model then Green and Laffont’s theorem ceases to apply. More specifically, we show that even if only a single player has budget constraints then there is no deterministic efficient mechanism that satisfies the individual rationality and DSIC properties. Furthermore, in a quasilinear utilities model with $k$ nonidentical items and $n$ players with multidimensional types, we characterize the sufficient and necessary conditions under which Green and Laffont’s theorem holds in the presence of budget-constrained players. Interestingly our characterization is similar in spirit to that of Maskin [(2000) Auctions, development and privatization: Efficient auctions with liquidity-constrained buyers, Eur. Econ. Rev. 44, 667–681] for Bayesian single-item constrained-efficiency auctions.