Marcin Pęski

Generalized risk-dominance and asymmetric dynamics

This paper proposes two (ordinal and cardinal) generalizations of [J.C. Harsanyi, R. Selten, A General Theory of Equilibrium Selection in Games, MIT Press, Cambridge, MA and London, 1988] risk-dominance to multi-player, multi-action games. There are three reasons why generalized risk-dominance (GR-dominance) is interesting. Extending the logic of risk-dominance, GR-dominant actions can be interpreted as best responses to conjectures that satisfy a certain type of symmetry. Second, in a local interaction game of [G. Ellison, Learning, local interaction, and coordination, Econometrica 61 (5) (1993) 1047], if an action is risk-dominant in individual binary interactions with neighbors, it is also GR-dominant in the large game on a network. Finally, we show that GR-dominant actions are stochastically stable under a class of evolutionary dynamics. The last observation is a corollary to new abstract selection results that applies to a wide class of so-called asymmetric dynamics. In particular, I show that a (strictly) ordinal GR-dominant profile is (uniquely) stochastically stable under the approximate best-response dynamics of [M. Kandori, G.J. Mailath, R. Rob, Learning, mutation, and long run equilibria in games, Econometrica 61 (1) (1993) 29]. A (strictly) cardinal GR-dominant equilibrium is (uniquely) stochastically stable under a class of payoff-based dynamics that includes [L.E. Blume, The statistical-mechanics of strategic interaction, Games Econ. Behav. 5 (3) (1993) 387-424]. Among others, this leads to a generalization of a result from [G. Ellison, Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution, Rev. Econ. Stud. 67 (230) (2000) 17] on the -dominant evolutionary selection to all networks and the unique selection to all networks that satisfy a simple, sufficient condition.

Comparison of information structures in zero-sum games

This note provides simple necessary and sufficient conditions for the comparison of information structures in zero-sum games. This solves an open problem of Grossner and Mertens [Gossner, O., Mertens, J.-F., 2001. The value of information in zero-sum games. http://ogossner.free.fr/Articles/abstract.pdf]. The conditions are phrased in terms of Blackwell garbling of information of each of the players.

Value of Persistent Information

We develop a theory of how the value of an agents information advantage depends on the persistence of information. We focus on strategic situations with strict conflict of interest, formalized as stochastic zero‐sum games where only one of the players observes the state that evolves according to a Markov operator. Operator Q is said to be better for the informed player than operator P if the value of the game under Q is higher than under P regardless of the stage game. We show that this defines a convex partial order on the space of ergodic Markov operators. Our main result is a full characterization of this partial order, intepretable as an ordinal notion of persistence relevant for games. The analysis relies on a novel characterization of the value of a stochastic game with incomplete information.

Large roommate problem with non-transferable random utility☆

We analyze a large roommate problem (i.e., marriage matching in which the marriage is not restricted solely to matchings between men and women) with non-transferable utility. It is well known that while a roommate problem may not have a stable proper matching, each roommate problem does have an stable improper matching. In a random utility model with types from Dagsvik (2000) and Menzel (2015), we show that all improper stable matchings are asymptotically close to being a proper stable matching. Moreover, the distribution of types in stable matchings (proper or not) converges to the unique maximizer of an expression that is a sum of two terms: the average “welfare” of the matching and the Shannon entropy of the distribution. In the noiseless limit, when the random component of the utility is reduced to zero, the distribution of types of matched pairs converges to the outcome of the transferable utility model.

Corrigendum to “Generalized risk-dominance and asymmetric dynamics” [J. Econ. Theory 145 (1) (2010) 216–248]

The authors regret that the above referenced paper imprecisely discusses the connection to the previous literature. That paper proposes two equilibrium refinements (cardinal and ordinal GR-dominance) that are related to Harsanyi–Selten’s risk-dominance [1] and that turn out to be useful in extending the previously known results about evolutionary equilibrium selection to larger class of games, including games on networks. In the same time, the paper contains two informal claims that are incorrect. Specifically,