Mathijs Jansen

Invariance properties of persistent equilibria and related solution concepts

Kohlberg and Mertens argued that a solution concept to a game should be invariant under the addition of deletion of an equivalent strategy and not require the use of weakly dominated strategies. In this paper we study which of these requirements are satisfied by Kalai and Samets concepts of persistent equilibria and persistent retracts. While none of these concepts has all the invariance properties, we show that a slight rephrasing of the notion of a persisent retract leads to a notion satisfying them all.

Player splitting in extensive form games

Abstract. By a player splitting we mean a mechanism that distributes the information sets of a player among so-called agents. A player splitting is called independent if each path in the game tree contains at most one agent of every player. Following Mertens (1989), a solution is said to have the player splitting property if, roughly speaking, the solution of an extensive form game does not change by applying independent player splittings. We show that Nash equilibria, perfect equilibria, Kohlberg-Mertens stable sets and Mertens stable sets have the player splitting property. An example is given to show that the proper equilibrium concept does not satisfy the player splitting property. Next, we give a definition of invariance under (general) player splittings which is an extension of the player splitting property to the situation where we also allow for dependent player splittings. We come to the conclusion that, for any given dependent player splitting, each of the above solutions is not invariant under this player splitting. The results are used to give several characterizations of the class of independent player splittings and the class of single appearance structures by means of invariance of solution concepts under player splittings.

On the Computation of Stable Sets and Strictly Perfect Equilibria

Summary. In this paper a procedure is described that computes for a given bimatrix game all stable sets in the sense of Kohlberg and Mertens (1986). Further the procedure is refined to find the strictly perfect equilibria (if any) of such a game.

On Quasi-Stable Sets

In this paper it is shown that for a bimatrix game each quasi-stable set is finite.

Independence of inadmissible strategies and best reply stability: a direct proof

Abstract.Hillas (1990) introduced a definition of strategic stability based on perturbations of the best reply correspondence that satisfies all of the requirements given by Kohlberg and Mertens (1986). Hillas et al. (2001) point out though that the proofs of the iterated dominance and forward induction properties were not correct. They also provide a proof of the IIS property, a stronger version of both iterated dominance and forward induction, using the results of that paper. In this note we provide a direct proof of the IIS property.

An ordinal selection of stable sets in the sense of Hillas

Abstract In this paper, it is shown in an example that the original definition of stable sets in Hillas [Econometrica 58 (1990) 1365–1391] does not satisfy (an even slightly weakenened version of) the invariance condition proposed in Mertens [Ordinality in noncooperative games, Core Discussion Paper 8728, CORE Louvain de la Neuve, Belgium, 1987]. However, it is also shown that the basic stability condition of Hillas underlying his definition of stable sets does admit a selection that is invariant in the strong sense, and even ordinal.

On stable sets of equilibria

A new kind of perturbations of normal form games is introduced and the stability concept related to these perturbations is investigated. The CQ-sets obtained in this way satisfy the properties of the Kohlberg-Mertens program except Invariance. In order to overcome this problem our solution concept is modified in such a way that all properties formulated by Kohlberg and Mertens are satisfied.

Consistency of assessments in infinite signaling games

Abstract In this paper we investigate possible ways to define consistency of assessments in infinite signaling games, i.e. signaling games in which the sets of types, messages and answers are complete, separable metric spaces. Roughly speaking, a consistency concept is called appropriate if it implies Bayesian consistency and copies the original idea of consistency in finite extensive form games as introduced by Kreps and Wilson ( Econometrica 1982, 50, 863–894) . We present a particular appropriate consistency concept, which we call strong consistency and give a characterization of strongly consistent assessments. It turns out that all appropriate consistency concepts are refinements of strong consistency. Finally, we define and characterize structurally consistent assessments in infinite signaling games.

Making solutions invariant

AbstractIn this paper a procedure is developed to modify a non-invariant solution in such a way that the resulting solution is invariant. Furthermore it is investigated which properties of the solution are inherited by the modified solution.