16 research outputs found
Shapley's "2 by 2" theorem for game forms
If a finite two person game form has the property that every 2-by-2 fragment is Nash consistent, then no derivative game admits an individual improvement cycle.
On continuous ordinal potential games
If the preferences of the players in a strategic game satisfy certain continuity conditions, then the acyclicity of individual improvements implies the existence of a Nash equilibrium. Moreover, starting from any strategy profile, an arbitrary neighborhood of the set of Nash equilibria can be reached after a finite number of individual improvements.potential game; compact-continuous game; finite improvement property
Dynamics and Coalitions in Sequential Games
We consider N-player non-zero sum games played on finite trees (i.e.,
sequential games), in which the players have the right to repeatedly update
their respective strategies (for instance, to improve the outcome wrt to the
current strategy profile). This generates a dynamics in the game which may
eventually stabilise to a Nash Equilibrium (as with Kukushkin's lazy
improvement), and we argue that it is interesting to study the conditions that
guarantee such a dynamics to terminate.
We build on the works of Le Roux and Pauly who have studied extensively one
such dynamics, namely the Lazy Improvement Dynamics. We extend these works by
first defining a turn-based dynamics, proving that it terminates on subgame
perfect equilibria, and showing that several variants do not terminate. Second,
we define a variant of Kukushkin's lazy improvement where the players may now
form coalitions to change strategies. We show how properties of the players'
preferences on the outcomes affect the termination of this dynamics, and we
thereby characterise classes of games where it always terminates (in particular
two-player games).Comment: In Proceedings GandALF 2017, arXiv:1709.0176
A Class of Best-Response Potential Games
We identify a class of noncooperative games in continuous strategies which are best-response potential games. We identify the conditions for the existence of a best-response potential function and characterize its construction, describing then the key properties of the equilibrium. The theoretical analysis is accompanied by applications to oligopoly and monetary policy games
Acyclicity of improvements in finite game forms
Game forms are studied where the acyclicity, in a stronger or weaker sense, of (coalition or individual) improvements is ensured in all derivative games. In every game form generated by an ``ordered voting'' procedure, individual improvements converge to Nash equilibria if the players restrict themselves to ``minimal'' strategy changes. A complete description of game forms where all coalition improvement paths lead to strong equilibria is obtained: they are either dictatorial, or voting (or rather lobbing) about two outcomes. The restriction to minimal strategy changes ensures the convergence of coalition improvements to strong equilibria in every game form generated by a ``voting by veto'' procedure.Improvement dynamics; Game form; Perfect information game; Potential game; Voting by veto
On continuous ordinal potential games
If the preferences of the players in a strategic game satisfy certain continuity conditions, then the acyclicity of individual improvements implies the existence of a Nash equilibrium. Moreover, starting from any strategy profile, an arbitrary neighborhood of the set of Nash equilibria can be reached after a finite number of individual improvements
A Semi-Potential for Finite and Infinite Sequential Games (Extended Abstract)
We consider a dynamical approach to sequential games. By restricting the
convertibility relation over strategy profiles, we obtain a semi-potential (in
the sense of Kukushkin), and we show that in finite games the corresponding
restriction of better-response dynamics will converge to a Nash equilibrium in
quadratic time. Convergence happens on a per-player basis, and even in the
presence of players with cyclic preferences, the players with acyclic
preferences will stabilize. Thus, we obtain a candidate notion for rationality
in the presence of irrational agents. Moreover, the restriction of
convertibility can be justified by a conservative updating of beliefs about the
other players strategies.
For infinite sequential games we can retain convergence to a Nash equilibrium
(in some sense), if the preferences are given by continuous payoff functions;
or obtain a transfinite convergence if the outcome sets of the game are
Delta^0_2 sets.Comment: In Proceedings GandALF 2016, arXiv:1609.0364
Well-founded extensive games with perfect information
We consider extensive games with perfect information with well-founded game trees and study the problems of existence and of characterization of the sets of subgame perfect equilibria in these games. We also provide such characterizations for two classes of these games in which subgame perfect equilibria exist: two-player zero-sum games with, respectively, two and three outcomes