Abstract
We consider a dynamic approach to games in extensive forms. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite sequential 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 games in extensive form 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.
Highlights
The Nash equilibria are the fixed points of the better response dynamics. In graph theory they would be called the sinks of these dynamics, and in computer science they may be called their terminal strategy profiles
– In a finite game with acyclic preferences, the dynamics stabilizes at a Nash equilibrium after a quadratic number of steps
A strategy profile s in S := a∈A Sa is a Nash equilibrium if it makes every Player a stable, i.e. v(s) ⊀a v(s ) for all s ∈ S that differ from s at most at the a-component
Summary
The Nash equilibria are the fixed points of the better (or best) response dynamics In graph theory they would be called the sinks of these dynamics, and in computer science they may be called their terminal strategy profiles. Better-response dynamics will always improve the potential, and terminates (at a Nash equilibrium) if the game is finite. – In a finite game with acyclic preferences, the dynamics stabilizes at a Nash equilibrium after a quadratic number of steps. The main advantage of our two proofs of termination is that they work on a per-player basis: Each player with acyclic preference will terminate, even in the presence of players with cyclic preferences This is far from obvious, as in most dynamics a single player who keeps altering their choices can induce the other players to keep changing, too. Outcome sets, and show that transfinite continuation of the dynamics still stabilizes
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