17,294 research outputs found
A Semi-Potential for Finite and Infinite Games in Extensive Form
We consider a dynamical approach to game 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 games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic (finite) 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 Δ02-sets
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
An Efficient Policy Iteration Algorithm for Dynamic Programming Equations
We present an accelerated algorithm for the solution of static
Hamilton-Jacobi-Bellman equations related to optimal control problems. Our
scheme is based on a classic policy iteration procedure, which is known to have
superlinear convergence in many relevant cases provided the initial guess is
sufficiently close to the solution. In many cases, this limitation degenerates
into a behavior similar to a value iteration method, with an increased
computation time. The new scheme circumvents this problem by combining the
advantages of both algorithms with an efficient coupling. The method starts
with a value iteration phase and then switches to a policy iteration procedure
when a certain error threshold is reached. A delicate point is to determine
this threshold in order to avoid cumbersome computation with the value
iteration and, at the same time, to be reasonably sure that the policy
iteration method will finally converge to the optimal solution. We analyze the
methods and efficient coupling in a number of examples in dimension two, three
and four illustrating its properties
Regular Boardgames
We propose a new General Game Playing (GGP) language called Regular
Boardgames (RBG), which is based on the theory of regular languages. The
objective of RBG is to join key properties as expressiveness, efficiency, and
naturalness of the description in one GGP formalism, compensating certain
drawbacks of the existing languages. This often makes RBG more suitable for
various research and practical developments in GGP. While dedicated mostly for
describing board games, RBG is universal for the class of all finite
deterministic turn-based games with perfect information. We establish
foundations of RBG, and analyze it theoretically and experimentally, focusing
on the efficiency of reasoning. Regular Boardgames is the first GGP language
that allows efficient encoding and playing games with complex rules and with
large branching factor (e.g.\ amazons, arimaa, large chess variants, go,
international checkers, paper soccer).Comment: AAAI 201
One for all, all for one---von Neumann, Wald, Rawls, and Pareto
Applications of the maximin criterion extend beyond economics to statistics,
computer science, politics, and operations research. However, the maximin
criterion---be it von Neumann's, Wald's, or Rawls'---draws fierce criticism due
to its extremely pessimistic stance. I propose a novel concept, dubbed the
optimin criterion, which is based on (Pareto) optimizing the worst-case payoffs
of tacit agreements. The optimin criterion generalizes and unifies results in
various fields: It not only coincides with (i) Wald's statistical
decision-making criterion when Nature is antagonistic, (ii) the core in
cooperative games when the core is nonempty, though it exists even if the core
is empty, but it also generalizes (iii) Nash equilibrium in -person
constant-sum games, (iv) stable matchings in matching models, and (v)
competitive equilibrium in the Arrow-Debreu economy. Moreover, every Nash
equilibrium satisfies the optimin criterion in an auxiliary game
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