8,802 research outputs found
Algorithms for generalized potential games with mixed-integer variables
We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches
Constant Rank Bimatrix Games are PPAD-hard
The rank of a bimatrix game (A,B) is defined as rank(A+B). Computing a Nash
equilibrium (NE) of a rank-, i.e., zero-sum game is equivalent to linear
programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an
FPTAS for constant rank games, and asked if there exists a polynomial time
algorithm to compute an exact NE. Adsul et al. (2011) answered this question
affirmatively for rank- games, leaving rank-2 and beyond unresolved.
In this paper we show that NE computation in games with rank , is
PPAD-hard, settling a decade long open problem. Interestingly, this is the
first instance that a problem with an FPTAS turns out to be PPAD-hard. Our
reduction bypasses graphical games and game gadgets, and provides a simpler
proof of PPAD-hardness for NE computation in bimatrix games. In addition, we
get:
* An equivalence between 2D-Linear-FIXP and PPAD, improving a result by
Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD.
* NE computation in a bimatrix game with convex set of Nash equilibria is as
hard as solving a simple stochastic game.
* Computing a symmetric NE of a symmetric bimatrix game with rank is
PPAD-hard.
* Computing a (1/poly(n))-approximate fixed-point of a (Linear-FIXP)
piecewise-linear function is PPAD-hard.
The status of rank- games remains unresolved
Computing equilibria of Cournot oligopoly models with mixed-integer quantities
We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called “2-groups partitionable” Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method
Computing all solutions of Nash equilibrium problems with discrete strategy sets
The Nash equilibrium problem is a widely used tool to model non-cooperative
games. Many solution methods have been proposed in the literature to compute
solutions of Nash equilibrium problems with continuous strategy sets, but,
besides some specific methods for some particular applications, there are no
general algorithms to compute solutions of Nash equilibrium problems in which
the strategy set of each player is assumed to be discrete. We define a
branching method to compute the whole solution set of Nash equilibrium problems
with discrete strategy sets. This method is equipped with a procedure that, by
fixing variables, effectively prunes the branches of the search tree.
Furthermore, we propose a preliminary procedure that by shrinking the feasible
set improves the performances of the branching method when tackling a
particular class of problems. Moreover, we prove existence of equilibria and we
propose an extremely fast Jacobi-type method which leads to one equilibrium for
a new class of Nash equilibrium problems with discrete strategy sets. Our
numerical results show that all proposed algorithms work very well in practice
Deflation for semismooth equations
Variational inequalities can in general support distinct solutions. In this
paper we study an algorithm for computing distinct solutions of a variational
inequality, without varying the initial guess supplied to the solver. The
central idea is the combination of a semismooth Newton method with a deflation
operator that eliminates known solutions from consideration. Given one root of
a semismooth residual, deflation constructs a new problem for which a
semismooth Newton method will not converge to the known root, even from the
same initial guess. This enables the discovery of other roots. We prove the
effectiveness of the deflation technique under the same assumptions that
guarantee locally superlinear convergence of a semismooth Newton method. We
demonstrate its utility on various finite- and infinite-dimensional examples
drawn from constrained optimization, game theory, economics and solid
mechanics.Comment: 24 pages, 3 figure
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