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

Proximal-like algorithms for equilibrium seeking in mixed-integer Nash equilibrium problems

We consider potential games with mixed-integer variables, for which we propose two distributed, proximal-like equilibrium seeking algorithms. Specifically, we focus on two scenarios: i) the underlying game is generalized ordinal and the agents update through iterations by choosing an exact optimal strategy; ii) the game admits an exact potential and the agents adopt approximated optimal responses. By exploiting the properties of integer-compatible regularization functions used as penalty terms, we show that both algorithms converge to either an exact or an $\epsilon$-approximate equilibrium. We corroborate our findings on a numerical instance of a Cournot oligopoly model

Selection methods for subgame perfect Nash equilibrium in a continuous setting

This thesis is focused on the issue of selection of Subgame Perfect Nash Equilibrium (SPNE) in the class of one-leader N-follower two-stage games where the players have a continuum of actions. We are mainly interested in selection methods satisfying the following significant features in the theory of equilibrium selection for such a class of games: obtaining an equilibrium selection by means of a constructive (in the sense of algorithmic) and motivated procedure, overcoming the difficulties due to the possible non-single-valuedness of the followers' best reply correspondence, providing motivations that would induce players to choose the actions leading to the designed selection, and revealing the leader to know the followers' best reply correspondence. Firstly, we analyze the case where the followers' best reply correspondence is assumed to be single-valued: in this case we show that finding SPNEs is equivalent to find the Stackelberg solutions of the Stackelberg problem associated to the game. Moreover, as regards to the related arising issue of the sufficient conditions ensuring the uniqueness of the followers' best reaction, we prove an existence and uniqueness result for Nash equilibria in two-player normal-form games where the action sets are Hilbert spaces and which allows the two compositions of the best reply functions to be not a contraction mapping. Furthermore, by applying such a result to the class of weighted potential games, we show the (lack of) connections between the Nash equilibria and the maximizers of the potential function. Then, in the case where the followers' best reply correspondence is not assumed to be single-valued, we examine preliminarily the SPNE selections deriving by exploiting the solutions of broadly studied problems in Optimization Theory (like the strong Stackelberg, the weak Stackelberg and the intermediate Stackelberg problems associated to the game). Since such selection methods, although behaviourally motivated, do not fit all the desirable features mentioned before, we focus on designing constructive methods to select an SPNE based on the Tikhonov regularization and on the proximal point methods (linked to the Moreau-Yosida regularization). After illustrated these two tools both in the optimization framework and in the applications to the selection of Nash equilibria in normal-form games, we present a constructive selection method for SPNEs based on the Tikhonov regularization in one-leader N-follower two-stage games (with N=1 and N>1), and a constructive selection method for SPNEs based on a learning approach which has a behavioural interpretation linked to the costs that players face when they deviate from their current actions (relying on the proximal point methods) in one-leader one-follower two-stage games