This objective is similar to Maximum Entropy Correlated Equilibrium (MECE) [48], and the proofs here are similar to the framework set out there. A drawback of MECE is that it is not easy to determine the minimum p permissible. If we choose p that does not permit a valid solution, then the parameters will diverge. We can circumvent this problem by optimizing the distance to a target ˆ p. And µis for balancing the linear objective.

Strategic interactions between a group of individuals or organisations can be modelled as games played on networks, where a player's payoff depends not only on their actions but also on those of their neighbours. Inferring the network structure from observed game outcomes (equilibrium actions) is an important problem with numerous potential applications in economics and social sciences. Existing methods mostly require the knowledge of the utility function associated with the game, which is often unrealistic to obtain in real-world scenarios. We adopt a transformer-like architecture which correctly accounts for the symmetries of the problem and learns a mapping from the equilibrium actions to the network structure of the game without explicit knowledge of the utility function. We test our method on three different types of network games using both synthetic and real-world data, and demonstrate its effectiveness in network structure inference and superior performance over existing methods.

We present a novel approach to maximize the communication-aware coverage for robots operating over large-scale geographical regions of interest (ROIs). Our approach complements the underlying network topology in neighborhood selection and control, rendering it highly robust in dynamic environments. We formulate the coverage as a multi-stage, cooperative graphical game and employ Variational Inference (VI) to reach the equilibrium. We experimentally validate our approach in an mobile ad-hoc wireless network scenario using Unmanned Aerial Vehicles (UAV) and User Equipment (UE) robots. We show that it can cater to ROIs defined by stationary and moving User Equipment (UE) robots under realistic network conditions.

Graphical games are one of the earliest examples of the impact that the general field of graphical models have had in other areas, and in this particular case, in classical mathematical models in game theory. Graphical multi-hypermatrix games, a concept formally introduced in this research note, generalize graphical games while allowing the possibility of further space savings in model representation to that of standard graphical games. The main focus of this research note is discretization schemes for computing approximate Nash equilibria, with emphasis on graphical games, but also briefly touching on normal-form and polymatrix games. The main technical contribution is a theorem that establishes sufficient conditions for a discretization of the players’ space of mixed strategies to contain an approximate Nash equilibrium. The result is actually stronger because every exact Nash equilibrium has a nearby approximate Nash equilibrium on the grid induced by the discretization. The sufficient conditions are weaker than those of previous results. In particular, a uniform discretization of size linear in the inverse of the approximation error and in the natural game-representation parameters suffices. The theorem holds for a generalization of graphical games, introduced here. The result has already been useful in the design and analysis of tractable algorithms for graphical games with parametric payoff functions and certain game-graph structures. For standard graphical games, under natural conditions, the discretization is logarithmic in the game-representation size, a substantial improvement over the linear dependency previously required. Combining the improved discretization result with old results on constraint networks in AI simplifies the derivation and analysis of algorithms for computing approximate Nash equilibria in graphical games.

Game theory has been extensively used as a framework to model and study the strategic interactions amongst rational but selfish individual players who are trying to maximize their payoffs. Game theory has been applied in many fields including but not limited to social and political science, economics, communication, system design and computer science. In non-cooperative games each player decides its action based on the actions of others players. These games are characterized by the equilibrium solution concept such as Nash equilibrium (NE) [18] which serves a descriptive role of the stable outcome of the overall behavior of self-interested players (e.g., people, companies, governments, groups or autonomous systems) interacting strategically with each other in distributed settings. Graphical games, introduced within the AI community about two decades ago, graphical games [16], are a representation of multiplayer games which capture and exploit locality or sparsity of direct influences. They are most appropriate for large-scale population games in which the payoffs of each player are determined by the actions of only a small number of other players. Indeed, graphical games played a prominent role in establishing the computational complexity of computing NE in normal-form games as well as in succinctly representable multiplayer games (see, e.g., [5, 6, 7] and the references therein). Graphical games have been studied for both discrete and continuous actions.

In this paper, we study the problem of learning the set of pure strategy Nash equilibria and the exact structure of a continuous-action graphical game with quadratic payoffs by observing a small set of perturbed equilibria. A continuous-action graphical game can possibly have an uncountable set of Nash euqilibria. We propose a $\ell_{12}-$ block regularized method which recovers a graphical game, whose Nash equilibria are the $\epsilon$-Nash equilibria of the game from which the data was generated (true game). Under a slightly stringent condition on the parameters of the true game, our method recovers the exact structure of the graphical game. Our method has a logarithmic sample complexity with respect to the number of players. It also runs in polynomial time.

Almost all of the work in graphical models for game theory has mirrored previous work in probabilistic graphical models. Our work considers the opposite direction: Taking advantage of recent advances in equilibrium computation for probabilistic inference. We present formulations of inference problems in Markov random fields (MRFs) as computation of equilibria in a certain class of game-theoretic graphical models. We concretely establishes the precise connection between variational probabilistic inference in MRFs and correlated equilibria. No previous work exploits recent theoretical and empirical results from the literature on algorithmic and computational game theory on the tractable, polynomial-time computation of exact or approximate correlated equilibria in graphical games with arbitrary, loopy graph structure. We discuss how to design new algorithms with equally tractable guarantees for the computation of approximate variational inference in MRFs. Also, inspired by a previously stated game-theoretic view of state-of-the-art tree-reweighed (TRW) message-passing techniques for belief inference as zero-sum game, we propose a different, general-sum potential game to design approximate fictitious-play techniques. We perform synthetic experiments evaluating our proposed approximation algorithms with standard methods and TRW on several classes of classical Ising models (i.e., with binary random variables). We also evaluate the algorithms using Ising models learned from the MNIST dataset. Our experiments show that our global approach is competitive, particularly shinning in a class of Ising models with constant, "highly attractive" edge-weights, in which it is often better than all other alternatives we evaluated. With a notable exception, our more local approach was not as effective. Yet, in fairness, almost all of the alternatives are often no better than a simple baseline: estimate 0.5.

We analyze the sample complexity of learning graphical games from purely behavioral data. We assume that we can only observe the players' joint actions and not their payoffs. We analyze the sufficient and necessary number of samples for the correct recovery of the set of pure-strategy Nash equilibria (PSNE) of the true game. Our analysis focuses on directed graphs with $n$ nodes and at most $k$ parents per node. Sparse graphs correspond to ${k \in O(1)}$ with respect to $n$, while dense graphs correspond to ${k \in O(n)}$. By using VC dimension arguments, we show that if the number of samples is greater than ${O(k n \log^2{n})}$ for sparse graphs or ${O(n^2 \log{n})}$ for dense graphs, then maximum likelihood estimation correctly recovers the PSNE with high probability. By using information-theoretic arguments, we show that if the number of samples is less than ${\Omega(k n \log^2{n})}$ for sparse graphs or ${\Omega(n^2 \log{n})}$ for dense graphs, then any conceivable method fails to recover the PSNE with arbitrary probability.

We provide the first fully polynomial time approximation scheme (FPTAS) for computing an approximate mixed-strategy Nash equilibrium in graphical multi-hypermatrix games (GMhGs), which are generalizations of normal-form games, graphical games, graphical polymatrix games, and hypergraphical games. Computing an exact mixed-strategy Nash equilibria in graphical polymatrix games is PPAD complete and thus generally believed to be intractable. In contrast, to the best of our knowledge, we are the first to establish an FPTAS for tree polymatrix games as well as tree graphical games when the number of actions is bounded by a constant. As a corollary, we give a quasi-polynomial time approximation scheme (quasi-PTAS) when the number of actions is bounded by a logarithm of the number of players.