Deployments of game-theoretic solution concepts in the real world have highlighted the necessity to consider human opponents' boundedly rational behavior. If subrationality is not addressed, the system can face significant losses in terms of expected utility. While there exist algorithms for computing optimal strategies to commit to when facing subrational decision-makers in one-shot interactions, these algorithms cannot be generalized for solving sequential scenarios because of the inherent curse of strategy-space dimensionality in sequential games and because humans act subrationally in each decision point separately. We study optimal strategies to commit to against subrational opponents in sequential games for the first time and make the following key contributions: (1) we prove the problem is NP-hard in general; (2) to enable further analysis, we introduce a non-fractional reformulation of the direct non-concave representation of the equilibrium; (3) we identify conditions under which the problem can be approximated in polynomial time in the size of the representation; (4) we show how an MILP can approximate the reformulation with a guaranteed bounded error, and (5) we experimentally demonstrate that our algorithm provides higher quality results several orders of magnitude faster than a baseline method for general non-linear optimization.

Computing Quantal Stackelberg Equilibrium in Extensive-Form Games.

We propose the first, to our knowledge, loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

pdf/approximating-nash-equilibria-in-normal-form-games-via-4zdbq7daav.pdf

Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization

Vaccine delivery in under-resourced locations with security risks is not just challenging but also life threat-ening. The current COVID pandemic and the need to vaccinate have added even more urgency to this issue. Motivated by this problem, we propose a general framework to set-up limited temporary (vacci-nation) centers that balance physical security and desired (vaccine) service coverage with limited resources. We set-up the problem as a Stackelberg game between the centers operator (defender) and an adversary, where the set of centers is not ﬁxed a pri-ori but is part of the decision output. This results in a mixed combinatorial and continuous optimization problem. As part of our scalable approximation of this problem, we provide a fundamental contribution by identifying general duality conditions of switching max and min when both discrete and continuous variables are involved. We perform detailed experiments to show that the solution proposed is scalable in practice.

Safe Delivery of Critical Services in Areas with Volatile Security Situation via a Stackelberg Game Approach

This work studies Stackelberg network interdiction games -- an important class of games in which a defender first allocates (randomized) defense resources to a set of critical nodes on a graph while an adversary chooses its path to attack these nodes accordingly. We consider a boundedly rational adversary in which the adversary's response model is based on a dynamic form of classic logit-based discrete choice models. We show that the problem of finding an optimal interdiction strategy for the defender in the rational setting is NP-hard. The resulting optimization is in fact non-convex and additionally, involves complex terms that sum over exponentially many paths. We tackle these computational challenges by presenting new efficient approximation algorithms with bounded solution guarantees. First, we address the exponentially-many-path challenge by proposing a polynomial-time dynamic programming-based formulation. We then show that the gradient of the non-convex objective can also be computed in polynomial time, which allows us to use a gradient-based method to solve the problem efficiently. Second, we identify a restricted problem that is convex and hence gradient-based methods find the global optimal solution for this restricted problem. We further identify mild conditions under which this restricted problem provides a bounded approximation for the original problem.

pdf/tackling-stackelberg-network-interdiction-against-a-2ra3jzn4.pdf

Tackling Stackelberg Network Interdiction against a Boundedly Rational Adversary

The combination of self-play and planning has achieved great successes in sequential games, for instance in Chess and Go. However, adapting algorithms such as AlphaZero to simultaneous games poses a new challenge. In these games, missing information about concurrent actions of other agents is a limiting factor as they may select different Nash equilibria or do not play optimally at all. Thus, it is vital to model the behavior of the other agents when interacting with them in simultaneous games. To this end, we propose Albatross: AlphaZero for Learning Bounded-rational Agents and Temperature-based Response Optimization using Simulated Self-play. Albatross learns to play the novel equilibrium concept of a Smooth Best Response Logit Equilibrium (SBRLE), which enables cooperation and competition with agents of any playing strength. We perform an extensive evaluation of Albatross on a set of cooperative and competitive simultaneous perfect-information games. In contrast to AlphaZero, Albatross is able to exploit weak agents in the competitive game of Battlesnake. Additionally, it yields an improvement of 37.6% compared to previous state of the art in the cooperative Overcooked benchmark.

Mastering Zero-Shot Interactions in Cooperative and Competitive Simultaneous Games

Solution concepts of traditional game theory assume entirely rational players; therefore, their ability to exploit subrational opponents is limited. One type of subrationality that describes human behavior well is the quantal response. While there exist algorithms for computing solutions against quantal opponents, they either do not scale or may provide strategies that are even worse than the entirely-rational Nash strategies. This paper aims to analyze and propose scalable algorithms for computing effective and robust strategies against a quantal opponent in normal-form and extensive-form games. Our contributions are: (1) we define two different solution concepts related to exploiting quantal opponents and analyze their properties; (2) we prove that computing these solutions is computationally hard; (3) therefore, we evaluate several heuristic approximations based on scalable counterfactual regret minimization (CFR); and (4) we identify a CFR variant that exploits the bounded opponents better than the previously used variants while being less exploitable by the worst-case perfectly-rational opponent.

Complexity and Algorithms for Exploiting Quantal Opponents in Large Two-Player Games

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Complexity and Algorithms for Exploiting Quantal Opponents in Large Two-Player Games