LP-based Covering Games with Low Price of Anarchy

We present a new class of vertex cover and set cover games. The price of anarchy bounds match the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs -- in contrast to all previously studied covering games, where the price of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In particular, we describe a vertex cover game with a price of anarchy of 2. The rules of the games capture the structure of the linear programming relaxations of the underlying optimization problems, and our bounds are established by analyzing these relaxations. Furthermore, for linear costs we exhibit linear time best response dynamics that converge to these almost optimal Nash equilibria. These dynamics mimic the classical greedy approximation algorithm of Bar-Yehuda and Even [3]

Cyclic game dynamics driven by iterated reasoning

Recent theories from complexity science argue that complex dynamics are ubiquitous in social and economic systems. These claims emerge from the analysis of individually simple agents whose collective behavior is surprisingly complicated. However, economists have argued that iterated reasoning--what you think I think you think--will suppress complex dynamics by stabilizing or accelerating convergence to Nash equilibrium. We report stable and efficient periodic behavior in human groups playing the Mod Game, a multi-player game similar to Rock-Paper-Scissors. The game rewards subjects for thinking exactly one step ahead of others in their group. Groups that play this game exhibit cycles that are inconsistent with any fixed-point solution concept. These cycles are driven by a "hopping" behavior that is consistent with other accounts of iterated reasoning: agents are constrained to about two steps of iterated reasoning and learn an additional one-half step with each session. If higher-order reasoning can be complicit in complex emergent dynamics, then cyclic and chaotic patterns may be endogenous features of real-world social and economic systems.Comment: 8 pages, 4 figures, and supplementary informatio

Asymptotic Convergence and Performance of Multi-Agent Q-Learning Dynamics

Achieving convergence of multiple learning agents in general $N$-player games is imperative for the development of safe and reliable machine learning (ML) algorithms and their application to autonomous systems. Yet it is known that, outside the bounds of simple two-player games, convergence cannot be taken for granted. To make progress in resolving this problem, we study the dynamics of smooth Q-Learning, a popular reinforcement learning algorithm which quantifies the tendency for learning agents to explore their state space or exploit their payoffs. We show a sufficient condition on the rate of exploration such that the Q-Learning dynamics is guaranteed to converge to a unique equilibrium in any game. We connect this result to games for which Q-Learning is known to converge with arbitrary exploration rates, including weighted Potential games and weighted zero sum polymatrix games. Finally, we examine the performance of the Q-Learning dynamic as measured by the Time Averaged Social Welfare, and comparing this with the Social Welfare achieved by the equilibrium. We provide a sufficient condition whereby the Q-Learning dynamic will outperform the equilibrium even if the dynamics do not converge.Comment: Accepted in AAMAS 202

Game Manipulators -- the Strategic Implications of Binding Contracts

Commitment devices are powerful tools that can influence and incentivise certain behaviours by linking them to rewards or punishments. These devices are particularly useful in decision-making, as they can steer individuals towards specific choices. In the field of game theory, commitment devices can alter a player's payoff matrix, ultimately changing the game's Nash equilibria. Interestingly, agents, whom we term game manipulators and who can be external to the original game, can leverage such devices to extract fees from players by making them contingent offers that modify the payoffs of their actions. This can result in a different Nash equilibrium with potentially lower payoffs for the players compared to the original game. For this scheme to work, it is required that all commitments be binding, meaning that once an offer is made, it cannot be revoked. Consequently, we analyse binding contracts as the commitment mechanism that enables game manipulation scenarios. The main focus of this study is to formulate the logic of this setting, expand its scope to encompass more intricate schemes, and analyse the behaviour of regret-minimizing agents in scenarios involving game manipulation

The graph structure of two-player games

In this paper we analyse two-player games by their response graphs. The response graph has nodes which are strategy profiles, with an arc between profiles if they differ in the strategy of a single player, with the direction of the arc indicating the preferred option for that player. Response graphs, and particularly their sink strongly connected components, play an important role in modern techniques in evolutionary game theory and multi-agent learning. We show that the response graph is a simple and well-motivated model of strategic interaction which captures many non-trivial properties of a game, despite not depending on cardinal payoffs. We characterise the games which share a response graph with a zero-sum or potential game respectively, and demonstrate a duality between these sets. This allows us to understand the influence of these properties on the response graph. The response graphs of Matching Pennies and Coordination are shown to play a key role in all two-player games: every non-iteratively-dominated strategy takes part in a subgame with these graph structures. As a corollary, any game sharing a response graph with both a zero-sum game and potential game must be dominance-solvable. Finally, we demonstrate our results on some larger games.Comment: 16 pages, 11 figure