Moderate Responder Committees Maximize Fairness in (NxM)-Person Ultimatum Games

We introduce and study a multiplayer version of the classical Ultimatum Game in which a group of N Proposers jointly offers a division of resources to a group of M Responders. In general, the proposal is rejected if the (average) proposed offer is lower than the (average) response threshold in the Responders group. A motivation for our work is the exchange of flexibilities between different smart energy communities, where the surplus of one community can be offered to meet the demand of a second community. We find that, in the absence of any mechanism, the co-evolving populations of Proposers and Responders converge to a state in which proposals and acceptance thresholds are low, as predicted by the rational choice theory. This is more evident if the Proposers' groups are larger (i.e., large N). Low proposals imply an unfair exchange that highly favors the Proposers. To circumvent this drawback, we test different committee selection rules which determine how Responders should be selected to form decision-making groups, contingent on their declared acceptance thresholds. We find that selecting the lowest-demanding Responders maintains unfairness. However, less trivially, selecting the highest-demanding individuals also fails to resolve this imbalance and yields a worse outcome for all due to a high fraction of rejected proposals. Selecting moderate Responders optimizes overall fitness

Efficient evolutionary dynamics with extensive-form games

Evolutionary game theory combines game theory and dynamical systems and is customarily adopted to describe evolutionary dynamics in multi-agent systems. In particular, it has been proven to be a successful tool to describe multi-agent learning dynamics. To the best of our knowledge, we provide in this paper the first replicator dynamics applicable to the sequence form of an extensive-form game, allowing an exponential reduction of time and space w.r.t. the currently adopted replicator dynamics for normal form. Furthermore, our replicator dynamics is realization equivalent to the standard replicator dynamics for normal form. We prove our results for both discrete-time and continuous-time cases. Finally, we extend standard tools to study the stability of a strategy profile to our replicator dynamics

A Generalised Method for Empirical Game Theoretic Analysis

This paper provides theoretical bounds for empirical game theoretical analysis of complex multi-agent interactions. We provide insights in the empirical meta game showing that a Nash equilibrium of the meta-game is an approximate Nash equilibrium of the true underlying game. We investigate and show how many data samples are required to obtain a close enough approximation of the underlying game. Additionally, we extend the meta-game analysis methodology to asymmetric games. The state-of-the-art has only considered empirical games in which agents have access to the same strategy sets and the payoff structure is symmetric, implying that agents are interchangeable. Finally, we carry out an empirical illustration of the generalised method in several domains, illustrating the theory and evolutionary dynamics of several versions of the AlphaGo algorithm (symmetric), the dynamics of the Colonel Blotto game played by human players on Facebook (symmetric), and an example of a meta-game in Leduc Poker (asymmetric), generated by the PSRO multi-agent learning algorithm.Comment: will appear at AAMAS'1

Evolution of coordination in pairwise and multi-player interactions via prior commitments

Upon starting a collective endeavour, it is important to understand your partners' preferences and how strongly they commit to a common goal. Establishing a prior commitment or agreement in terms of posterior benefits and consequences from those engaging in it provides an important mechanism for securing cooperation. Resorting to methods from Evolutionary Game Theory (EGT), here we analyse how prior commitments can also be adopted as a tool for enhancing coordination when its outcomes exhibit an asymmetric payoff structure, in both pairwise and multiparty interactions. Arguably, coordination is more complex to achieve than cooperation since there might be several desirable collective outcomes in a coordination problem (compared to mutual cooperation, the only desirable collective outcome in cooperation dilemmas). Our analysis, both analytically and via numerical simulations, shows that whether prior commitment would be a viable evolutionary mechanism for enhancing coordination and the overall population social welfare strongly depends on the collective benefit and severity of competition, and more importantly, how asymmetric benefits are resolved in a commitment deal. Moreover, in multiparty interactions, prior commitments prove to be crucial when a high level of group diversity is required for optimal coordination. The results are robust for different selection intensities. Overall, our analysis provides new insights into the complexity and beauty of behavioral evolution driven by humans' capacity for commitment, as well as for the design of self-organised and distributed multi-agent systems for ensuring coordination among autonomous agents

A Generalized Training Approach for Multiagent Learning

This paper investigates a population-based training regime based on game-theoretic principles called Policy-Spaced Response Oracles (PSRO). PSRO is general in the sense that it (1) encompasses well-known algorithms such as fictitious play and double oracle as special cases, and (2) in principle applies to general-sum, many-player games. Despite this, prior studies of PSRO have been focused on two-player zero-sum games, a regime wherein Nash equilibria are tractably computable. In moving from two-player zero-sum games to more general settings, computation of Nash equilibria quickly becomes infeasible. Here, we extend the theoretical underpinnings of PSRO by considering an alternative solution concept, $\alpha$-Rank, which is unique (thus faces no equilibrium selection issues, unlike Nash) and applies readily to general-sum, many-player settings. We establish convergence guarantees in several games classes, and identify links between Nash equilibria and $\alpha$-Rank. We demonstrate the competitive performance of $\alpha$-Rank-based PSRO against an exact Nash solver-based PSRO in 2-player Kuhn and Leduc Poker. We then go beyond the reach of prior PSRO applications by considering 3- to 5-player poker games, yielding instances where $\alpha$-Rank achieves faster convergence than approximate Nash solvers, thus establishing it as a favorable general games solver. We also carry out an initial empirical validation in MuJoCo soccer, illustrating the feasibility of the proposed approach in another complex domain