The Evolutionary Price of Anarchy: Locally Bounded Agents in a Dynamic Virus Game

The Price of Anarchy (PoA) is a well-established game-theoretic concept to shed light on coordination issues arising in open distributed systems. Leaving agents to selfishly optimize comes with the risk of ending up in sub-optimal states (in terms of performance and/or costs), compared to a centralized system design. However, the PoA relies on strong assumptions about agents\u27 rationality (e.g., resources and information) and interactions, whereas in many distributed systems agents interact locally with bounded resources. They do so repeatedly over time (in contrast to "one-shot games"), and their strategies may evolve. Using a more realistic evolutionary game model, this paper introduces a realized evolutionary Price of Anarchy (ePoA). The ePoA allows an exploration of equilibrium selection in dynamic distributed systems with multiple equilibria, based on local interactions of simple memoryless agents. Considering a fundamental game related to virus propagation on networks, we present analytical bounds on the ePoA in basic network topologies and for different strategy update dynamics. In particular, deriving stationary distributions of the stochastic evolutionary process, we find that the Nash equilibria are not always the most abundant states, and that different processes can feature significant off-equilibrium behavior, leading to a significantly higher ePoA compared to the PoA studied traditionally in the literature

Evolutionary Games on Networks and Payoff Invariance Under Replicator Dynamics

The commonly used accumulated payoff scheme is not invariant with respect to shifts of payoff values when applied locally in degree-inhomogeneous population structures. We propose a suitably modified payoff scheme and we show both formally and by numerical simulation, that it leaves the replicator dynamics invariant with respect to affine transformations of the game payoff matrix. We then show empirically that, using the modified payoff scheme, an interesting amount of cooperation can be reached in three paradigmatic non-cooperative two-person games in populations that are structured according to graphs that have a marked degree inhomogeneity, similar to actual graphs found in society. The three games are the Prisoner's Dilemma, the Hawks-Doves and the Stag-Hunt. This confirms previous important observations that, under certain conditions, cooperation may emerge in such network-structured populations, even though standard replicator dynamics for mixing populations prescribes equilibria in which cooperation is totally absent in the Prisoner's Dilemma, and it is less widespread in the other two games.Comment: 20 pages, 8 figures; to appear on BioSystem

Coalitions, tipping points and the speed of evolution

This study considers pure coordination games on networks and the waiting time for an adaptive process of strategic change to achieve efficient coordination. Although it is in the interest of every player to coordinate on a single globally efficient norm, coalitional behavior at a local level can greatly slow, as well as hasten convergence to efficiency. For some networks, when one action becomes efficient enough relative to the other, the effect of coalitional behavior changes abruptly from a conservative effect to a reforming effect. These effects are confirmed for a variety of stylized and empirical social networks found in the literature. For coordination games in which the Pareto efficient and risk dominant equilibria differ, polymorphic states can be the only stochastically stable states

Characterizing Strategic Cascades on Networks

Transmission of disease, spread of information and rumors, adoption of new products, and many other network phenomena can be fruitfully modeled as cascading processes, where actions chosen by nodes influence the subsequent behavior of neighbors in the network graph. Current literature on cascades tends to assume nodes choose myopically based on the state of choices already taken by other nodes. We examine the possibility of strategic choice, where agents representing nodes anticipate the choices of others who have not yet decided, and take into account their own influence on such choices. Our study employs the framework of Chierichetti et al. [2012], who (under assumption of myopic node behavior) investigate the scheduling of node decisions to promote cascades of product adoptions preferred by the scheduler. We show that when nodes behave strategically, outcomes can be extremely different. We exhibit cases where in the strategic setting 100% of agents adopt, but in the myopic setting only an arbitrarily small epsilon % do. Conversely, we present cases where in the strategic setting 0% of agents adopt, but in the myopic setting (100-epsilon)% do, for any constant epsilon > 0. Additionally, we prove some properties of cascade processes with strategic agents, both in general and for particular classes of graphs.Comment: To appear in EC 201

A Note on the KKT Points for the Motzkin-Straus Program

In a seminal 1965 paper, Motzkin and Straus established an elegant connection between the clique number of a graph and the global maxima of a quadratic program defined on the standard simplex. Since then, the result has been the subject of intensive research and has served as the motivation for a number of heuristics and bounds for the maximum clique problem. Most of the studies available in the literature, however, focus typically on the local/global solutions of the program, and little or no attention has been devoted so far to the study of its Karush-Kuhn-Tucker (KKT) points. In contrast, in this paper we study the properties of (a parameterized version of) the Motzkin-Straus program and show that its KKT points can provide interesting structural information and are in fact associated with certain regular sub-structures of the underlying graph

A Game-Theoretic Approach to Pairwise Clustering and Matching

Clustering refers to the process of extracting maximally coherent groups from a set of objects using pairwise, or high-order, similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a predetermined number of classes, thereby obtaining the clusters as a by-product of the partitioning process. In this chapter, we provide a brief review of our recent work which offers a radically different view of the problem and allows one to work directly on non-(geo)metric data. In contrast to the classical approach, in fact, we attempt to provide a meaningful formalization of the very notion of a cluster in the presence of non-metric (even asymmetric and/or negative) (dis)similarities and show that game theory offers an attractive and unexplored perspective that serves well our purpose. To this end, we formulate the clustering problem in terms of a non-cooperative “clustering game” and show that a natural notion of a cluster turns out to be equivalent to a classical (evolutionary) game-theoretic equilibrium concept. Besides the game-theoretic perspective, we exhibit also characterizations of our cluster notion in terms of optimization theory and graph theory. As for the algorithmic issues, we describe two approaches to find equilibria of a clustering game. The first one is based on the classical replicator dynamics from evolutionary game theory, the second one is a novel class of dynamics inspired by infection and immunization processes which overcome their limitations. Finally, we show applications of the proposed framework to matching problems, where we aim at finding correspondences within a set of elements. In particular, we address the problems of point-pattern matching and surface registration