Dresden 2017 – scientific programme

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SOE: Fachverband Physik sozio-ökonomischer Systeme

SOE 9: Evolutionary Game Theory (joint session SOE / BP / DY)

SOE 9.4: Talk

Tuesday, March 21, 2017, 11:45–12:00, GÖR 226

Family-friendly zero-sum games — •Philipp M. Geiger, Johannes Knebel, Markus F. Weber, and Erwin Frey — Ludwigs-Maximilians-Universität, München, Deutschland

Here we study how network topology determines the long-time coexistence in the antisymmetric Lotka-Volterra equation (ALVE). The ALVE is the replicator equation of zero-sum games, in which interactions are defined by an antisymmetric matrix such that the gain of one strategy equals the loss of a dominated one. The interactions are represented by a weighted network: nodes correspond to strategies, the topology of directed links indicate their dominance relations, and the weights of links define their interaction strengths. Although one generically observes extinction of some nodes, there are network topologies in which all nodes coexist irrespective of the chosen weights. For example, in the rock-paper-scissors game, the network topology is a directed cycle of three nodes. This topology ensures coexistence of all nodes irrespective of the chosen weights.

In our work, we systematically construct nontrivial coexistence networks of the ALVE by mapping its long-time dynamics to an algebraic problem that we analyze by using concepts from graph theory. In particular, we characterize the kernel of an antisymmetric matrix in terms of Pfaffians and their relation to near-perfect matchings. We understand these coexistence networks as ``family-friendly zero-sum games'' in which all strategies coexist due to network topology.