Evolutionary games on graphs and discrete dynamical systems

Abstract

Evolutionary games on graphs play an important role in the study of evolution of cooperation in applied biology. Using rigorous mathematical concepts from a dynamical systems and graph theoretical point of view, we formalize the notions of attractor, update rules and update orders. We prove results on attractors for different utility functions and update orders. For complete graphs we characterize attractors for synchronous and sequential update rules. In other cases (for k-regular graphs or for different update orders) we provide sufficient conditions for attractivity of full cooperation and full defection. We construct examples to show that these conditions are not necessary. Finally, by formulating a list of open questions we emphasize the advantages of our rigorous approach.

Unpleasant Monetarist Arithmetic: Macroprudential Edition

“…It has been shown extensively that discrete settings may exhibit richer behaviour and be more relevant for many real-world applications, see e.g. Epperlein et al (2015). Our analysis provides one such example in the area of macroeconomics, showing that even a small degree of discreteness may qualify conventional findings substantially.…”

Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics

“…In standard evolutionary game theory [8,9], infinite homogeneous populations are considered and cooperation and defection coexist in the case of Stag hunt and 804 EPPERLEIN, SIEGMUND, STEHLÍK ANDŠVÍGLER Hawk and dove games (whereby the coexistence equilibrium is unstable in the former and stable in the latter case). In recent years, numerous studies of finite and heterogenous populations (modelled by evolutionary games on graphs) revealed that the introduction of spatial structure could extend the areas of coexistence of cooperation and defection to other social-dilemma games, especially prisoner's dilemma [5,7,13,14,15,16]. We contribute to this line of research and provide constructive proofs showing that, under deterministic imitation dynamics, for every social dilemma parameter cooperation and defection can coexist.…”

“…Putting those ideas together, we can formulate formally an evolutionary game on a graph as a dynamical system in the following way (see [5] for more details):…”

“…In the case of regular graphs, the dynamics is the same, but it differs for irregular graphs, see [5]. (c) Two major examples of update orders (for others see [5,Section 5]) are synchronous (T (t) = V for each t ∈ N 0 ), and sequential (vertices can be ordered so that T (t) = {(t + 1)(mod n)}). However, in this paper we deal with fixed points and our results apply to any update order.…”

Unpleasant Monetarist Arithmetic: Macroprudential Edition

“…In applied mathematics it has long been recognised that various dynamic properties differ in continuous and discrete settings. It has been shown that discrete settings may exhibit richer behaviour and be more relevant for many real‐world applications; for a discussion and references, see Epperlein et al (). Our analysis provides one such example in the area of macroeconomics.…”