Evolutionary games combining two or three pair coordinations on a square lattice

Abstract

We study multiagent logit-rule-driven evolutionary games on a square lattice whose pair interactions are composed of a maximal number of nonoverlapping elementary coordination games describing Ising-type interactions between just two of the available strategies. Using Monte Carlo simulations we investigate the macroscopic noise-level-dependent behavior of the two-and three-pair games and the critical properties of the continuous phase transtitions these systems exhibit. The four-strategy game is shown to be equivalent to a system that consists of two independent and identical Ising models.

Entropy Affects the Competition of Ordered Phases

“…Additionally, we have observed non-universal behavior in a narrow range of parameters where the interplay between these two different transitions seems to be relevant. It is worth mentioning that similar non-universal critical phase transitions were also reported in several versions of the Ashkin-Teller model [32,[52][53][54] which are also combinations of elementary coordination games [16]. The systematic analysis of the non-universal phase transitions goes beyond the scope of the present work.…”

“…These results support the robustness of Ising-type behavior in composite games with independent elementary coordination components (cf. Reference [15,16] for a counterexample). If α = 1.1, the critical exponent for ρ 3 − (ρ 4 + ρ 5 )/2 was found to be β P = 0.102 (10) [K c = 1.1777 (10)] that is remarkably close to β = 1/9 characteristic to the two-dimensional three-state Potts model [37].…”

“…This payoff matrix contains a coordination-type (Ising) interaction between strategies 1 and 2 with unit strength while the strategy pairs (3,4), (3,5), and (4,5) are coordinated similarly with a strength of α/2 [16]. Our analysis is restricted to α > 0.…”

“…The linear decomposition of payoff matrices [11][12][13][14] can easily reveal such symmetry relations, provides a classification scheme for payoff matrices, and allows us to distinguish fundamentally different types of elementary games [15,16]. This approach generalizes the two-strategy coordination game that enforces players choosing the same strategy.…”

“…These latter models are equivalent to the kinetic Ising model if n = 2 and the logit rule controls the evolution [17][18][19][20][21][22][23][24][25][26][27]. For n strategies, generalized coordination games are composed of two-strategy coordination subgames between all possible strategy pairs [16]. When the logit rule is used [28][29][30], some lattice versions of these models become equivalent to spin systems or lattice gas models (e.g., Potts models [31] or the Ashkin-Teller model [32]) that exhibit thermodynamic behavior [33][34][35].…”