Evolutionary games with coordination and self-dependent interactions

Critical phenomena and strategy ordering with hub centrality approach in the aspiration-based coordination game

We study the coordination game with an aspiration-driven update rule in regular graphs and scale-free networks. We prove that the model coincides exactly with the Ising model and shows a phase transition at the critical selection noise when the aspiration level is zero. It is found that the critical selection noise decreases with clustering in random regular graphs. With a non-zero aspiration level, the model also exhibits a phase transition as long as the aspiration level is smaller than the degree of graphs. We also show that the critical exponents are independent of clustering and aspiration level to confirm that the coordination game belongs to the Ising universality class. As for scale-free networks, the effect of aspiration level on the order parameter at a low selection noise is examined. In model networks (the Barabási-Albert network and the Holme-Kim network), the order parameter abruptly decreases when the aspiration level is the same as the average degree of the network. In contrast, in real-world networks, the order parameter decreases gradually. We explain this difference by proposing the concepts of hub centrality and local hub. The histogram of hub centrality of real-world networks separates into two parts unlike model networks, and local hubs exist only in real-world networks. We conclude that the difference of network structures in model and real-world networks induces qualitatively different behavior in the coordination game.

Entropy Affects the Competition of Ordered Phases

The effect of entropy at low noises is investigated in five-strategy logit-rule-driven spatial evolutionary potential games exhibiting two-fold or three-fold degenerate ground states. The non-zero elements of the payoff matrix define two subsystems which are equivalent to an Ising or a three-state Potts model depending on whether the players are constrained to use only the first two or the last three strategies. Due to the equivalence of these models to spin systems, we can use the concepts and methods of statistical physics when studying the phase transitions. We argue that the greater entropy content of the Ising phase plays an important role in its stabilization when the magnitude of the Potts component is equal to or slightly greater than the strength of the Ising component. If the noise is increased in these systems, then the presence of the higher entropy state can cause a kind of social dilemma in which the players’ average income is reduced in the stable Ising phase following a first-order phase transition.

Social dilemmas in multistrategy evolutionary potential games

The nature of social dilemmas is studied in n-strategy evolutionary potential games on a square lattice with nearest-neighbor interactions and the logit rule. For symmetric games with symmetric payoff matrices there are no dilemmas because of the coincidence of individual and common interests. The dilemmas are caused by the antisymmetric parts of the self- and cross-dependent payoff components if it modifies the preferred Nash equilibrium. The contentment of players and the emergence of dilemmas in the preferred Nash equilibria are illustrated on some two-dimensional cross sections of the parameter space.

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

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.

Separation of cyclic and starlike hierarchical dominance in evolutionary matrix games

We study antisymmetric components of matrices characterizing pair interactions in multistrategy evolutionary games. Based on the dyadic decomposition of matrices we distinguish cyclic and starlike hierarchical dominance in the appropriate components. In the symmetric matrix games the strengths of these elementary components are determined. The general features and intrinsic symmetries of these interactions are represented by directed graphs. It is found that the variation of a single matrix component modifies simultaneously the strengths of two starlike hierarchical basis games and many other independent rock-paper-scissors type cyclic basis games. The application of the related concepts is illustrated by discussing the three-strategy voluntary prisoner's dilemma.