Nash equilibria for symmetrix 2 x 2 games

Nash equilibria for symmetrix 2 x 2 games: strategies in social dilemma notation: D: defection (black bullet) Lecture 5 Payoff matrix: C: cooperation (white bullet) R: Reward for mutual cooperation Four regions: 1. ) Harmony games (H): T<1, S>0 one pure Nash equilibrium: CC (no problems) 2. ) Hawk-dove games (HD): T>1, S>0 two equivalent pure Nash equilibria: CD and DC 3. ) Stag hunt game (SH): T<1, S<0 preferred Nash equilibrium: DD if S-T+1<0 CC if S-T+1> 0 4. ) Prisoner’s dilemma (PD): T>1, S<0 one preferred Nash equilibrium: DD representing the tragedy of community Only four types of flow graphs exist! P: Punishment for mutual defection T: Temptation to choose defection S: Sucker’s payoff Simplification: P=0 and R=1

Additional stability criteria in populations for repeated games with Darwinian selection n strategies (species), with N players (N→∞) Strategy i (i=1, …, n) is followed by Ni players. Density vector: ρ=(ρ1, ρ2, …, ρn), where This population dynamics is equivalent to mean-field approximation in physics. The payoff A describes fitness characterizing the capability to create offspring as a result of interactions among them. ρ* is and Evolutionarily Stable Strategy (ESS), if a mutant ρ’ cannot survive, that is, if its fitness is lower, namely Small perturbation in the population ε→ 0 limit Equality: weak ESS This concept considers the stability without defining dynamical details. The concept of ESS was introduced by Maynard Smith and Price, Nature 246 (1973) 15.

After rearranging the previous expression: This can be satisfied for arbitrary ε, if Fitness of ρ* exceeds the fitness of mutants both in its own or in the mutant populations. ρ* ESS, if it exhibits a local maximum in the population (as a function of densities) There can exist many ESSs and there are systems without ESS

Determination of ESS for the Hawk-dove game Three NE: Let us consider the evolutionary stability of the mixed NE Payoff of the second player choosing ρ while her co-player follows the mixed NE: Independent of ρ It is a weak NE

The second criterium of ESS: Thus ρ* is an ESS.

Payoffs for symmetric two-player, two-strategy games for mixed strategies: Social dilemma notation: Two mixed strategies: Payoffs (Ux and Uy) for player x and y as a function of α and β, if T=1. 5 and S=0. 5 (HD game) The mixed NE: α*=0. 5 and β*=0. 5 Notice: if one of the players chooses mixed NE then the other’s payoff is constant (see dashed lines)

Replicator dynamics Taylor and Jonker, Math. Bio. Sci. 40 (1978) 145 In biology: strategy = species and payoff = fitness (offspring creation capability) N strategies (species): ρi (i=1, …, N) is the portion of strategy i in the population. According to the selection rule of Darwin: ρi(t) depends on time and portion of successful species increases at the expense of unsuccessful ones. Taylor’s formula: Maynard-Smith’s formula (difference in time scale) The stationary states (fix points) are the same. Each homogeneous state (when only one species exists, e. g. , ) is a stationary state as there is no payoff difference for one species and all ρi=0 remains zero.

Classification of stationary states Stationary solution (fix point) ρ*: ρ* is stable, if for all open neighborhood U of ρ* there is ρ* is unstable, if it is not stable ρ* is attractive, if there is an open neighborhood U thus Basin of attraction: maximum of U Unstable fixpoint may be attractive see L in the simplex

Replicator dynamics SH=stag hunt, HD=hawk-dove, PD=prisoner’s dilemma, H=harmony Tipical solutions for n=2: Social dilemmas Nash-equilibria Solutions for population dynamics

Replicator dynamics Rock-paper-scissors model (n=3) in the presence of cyclic dominance: Concentric trajectories spiral in spiral out Common features Four stationary solutions: 1 -3) ρ1=1; ρ2=ρ3=0, etc. (homogeneous, unstable) 4) ρ1=ρ2=ρ3=1/3 (stable or unstable) The classification of all solutions is prevented by the large number of possibilities Chaotic solutions can also occur for n>3

Comparison of the stability criteria Dynamical stability versus NE a. ) NEs are fix points b. ) Strict NEs are attractors. (there exists attractor that is not a strict NE) Example with many NEs and fix points strict NE ESS ↓ c. ) If an internal trajectory converges to ρ*, then ρ* is a NE. d. ) Stable fix points are NEs. Dinamical stability vs. ESS a. ) ESSs are attractors. b. ) Internal ESS is a global attractor. (→) c. ) For potential games a fix point is an ESS if and only if it is an attractor d. ) for 2 x 2 matrix games a fix point is an ESS if and only if it is an attractor The above features help the discussion and classification of models

Further examples in the zoo (n=3) [figures made by Dynamo (Sandholm & Dokumaci, 2006)] Black bullets: stable solutions (attractors); White bullets: unstable solutions