# Evolutionary Game Theory: A Renaissance

Structure of this survey. A model of behavior answers the question who does what to whom and in what circumstances? Explicit rules of behavior address the circumstances of every state. Equilibrium models address the circumstances of equilibrium. The remaining parts of the question are addressed by the sections of this survey as given above.

Coalitional Stochastic Stability. There are three states, x , y and z , all assumed to be rest points of an individualistic best response dynamic. Arrows between states indicate, for the most probable transition path from one state to another, e , the number of random errors on this transition path, and c , the size of the largest coalition that makes a coalitional response on this path.

Two player prisoner’s dilemma. Let β , γ > 0 . For each combination of C and D , entries give payoffs for the row player. β is the payoff advantage of ( C , C ) over ( D , D ) . γ is the gain from defecting on a cooperator. The gain from defecting on a defector is normalized to 1.

Two player coordination game. For each combination of A and B , entries give payoffs for the row player. Let α > 0 so that ( A , A ) is the risk dominant Nash equilibrium. Panel ( i ): general payoffs up to affine transformation; Panel ( ii ): zero payoff for miscoordination; Panel ( iii ): a stag hunt.

Conservative and reforming effects of collective agency. Grey vertices are playing strategy A , white vertices are playing B , as are all vertices not shown. Payoffs are given by the game in Figure 4 ii. Panel ( i ): an initial strategy profile. Without coalitional behavior, it is not a best response for any player to change strategy as long as α 2 . Panel ( ii ): Conservative effect. From the profile in Panel (i), if α 1 , then all of the members of coalition T 1 gain by switching to B . Panel ( iii ): Reforming effect. From the profile in Panel (i), if α > 1 2 , then all of the members of coalition T 2 gain by switching to A .

Assortativity in matching. From a population, individuals are matched into groups to interact ( Section 3.3 ). This matching may be affected by the traits of the individuals ( Section 3.2.1 ) and also by institutions at a societal level ( Section 3.2.2 ). Individuals may be able to choose to join institutions ( Section 3.2.3 ) and the institutions may have their own preferences over their membership. Note that the institution in the Figure is highly positively assortative in that it matches individuals with other individuals of a similar type. Individuals may have the option to interact multiple times with those with whom they are matched, or to leave them and seek new partners ( Section 3.4 ).

The evolution of behavior. From a population, individuals are matched into groups to interact ( Section 3 ). In some cases, the entire population will constitute a single group. Groups of matched players then play a game. How the game is played within a group may depend on the traits of individuals within the group, which may be genetic ( Section 4.1 ) or cultural ( Section 4.3 ). How the game is played may also depend on cultural conventions based on how the game has been played in the past ( Section 4.2 ). Strategies are reproduced through intergenerational transmission or through individuals following some rule of strategic adjustment such as imitation or best response.

Length, steepness and cost of transitions between conventions. The state space is the set of integers from 0 to 11. The only transitions that occur with positive probability are between adjacent states. The cost of a transition, the exponential decay rate of its probability, equals the change in height on the vertical axis if this quantity is positive. Otherwise, the cost is zero. For example, the cost of 9 → 10 is two and the cost of 10 → 9 is zero. The length of the path from x to y is one, as only a single transition on the path, 2 → 3 , has strictly positive cost and is therefore an error . However, this transition is relatively steep , with a cost of three. In contrast, the path from y to z has length two as both 6 → 7 and 7 → 8 are errors, but is less steep as each error only has a cost of one, thus the total cost of the path is two. States x and y minimize stochastic potential and are thus stochastically stable .

Two player coordination game with heterogeneous preferences. Let γ i , γ j ∈ ( 0 , 1 ) . For each combination of A and B , entries give payoffs for the row player and column player respectively. Note that if γ i 1 2 γ j or vice versa, this game is a Battle of the Sexes.

Two player coordination game with payoff restrictions. For each combination of A and B , entries give payoffs for the row player. Let a > b > c > d and a − c > b − d so that ( A , A ) is the risk dominant and payoff dominant Nash equilibrium, but B is the maximin strategy. This corresponds to setting β > α > 0 in our general coordination game in Figure 4 [i].

Stochastic potential under perturbed best response for coordination games under infrequent rematching. The game in Figure 10 is played by a population of size n = 8 . As rematching is infrequent, any updating individual chooses the same action as his current partner, so a path escaping the basin of attraction of any convention only requires a single error. In Panel ( i ), as all errors have equal steepness, this results in both conventions being stochastically stable. In Panel ( ii ), payoff losses b − d and a − c are compared, resulting in the risk dominant convention being stochastically stable. In Panel ( iii ), current conventional payoffs b and a are compared, resulting in the payoff dominant convention being stochastically stable.

Stochastic potential under perturbed best response for coordination games under rematching every period. The game in Figure 10 is played by a population of size n = 8 . Updating players maximize their expected payoff over all possible opponents and the threshold between basins of attraction is approximated by n b − d a − c + b − d as population size n becomes large. In Panel ( i ), as all errors have equal steepness, the convention with the longer basin of attraction, the risk dominant convention, is stochastically stable. In Panel ( ii ), the path exiting the risk dominant convention is not only longer, but also steeper, so it remains stochastically stable. In Panel ( iii ), for large enough populations, we can ignore the g ( a ) and g ( b ) terms, so that stochastic stability is determined by comparing g ( c ) b − d a − c + b − d and g ( d ) a − c a − c + b − d . In this example, the difference in steepnesses g ( c ) and g ( d ) is large enough that steepness dominates length and the maximin convention is stochastically stable.

The Evolutionary Nash Program. Connections are made between evolutionary game theory and cooperative game theory. For example, sometimes a state space can be derived from an underlying cooperative game. For some evolutionary dynamics, the rest points will correspond to the core of the associated cooperative game. When these dynamics are perturbed, the stochastically stable states will then correspond to a (possibly strict) subset of the core.

Transition costs between conventions. For transitions between conventions in the set , the table gives the cost of a transition from the convention specified by the row to the convention specified by the column. For example, a transition from v to x has a cost of 4.

Cyclic decomposition. The unperturbed dynamic converges to conventions in the set . A directed edge from a convention corresponds to a least cost transition from this convention as per Figure 15 . This decomposes the set of conventions into two sets, and . Transition costs between these two sets are determined as described in the main text.

Information and context in an experiment. If only the shaded elements pertain, this is sufficient to make it possible that players follow an individualistic best response dynamic. However, all of the elements are compatible with players following such a dynamic, so there is nothing wrong with including any of them in a situational context that is being explored. Some of the elements, such as telling subjects individual best responses, might be expected to work towards inducing such a dynamic, whereas others, such as allowing subjects to talk, might be expected to work against it and in favor of some other dynamic such as coalitional best response.

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