# Evolutionary Game Theory: A Renaissance

## Abstract

**:**

## Table of contents

**1 Introdiction**P. 2

**2 Agency—Who Makes Decisions?**p. 3

**4 Evolution of Behavior**p. 17

**5 Economic Applications**p. 29

**6 The Evolutionary Nash Program**p. 30

**7 Behavioral Dynamics**p. 36

**8 General Methodology**p. 44

**9 Empirics**p. 50

**10 Conclusions**p. 54

## 1. Introduction

#### 1.1. The Shadow of Nash Equilibrium

#### 1.2. Renaissance and the Scope of This Survey

#### 1.3. Structure of the Survey

## 2. Agency—Who Makes Decisions?

“... humans are able to coordinate with others, in a way that other primates seemingly are not, to form a “we” that acts as a kind of plural agent to create everything from a collaborative hunting party to a cultural institution.”—Tomasello[342]

**Open**

**Topic 1.**

#### 2.1. Advances in Methodology

#### 2.1.1. Coalitional Stochastic Stability

#### 2.1.2. Coalitional Logit Choice

#### 2.1.3. Frequent or Infrequent Coalitional Behavior

#### 2.2. Implications of Collective Agency

#### 2.2.1. Coordination Games on Networks.

#### 2.2.2. Matching

#### 2.2.3. Social Choice Rules

#### 2.3. The Evolution of Collective Agency

#### 2.4. Links between Individual and Collective Agency

**Open**

**Topic 2.**

## 3. Assortativity—With Whom Does Interaction Occur?

#### 3.1. Assortativity and Preferences

#### 3.2. Evolution of Assortativity

#### 3.2.1. Individual Types and Assortativity

#### 3.2.2. Institutions that Determine Assortativity

#### 3.2.3. Choosing an Institution

**Open**

**Topic 3.**

#### 3.3. Generalized Assortative Matching Protocols

#### 3.4. Conditional Dissociation

#### 3.5. Network Formation

## 4. Evolution of Behavior

#### 4.1. Evolution of Traits

#### 4.1.1. Self-Confirming Beliefs

#### 4.1.2. Level k Thinking

#### 4.1.3. Foresight

#### 4.1.4. Competing Cognition

#### 4.1.5. Biases: Overconfidence and Endowment Effects

**Open**

**Topic 4.**

**Open**

**Topic 5.**

#### 4.2. Conventions—Culture Embodied in Society

#### 4.2.1. Perturbed Dynamics and Stochastic Stability

#### 4.2.2. Coordination Games

#### 4.2.3. Communication and Language

#### 4.2.4. Best Shot and Minimum Effort Games

#### 4.2.5. Cournot Oligopoly

#### 4.2.6. Prisoner’s Dilemmas

#### 4.3. Culture Embodied in Individuals

#### 4.4. Interaction of Culture Embodied in Individuals and Society

## 5. Economic Applications

#### 5.1. Macroeconomics, Market Selection and Finance

#### 5.2. Industrial Organization

## 6. The Evolutionary Nash Program

#### 6.1. Recontracting and Nash Demand Games

**Open**

**Topic 6.**

#### 6.2. Transferable Utility Matching—The Assignment Game

#### 6.3. Non-Transferable Utility Matching—Marriage, College Admissions

**Open**

**Topic 7.**

#### 6.4. Bargaining Solutions and Coordination Games

## 7. Behavioral Dynamics

#### 7.1. Reinforcement Learning

#### 7.2. Imitation

#### 7.3. Sampling Equilibrium and Best Experienced Payoff Dynamics

#### 7.4. Best and Better Response

**Open Topic 8.**There is broad scope to develop dynamics that incorporate multiple agency (Section 2) for continuum populations. Such models will answer questions such as: (i) What are sensible ways to model multiple agency (e.g., coalitional updating) in an environment in which interaction is driven by random matching? (ii) When will the presence of multiple agency give different results to individualistic models and when will it make no difference?

#### 7.5. Continuous Strategy Sets

#### 7.6. Completely Uncoupled Dynamics

**NP**problems are the class of problems whose answers can be verified in polynomial time. Daskalakis et al. [95] show that finding a (possibly mixed) Nash equilibrium belongs to a class of problems that they call

**PPAD**-complete, a subset of

**NP**problems. One distinguishing feature of

**PPAD**is that a solution is known to exist for such problems. For example, it is known that a mixed strategy Nash equilibrium exists for finite games (Nash [247]). It is very likely that

**PPAD**is not a subset of the class of problems $\mathbf{P}$ that can be solved by an algorithm that runs in polynomial time (see discussion in Daskalakis et al. [95]).

## 8. General Methodology

#### 8.1. Perturbed Dynamics and Stochastic Stability

#### 8.1.1. Least Cost Transition Paths

#### 8.1.2. Cyclic Decomposition

#### 8.1.3. Convergence Time

#### 8.1.4. Elimination of Weakly Dominated Strategies

#### 8.2. Further Stability Results

#### 8.3. Further Convergence Results

#### 8.4. Distributed control

**Open**

**Topic 9.**

#### 8.5. Software and Simulations

^{®}code (GNU Octave Version 3.4.0 and Gnuplot 4.2.5, Massachusetts Institute of Technology, Cambridge, MA, USA) and documentation publically available for simulating games on networks under coalitional better response dynamics. The full model is a multi-generational group selection model of the evolution of the ability to participate in collective agency (discussed in Section 2.3), but the subroutines that deal with coalitional updating on networks within a single generation build on code written for Newton and Angus [259], which was published without simulations as Newton and Angus [260].

## 9. Empirics

#### 9.1. Best and Better Response

#### 9.2. Imitation

#### 9.3. Completely Uncoupled Dynamics

#### 9.4. Errors in Perturbed Dynamics

**Open**

**Topic 10.**

- (a)
- That the behavior of subjects in context X is approximated by dynamic Y may be non-generic in the sense that small changes to X may lead to the connection being broken. This is one reason, beyond the usual reasons, that replication is important, as any replication will never replicate X exactly (e.g., the weather outside the laboratory will be different). An important question is then the size of the set of contexts containing X that can be approximated by Y. This can be examined by including or excluding elements such as those of Figure 17. Resources permitting, for any positive results (i.e., $X\to Y$ mapping), X can be adjusted until the mapping fails.
- (b)
- Further study of separate attributes and features of decision rules, as discussed in several papers in Section 9.3 and Section 9.4, could be promising. In particular, the cues and information that influence each feature could be studied.
- (c)
- There is much real world time series data that could be considered using evolutionary models.
- (d)
- Theories of the evolution of traits, including preferences, should be tested, as suggested in Open Topic 5.

## 10. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**

**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.

**Figure 2.**

**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.

**Figure 3.**

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

**Figure 4.**

**Two player coordination game.**For each combination of A and B, entries give payoffs for the row player. Let $\alpha >0$ 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.

**Figure 5.**

**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 4ii. Panel (

**i**): an initial strategy profile. Without coalitional behavior, it is not a best response for any player to change strategy as long as $\alpha <2$. Panel (

**ii**): Conservative effect. From the profile in Panel (i), if $\alpha <1$, 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 $\alpha >\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, then all of the members of coalition ${T}_{2}$ gain by switching to A.

**Figure 6.**

**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).

**Figure 7.**

**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.

**Figure 8.**

**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 $9\to 10$ is two and the cost of $10\to 9$ is zero. The length of the path from x to y is one, as only a single transition on the path, $2\to 3$, 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 $6\to 7$ and $7\to 8$ 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.

**Figure 9.**

**Two player coordination game with heterogeneous preferences.**Let ${\gamma}_{i},{\gamma}_{j}\in (0,1)$. For each combination of A and B, entries give payoffs for the row player and column player respectively. Note that if ${\gamma}_{i}<\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.<{\gamma}_{j}$ or vice versa, this game is a Battle of the Sexes.

**Figure 10.**

**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 $\beta >\alpha >0$ in our general coordination game in Figure 4[i].

**Figure 11.**

**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.

**Figure 12.**

**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 $n\frac{b-d}{a-c+b-d}$ 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 $g\left(a\right)$ and $g\left(b\right)$ terms, so that stochastic stability is determined by comparing $g\left(c\right)\frac{b-d}{a-c+b-d}$ and $g\left(d\right)\frac{a-c}{a-c+b-d}$. In this example, the difference in steepnesses $g\left(c\right)$ and $g\left(d\right)$ is large enough that steepness dominates length and the maximin convention is stochastically stable.

**Figure 13.**

**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.

**Figure 14.**

**Conventions and bargaining solutions.**Each bargaining solution can be justified as the stochastically stable state of a coordination game under the corresponding behavioral rule and without reference to any appealing ex-post properties that the solution might have.

Unintentional | Intentional | |
---|---|---|

Uniform | Kalai-Smorodinsky Young [364] | Nash bargaining Naidu et al. [244] |

Logit | Logit bargaining Hwang et al. [179] | Egalitarian Hwang et al. [179] |

**Figure 15.**

**Transition costs between conventions.**For transitions between conventions in the set $\{v,w,x,y,z\}$, 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.

**Figure 16.**

**Cyclic decomposition.**The unperturbed dynamic converges to conventions in the set $\{v,w,x,y,z\}$. 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, $\{v,w,x\}$ and $\{y,z\}$. Transition costs between these two sets are determined as described in the main text.

**Figure 17.**

**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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Evolutionary Game Theory: A Renaissance. *Games* **2018**, *9*, 31.
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Evolutionary Game Theory: A Renaissance. *Games*. 2018; 9(2):31.
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