EVOLUTIONARY GAMES

Jorgen W. Weibull

New approaches to economic challenges

OECD Headquarters, ParisApril 25, 2014

Themes

1. The economics paradigm and evolutionary game theory

2. Evolutionarily stable strategies

3. Evolutionarily stable family ties: Max Weber meets Charles Darwin

4. Evolutionarily stable balance between self-interest and morality

5. Implications for economic analysis and policy

Background texts

[with lots of references to the literature]

• Weibull: Evolutionary Game Theory. MIT Press, 1995.

• Alger and Weibull: ”Kinship, incentives and evolution”,

American Economic Review, 2010.

• Alger and Weibull: ”Homo moralis - preference evolution under incom-plete information and assortative matching”, Econometrica, 2013.

1 The economics paradigm

• The main paradigm in economics is Bayesian and rationalistic. Foun-

dations:

— John von Neumann and Oskar Morgenstern (1944): Games and

Economic Behavior

— John Nash (1950): “Non-cooperative games”, Ph D thesis (Prince-

ton University)

— Leonard Savage (1954): The Foundations of Statistics

• Each economic agent’s behavior derived from maximization of some

goal function (utility, profit), under given constraints and information

• The ”as if” defence of this paradigm is evolutionary:

— Milton Friedman (1953): The Methodology of Positive Economics

— Firms that do not take profit-maximizing actions are selected against

in the market

— Is this claim right? Under perfect competition? Under imperfect

competition?

2 The evolutionary paradigm

• Formulated by Charles Darwin and combined with game theory by JohnMaynard Smith

— Darwin: non-strategic interactions, like perfect competition in eco-

nomics

— Maynard Smith: strategic interactions, like imperfect competition

in economics

3 Three branches of game theory

• A mathematically formalized theory of strategic interaction

• Applications abound, in economics, political science, biology, and com-puter science

• Non-cooperative, cooperative, and evolutionary game theory

• John Nash’s (1950) Ph D thesis in mathematics at Princeton (”A

Beautiful Mind”)

• Nash’s two interpretations: one rationalistic/individualistic,

one evolutionary/population-statistical

John Nash

(born 1928, PhD 1950)

Director: Ron Howard.

Main actors: Russell Crowe and Jennifer Connelly

Inspired by the 1998 book of the same name by Sylvia Nasar

The film won 4 Academy Rewards

A Beautiful Mind (2001)

Citation from Nash’s Ph D thesis∞

”We shall now take up the ”mass-action” interpretation of equi-

librium points. [...] It is unnecessary to assume that the partic-

ipants have full knowledge of the total structure of the game, or

the ability and inclination to go through any complex reasoning

processes. But the participants are supposed to accumulate em-

pirical information on the relative advantages of the various pure

strategies at their disposal.

To be more detailed, we assume that there is a population (in

the sense of statistics) of participants for each position of the

game. Let us also assume that the ”average playing” of the game

involves participants elected at random from the populations,

and that there is a stable average frequency with which each pure

strategy is employed by the ”average member” of the appropriate

population.

Since there is to be no collaboration between individuals playing

in different positions of the game, the probability that a particular

-tuple of pure strategies will be employed in a playing of the game

should be the product of the probabilities indicating the chance of

each of the pure strategies to be employed in a random playing.

[...]

Thus the assumptions we made in this ”mass-action” interpreta-

tion lead to the conclusion that the mixed strategies representing

the average behavior in each of the populations form an equilib-

rium point.”

4 Evolutionary game theory

Evolutionary process = mutation process + selection process

Unit of selection: usually strategies (”strategy evolution”), sometimes goal

functions (”preference evolution”, ”indirect evolution”)

Analytical tools for the researcher:

1. Evolutionary stability: focus on mutations

2. Replicator dynamic: focus on selection

3. Stochastic stability: both selection and mutations

5 Evolutionarily stable strategies

[Maynard Smith and Price (Nature, 1973)]

Here the unit of selection, the heritable trait, is a behavior, a pure or mixed

strategy in a finite and symmetric two-player game

• ESS = evolutionarily stable strategy

— ”ESS” ≈ “a strategy that ‘cannot be overturned’ once it has be-

come the ‘convention’ in a population

Heuristically

1. A large population of individuals who are recurrently and uniformly

randomly matched in pairs to play a finite and symmetric two-player

game

2. Initially, all individuals always use the same pure or mixed strategy, ,

the incumbent (pure or mixed) strategy

3. Suddenly, a small population share switch to another pure or mixed

strategy, , the mutant (pure or mixed) strategy

4. If the residents/incumbents on average do better (in material payoffs,

fitness) than the mutants, then is evolutionarily stable against

5. is evolutionarily stable if it is evolutionarily stable against all 6=

Formally

A (pure or mixed) strategy is an ESS if

(i) is a best reply to itself, and

(ii) is a better reply to all other best replies to (than they are to

themselves)

⇒ ( ) must constitute a Nash equilibrium, and, in addition, ”fight back”

other best replies

5.1 Examples

5.1.1 Prisoner’s dilemma

- To cooperate or defect?

- To fish aggressively in the common pool, or fish modestly?

3 3 0 4 4 0 2 2

• One ESS: play D. Cooperation is ruled out

5.1.2 Coordination game

- To meet at the good restaurant or at the bad restaurant?

- To stick to the more efficient industrial standard or to the less efficient?

2 2 0 0 0 0 1 1

• Two ESSs: play A, or, alternatively, play B. The inefficient industrialstandard is not ruled out (but the mixed Nash-equilibrium strategy is

ruled out)

5.1.3 Hawk-dove game

- Start-up business with two partners

- Pairs of researchers or workers assigned a common task

To work or shirk?

3 3 0 4 4 0 −1−1

What will happen?

A unique strategy that is a best reply to itself: randomize 50/50 between

”work” and ”shirk” ∗ = (12 12)

This is an ESS if it is also a better reply to all other (pure or mixed)

strategies than they are to themselves

Can be verified that this is the case, by way of calculus

• One ESS: randomize 50/50 between work and shirk

6 Extensions and generalizations

Based on joint work with Ingela Alger (Toulouse School of Economics and

Institute for Advanced Study in Toulouse),

Extend and generalize the notion of evolutionary stability !

(a) from a property of strategies (behaviors) to a property of preferences

and moral values (goal functions), and

(b) from uniform random matching to assortative random matching (here

mutants may be more likely to be matched with mutants)

• Evolutionary stability of family ties in symmetric pairwise interactionsbetween siblings (who know each other)

• Evolutionary stability of preferences and/or moral values in symmetricpairwise interactions between strangers (who do not know each other)

• Evolutionary stability of preferences and/or moral values in symmetric-player interactions between strangers

7 Kinship, incentives and evolution (AER, 2010)

– or ”Max Weber meets Charles Darwin”

How much should one expect siblings to care for each other?

How does their caring influence their economic incentives?

• Particularly important when formal insurance institutions are absent orweak

• Preferences inherited from biological or ”cultural” parents

• Represent family ties between siblings as a degree of altruism/spite

• Assume a sibling attaches unit weight to his/her own material well-being and weight to the material well-being of sibling

• Assume −1 +1

• Evolutionary biology (Hamilton’s rule), suggests siblings behave as if = 12 (their degree of relatedness)

• However, biologists then treat resources as exogenous (exchange econ-omy), while in many situations resources are endogenous (production

economy)

Our model:

• sequential interactions between sibling pairs: individual production,random outputs, voluntary transfers

• material outcomes drive evolutionary selection

• complete information: siblings know each other’s degree of altruism

*Note that there is assortative matching: if a sibling is a rare mutant, and altruism is

inherited from mother or father (with equal probability), then also the other sibling is a

mutant with proba. 1/2

Q: In a given environment: Is there an evolutionarily stable degree of

altruism between siblings? If so, how large and on what does it depend?

A: There is anvolutionarily stable degree ∗ of sibling altruism, and ∗ 12. Moreover, ∗ depends on the ”environment”: lower in harsher (Swe-den) than in milder (Italy)

8 Homo moralis

• Now consider evolutionary stability of preferences and/or moral valueswhen these are private information

• Unlike in the sibling study: make no assumption about the form of

preferences or moral values

• Assume that individuals adjust their behavior according to their per-sonal preferences or moral values, so that play reaches a Nash equilib-

rium under incomplete information

• Allow for arbitrary assortative matching (with uniform matching and

siblings as special cases)

Q: What social preferences and/or moral values should one expect humansto have from first principles?

A: The mathematics leads to a new class of social preferences cum moralvalues, those of homo moralis

• Homo moralis is torn between

— self-interest and

— morality in line with Kant’s categorical imperative

• Homo oeconomicus the special case when all focus is on self interest,with no regard to morality

• We will show that, in a theoretical sense, homo oeconomicus is in factrare

Kant’s categorical imperative

“Act only according to that maxim whereby you can,

at the same time, will that it should become a universal law”

[Grundlegung zur Metaphysik der Sitten, 1785 ]

Immanuel Kant

(1724 – 1804)

8.1 Pairwise interactions

• Individuals randomly matched into pairs

• Each pair plays a symmetric game in material payoffs

• Material payoff ( ) from using strategy against (where is

continuous)

• Material payoff outcomes drive evolution

• Each individual has a type , which defines a continuous goal function ( )

— The goal function may, but need not, depend on (own and others’)

material payoffs

• The type set Θ is rich: it contains all continuous goal functions, in-

cluding that of homo oeconomicus, =

• Each individual’s type is his/her private information

• Each matched pair plays a game of incomplete information

• The probabilistic type-profile in a given individual’s matches may de-pend on whether she is a mutant or not

• Let ∈ [0 1] be the probability, for a given mutant, that the otherparty is another mutant, when mutants are vanishingly rare

• is called the index of assortativity (Bergstrom, American Economic

Review 2003)

— Uniform random matching: = 0

— Siblings: = 05

— ”Cultural parents,” and homophyly: 0 1

Definitions from Alger and Weibull (2013):

• A type is evolutionarily stable if rare mutants fare strictly worse (inmaterial payoffs) than residents in all (Bayesian) Nash equilibria

• A type is evolutionarily unstable if ∃ a mutant type that fares

strictly better (in material payoffs) in all (Bayesian) Nash equilibria

• Given any type ∈ Θ, a behavioral clone is a type 0 ∈ Θ that, as rare

mutant among -individuals, behaves exactly like

8.2 Main result

[Alger and Weibull, 2013]

Theorem 8.1 Suppose the equilibrium behavior of homo moralis, in the

absence of mutants, is uniquely determined. Then

(a) Homo moralis with degree of morality is evolutionarily stable against

all types that are not its behavioral clones.

(b) All types that are not its behavioral clones are evolutionarily unstable.

• So, what, exactly, is a ”homo moralis”? And what is the ”degree of

morality”?

Definition 8.1 A homo moralis is an individual with utility function

( ) = (1− ) · ( ) + · ( )

for some ∈ [0 1], her degree of morality.

• Homo oeconomicus: = 0

• Homo kantientis: = 1

• Homo moralis is torn between selfishness and Kantian morality:

— maximization of own material payoff

— “doing the right thing, in terms of material payoffs, if upheld as a

universal law”

• Intuition for the stability result: HM with = preempts mutants;

does what the most threatening mutant would do

• Intuition for the instability of other types: for any other type there willexist a mutant type who is ”committed” to a strategy/behavior that

fares better (in material terms) and can thus ”break in”

8.3 Taking homo moralis for a short ride

Prisoner’s dilemma

Two homo moralis of equal degree of morality

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

kappa

Pr(C)

Dictator game

Two homo moralis of equal degree of morality

• Random resource allocation so that one is “rich” and one “poor”, with

equal probability for both

• The rich individual decides (dictatorially) how much to give (if at all)to the poor individual

• A strategy is the amount to give if rich

• Continuous, strictly increasing and strictly concave indirect utility ofmoney (=material payoff)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

1

2

3

4

5

6

kappa

x

9 Interactions in groups of arbitrary size

• The same result seems to hold for symmetric interactions for any num-ber ≥ 2 of players

• The notions of symmetry, assortative matching and the definition ofhomo moralis need to be worked out

• Work in progress, see our WP (2014)

10 Implications for economic analysis and policy

1. Evolutionary game theory asks for more than equilibrium; also robust-

ness against ”rare mutations” is asked for, and this can drastically

reduce the set of outcomes.

(a) The efficient equilibrium in coordination games: (i) evolutionary

stability and pre-play communication (Arthur Robson), (ii) ”sto-

chastic stability” (Peyton Young)

(b) The Nash bargaining solution as a result of evolution (”stochastic

stability”, Peyton Young)

2. Evolutionary stability of family ties, with implications for economic

incentives, can be relevant for economic history and development eco-

nomics. (Ingela Alger is working on such a project with economists in

Mexico.)

3. Our homo moralis can potentially make a difference in many areas of

economics and social science, and for policy analysis:

(a) Environmental economics: conventional analysis assumes that in-

dividuals may care about their marginal effects on the environment,

but not what would be ”the right thing to do” if others did likewise

(b) Bargaining, contracts, moral hazard: conventional analysis assumes

pure self-interest, not that parties might, to some extent, also care

about what is ”the right thing to do” [However: see Edgeworth

1881!]

(c) Participation and voting in elections: conventional analysis assumes

that voters only consider the probability of being pivotal and the

cost of voting, not what would be ”the right thing to do” etc.

11 Conclusions

• Our analysis suggests that selfishness is evolutionarily stable only inspecial circumstances, while homo moralis with degree of morality

equal to the index of assortativity is always evolutionarily stable

• Moral preferences may thrive, even under incomplete information andeven in interactions in large groups (even infinite)

• Our analysis permits a new interpretation of Maynard Smith’s and

Price’s ESS - as equilibrium play by homo moralis under incomplete

information

• Lots of new challenges: extensions, applications, tests in laboratory

experiments and on field data...

THE END