Evolutionary dynamics: an introduction

Review: Why are coordination failures common? Why are P-improvements not implemented?

• No NE is P-efficient (PD)

• There exists a P-efficient NE that is a P-improvement over the status quo but it is inaccessible (AG)– It is not risk dominant

– Coordinating actions is impossible (‘if we knew how to do that, we would not be poor.’)

• The transformation of the game to support an accessible P-efficient NE may require institutional innovations that subject one or more parties to the risk of a utility loss.

• There may be no mutually acceptable process to determine the sharing of the gains to cooperation. (UG: process-based utility)

• Key idea: Differential replication.

• Selection has been studied mainly in genetics, but of course there is much more to selection than genetical selection. In psychology, for example, trial-and-error learning is simply learning by selection. …In linguistics, selection unceasingly shapes and reshapes phonetics, grammar and vocabulary. In history we see political selection in the rise of Macedonia, Rome, and Muscovy. Similarly economic selection ...causes the rise and fall of firms and products. George Price (1995)

• Key idea: spontaneous order• When I was a child I pictured our language as

settled and passed down by a board of syndics, seated in grave convention along a table in the style of Rembrandt. The picture remained for a while undisturbed by the question what language the syndics might have used in their deliberation.... W.W Quine Preface to Lewis (1969):xi

The big idea

• Q In large populations how do the preferences and institutions co-evolve in the absence of deliberate design: how does (Richard Dawkins’) The Blind Watchmaker work?

– absence of deliberate design means that no individual has preferences over the aggregate outcomes that we study

• We will attempt an answer using the key ideas of co-evolutionary modeling – Group structured populations and multi level selection

– Differential replication of individual and group level characteristics

– Chance (non best response play, mutation, other perturbations) plays a central role in equilibrium selection

– Population level analysis

A puzzle:

• Half of the white residents of LA and a few other U.S. cities prefer neighborhoods with 20 percent or more black residents

• The white ‘ideal’ neighborhood in a U.S sample was 57% white and 16% black.

• Almost none live in such neighborhoods

• Why do we observe equilibrium racial segregation among only weakly discriminatory individuals?

• Objective : to illustrate the explicit analysis of market dynamics (without auctioneer) as a general case of the dynamics of social interactions.

Racial segregation in Manhattan

A model of why greens and blues are not neighbors

• Homes are of identical value other than neighborhood composition effects (f = % green).

• The value of a house for blues and greens, respectively is given by

pb(f) = ½(f + ) - ½(f + )2 + p

pg(f) = ½(f - ) - ½(f - )2 + p

with (0,½) where p is a positive constant.• So the ideal neighborhood for greens (that which

maximizes pg) is composed of ½ + greens, while blues prefer a neighborhood with ½ - greens.

• is thus a measure of discriminatory preferences (because 2is the difference between the two groups’ ideal p.)

The demand for housing:

Pb(0) Pg(1)

Pb Pg

0 1

f

f*

pb(f) = ½(f + ) - ½(f + )2 + p

pg(f) = ½(f - ) - ½(f - )2 + p

A model..cont.: the social interactions and transactions• Those seeking new housing visit the neighborhood in the

same proportions as those currently residing there.• They are randomly paired with a home owner. • If the the current owner values the home less than to the

prospective buyer, it is sold with a probability monotonic in the difference in the two values.

• Only a fraction of the current owners are interested in selling

Market dynamics based on market interactions:

Let b = 1 if pb > pg and is zero otherwise, and g = 1 if pg pb and is zero otherwise. (b + g = 1) Then.

f = f - f(1-f)b(pb - pg) + (1-f)fg (pg - pb)

f next period = f now - greens who sold to a blue + blues who sold to a green

• In the second term on the right hand side f is the number of greens seeking to sell, of these (1-f) will be matched with a blue, and if the blue's price exceeds the greens’ price the sale will take place with probability (pb - pg).

Market dynamics:

f = f - f(1-f )b(pb - pg) + (1-f )fg (pg - pb)• Then rearranging the previous equation, the change in f

is given by the replicator equation

f = f(1-f)(pg - pb)

What are the stationary values of f in this dynamic?

f = 0 if pg = pb (no sales take place among those prospective buyers and sellers of different types who do meet) or f = 0 or f=1 (the neighborhood is visited only by prospective buyers of the same type as the homogeneous population already there.)

Residential segregation in the equilibria of a housing market with non discriminatory preferences: which values of f are stationary and stable?

Pb(0) Pg(1)

Pb Pg

0 1

f

f*

Results of the underlying strategic complementarity:

• The only asymptotically stable equilibria are segregated (true even for arbitrarily small differences in tastes): local homogeneity and global heterogeneity

• Which composition obtains is historically contingent.• There exist values of 0<f<1 which are Pareto superior to

the segregated equilibria but none of these values are stable equilibria: Pareto inferior outcomes.

Pb(0) Pg(1)

Pb Pg

0 1

f

f*

What accounts for the coordination failure in this case?What accounts for the local homogeneity/global heterogeneity (and history-dependence)?

The segregation example illustrates evolutionary modeling

• Non-market social interactions (valuation of housing depends on others’ locational choices)

• Strategic complementarities (positive feedbacks)• Adaptive agents (they knew only their own preferences

and the prices being offered, not the whole system)• Multiple equilibria and path dependence: divergent

dynamics (local homogeneity, global heterogeneity)• Pareto inferior outcomes • Explicit representation of dynamics (no auctioneer) • Punctuated equilibria

Other examples of evolutionary dynamics?

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UN

ION

IZA

TIO

N

RA

TE

BELCANFRAGERITANETNORSWEUK USAFINJPN

Sweden

Finland

Norway

Belgium

United Kingdom Italy

Canada

France

USA

Japan

Germany

Netherlands

1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992

UN

ION

IZA

TIO

N R

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EAn Example of Divergent Dynamics: Union Density

What might account for these trends?

Divergent development: real gdp per capital relative to 1900

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Years

Bulgaria

Czech Republic

Estonia

Hungary

Kazakhstan

Lithuania

Kyrgyz Republic

Moldova

Poland

Romania

Russian Federation

Slovak Republic

Slovenia

Croatia

OECD average

H

P

O

R

M

Source: World Bank SMIA data base

Another example: Winner take all markets, lock-ins, and inefficient outcomes

America Online’s market share is measured from the left, Prodigy’s from the right. Which one wins is path dependent, not based on which one would maximize the value of the network.

Total size of the market at time t0 N

Value of the network = V = n(n-1)

VaVp

Suppose the value of a network to a member is proportional to the number of other members, then the total value of a network with n members is n(n-1).

Sources of evolutionary modeling of economic institutions and behaviors

• Roots 1: Classical game theory: explicit representation of interaction structure, but exaggerated cognitive capacities, overly simple (payoff monotonic) replication, little analysis of institutional ecology.

• Roots 2: Population biology: adaptive agents lacking social learning, limited model of transmission

• Cultural evolution: Boyd and Richerson, Cavalli-Sforza and Feldman, Durham

• Evolutionary game theory: Sugden, Hamilton & Axelrod, Weibull, Young.

Classical and evolutionary game theory• Because generalized increasing returns is a common

phenomenon there are often a great many equilibria, so to make predictions about the world, a principle of equilibrium selection is necessary.

• Classical game theory uses refinements to eliminate some of the equilibria (e.g. the exclusion of treats that are not credible).

• Evolutionary game theory introduces idiosyncratic play and focuses on the accessibility and likely persistence of equilibria (e.g. stochastically stable states are those that will be observed most of the time as idiosyncratic play goes to zero).

• The former selects equilibria by enhancing agents’ cognitive capacities, while the latter does the same task by introducing random behaviors.

A view from biology: Replicators and levels of selection (cultural and genetic transmission; individual and group as units of selection.) what is a replicator?

NB: genes and culture (learned behavior’s) co-evolve, as do individual and group level characteristics (chapters 7, 11-13).

Replicators: a basic tool for studying payoff monotone dynamics • A large population randomly paired in a 2 person

symmetrical game• Two behaviors: x and y with expected payoffs (p=%x)

bx(p) = p(x,x) + (1-p)(x,y)

by(p) = p(y,x) + (1-p)(y,y)

• Updating. Following play, with probability each is paired with a cultural model; if the model is a different type the person switches her trait with probability (B’-B) where B’ is the realized payoff of the model in the previous period if B’-B > 0 and does not switch otherwise.

• Why are realized not equal to expected payoffs?• Matching noise.

The replicator equation• Expected discrete time dynamics (large population)

• p' = p - p(1-p)y>x(by-bx) + p(1-p)(1-y>x )(bx-by)

• Explain the second term.

• p = p'-p = p(1-p)(bx-by)

• p = p(bx-b)

• The replicator gives us the vector field defining for each state the change from that state.

• Stationarity?

• p= 0, p = 1 and bx-by= 0

• Basin of attraction of p* when bx(p*)-by(p*) = 0

• states from which the unperturbed dynamic moves to p*• We postpone stability until next time.

Co-evolution of preferences, beliefs and institutions• Recall

– Beliefs: an individual’s understanding of the relationship between her actions and consequent outcomes.

– Preferences: an individual’s evaluation of the outcomes.

– Institutions: the group level laws, informal rules, and conventions which give a durable structure to social interactions (in part by defining constraints facing individuals)

• Co-evolution: institutions influence individual beliefs and preferences and their evolution; individual beliefs and preferences affect the evolution of institutions

Summing up: Modeling choices

• Content of transactions: complete contracting of all effects or non-market social interactions

• Technology of interactions: non increasing returns to scale or generalized increasing returns (strategic complementarities, positive feedbacks).

• Individual action: rational action or adaptive agents.• Characterization of outcomes: unique asymptotically stable

equilibria or multiple equilibria and path dependent outcomes.

• Representation of dynamics: hypothetical (as if) or explicit (punctuated equilibria).

NB: the italicized choices are characteristic of evolutionary approaches; an example of the alternative choices is the conventional model of Walrasian general equilibrium.

Next time

• Read the remainder of ch 2.

• Time remaining? A practice problem?

Residential Segregation.

pb(f) = ½(f + ) - ½(f + )2 + p (f = % green)

pg(f) = ½(f - ) - ½(f - )2 + p• Confirm that greens prefer f = ½ + show that for = ¼, the

outcome with completely segregated neighborhoods (f=1, f=0) yields the same level of home values as the completely integrated neighborhoods (f=½).

• For < ¼, show that there exists a value of > 0 such that a law prohibiting house sales unless f (½- , ½+) would implement a outcome which is Pareto superior to the competitive equilibrium.

• Suppose that the greens have the same preferences as in the text, with = 0.1 and p = 1 but the blues have all converted to the Love Everybody Equally religion and as a consequence are indifferent to the types of their neighbors and simply value all homes at pb = 1.1. Indicate all the equilibria of the resulting housing market and indicate which are stable in the replicator dynamic in the text (that is, determine the sign of df/df for each stationary value of f).