Maynard Smith Revisited: Spatial Mobility and Limited Resources Shaping Population Dynamics and Evolutionary Stable Strategies

Pedro Ribeiro de Andrade

October, 2010

Game Theory

“Game theory is math for how people behave strategically” (Bueno de Mesquita)

Raftsmen Playing Cards, 1847, George C. Bingham

Non-cooperative Games Each player has

A finite set of pure strategies A payoff function A mixed strategy

Nash equilibrium

Even OddEven (+1, 0) (0, +1)Odd (0, +1) (+1, 0)

Even Player

Odd Player

Nash equilibrium: 50% of probability for each strategy

Chicken Game

Escalate Not to escalate

Escalate (–10, –10) (+1, –1)Not to

escalate (–1, +1) (0, 0)Player A

Player B

Mixed strategy equilibrium: escalate with 10% of probability

Chicken Game – Expected PayoffsPlayer

Agai

nst

Chicken Game – Expected PayoffsPlayer

Agai

nst

Chicken Game – Expected PayoffsPlayer

Agai

nst

Evolutionary Stable Strategy (ESS)

“An ESS is a strategy such that, if all members of a population adopt it, then no mutant strategy could invade the population under the influence of natural selection.” (Maynard Smith)

Competition among members of a population

Refinement of Nash equilibrium

ESS Interpretation

Two interpretations: Every member fighting 10% of the time 10% always fighting and 90% never fighting

Fight Not to fight

Fight (–10, –10) (+1, –1)

Not to fight (–1, +1) (0, 0)Player A

Player B

Mixed strategy equilibrium: escalate with 10% of probability

ESS: Assumptions Infinite population Pairwise contests Finite set of alternative strategies Symmetric contests Asexual reproduction

“An obvious weakness of the [...] approach [...] is that it places great emphasis on equilibrium states, whereas evolution is a process of continuous, or at least periodic, change.” (Maynard Smith, 1982)

“Is equilibrium attainable?” (Epstein and Hammond, 2002)

Instability of ESS in Small Populations

Source: (Fogel et al 1998)

Compete for What?

Territory owners

Fluctuating

Lose Gain

Freq

uenc

y (%

)

Source: (Odum, 1983; Riechert, 1981)

Scientific Question

Does spatial mobility affect equilibrium?

Proposal

Add space as the resource players compete for

Study the dynamics of this model Mobility as result

of the interaction: satisfaction (s)

Limited fitness (f) Mixed strategies

Players distributed in a 20x20 grid.

1200 players in each of the following classes:

(10%; 200f; 0s)(50%; 200f; 0s)(100%; 200f; 0s)

The Initial Model

The Initial Model

A = (10%; 150f; -5s)B = (50%; 139f; -19s)C = (100%; 209f; 0s)*

B x C

B does not fightC fights

B = (50%; 138f; -20s)C = (100%; 210f; 1s)

B = (50%; 138f; -20s)

B is unsatisfied and decides to move

B = (50%; 138f; 0s)

The Initial Model

The Initial Model

400 cells (20x20) 3600 players 3 mixed strategies (10%, 50%, 100%) 200f initial fitness per player Movement threshold: –20s A player with 0 or less fitness leaves the game The owner of a cell is the one with higher fitness

Fight Not to fight

Fight (–10, –10) (+1, –1)

Not to fight (–1, +1) (0, 0)

The Initial Model: Results

Variation 1: “Infinite” Fitness (1)

Variation 1: “Infinite” Fitness (2)

Extra Fitness

Nash equilibrium does not change

Gains of 0.1, 0.2, 0.4, 0.8, 1.6, and 3.2

Play

ers

after

the

6000

turn

Extra Gain: Owners

Eleven Strategies: Owners in the First Turns

Each player has a strategy randomly chosen from {0%; 10%; ...; 90%; 100%}

Without extra gain

Eleven Strategies: Convergence

Eleven Strategies: Distribution

The Evolutive Model

40%

70%

30%

The Evolutive Model

30%40%

70%

30%

Basic model

The Evolutive Model

Basic model

30%40%

70%

30%

Asexual Non-overlapping Three descendants Mutation of ±10%

40%

30%

20%

Next generation

...

First Simulation: Initially only 1.0 players

0.01.0

Initial Simulation: Stable Point

AnalyticEquilibrium

New ESS Interpretation

Maynard Smith interpretations: Every member fighting 10% of the time 10% always fighting and 90% never fighting

It is possible to have different strategies as long as the average population is stable.

Convergence and the Initial Population

Mutation Change and Probability

Initial Fitness

Extra gain

S ~S

S (–10,–10) (+2,–2)

~S (–2, +2) (0, 0)S ~S

S (–10,–10) (+4,–4)

~S (–4, +4) (0, 0)S ~S

S (–10,–10) (+6,–6)

~S (–6, +6) (0, 0)

Four Different Games

X ESS = X/10

X {2, 4, 6, 8}

S ~S

S (–10,–10) (+8,–8)

~S (–8, +8) (0, 0)

Four Different Games

S ~S

S (–10,–10) (+8,–8)~S (–8, +8) (0, 0)

S ~S

S (–10,–10) (+4,–4)~S (–4, +4) (0, 0)

S ~S

S (–10,–10) (+2,–2)~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+6,–6)~S (–6, +6) (0, 0)

Four Different Games

S ~S

S (–10,–10) (+8,–8)~S (–8, +8) (0, 0)

S ~S

S (–10,–10) (+2,–2)~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+4,–4)~S (–4, +4) (0, 0)

S ~S

S (–10,–10) (+6,–6)~S (–6, +6) (0, 0)

S ~S

S (–10,–10) (+2,–0.2)

~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+2,–0.4)

~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+4,–0.6)

~S (–4, +4) (0, 0)

S ~S

S (–10,–10) (+6,–0.8)

~S (–6, +6) (0, 0)

Range of Analytical Equilibrium Points

X ESS = X/10

X {2, 4, 6, 8}S ~S

S (–10,–10) (+2,–2)

~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+4,–4)

~S (–4, +4) (0, 0)

S ~S

S (–10,–10) (+6,–6)

~S (–6, +6) (0, 0)

S ~S

S (–10,–10) (+8,–8)

~S (–8, +8) (0, 0)

S ~S

S (–10,–10) (+9.2,–9.2)

~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+9.4,–9.4)

~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+9.6,–9.6)

~S (–2, +2) (0, 0)

S ~S

S (–10,–10) (+9.8,–9.8)

~S (–9.8, +9.8) (0, 0)

Range of Analytical Equilibrium Points

Parameter Maynard Smith’s model

Effect on the stable state Simulation

Initial strategy Not applicable. No effect.

Initial fitness Infinite, as resources are not limited.

Inversely proportional to the distance to the theoretical point.

Mutation probability

Zero, although it is considered that a mutation may emerge.

Proportional to the distance to the theoretical equilibrium point.

Mutation change

Zero, although it is considered that a mutation may emerge.

Proportional to the distance to the theoretical point.

Extra gain Not applicable. Proportional to the distance to the theoretical equilibrium point with positive gain. Inversely proportional with negative gain.

Equilibrium point

Not applicable. Logistic curve in the range of theoretical points, with usual stabilization above the theoretical equilibrium point. Cooperation works as an attractor and defection as a repulser.

Summary of the Results

Conclusions More parameters than the equilibrium affect the

model More robust model

Average strategy always converges to a stable state Stable state independent of the initial population

Instead of having only one or two strategies in the population, lots of different strategies can live together

When the parameters tend to infinity, the model converges to the theoretical equilibrium point

Conclusions

“We can only expect some sort of approximate equilibrium, since […] the stability of the average frequencies will be imperfect” (Nash, 1950)

“... [A]gents are naturally heterogeneous... It is not in competition with equilibrium theory... It is economics done in a more general, out-of-equilibrium way. Within this, standard equilibrium behavior becomes a special case.” (Brian Arthur, 2006)

Maynard Smith Revisited: Spatial Mobility and Limited Resources Shaping Population Dynamics and Evolutionary Stable Strategies

Pedro Ribeiro de Andrade

October, 2010