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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