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 EvenOdd Even(+1, 0)(0, +1) Odd(0, +1)(+1, 0) Even Player Odd Player Nash equilibrium: 50% of probability for each strategy

Chicken Game EscalateNot 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 Payoffs Player Against

Chicken Game – Expected Payoffs Player Against

Chicken Game – Expected Payoffs Player Against

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 FightNot 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 Fluctuati ng Lose Gain Frequency (%) 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 players in each of the following classes: (10%; 200f; 0s)‏ (50%; 200f; 0s)‏ (100%; 200f; 0s)‏ The Initial Model

A = (10%; 150f; -5s)‏ B = (50%; 139f; -19s)‏ C = (100%; 209f; 0s)* B x C B does not fight C 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

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 FightNot 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 Players 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

Initial Simulation: Stable Point Analytic Equilibrium

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

ParameterMaynard Smith’s modelEffect on the stable stateSimulation Initial strategyNot applicable.No effect. Initial fitnessInfinite, 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 gainNot 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