Evolution of Cooperation, Part 1: Cooperators vs. Defectors

The evolution of cooperation is a puzzle at first sight. In a world of the survival of the fittest, it would seem that there is no way that something even resembling cooperation could evolve. And yet, nature is full of cooperation.



As explained in the book, game theory is a great framework to think about the evolution of cooperation. Here, we implement a stochastic, individual-based simulation of two strategies competing against each other. The first strategy, ALL_C, always cooperates. The second strategy, ALL_D, always defects.



We use the following prisoner's dilemma payoff matrix:

Individual 2 Defecting Individual 2 Cooperating Individual 1 Defecting Payoff Individual 1: 0

Payoff Individual 2: 0 Payoff Individual 1: b

Payoff Individual 2: -c Individual 1 Cooperating Payoff Individual 1: -c

Payoff Individual 2: b Payoff Individual 1: b-c

Payoff Individual 2: b-c

cooperators

defectors

Unfortunately, as we can see (and as we should expect), thedon't stand a chance against. Thankfully, as we'll see in the next part , there is a way out of this.

RELOAD SIMULATION

Code

var c = 1; var b = 4; var population = []; var population_size = 100; var number_of_time_steps = 1000; var mutation_rate = 0.001; var strategies = ["ALL_C","ALL_D"]; var data = []; function Individual(strategy, payoff) { this.strategy = strategy; this.payoff = payoff; this.compute_move = function() { return this.strategy == "ALL_C" ? "C" : "D"; }; this.add_to_payoff = function(game_payoff) { this.payoff += game_payoff; }; this.mutate = function() { if (this.strategy == "ALL_C") { this.strategy = "ALL_D"; } else { this.strategy = "ALL_C"; } }; } function play_game(individual1, individual2) { var move_individual1 = individual1.compute_move(); var move_individual2 = individual2.compute_move(); if (move_individual1 == "C") { if (move_individual2 == "C") { individual1.add_to_payoff(b - c); individual2.add_to_payoff(b - c); } else { individual1.add_to_payoff(-c); individual2.add_to_payoff(b); } } else { if (move_individual2 == "C") { individual1.add_to_payoff(b); individual2.add_to_payoff(-c); } } } function init_simulation() { for (var i = 0; i < population_size; i++) { population.push(new Individual("ALL_C",0)); } for (i = 0; i < strategies.length; i++) { data.push([]); } } function run_time_step() { games(); selection(); mutation(); for (i = 0; i < strategies.length; i++) { data[i].push(get_number_with_strategy(strategies[i]) / population_size); } } function games() { for (var i = 0; i < population_size; i++) { var current_individual = population[i]; var other_individual = get_other_individual(current_individual); play_game(current_individual, other_individual); } } function selection() { var temp_population = []; for (var i = 0; i < population_size; i++) { var current_individual = population[i]; var other_individual = get_other_individual(current_individual); if (other_individual.payoff > current_individual.payoff) { temp_population[i] = other_individual; } else { temp_population[i] = current_individual; } } for (i = 0; i < population_size; i++) { current_individual = population[i]; current_individual.strategy = temp_population[i].strategy; current_individual.payoff = 0; } } function mutation() { for (var i = 0; i < population_size; i++) { if (Math.random() < mutation_rate) { var current_individual = population[i]; current_individual.mutate(); } } } function get_other_individual(individual) { do { var random_index = get_random_int(0,population_size-1); var other_individual = population[random_index]; } while (other_individual == individual); return other_individual; } function get_number_with_strategy(strategy) { var count = 0; for (var i = 0; i < population_size; i++) { if (population[i].strategy == strategy) { count++; } } return count; } function get_random_int(min, max) { return Math.floor(Math.random() * (max - min + 1)) + min; } function run_simulation() { init_simulation(); for (var i = 0; i < number_of_time_steps; i++) { run_time_step(); } } run_simulation(); for (var i = 0; i < number_of_time_steps; i++) { console.log(i,"ALL_C",data[0][i],"ALL_D",data[1][i]); } draw_line_chart(data,"time step","frequency",[]);

Note: the draw_line_chart function is built with D3.js and can be found here