Evolution of Cooperation, Part 3: Introducing Win-Stay, Lose-Shift
Win-Stay, Lose-Shift (WSLS) is based on a simple idea: if a player does well,
it will keep playing the previous move; it doesn't, it will switch.
By playing this strategy, a player simply monitors its own payoff,
given its current move. Like TFT, it
begins a game by cooperating.
Let's remind ourselves briefly of the payoff matrix of the prisoner's dilemma:
| 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 |
Let's take the perspective of Individual 1 in this table. If the individual is defecting, it "wins" whenever the other individual is cooperating (payoff b), and it "loses" whenever the other individual is defecting (payoff 0). If the individual is cooperating, it "wins" whenever the other individual is cooperating (payoff b-c), and it "loses" whenever the other individual is defecting (payoff -c). Note that winning and losing here are simply assessed by comparing the payoff earned to the one that would have been earned if the opponent had made another move.
If you re-read the preceding paragraph, you'll note that the strategy can be simplified to the following simple rule: If my opponent cooperated in the previous interaction, I will continue doing what I was doing, otherwise I'll switch. In some sense, this is quite a strange strategy - the player is evaluating what would have happened if the other player had played differently, but then goes on to change its own strategy! For example, if a WSLS player defected against another defector, it will in the next round switch to cooperating, which is not how most of us would react.
As we can see in the simulation below, WSLS is the dominating strategy, and is never invaded. Because it starts out by cooperating, almost all interactions are cooperative.
The major reason why WSLS cannot be invaded ALL_C through drift is that it is switches to defection when ALL_C accidentally plays the wrong strategy. All of this, and more, is explained in great detail the book.
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 error_rate = 0.002;
var strategies = ["ALL_C","ALL_D","TFT","WSLS"];
var data = [];
var number_of_game_repeats = 100;
function Individual(strategy, payoff) {
this.strategy = strategy;
this.payoff = payoff;
this.last_move_by_opponent = "";
this.last_move = "";
this.compute_move = function() {
if (Math.random() < error_rate) {
return (Math.random() < 0.5) ? "C" : "D";
}
if (this.strategy == "ALL_D") {
return "D";
}
else if (this.strategy == "ALL_C") {
return "C";
}
else if (this.strategy == "TFT") {
if (this.last_move_by_opponent == "") {
return "C";
}
else {
return this.last_move_by_opponent;
}
}
else if (this.strategy == "WSLS") {
if (this.last_move == "") {
return "C";
}
else {
if (this.last_move_by_opponent == "D") {
return (this.last_move == "C" ? "D" : "C");
}
else {
return this.last_move;
}
}
}
};
this.add_to_payoff = function(game_payoff) {
this.payoff += game_payoff;
};
this.mutate = function() {
do {
var new_s = strategies[get_random_int(0, strategies.length - 1)];
}
while (new_s == this.strategy);
this.strategy = new_s;
};
}
function play_game(individual1, individual2) {
for (var i = 0; i < number_of_game_repeats; i++) {
var move_individual1 = individual1.compute_move();
var move_individual2 = individual2.compute_move();
individual1.last_move_by_opponent = move_individual2;
individual2.last_move_by_opponent = move_individual1;
individual1.last_move = move_individual1;
individual2.last_move = move_individual2;
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("WSLS",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;
current_individual.last_move_by_opponent = "";
current_individual.last_move = "";
}
}
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();
draw_line_chart(data,"time step","frequency",[]);