| % % % % % % % % % % % % % % % % % % % | |
| % CODE FOR BEAR, KAGAN & RAND (2017) PROCB | |
| % | |
| % This MATLAB script runs the low mutation limit (steady state) calculations | |
| % shown in Figures S1-S12 | |
| % | |
| % This script calls the following functions: | |
| % | |
| % calculate_payoff_YDYI(strategy1,strategy2,b,c,d,p,YD,YI) | |
| % - returns the payoff of strategy1 against strategy2 given | |
| % the specified values of b, c, d, p, YD, and YI | |
| % | |
| % steady_state_calc(payoffTable,N,w,a) | |
| % - returns the steady state distribution of the population in | |
| % the limit of low mutation, where payoffTable is an MxM matrix | |
| % with payoffTable(i,j) being the payoff of strategy i against | |
| % strategy j; N is the population size; w is the selection | |
| % strength; and a is the amount of assortment | |
| % | |
| % vary_YD(N,b,c,d,w,a) | |
| % - returns and plots the average results from steady_state_calc | |
| % when varying YD and p (with YI fixed at .5) | |
| % | |
| % vary_YI(N,b,c,d,w,a) | |
| % - returns and plots the average results from steady_state_calc | |
| % when varying YI and p (with YD fixed at 1) | |
| % % % % % % % % % % % % % % % % % % % | |
| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |
| %% STEADY STATE CALCULATION %% | |
| %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | |
| % parameters used in Figures S1-S12 | |
| N = 50; | |
| w = 3; | |
| a = 0; | |
| bs = [4,2,8,4,4,4,4]; | |
| cs = [1,1,1,2,.5,1,1]; | |
| ds = [1,1,1,1,1,.75,2]; | |
| for i=1:length(bs) | |
| vary_YD(N,b,c,d,w,a); | |
| vary_YI(N,b,c,d,w,a); | |
| end | |
| % Note: These functions may take a long time to run. |
| function payoff = calculate_payoff_YDYI(s1,s2,b,c,d,p,YD,YI) | |
| %% SETUP %% | |
| % s1 and s2 are agent 1's and agent 2's respective strategy profiles | |
| % where a strategy profile consists of [pR,p1,pIR,pI1,T] | |
| % STRATEGY PARAMETERS % | |
| % pR is probability of cooperating when agent deliberates and realizes the | |
| % game is repeated/coordination game | |
| % p1 is probability of cooperating when agent deliberates and realizes the | |
| % game is 1-shot/social dilemma | |
| % pIR is probability of cooperating when agent uses intuition in a repeated | |
| % game | |
| % pI1 is probability of cooperating when agent uses intuition in a 1 shot | |
| % game | |
| % T is (threshold) max cost in [0,d] agent is willing to pay to deliberate | |
| % ENVIRONMENT/GAME PARAMETERS % | |
| % b is benefit of cooperation | |
| % c is cost of cooperation | |
| % d is max possible cost of deliberation (such that the cost of | |
| % deliberation varies uniformly in the interval [0,d]) | |
| % p is probability of repeated game | |
| % YD is the sensitivity of deliberation | |
| % YI is the sensitivity of intuition | |
| % OUTPUTS % | |
| % payoff is s1's payoff | |
| %%%%%%%%%%%%%%%%%%%%%%% | |
| %% CALCULATE PAYOFFS %% | |
| %%%%%%%%%%%%%%%%%%%%%%% | |
| % Rename each agent's threshold to T1 and T2, respectively | |
| T1 = s1(5); | |
| T2 = s2(5); | |
| %%%% | |
| % Calculate repeated game payoffs | |
| % Repeated Game Payoff Matrix | |
| repPayoffMat=[0 0 | |
| 0 (b-c)]; | |
| % Rows correspond to agent 1's strategy | |
| % where row 1 = agent 1 defects, row 2 = agent 1 cooperates | |
| % Columns correspond to agent 2's strategy | |
| % where col 1 = agent 2 defects, col 2 = agent 2 cooperates | |
| % Values correspond to agent 1's payoffs | |
| % e.g., if agent 1 cooperates and agent 2 defects, | |
| % agent 1 gets a payoff of 0 because repPayoffMat(2,1) = 0 | |
| % We consider repeated-game payoffs in 4 situations: | |
| % rPayDD: both agents deliberate | |
| % rPayDI: only agent 1 deliberates | |
| % rPayID: only agent 2 deliberates | |
| % rPayII: neither agent deliberates | |
| %Given the possibility of innaccurate deliberation and sensitivity of intuition | |
| % rPayDD: | |
| % rPayDtDt: Both agents deliberate, both deliberations track game type | |
| % rPayDtDf: Both agents deliberate, agent 1's deliberation tracks, agent | |
| % 2's fails | |
| % rPayDfDt: Both agents deliberate, agent 1's deliberation fails, | |
| % agent 2's tracks | |
| % rPayDfDf: Both agents deliberate, both deliberations fail | |
| % rPayDI: | |
| % rPayDtIt: 1 deliberates (tracks) 1 intuits (tracks) | |
| % rPayDtIf: 1 deliberates (tracks) 1 intuits (fails) | |
| % rPayDfIt: 1 deliberates (fails) 1 intuits (tracks) | |
| % rPayDfIf: 1 deliberates (fails) 1 intuits (fails) | |
| % rPayID: | |
| % rPayItDt: agent 2 deliberates (tracks), agent 1 intuits (tracks) | |
| % rPayIfDt: agent 2 deliberates (tracks), agent 1 intuits (fails) | |
| % rPayItDf: agent 2 deliberates (fails), agent 1 intuits (tracks) | |
| % rPayIfDf: agent 2 deliberates (fails), agent 1 intuits (fails) | |
| % rPayII: | |
| % rPayItIt: both intuit (both track) | |
| % rPayItIf: both intuit (one tracks, 1 fails) | |
| % rPayIfIt: both intuit (one tracks, 1 fails) | |
| % rPayIfIf: both intuit (both fail) | |
| % E.g., to calculate rPayDfIf, | |
| % We know agent 1 deliberates but fails to reailze they're in a repeated game => | |
| % their prob. of cooperating is p1, i.e., s1(2) | |
| % We know agent 2 uses their intuitive response but fails to realize they're in a repeated game => | |
| % their prob. of cooperating is pI1, i.e., s2(4) | |
| % Thus, rPayDfIf is a weighted average of the payoff agent 1 gets when both | |
| % cooperate (repPayoffMat(2,2)), when only player 1 cooperates | |
| % (repPayoffMat(2,1)), and so on | |
| % where the prob. both cooperate is given by s1(2)*s2(4), prob. only | |
| % player 1 cooperates is given by s1(2)*(1-s2(4)), and so on | |
| rPayDtDt=s1(1)*s2(1)*repPayoffMat(2,2) + s1(1)*(1-s2(1))*repPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(1)*repPayoffMat(1,2) + (1-s1(1))*(1-s2(1))*repPayoffMat(1,1); | |
| rPayDtDf=s1(1)*s2(2)*repPayoffMat(2,2) + s1(1)*(1-s2(2))*repPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(2)*repPayoffMat(1,2) + (1-s1(1))*(1-s2(2))*repPayoffMat(1,1); | |
| rPayDfDt=s1(2)*s2(1)*repPayoffMat(2,2) + s1(2)*(1-s2(1))*repPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(1)*repPayoffMat(1,2) + (1-s1(2))*(1-s2(1))*repPayoffMat(1,1); | |
| rPayDfDf=s1(2)*s2(2)*repPayoffMat(2,2) + s1(2)*(1-s2(2))*repPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(2)*repPayoffMat(1,2) + (1-s1(2))*(1-s2(2))*repPayoffMat(1,1); | |
| rPayDtIt=s1(1)*s2(3)*repPayoffMat(2,2) + s1(1)*(1-s2(3))*repPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(3)*repPayoffMat(1,2) + (1-s1(1))*(1-s2(3))*repPayoffMat(1,1); | |
| rPayDtIf=s1(1)*s2(4)*repPayoffMat(2,2) + s1(1)*(1-s2(4))*repPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(4)*repPayoffMat(1,2) + (1-s1(1))*(1-s2(4))*repPayoffMat(1,1); | |
| rPayDfIt=s1(2)*s2(3)*repPayoffMat(2,2) + s1(2)*(1-s2(3))*repPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(3)*repPayoffMat(1,2) + (1-s1(2))*(1-s2(3))*repPayoffMat(1,1); | |
| rPayDfIf=s1(2)*s2(4)*repPayoffMat(2,2) + s1(2)*(1-s2(4))*repPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(4)*repPayoffMat(1,2) + (1-s1(2))*(1-s2(4))*repPayoffMat(1,1); | |
| rPayItDt=s1(3)*s2(1)*repPayoffMat(2,2) + s1(3)*(1-s2(1))*repPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(1)*repPayoffMat(1,2) + (1-s1(3))*(1-s2(1))*repPayoffMat(1,1); | |
| rPayIfDt=s1(4)*s2(1)*repPayoffMat(2,2) + s1(4)*(1-s2(1))*repPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(1)*repPayoffMat(1,2) + (1-s1(4))*(1-s2(1))*repPayoffMat(1,1); | |
| rPayItDf=s1(3)*s2(2)*repPayoffMat(2,2) + s1(3)*(1-s2(2))*repPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(2)*repPayoffMat(1,2) + (1-s1(3))*(1-s2(2))*repPayoffMat(1,1); | |
| rPayIfDf=s1(4)*s2(2)*repPayoffMat(2,2) + s1(4)*(1-s2(2))*repPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(2)*repPayoffMat(1,2) + (1-s1(4))*(1-s2(2))*repPayoffMat(1,1); | |
| rPayItIt=s1(3)*s2(3)*repPayoffMat(2,2) + s1(3)*(1-s2(3))*repPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(3)*repPayoffMat(1,2) + (1-s1(3))*(1-s2(3))*repPayoffMat(1,1); | |
| rPayItIf=s1(3)*s2(4)*repPayoffMat(2,2) + s1(3)*(1-s2(4))*repPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(4)*repPayoffMat(1,2) + (1-s1(3))*(1-s2(4))*repPayoffMat(1,1); | |
| rPayIfIt=s1(4)*s2(3)*repPayoffMat(2,2) + s1(4)*(1-s2(3))*repPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(3)*repPayoffMat(1,2) + (1-s1(4))*(1-s2(3))*repPayoffMat(1,1); | |
| rPayIfIf=s1(4)*s2(4)*repPayoffMat(2,2) + s1(4)*(1-s2(4))*repPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(4)*repPayoffMat(1,2) + (1-s1(4))*(1-s2(4))*repPayoffMat(1,1); | |
| %Thus: | |
| rPayDD=YD*YD*rPayDtDt + YD*(1-YD)*rPayDtDf + (1-YD)*YD*rPayDfDt + (1-YD)*(1-YD)*rPayDfDf; | |
| rPayDI=YD*YI*rPayDtIt + (1-YD)*YI*rPayDfIt + YD*(1-YI)*rPayDtIf +(1-YD)*(1-YI)*rPayDfIf; | |
| rPayID=YI*YD*rPayItDt + (1-YI)*(1-YD)*rPayIfDf + (1-YI)*YD*rPayIfDt +YI*(1-YD)*rPayItDf; | |
| rPayII=YI*YI*rPayItIt + (1-YI)*YI*rPayIfIt + YI*(1-YI)*rPayItIf + (1-YI)*(1-YI)*rPayIfIf; | |
| %%% | |
| % Calculate 1-shot game payoffs | |
| % (these work the same way as the repeated game payoffs) | |
| % 1-shot Game Payoff Matrix | |
| onesPayoffMat=[0 b | |
| -c (b-c)]; | |
| oPayDtDt=s1(2)*s2(2)*onesPayoffMat(2,2) + s1(2)*(1-s2(2))*onesPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(2)*onesPayoffMat(1,2) + (1-s1(2))*(1-s2(2))*onesPayoffMat(1,1); | |
| oPayDtDf=s1(2)*s2(1)*onesPayoffMat(2,2) + s1(2)*(1-s2(1))*onesPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(1)*onesPayoffMat(1,2) + (1-s1(2))*(1-s2(1))*onesPayoffMat(1,1); | |
| oPayDfDt=s1(1)*s2(2)*onesPayoffMat(2,2) + s1(1)*(1-s2(2))*onesPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(2)*onesPayoffMat(1,2) + (1-s1(1))*(1-s2(2))*onesPayoffMat(1,1); | |
| oPayDfDf=s1(1)*s2(1)*onesPayoffMat(2,2) + s1(1)*(1-s2(1))*onesPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(1)*onesPayoffMat(1,2) + (1-s1(1))*(1-s2(1))*onesPayoffMat(1,1); | |
| oPayDtIt=s1(2)*s2(4)*onesPayoffMat(2,2) + s1(2)*(1-s2(4))*onesPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(4)*onesPayoffMat(1,2) + (1-s1(2))*(1-s2(4))*onesPayoffMat(1,1); | |
| oPayDtIf=s1(2)*s2(3)*onesPayoffMat(2,2) + s1(2)*(1-s2(3))*onesPayoffMat(2,1) + ... | |
| (1-s1(2))*s2(3)*onesPayoffMat(1,2) + (1-s1(2))*(1-s2(3))*onesPayoffMat(1,1); | |
| oPayDfIt=s1(1)*s2(4)*onesPayoffMat(2,2) + s1(1)*(1-s2(4))*onesPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(4)*onesPayoffMat(1,2) + (1-s1(1))*(1-s2(4))*onesPayoffMat(1,1); | |
| oPayDfIf=s1(1)*s2(3)*onesPayoffMat(2,2) + s1(1)*(1-s2(3))*onesPayoffMat(2,1) + ... | |
| (1-s1(1))*s2(3)*onesPayoffMat(1,2) + (1-s1(1))*(1-s2(3))*onesPayoffMat(1,1); | |
| oPayItDt=s1(4)*s2(2)*onesPayoffMat(2,2) + s1(4)*(1-s2(2))*onesPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(2)*onesPayoffMat(1,2) + (1-s1(4))*(1-s2(2))*onesPayoffMat(1,1); | |
| oPayIfDt=s1(3)*s2(2)*onesPayoffMat(2,2) + s1(3)*(1-s2(2))*onesPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(2)*onesPayoffMat(1,2) + (1-s1(3))*(1-s2(2))*onesPayoffMat(1,1); | |
| oPayItDf=s1(4)*s2(1)*onesPayoffMat(2,2) + s1(4)*(1-s2(1))*onesPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(1)*onesPayoffMat(1,2) + (1-s1(4))*(1-s2(1))*onesPayoffMat(1,1); | |
| oPayIfDf=s1(3)*s2(1)*onesPayoffMat(2,2) + s1(3)*(1-s2(1))*onesPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(1)*onesPayoffMat(1,2) + (1-s1(3))*(1-s2(1))*onesPayoffMat(1,1); | |
| oPayItIt=s1(4)*s2(4)*onesPayoffMat(2,2) + s1(4)*(1-s2(4))*onesPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(4)*onesPayoffMat(1,2) + (1-s1(4))*(1-s2(4))*onesPayoffMat(1,1); | |
| oPayItIf=s1(4)*s2(3)*onesPayoffMat(2,2) + s1(4)*(1-s2(3))*onesPayoffMat(2,1) + ... | |
| (1-s1(4))*s2(3)*onesPayoffMat(1,2) + (1-s1(4))*(1-s2(3))*onesPayoffMat(1,1); | |
| oPayIfIt=s1(3)*s2(4)*onesPayoffMat(2,2) + s1(3)*(1-s2(4))*onesPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(4)*onesPayoffMat(1,2) + (1-s1(3))*(1-s2(4))*onesPayoffMat(1,1); | |
| oPayIfIf=s1(3)*s2(3)*onesPayoffMat(2,2) + s1(3)*(1-s2(3))*onesPayoffMat(2,1) + ... | |
| (1-s1(3))*s2(3)*onesPayoffMat(1,2) + (1-s1(3))*(1-s2(3))*onesPayoffMat(1,1); | |
| oPayDD=YD*YD*oPayDtDt + YD*(1-YD)*oPayDtDf + (1-YD)*YD*oPayDfDt + (1-YD)*(1-YD)*oPayDfDf; | |
| oPayDI=YD*YI*oPayDtIt + (1-YD)*YI*oPayDfIt + YD*(1-YI)*oPayDtIf +(1-YD)*(1-YI)*oPayDfIf; | |
| oPayID=YI*YD*oPayItDt + (1-YI)*(1-YD)*oPayIfDf + (1-YI)*YD*oPayIfDt +YI*(1-YD)*oPayItDf; | |
| oPayII=YI*YI*oPayItIt + YI*(1-YI)*oPayItIf + (1-YI)*YI*oPayIfIt + (1-YI)*(1-YI)*oPayIfIf; | |
| %%% | |
| % Calculate agent 1's final payoff based on | |
| % Prob. of repeated game (p) vs. 1-shot game (1-p) | |
| % Prob. agents 1 and 2 deliberate (T1/d and T2/d) vs. use intuition (1-T1/d and 1-T2/d) | |
| % Average costs agent 1 and 2 pay when deliberating (T1/2 and T2/2) | |
| % (because the cost of deliberation varies uniformly, an agent who's | |
| % willing to pay max cost of deliberation T pays an average cost of | |
| % T/2 to deliberate) | |
| % We take a weighted avg. of agent 1's payoff in 4 situations: | |
| % 1. payoff when both agents deliberate = (p*rPayDD + (1-p)*oPayDD - T1/2) | |
| % this happens with probability (T1/d)*(T2/d) | |
| % 2. payoff when only agent 1 deliberates = (p*rPayDI + (1-p)*oPayDI - T1/2) | |
| % this happens with probability (T1/d)*(1 - T2/d) | |
| % ... and so on | |
| payoff = (T1/d)*(T2/d)*(p*rPayDD + (1-p)*oPayDD - T1/2) + ... | |
| (T1/d)*(1 - T2/d)*(p*rPayDI + (1-p)*oPayDI - T1/2) + ... | |
| (1 - T1/d)*(T2/d)*(p*rPayID + (1-p)*oPayID) + ... | |
| (1 - T1/d)*(1 - T2/d)*(p*rPayII + (1-p)*oPayII); | |
| end |
| function [ssDist,transitionPs]=steady_state_calc(payoffTable,N,w,a) | |
| % Calculate the steady state distribution of the Moran process with | |
| % exponential fitness, assuming the limit of low mutation | |
| % | |
| % INPUTS | |
| % payoffTable is an MxM matrix of payoffs, where M is the number of strategies | |
| % N is the population size | |
| % w is the selection strength (0=neutral drift, ->Inf=deterministic) | |
| % a is assortment (probability of interacting with a player having the same | |
| % strategy as you, rather than a randomly selected player) | |
| % | |
| % OUTPUTS | |
| % ssDist is an Mx1 vector with the steady state frequency of each strategy | |
| % transitionPs is an MxM vector with transition probilities between | |
| % homogeneous states of the population | |
| numStrats=length(payoffTable); | |
| transitionPs=zeros(numStrats); | |
| % To do the steady state calc, we need to assemble a transition matrix | |
| % betewen homogeneous states of the population | |
| for s1=1:numStrats | |
| for s2=1:numStrats | |
| if s1~=s2 | |
| % for each combination of strategies s1 and s2, the transition prob | |
| % going from s1 to s2 is the product of the probability a s1 mutant | |
| % arises, and the probability that mutant goes to fixation | |
| %prob s1 mutant arises %prob that s1 goes to fixation | |
| %(assuming uniform mutation) | |
| transitionPs(s1,s2)=(1/(numStrats-1)) *calc_invasionprob(payoffTable,N,w,s1,s2,a); | |
| end | |
| end | |
| end | |
| % The diagonal entries of the transition matrix are the probability you | |
| % stay in the given state - which is just 1 minus all the transition | |
| % probabilities out of that state | |
| for i=1:numStrats | |
| transitionPs(i,i)=1-sum(transitionPs(:,i)); | |
| end | |
| % Now that you've got the transition matrix, find the eigenvector with | |
| % eigenvalue 1, and that's your steady state distribution! | |
| [vec,val]=eig(transitionPs); | |
| [~, idxofEigWithVal1]=min(abs(diag(val)-1)); % For some reason, sometimes Matlab gets the eigenvalue wrong s/t its not literally 1, just very close | |
| % normalize the eigenvector corresponding to eigenvalue so that it sums to 1, and you're done! | |
| ssDist=vec(:,idxofEigWithVal1)./sum(vec(:,idxofEigWithVal1)); | |
| end | |
| function rhoBA=calc_invasionprob(payoffTable,N,w,sA,sB,a) | |
| % here we calculate rhoAB, the probability that a sA mutant goes to fixation in | |
| % a sB population | |
| % this probability is the inverse of (1 + a sum of products) so I | |
| % start by intitalizing rhoInv to 1, and then add on the sum of series of | |
| % products, and then take the inverse | |
| rhoInv=1; | |
| % loop through each possible number of steps the sA mutant would need to go through in order to take over | |
| % (this ranges from 1, if there are already N-1 sA players, | |
| % to N-1, if there is only the 1 initial sA mutant) | |
| for k=1:N-1 | |
| % same thing - initialize the eventual product of various terms as 1, | |
| % then multiply it by stuff | |
| thisProd=1; | |
| % loop through each possible make-up of the population (indexed by k=the | |
| % number of sA players) that would need to be traversed to get from | |
| % there being only 1 sA player to there being k sA players | |
| for i=1:k | |
| % for each population make-up, multiply by the ratio of the fitness | |
| % of an sA player in the current pop to the fitness of an sB player | |
| % in the current pop (where there are i sA players) | |
| % fitness = exp( w * expectedpayoff) | |
| % expectedpayoff = a * payoff against your own strategy | |
| % + (1-a) *( (payoff against sB * fraction of other players who are sB) | |
| % +(payoff against sA * fraction of other players who are sA) | |
| % ) | |
| fi=exp(w* (a*payoffTable(sA,sA)+(1-a)*(payoffTable(sA,sB)*(N-i)/(N-1) + payoffTable(sA,sA)*(i-1)/(N-1)))); | |
| gi=exp(w* (a*payoffTable(sB,sB)+(1-a)*(payoffTable(sB,sB)*(N-i-1)/(N-1) + payoffTable(sB,sA)*(i)/(N-1) ))); | |
| % multiple the current state of the product by the ratio for this value of i | |
| thisProd=thisProd*(gi/fi); | |
| end | |
| % now that we've finished with the product for k, add it onto the sum | |
| rhoInv=rhoInv+thisProd; | |
| end | |
| % now that we've finished with the sum, take the inverse | |
| rhoBA=1/rhoInv; | |
| end |
| function [ssDists,avgStrats] = vary_YD(N,b,c,d,w,a) | |
| % Calculates and plots the average results from steady_state_calc | |
| % when varying YD and p (with YI fixed at .5) | |
| % INPUTS | |
| % N is the population size | |
| % b is benefit of cooperation | |
| % c is cost of cooperation | |
| % d is max possible cost of deliberation (such that the cost of | |
| % deliberation varies uniformly in the interval [0,d]) | |
| % w is the selection strength (0=neutral drift, ->Inf=deterministic) | |
| % a is assortment (probability of interacting with a player having the same | |
| % strategy as you, rather than a randomly selected player) | |
| % | |
| % OUTPUTS | |
| % ssDists stores the ssDist steady state frequencies of each strategy for | |
| % each p x YD combination. | |
| % avgStrats stores the average strategy of of the steady state populations | |
| % for each p x YD combination. | |
| ps_steadyState = 0:.01:1; | |
| YD_steadyState= 0.5:0.01:1; | |
| %% Create strat space | |
| % generate all possible combinations of the first three strat parameters (sR,s1,sI) | |
| % set to be 0 or 1 | |
| [x1,x2,x3] = ind2sub([2,2,2], find(zeros(2,2,2)==0)); | |
| strats = [x1 x2 x3] - 1; | |
| % we now combine the above with the 11 possible deliberation thresholds, | |
| % running from 0 to d in increments of d/10 | |
| Tvals = 0:d/10:d; | |
| strats = repmat(strats,length(Tvals),1); | |
| Tvals = sort(repmat(Tvals,1,length(strats)/length(Tvals))); | |
| strats = [strats Tvals']; | |
| % because we're using the YIYD calc payoff function, have to add in another | |
| % column (4th column) for second intuitive response | |
| strats = [strats(:,1:3) strats(:,3) strats(:,4)]; | |
| % ... to create a total of 2*2*2*11 (88) possible strategies | |
| M = size(strats,1); % define M to be number of strats | |
| %% Calculate steady state distribution | |
| % initialize matrix to store steady state distributions for each p value | |
| % and each YD value | |
| ssDists = zeros(M,length(ps_steadyState),length(YD_steadyState)); | |
| %ssDists = zeros(M ,length(ps_steadyState)); | |
| % initialize matrix to store average strat values for each p value and each | |
| % YD value | |
| avgStrats = zeros(length(ps_steadyState),5,length(YD_steadyState)); | |
| %avgStrats = zeros(length(ps_steadyState),4); | |
| for YD=YD_steadyState | |
| YD % display YD value that's running | |
| for p=ps_steadyState | |
| p % display p value that's running | |
| % initialize payoff table | |
| payoffTable=zeros(M); | |
| % create payoff table | |
| for i=1:M | |
| for j=1:M | |
| payoffTable(i,j)=calculate_payoff_YDYI(strats(i,:),strats(j,:),b,c,d,p,YD,.5); | |
| end | |
| end | |
| % run steady state using this payoff table and store result in ssDists | |
| ssDists(:,p==ps_steadyState,YD==YD_steadyState)=steady_state_calc(payoffTable,N,w,a); | |
| % use ssDists to get average strategy values by taking a weighted average of | |
| % the strat values based on the distribution in ssDists | |
| avgStrats(p==ps_steadyState,:,YD==YD_steadyState) = sum(repmat(ssDists(:,p==ps_steadyState,YD==YD_steadyState),1,5)... | |
| .*strats); | |
| end | |
| end | |
| %% Create Surface Plots | |
| % create a surface showing avg. amount of deliberation in p x YD space | |
| T = squeeze(avgStrats(:,5,:)); % stores this average value | |
| figure | |
| surf(YD_steadyState,ps_steadyState,T) | |
| xlabel('Sensitivity of Deliberation y_d') | |
| ylabel('Probability of Coordination p') | |
| colormap(parula) | |
| view(2); view(-90,90) % change the orientation of the plot... | |
| set(gca,'ydir','reverse') % ... and reverse axis direction for clearer presentation | |
| % create a surface showing closest intuitive profile in p x YD space | |
| intuitiveProfile = round(squeeze(avgStrats(:,3,:))).*round(squeeze(avgStrats(:,4,:))); % stores the intuitive responses as 1,2 | |
| figure | |
| surf(YD_steadyState,ps_steadyState,intuitiveProfile) | |
| xlabel('Sensitivity of Deliberation y_d') | |
| ylabel('Probability of Coordination p') | |
| colormap([.8 0 0; 1 1 0]) % we create a custom colormap where... | |
| % red always intuitively defects and yellow always intuitively cooperates | |
| view(2); view(-90,90) % change the orientation of the plot... | |
| set(gca,'ydir','reverse') % ... and reverse axis direction for clearer presentation |
| function [ssDists,avgStrats] = vary_YI(N,b,c,d,w,a) | |
| % Calculates and plots the average results from steady_state_calc | |
| % when varying YI and p (with YD fixed at 1) | |
| % INPUTS | |
| % N is the population size | |
| % b is benefit of cooperation | |
| % c is cost of cooperation | |
| % d is max possible cost of deliberation (such that the cost of | |
| % deliberation varies uniformly in the interval [0,d]) | |
| % w is the selection strength (0=neutral drift, ->Inf=deterministic) | |
| % a is assortment (probability of interacting with a player having the same | |
| % strategy as you, rather than a randomly selected player) | |
| % | |
| % OUTPUTS | |
| % ssDists stores the ssDist steady state frequencies of each strategy for | |
| % each p x YI combination. | |
| % avgStrats stores the average strategy of of the steady state populations | |
| % for each p x YI combination. | |
| ps_steadyState = 0:.01:1; | |
| YI_steadyState= 0.5:0.01:1; | |
| %% Generate Strategy Space | |
| % For the steady state calculation, we must generate a discrete strategy | |
| % set. To do so: | |
| % pR, p1, pIR and pI1 are all restricted to being either 0 or 1; | |
| % we discretize the deliberation threshold (T) in intervals of d/10 | |
| % generate all possible combinations of the first four strat parameters (pR,p1,pIR SI1) | |
| % set to be 0 or 1 | |
| [x1,x2,x3,x4] = ind2sub([2,2,2,2], find(zeros(2,2,2,2)==0)); | |
| strats = [x1 x2 x3 x4] - 1; | |
| % we now combine the above with the 11 possible deliberation thresholds, | |
| % running from 0 to d in increments of d/10 | |
| Tvals = 0:d/10:d; | |
| strats = repmat(strats,length(Tvals),1); | |
| Tvals = sort(repmat(Tvals,1,length(strats)/length(Tvals))); | |
| strats = [strats Tvals']; | |
| % ... to create a total of 2*2*2*2*11 (176) possible strategies | |
| M = size(strats,1); % define M to be number of strats | |
| %% Main Steady State Loop | |
| % initialize matrix to store steady state distributions for each p value | |
| % each 1 value and each YI value | |
| ssDists = zeros(M,length(ps_steadyState),length(YI_steadyState)); | |
| % initialize matrix to store average strat values for each p value, YI val and 1 val | |
| avgStrats = zeros(length(ps_steadyState),5,length(YI_steadyState)); | |
| for YI=YI_steadyState | |
| YI % display YI value that's running | |
| for p=ps_steadyState | |
| p % display p value that's running | |
| % initialize payoff table | |
| payoffTable=zeros(M); | |
| % create payoff table | |
| for i=1:M | |
| for j=1:M | |
| payoffTable(i,j)=calculate_payoff_YDYI(strats(i,:),strats(j,:),b,c,d,p,1,YI); | |
| end | |
| end | |
| % run steady state using this payoff table and store result in ssDists | |
| ssDists(:,p==ps_steadyState,YI==YI_steadyState)=steady_state_calc(payoffTable,N,w,a); | |
| % use ssDists to get average strategy values by taking a weighted average of | |
| % the strat values based on the distribution in ssDists | |
| avgStrats(p==ps_steadyState,:,YI==YI_steadyState) = sum(repmat(ssDists(:,p==ps_steadyState,YI==YI_steadyState),1,5).*strats); | |
| end | |
| end | |
| %% Create Surface Plots | |
| % create a surface showing avg. amount of deliberation in p x YI space | |
| T = squeeze(avgStrats(:,5,:)); % stores this average value | |
| figure | |
| surf(YI_steadyState,ps_steadyState,T) | |
| xlabel('Sensitivity of Intuition y_i') | |
| ylabel('Probability of Coordination p') | |
| colormap(parula) | |
| view(2); view(-90,90) % change the orientation of the plot... | |
| set(gca,'ydir','reverse') % ... and reverse axis direction for clearer presentation | |
| % create a surface showing closest intuitive profile in p x YI space | |
| intuitiveProfile = round(squeeze(avgStrats(:,3,:))).*round(squeeze(avgStrats(:,4,:))) ... | |
| + 2*(round(squeeze(avgStrats(:,3,:))) - round(squeeze(avgStrats(:,4,:)))); % stores the intuitive responses as 1,2,3 | |
| figure | |
| surf(YI_steadyState,ps_steadyState,intuitiveProfile) | |
| xlabel('Sensitivity of Intuition y_i') | |
| ylabel('Probability of Coordination p') | |
| colormap([.8 0 0; 1 1 0; 0 .8 0]) % we create a custom colormap where... | |
| % red always intuitively defects, yellow always intuitively cooperates, and | |
| % green discriminates its response (defect in SD; cooperate in CG) | |
| view(2); view(-90,90) % change the orientation of the plot... | |
| set(gca,'ydir','reverse') % ... and reverse axis direction for clearer presentation |
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