Brief history of The Prisoner s Dilemma (From Harman s The Price of Altruism) 1948 The RAND Corporation. a civilian nonprofit think tank of an elite cadre of physicists, mathematicians, economists, and political scientists, it was the Cold War equivalent of Los Alamos. Von Nueman and John Nash were there. Albert Tucker (Princeton Mathematician) develops the prisoner s dilemma. 1. Two people are arrested and placed into separate cells. They cannot communicate. 2. Police admit that they do not have enough evidence to convict either on the principal charge, so they decide to sentence both to a lesser charge to one year in prison. 3. But then the police offer each one a bargain. a. if you testify against your partner, you will go free, but he/she will get 3 years in prison, provided that he/she does not testify against you. b. But if you both testify against each other, both of you will get 2 years.
---------------------------------------------------------------------------------------- Given that your partner testifies against you does not testify against you jail time for you if: you testify against 2 years prison 0 year prison your partner you do not testify 3 years prison 1 year prison against your partner --------------------------------------------------------------------------------------- What would you do?
---------------------------------------------------------------------------------------- Given that your partner testifies against you does not testify against you jail time for you if: you testify against 2 years prison 0 year prison your partner you do not testify 3 years prison 1 year prison against your partner --------------------------------------------------------------------------------------- What would you do? From Harmon: The problem was that if both prisoners were rational and selfseeking, both would reason in the same way. What that meant was that they would both defect and get two years in jail, whereas had they cooperated and kept their mouths shut, they d only have to serve 1 year a better solution for everybody. It was a maddening contradiction of Adam Smith s Invisible Hand: The pursuit of self interest does not necessarily promote the collective good. 3 Neither Nash nor von Nuemann were able to solve the paradox. The conflict between individual and collective rationality was real.
EVOLUTION OF SOCIAL BEHAVIOR 1. Suppose that individuals mate randomly, and then randomly break into small, interacting groups. 2. Let there be two kinds of individuals: A Altruists, which perform acts at cost c to themselves, and benefit b to the recipient, where p is the probability of the benefit. and a non-altruists 3. If W 0 is the baseline fitness, then The fitness of the altruist is: W A = W 0 c + p b And the fitness of the non-altruist is: W a = W 0 + p b Altruism spreads if W A > W a, which is when c < 0 (which means the act is not costly). The altruist is not really an altruist.
Suppose from the previous page, we have W 0 = 2, b = 2, c = 1. We get Partner Defects Cooperates Payoff to Defect W 0 =2 W 0 +b=4 Cooperate W 0 -c=1 W 0 +b-c=3 Clearly the only ESS is to Defect. But, both would do better if they cooperated. This is a modified PRISONER S DELIMMA. (note: see Harman s The Price of Altruism page 134 for a historical view of the Prisoner s Dilemma) Relaxing the assumptions. Natural selection would take the population to 100% defect, if the animals interacted only once, but what if the game is repeated many times?
Axelrod and Hamilton (1981). The evolution of cooperation. Science 211 The tit for tat (TFT) strategy: a) cooperate on the first move b) then do what ever your partner did in the previous move. For 10 trials with the same partner, we get: Partner Defect TFT Payoff to Defect TFT Fill in the blanks
The ESS is tit-for-tat (TFT) a) cooperate on the first move b) then repeat what ever your partner does on your next move. For 10 trials, we get: Partner Defect TFT Payoff to Defect 20 = 2*10 22 = 4 + 9*2 TFT 19 = 1 + 9*2 30 = 10*3 HENCE IN REPEATED CONTESTS, CONDITIONAL COOPERATION (TFT) CAN BE AN ESS. But note that TFT cannot invade defect, as defect is also an ESS. How could TFT ever become common?
4. Suppose that groups are not assembled at random. As a direct result, individuals can interact with close relatives. Let P = the probability that our target individual feels cooperative, and P res = the same probability in the resident population. And r = the relatedness among individuals b = benefit of cooperative act to recipient k = (1 - c) the fraction of fitness remaining to cooperative actor, where c is the cost of the cooperative act
The fitness of our target individual as both actor and recipient is: As ACTOR As RECIPIENT / \ / \ / \ / \ P k (1 - P) r P b (1 - r) b P res W i = P k + (1 - P) + r P b + (1 - r) b P res The costly, but cooperative, allele will spread when W i P > 0, which is when r b + k 1 > 0. Substituting c = 1 k, we get HAMILTON S RULE: r b > c Hamilton s rule: r b > c (Altruism will spread if relatedness, r, times the benefit, b, is greater than the cost, c.
DARWIN, on cooperation in bees and ants: STERILE CASTES ARE AN INSUPERABLE* DIFFICULTY FOR MY THEORY Hamilton suggested that haplodiploidy would tend to favor the evolution of sterile castes, because females are more related to their sisters (r = ¾) than to their own offspring. (note: brothers and sisters have r = 1/2) (note: see Harman s The Price of Altruism page 160-162 for a historical view of the hymenoptera.) But Hamilton did not claim that haplodiploidy (haploid males and diploid females) was either necessary or sufficient for the evolution of sterile castes (eusociality). Did Hamilton s model overcome Darwin s insuperable difficulties? ANY PROBLEMS WITH Hamilton s THEORY? IT HAS BEEN CALLED THE MOST IMPORTANT ADVANCE IN EVOLUTIONARY BIOLOGY SINCE THE WORK OF CHARLES DARWIN AND GREGOR MENDEL. But it also has many critics * in su per a ble inˈso op(ə)rəbəl/ adjective: insuperable 1. (of a difficulty or obstacle) impossible to overcome. "insuperable financial problems"
TAYLOR AND FRANK S (1996) MODEL FOR KIN SELECTION (on web site). Let y be the phenotype of our focal (or target) individual Let z be the mean phenotype in the population that contains our focal individual. So a change in y might also change z. Also keep in mind that individual fitness is affected by the mean phenotype in the population, z. Thus, individual fitness could be a function of both y and z. W i = f (y,z). Now let x be the genic value or breeding value underlying the phenotype, y, of our focal individual. We want to know how is individual fitness (W i ) affected by changing x. In other words we want to know dx which gives the change in fitness with respect to the change in x. (Note we have the total derivative here.)
By the chain rule, we get dx = W i y dy dx + W i z dz dx Have a close look at the right-hand side of the equation. The red part (dy/dx) takes into account how the phenotype, y, changes with the genotype, x. Note that it is multiplied by how individual fitness changes with a change in y. So y is affected by a change in x, and fitness is affected by a change in y. Now look at the blue part (dz/dx). It gives the change in the mean phenotype with respect to a change in x. And it (the blue part) is multiplied by the change in fitness with respect to the change in population mean phenotype, z. As we already know, the red and blue derivatives can be written as regression coefficients, giving. dx = W i y β y,x + W i z β z,x
By reorganization, we can write the equation as dx = W i y + W i β z,x z β y,x β y,x Finally, since the regressions coefficients are ratios of covariances to variances, the expression becomes dx = W i y + W i z cov(z,x) / var(x) cov(y,x) / var(x) β y,x which simplifies to dx = W i y + W i z cov(z, x) β cov(y, x) y,x. Queller (1992) has shown that this ratio of covariances is, in fact, relatedness, R, giving:
dx = W i y + W i z R β y,x Can you see how the ratio of covariances is relatedness? Is it the same thing as genealogical relatedness? Now let us ask: when will an increase in altruistic behavior in our focal individual spread in the population given that it decreases the fitness of the focal individual? In other words, when will the trait spread, given that W i y < 0? Which means that there is a cost to the altruistic act. This will happen when the term in square brackets is positive (assuming β y,x > 0 ), which is when W i y + W i z R > 0
R W i z > W i y Note the relatedness (no pun intended) to Hamilton s rule: rb > c. But also note how the R falls out of the model by Taylor and Frank. There is one other thing to see in the model. From above we have dx = W i y dy dx + W i z dz dx If z does not change with x, or W i does not change with z, we have dx = W i y dy dx Candidates for the ESS are found by setting dx = W i y dy dx = 0
If y varies positively with x, in other words if there is a genetic basis for the phenotype, y, then W i y = 0 at the candidate ESS. As such, Taylor and Frank s model collapses on our previously studied model for finding the ESS when the phenotype is continuously distributed, such as for sex ratio. Taylor and Frank s model is more general. This finding might clarify some of the implicit assumptions that went into our method for finding the ESS. One more thing Relatedness in this model does not have much to do with genealogy. Relatedness at the locus of interest is all that matters. Other loci do not figure in. Question: taking it back to Hamilton s rule: rb > c. Under what condition would you get spite, which is an act that harms the actor as well as the recipient?