Toulouse Economics and Biology Workshop: Evolution and Morality University of California Santa Barbara June, 2015
A Dark Tale from the Big Apple In 1964, as she returned home from work late at night, Kitty Genovese was assaulted and murdered, near her apartment in Queens, New York City. According to a story in the New York Times: For more than half an hour, 38 respectable, law-abiding citzens in Queens watched a killer stalk and stab a woman in three separate attacks.... Not one person telephoned the police during the assault
Pundits reaction Pundits found this emblematic of the callousness or apathy of life in big cities, particularly New York. The incident was taken as evidence of moral decay and of dehumanization caused by the urban environment.
In Defense of New Yorkers Sociologists, John Darley and Bibb Latane suggested an alternative theory. City dwellers might not be callous or dehumanized. They know that many others are present and believe it likely that someone else will act. Darley and Latane called this the bystander-effect. They found this effect in lab experiments: Someone pretended to be in trouble, When subjects believed nobody else could help, they did so with probability.8. When they believed that 4 others observed the same events, they helped with probability.34.
Further Defense of New Yorkers Not central to the theory, but facts deserve respect. Fact checkers found the journalists story to be partly fabricated (albeit by New York journalists.) No evidence that 38 people observed the murder. (let alone 38 respectable, law abiding people) It was 3 am, a cold night. Windows were shut. More than one person tried to help.
Volunteer s Dilemma Game Andreas Diekmann, another sociologist, created a game theoretic model, the Volunteer s Dilemma, to capture the bystander effect N-player simultaneous move game: Strategies Act or Not. All who act pay C. If at least one acts, those who acted get B-C.. Those who didn t act get B. If nobody acts, all get 0. There is no pure strategy equilibrium. If everybody helps, nobody would want to help. If nobody helps everybody would want to help. In symmetric mixed strategy Nash equilibrium, as N increases, it less likely that any one person calls In fact, as N increases it becomes more likely that nobody calls.
Ethics and the Volunteer s Dilemma Diekmann s Volunteer s Dilemma captures the coordination problem for competing altruists who must move simultaneously. It does not offer a theory of how much players value or should value rescuing another person.
Kant s Categorical Imperative Act only according to that maxim whereby you can at the same time will that it should become a universal law.
Confucian ethics Be trustworthy and honest: don t befriend those who are not on the same level as you in these virtues. Analects [1:8] Virtue always finds you good neighbors. Analects [4:25]
Hamilton s Kin-Selection Theory Altruism arises when genetically determined strategies of relatives are correlated. Similar logic applies when unrelated individuals match assortatively with respect to strategies.
Norms as a Coordination Device... As Explained by Wise Philosophers
Kant: Practical Advice or Just Hectoring? Kant claims that his categorical imperative is an ultimate commandment of reason which must be always be obeyed by rational people. He distinguishes this from hypothetical imperatives, which take the form If I want this to happen, I need to do that. But if it were true that you were always matched in symmetric games with people who used the same maxim as you, choosing a strategy that satisfies the categorical imperative would be in your self-interest.
Imperfect Assortativity In the world where we live, we don t have complete knowledge of each others strategies, nor is it costless to enforce exclusion of those whose strategies we don t approve of. But we do have norms and people who usually behave well toward each other tend to associate and tend to avoid those who do not. Confucius recognized that humans are imperfect moral beings and urged people to practice proper behavior in ordinary life to gradually become a person of virtue. Hamilton noted that for sexual diploids, a dominant mutant gene that controls behavior towards one s sibling, will with probability 1/2 be shared with this sibling.
r-assortative (semi-kantian) Ethical Equilibrium Definition: For an n-player game, strategies are matched r-assortatively with equilibrium strategy s if a player who uses strategy s s is assigned to a group in which the strategy of each of the other n 1 players is an independent random draw that turns up s with probability r and s with probability 1 r. Definition: An r-assortative ethical equilibrium for a symmetric game is a strategy profile in which all play strategy s and if when players are matched r-assortatively, no player can get a higher expected payoff by playing s s.
Mutual Aid and the Three Brothers Problem Petr Kropotkin, 1902, Mutual Aid: A Factor of Evolution Ilan Eshel and Uzi Motro The Three Brothers Problem: Kin Selection with more than one Potential Helper American Naturalist, 1988
A Mutual Aid Game A group of n players. Each gets in trouble with probability π. Common knowledge that all members know when another is in trouble Players choose simultaneously whether to help the one in trouble. If player in trouble gets no help, his loss is b. If at least one other player helps, his loss is 0. Players who help must pay a cost c.
Strategies and payoffs A strategy for any player is a probability p of helping any other player who is in trouble. If all play p, then total costs include expected cost of being in trouble and getting no help plus expected cost of helping each other victim with probability p. Thus expected total costs are ( ) π b (1 p) n 1 + p (n 1) c.
With r-assortative matching With r-assortative matching, if others are playing p and a deviant player plays p, the probability that the deviant player will get no help when in trouble is (1 (1 r) p + rp) n 1 = (1 p + r( p p)) n 1. Expected total costs for the deviant player are T (p, p) = b (1 p + r( p p)) n 1 + p(n 1)c.
r-assortative ethical equilibrium In an equilibrium, p, no deviant strategy p has lower total costs than p. Thus T ( p, p) T ( p, p) for all p p. Calculus first order condition implies that ( c ) 1 n 2 p = 1 rb
Some implications For any n, Equilibrium total costs are increasing in c decreasing in r decreasing in b As n increases, The equilibrium expected cost of getting no help when in trouble increases and asymptotically approaches πc/r. The equilibrium expected cost of helping others who are in trouble increases and asymptotically approaches -πc ln c/rb. Total costs increase.
Note also: Although the r-assortative equilibrium strategy is a mixed strategy, in equilibrium players are not indifferent between pure strategies. Payoff functions are non-linear in probabilities. The equilbrium mixed strategies that we find with r-assortative matching are the same that we would find if matching were random and players cared about each others payoffs in ratio r to their own. More about this later.
Mutual aid when costs vary Suppose that players costs of helping others vary from time to time. Could it then be that larger groups could attain more efficiency because players could choose to help only when their costs are low? Let s try as simple model that shows the forces at work here.
A simple model A group has n players. Each has a Poisson probability λ of getting into trouble at any time. At the time when any group member is in trouble, each other player independently draws a random cost of helping, which is c L > 0 with probability l and c H > c L with probability 1 l. Each player is informed of his own cost of helping but does not know the realizations for others. Players move simultaneously in deciding whether or not to help a player in trouble. A player in trouble will suffer no costs if at least one other player offers help. If none offer help, the player in trouble suffers a cost of b.
Possible strategies (maxims) Possible strategies for any player are mixed strategies in which one s probability of helping at any time may depend on one s cost of helping at that time. A strategy is denoted p = (p L, p H ) where p L and p H are the probabilities that one helps a group member in trouble if one s costs are, respectively, p L and p H.
r-assortative ethical maxims There are 3 possible types of equilibria in which help is sometimes offered. These are: p = ( p L, 0) where 0 < p L < 1. p = (1, 0) p = (1, p H ) where 0 < p H, 1. Which regime applies depends on the probability l and on the cost parameters b, c L, c H and on r and n.
The case of large l Where n 3, c L < rb and l > 1 c L rb the r-assortative equilibrium is of the form p = ( p L, 0), where 0 < p L < 1. Then the probability that a player in trouble gets no help is ( cl ) n 1 n 2 rb which increases with n and approaches c L rb. The equilibrium expected cost of helping also increases with n and asymptotically approaches ln ( c L rb ). So for this range of l, total costs are greater, the larger the group size.
The case of intermediate l Where c L < rb and 1 c H rb < l < 1 c L rb, there is a largest integer n 3 such that c L < (1 l) n 2. For all n such that 3 n n, the r assortative equilibrium is p = (1, 0). For 3 n n, the probability of receiving help when in trouble is 1 (1 l) n 1 which is increasing with n. Over this range, each player s expected total costs of helping others is l(n 1) which is increasing in n. Over this range, total expected costs, including the expected cost of being in trouble with no help and the expected cost of helping others is decreasing with n.
The case small l Where c L < rb and l < 1 c H rb, we have the following. For n = 3, p = (1, p H ) where 0 < p H < 1. There will be a threshold value n such that for n < n, 0 < p H < 1. Over this range of n, the probability of receiving help if in trouble diminishes and expected costs of helping increase with n. Then there is some n > n such that for n < n < n, p = (1, 0), the probability of receiving help rises with n and expected costs of helping fall with n. Finally for n > n, p = ( p L, 0) where 0 < p L. Over this range, the probabilty of getting help when in trouble falls with n and the expected cost of helping increases with n.
Confucius, Kant, Hamilton and Empathy Neither the Confucian nor the Kantian recommendations that we have discussed ask individuals to empathize with their fellows. Instead they ask you to act in the way that you would like others to behave toward you. Similarly the genetic foundation of Hamilton s kin selection theory relies on the correlation between one s behavior toward ones relatives and one s relatives behavior toward oneself. For n > n, p = ( p L, 0) where 0 < p L < 1. Over this range, the probability of being helped when in trouble diminishes and the expected cost of helping others increases with n.
Assortativity and Brotherly Love Hamilton, however, presents this theory as one in which individuals maximize inclusive fitness which is a weighted average of their own reproductive success and that of their kin. Hamilton s rule could be stated as Love your relative r times as well as yourself where r is your coefficient of relationship. The r-assortative versions of Confucius and Kant could be stated as Act as you would if you believed that your relative will mimic your action with probability r. It is well known that for games in which payoffs take a simple additive form these two rules are equivalent. They are also predict the same equilibrium for our Mutual Aid game.
Equivalence of Love and Assortativity, more Generally The first order calculus conditions for a symmetric Nash equilibrium in a symmetric n-player game with r-assortative matching are the same as those for the Nash equilibrium in a game where matching is not assortative but where each player player weights the payoffs of all other players at r times his own payoff. Thus appropriate levels of concern for others welfare will often work just as well as awareness of matching. But they aren t always equivalent. Even where first order conditions are identical, second order conditions may have opposite signs.
Weakest link with Love thy neighbor Two players: Possible strategies are x i [0, 1]. Payoffs are U i (x 1, x 2 ) = min{x 1, x 2 } cx i. With the love-thy-neighbor-r-times-as-much-as-thyself ethic, payoff to i is U i (x 1, x 2 ) = (1 + r) min{x 1, x 2 } cx i. For this game any strategy profile ( x, x) where 0 x 1 is a symmetric Nash equilibrium.
Weakest link with assortative ethic If ( x, x) is an r-assortative ethical equilibrium, then for each i x i = x maximizes on the set [0, 1] (1 r) min{x i, x} + r min{x i, x i } cx i If r > c, this implies x i = 1. Thus the only symmetric Nash equilibrium is ( x 1, x 2 ) = (1, 1).
Best shot with Love thy neighbor Two players: Possible strategies are x i [0, 1]. Payoffs are U i (x 1, x 2 ) = max{x 1, x 2 } cx i. With the love-thy-neighbor-r-times-as-much-as-thyself ethic, payoff to i is U i (x 1, x 2 ) = (1 + r) max{x 1, x 2 } cx i. For this game, if 0 < c, there is no symmetric Nash equilibrium in pure strategies.
Best Shot with assortative ethic If ( x, x) is an r-assortative ethical equilibrium, then for each i x i = x maximizes on the set [0, 1] (1 r)max{x i, x} + r max{x i, x i } cx i If 0 < c < r, then this expression is maximized when x i = 1. Thus the only symmetric Nash equilibrium is ( x 1, x 2 ) = (1, 1).
Observations In the weakest link game, every assortative ethical equilibrium is a symmetric love-thy-neighbor equilibrium, but not conversely. In the best shot game where 0 < c < r, there are no (pure strategy) symmetric love-thy-neighbor equilibria, but there is exactly one assortative ethical equilibrium. More generally,: With symmetric complementarity every assortative ethical equilibrium is a love-thy-neighbor equilibrium. With symmetric substitutabilty every love-thy-neighbor equilibrium is an asortative equilibrium. Sometimes, but not always, the two sets of equilibria are the same.
That s all, folks