Strong reciprocity

We follow B&G ch.9

Unconditional altruism can coevolve with parochialism, and the spreading of altruist/parochialists is likely if conflict among groups is frequent, as it was because of climate fluctuations in the pre-Holocene period.

However, experiment and ethological record show that:

1. Altruism is conditional. For example, in Public Good games cooperators eventually defect if too much free riding occurs

2. Altruistic punishers cooperate when punishing transgressors, and punish only when they reach a certain number because they have a quorum sensing ability

These two features are present in B&G’s model.

First model features:

Large population randomly divided into groups of n individuals.

Two heritable strategies: Punisher and Non-punisher

Groups initially undergo 3 stages:

1. Signaling stage:

· Punishers signal their presence at a cost q high enough that it does not pay to signal and not punish. If at least N (0<N<n-1) other group members signal, a punisher joins in punishing, if defectors present. Otherwise, punishers don’t punish.

· Non-punishers don’t signal and don’t punish

2. Cooperation stage: individuals in group cooperate or defect. Cooperation costs c and produces benefit b divided among all n members of group: b>c>b/n. (This is a public good game).

· Punishers: If at least N signaling punishers, then punishers defect with probability E (error) and cooperate with probability 1-E. If signaling punishers fewer than N, then punishers defect

· Non-punishers: defect in first period; in subsequent periods defect unless cooperating has better payoff than defecting; however, defecting occurs with probability E (error) even if cooperating has better payoff than defecting

3. Punishment stage:

· Cost of being punished is BP> (c-b)/n.

· If at least N punishers in group, punishers punish defectors. Punishers with a threshold of N as N–punishers.

· Cost of punishing is k iff the punishment encounter is a standoff. The probability of a standoff is 1/#PU, where #PU is the number of punishers. However the target pays k alone, while each punisher pays k/#PU (#P U= number of punishers). Hence, the expected cost for a punisher is (k/#PU)(1/#PU)= k/#PU².

Reproduction: proportional to payoff compared to population average payoff.

After the first period, the signaling stage is dropped and only the cooperation and the punishing state are played, followed by reproduction. Each period is followed by another period with probability p, so that on average the number of periods of a group is 1/(1-p). Defectors who got punished in the previous period cooperate in the next one with probability 1-E . Punishers punish defectors if at least N individuals punished defectors in the previous period.

After the last period, the group disbands. When no group is left, the game is restarted by randomly re-dividing the population into groups. Frequent disbanding of groups mimics large inter-group gene flow causing low intra-group relatedness.

Benchmark parameters: (cost of punishing) k=(cost of being punished) BP= (cost of signaling) q=1.5c (cost of cooperating); (error rate) E=.1; n=18.

Results: (for graphs, see B&G, 152).

Outcomes depend on

· The relation of b to c

· The value of threshold N.

In general:

· In a group, Punishers are advantaged over Non-punishers only when there are exactly N punishers. Rationale:

o If they are fewer than N, then punishers pay cost of signaling while Non-punishers do not

o If they are more than N, then it would pay to switch to Non-punisher because punishing and cooperation would occur anyway.

· Hence, the spreading of Punishers in the population depends on the fraction of threshold groups. Consequently, when fraction of Punishers in the population is large enough, the frequency of threshold groups diminishes; this results in a stable polymorphic equilibrium.

· For many threshold values there is a polymorphic Punishers/Non-punishers stable equilibrium; the higher the value of N, the closer to 1 the ratio Punishers/Non-punishers is. However, the greater N is, the less accessible the stable equilibrium is.

Addition of Liars.

· In the first period: Liars turn on the punishing process resulting in cooperation by falsely signaling they are punishers and then cooperating but not punishing, thus avoiding the cost k/#PU. (They are a type of second level defectors). Hence, in the first period Liars do better than Punishers.

· In subsequent periods: Liars cooperate if they cooperated in the first period, thus receiving the same payoff as Non-Punishers.

Punishers are marginally affected because punishing benefits them with respect to the general population. However, Liars can invade and replace Non-punishers when the cost of signaling q is small enough that they do better than Non-punishers in the first period; q=1.5c is high enough that Liars do worse than Non-punishers in the first period.

Addition of Contingent-Cooperators

· First period: Contingent-Cooperators cooperate if the punishment threshold value is reached but don’t signal and don’t punish. Hence, if threshold is reached they do better than Non-punishers who defect and are punished and of Punishers.

· Later periods: do as well as Non-Punishers.

Punishers are marginally affected because punishing benefits them with respect to the general population. However, Contingent-Cooperators replace Non-Punishers and a Punisher/Contingent-Cooperator equilibrium is reached. At equilibrium, the ratio Punisher/Contingent-Cooperator is close to Punisher/Non-Punisher.

Second model.

Agent model: See B&G or online model.

Notes:

· B&G’ agent model assumes a degree of truthful communication regarding transgression. Hence a norm of truth telling either preexisted or coevolved with strong reciprocity. This is a problem for the model. However, since agents are allowed to make mistakes, communication need not be fully public or perfect. In this context, many have emphasized the importance of gossip.

· Truth telling norms can be supported by social emotions, which are also needed to avoid excessive discounting of future punishment, as B&G argue.