Appendix 2. Solving the prisoner's dilemma with generosity.

The game of prisoner's dilemma is often presented to illustrate the fundamental puzzle that faces any two or more persons that are harvesting from a common-pool resource (Binmore 1994, Ostrom et al. 1994). Notation varies. There are two players, Player 1 and Player 2. Each has a choice of two strategies: Cooperate or Defect. Their maximum joint payoff occurs when both cooperate. But the payoff structure of the game is set up in such a way that neither player will cooperate unless some condition outside of the game provides inducement to do so.

Let

C = The payoff each receives if both cooperate

D = The payoff each receives if both defect

b = The amount added to C that a defector receives if the other player cooperates

a = The reduction in D received by the cooperator if the other player defects.

Table 1. Abstract payoff matrix for the prisoner's dilemma.

In abstract form, the payoff matrix for the prisoner's dilemma can be written as follows in Table 1. The first amount in each parenthetical entry is the return to Player One; the second is the return to Player Two.
Player Two
 Player Two
Cooperate
 Defect
Player One
 Cooperate
 (C,C)
 (D-a,C+b)
Player One
 Defect
 (C+b, D-a)
 (D,D)

For C = 10, D = 5, a = 1, and b = 2, the prisoner's dilemma payoff matrix is as follows in Table 2:

Table 2. Numerical example matrix for the prisoner's dilemma.



Player Two
Player Two
Cooperate
Defect
Player One
Cooperate
(10,10)
(4,12)
Player One
 Defect
(12,4)
(5,5)

Using the the numerical example, one can demonstrate the consequence of imposing the symmetric generosity rule (defined in Appendix 1). If we interpret each of the payoffs as the surplus received by each player from an economic activity, such as harvesting salmon, then imposition of a symmetric generosity rule changes the payoff matrix to the contents of Table 3:

Table 3. Prisoner's dilemma removed with the symmetric generosity rule.



Player Two
Player Two
Cooperate
Defect
Player One
Cooperate
(10,10)
(8,8)
Player One
 Defect
(8,8)
(5,5)

In this matrix, the incentive to defect has been removed for each player.

If we apply the Kwakiutl generosity rule (defined in Appendix 1) to this two person game, then each player's payoff becomes the payoff of the other player. The new payoff matrix is given in Table 4:

Table 4. Prisoner's dilemma removed with the Kwakiutl generosity rule.



Player Two
Player Two
Cooperate
Defect
Player One
Cooperate
(10,10)
(12,4)
Player One
 Defect
(4,12)
(5,5)

In this matrix, the incentives also support cooperation by both players. If the values of a and b are large in comparison to C and D, then the application of the symmetric generosity rule will generate either the game of chicken or the assurance game. In all cases, the application of a symmetric generosity rule will eliminate the prisoner's dilemma, replacing it with simpler games. The Kwakiutl generosity rule always transforms the prisoner's dilemma into a game with a clear solution.

Because the prisoner's dilemma game captures the essence of many common-pool problems, the generality of the results just given are potentially very great. If this answer is so simple, why have so many commentators not stressed it? The answer is that few commentators realize the possibility of forced generosity. Most assume that an agreement to share the outcomes of a game such as the prisoner's dilemma is not enforceable. Even if players were to agree before playing the game to divide the returns, there is assumed to be no enforcement mechanism. In a society such as one that requires give-aways, however, the enforcement mechanism is credible.