4b) The N-Person Prisoner's Dilemma Game
The left 2x2 matrix below is the prisoner's dilemma game matrix considered in section 3d. The three-dimensional matrix to the right is a three-person prisoner's dilemma game matrix (2 rows x 2 collumns x 2 layers). The layers, which would be situated in front of and behind each other in a proper three-dimensional representation have been placed diagonally besides each other in order to simplify the drawing. The entries were derived from the left 2x2 matrix by adding the pay-offs that each player would have received from the two other players when playing the left 2x2 matrix simulteneously. The (summed) pay-offs of the row- and collumn players are entered in the cells as usual, the pay-offs of the layer player are entered in the lower right corners of the cells.
A more illustrative representation of the pay-offs in the three-dimensional matrix above is the graph below.
Cooperation and defection, here, denote the selection of the cooperative first alternative or the defecting second alternative respectively as explained in section 3d.
The graph is extended to a N(=7)-players prisoner's dilemma game below.
Clearly, whatever the number of cooperating (resp. defecting) players is, every player is always two pay-off units better off if he defects (= individual advantage of defection). It is typical for N-person prisoner's dilemma games that the above pay-off function of a cooperating player never exceeds that of a defecting player, a property that reflects the dominance of the defecting alternative in prisoner's dilemma games. On the other hand, due to the positive slope of the pay-off functions for N-person prisoner's dilemma games there is a collective advantage for every player that increases with the number of cooperating players (= collective advantage of cooperation). The collective advantage, as above, may be much larger than the individual advantage, but for every player there is the temptation to add the individual advantage to the collective advantage by defecting. The more players yield to that temptation the further to the left of the pay-off functions the two possible pay-offs of every player move, i.e. the more the collective advantage disappears. Hence the dilemma.
The subsequent graph shows conceivable pay-off functions for the well known fisheries dilemma, a typical social dilemma of an N-person prisoner's dilemma type. If unrestrained fishing is not punished, every fisher suffers individual losses of income from (self-imposed) fishing restrictions, but all suffer losses from overfishing due to unrestrained fishing.
If useful public services or projects are funded by compulsary or voluntary contributions, free riders who use the services without contributing save expenses and, thus, secure a higher pay-off than contributors for whom the utility of the contribution must be deduced from the utility of the service. If, however, the number of contributors is not sufficiently large, the services or projects will suffer in quality or even cease to exist (see below).
A critical battle situation may result in victory or defeat depending on the number of engaged fighters vs. self-protecting defectors. Assuming sanctions for defection being les severe than the dangers involved in fighting, the utility functions may look, somehow, like in the graph below.
In all real life situations mentioned we observe the dilemma discussed in the analysis of the N-person prisoner's dilemma game that defection secures a self-controlled individual advantage but reduces a, possibly larger, collective advantage that is mainly controlled by the co-actors, since each individual actor contributes only 1/N to its size.
Franzen (1994) summarized research on the effect of group size on rate of cooperation in the N-person prisoner's dilemma game. Contrary to what was believed hitherto, he concluded that group size does not affect cooperation rate in one-shot games or finite series of games. It only decreased with group size in game series of unknown, possibly infinite length. Earlier, that effect was attributed to the fact that in larger groups it is more likely to have some competitors (maximizing pay-off difference) which defect and force the others to defect as well in order not to be exploited. Also a general concept of social psychology, the diffusion of responsibility in large groups, was employed to explain that effect. Any interpretation, however, must take into account that group size cannot be manipulated without manipulating the ratio between pay-offs for outcomes achievable by cooperation and defection respectively. Depending of the ratio of the slopes of the pay-off functions for cooperative and defecting choices, either the collective advantage or the individual advantage or the ratio of these two changes and makes one or the other alternative comparatively more attractive.
Group size also affects the possibility of organizing cooperation inducing strategies in the N-person prisoner's dilemma game. Clearly, a player cannot play tit for tat in an N-person game, because cooperative choices would not only reward those who cooperated but also those who defected on the preceding trial. Similarly, a defective choice would punish both. Of course, if N-1 players form a coalition and coordinate their choice behavior, they can play tit for tat against the N-th player. A smaller number, i, 1£ i£ N-2, too, can form a tit for tat coalition and play a modified tit for tat-type of strategy: A tit for tat coalition cooperates on the first trial. In subsequent trials it selects the cooperative alternative, same alternative as before or defecting alternative depending on whether more, the same number or less of the none-coalition players selected the cooperative alternative on the last trial as compared with the trial before the last. That strategy was not yet tested against human players, but in simulations with neural networks, which were trained to maximize pay-off in N-person prisoner's dilemma games, the effectiveness of tit tat coalitions in securing a higher collective advantage decreased fast as N increased and i decreased within the range 0<i<N£ 20 (Stefanutti & Manarini, 1997).