| Buy the book! It has so much more .... at Amazon, Game Theory: A Non-Technical Introduction to the Analysis of Strategy at Barnes and Noble -- of course! Game Theory: A Non-Technical Introduction to the Analysis of Strategy |
Two-person games don't get us very far. Many of the "games" that are most important in the real world involve considerably more than two players -- for example, economic competition, highway congestion, overexploitation of the environment, and monetary exchange. So we need to explore games with more than two players.
Von Neumann and Morgenstern spent a good deal of time on games with three players, and some more recent authors follow their example. This serves to illustrate how even one more player can complicate things, but it does not help us much with realism. We need an analysis of games with N>3 players, where N can be quite large. To get that, we will simply have to pay our way with some simplifying assumptions. One kind of simplifying assumption is the "representative agent model." In this sort of model, we assume that all players are identical, have the same strategy options and get symmetrical payoffs. We also assume that the payoff to each player depends only on the number of other players who choose each strategy, and not on which agent chooses which strategy.
This "representative agent" approach shouldn't be pushed too far. It is quite common in economic theory, and economists are sometimes criticized for overdoing it. But it is useful in many practical examples, and the next few sections will apply it.
This section presents a "game" which extends the Prisoners' Dilemma in some interesting ways. The Prisoners' Dilemma is often offered as a paradigm for situations in which individual self-interested rationality leads to bad results, so that the participants may be made better off if an authority limits their freedom to choose their strategies independently. Powerful as the example is, there is much missing from it. Just to take one point: the Prisoners' Dilemma game is a two-person game, and many of the applications are many-person interactions. The game considered in this example extends the Prisoners' Dilemma sort of interaction to a group of more than two people. I believe it gives somewhat richer implications about the role of authority, and as we will see in a later section, its N-person structure links it in an important way to cooperative game theory.
As usual, let us begin with a story. Perhaps the story will call to mind some of the reader's experience. We suppose that six people are waiting at an airline boarding gate, but that the clerks have not yet arrived at the gate to check them in. Perhaps these six unfortunates have arrived on a connecting flight with a long layover. Anyway, they are sitting and awaiting their chance to check in, and one of them stands up and steps to the counter to be the first in the queue. As a result the others feel that they, too, must stand in the queue, and a number of people end up standing when they could have been sitting.
Here is a numerical example to illustrate a payoff structure that might lead to this result. Let us suppose that there are six people, and that the gross payoff to each passenger depends on when she is served, with payoffs as follows in the second column of Table 7-1. Order of service is listed in the first column.
| Order served | Gross Payoff | Net Payoff |
| First | 20 | 18 |
| Second | 17 | 15 |
| Third | 14 | 12 |
| Fourth | 11 | 9 |
| Fifth | 8 | 6 |
| Sixth | 5 | 3 |
These payoffs assume, however, that one does not stand in line. There is a two-point effort penalty for standing in line, so that for those who stand in line, the net payoff to being served is two less that what is shown in the second column. These net payoffs are given in the third column of the table.
Those who do not stand in line are chosen for service at random, after those who stand in line have been served. (Assume WLOG that these six passengers are risk neutral.) If no-one stands in line, then each person has an equal chance of being served first, second, ..., sixth, and an expected payoff of 12.5. In such a case the aggregate payoff is 75.
But this will not be the case, since an individual can improve her payoff by standing in line, provided she is first in line. The net payoff to the person first in line is 18>12.5, so someone will get up and stand in line.
This leaves the average payoff at 11 for those who remain. Since the second person in line gets a net payoff of 15, someone will be better off to get up and stand in the second place in line.
This leaves the average payoff at 9.5 for those who remain. Since the third person in line gets a net payoff of 12, someone will be better off to get up and stand in the third place in line.
This leaves the average payoff at 8 for those who remain. Since the fourth person in line gets a net payoff of 9, someone will be better off to get up and stand in the fourth place in line.
This leaves the average payoff at 6.5 for those who remain. Since the fifth person in line gets a net payoff of 6, no-one else will join the queue. With 4 persons in the queue, we have arrived at a Nash equilibrium of the game. The total payoff is 67, less than the 75 that would have been the total payoff if, somehow, the queue could have been prevented.
Two people are better off -- the first two in line -- with the first gaining an assured payoff of 5.5 above the uncertain average payoff she would have had in the absence of queuing and the second gaining 2.5. But the rest are worse off. The third person in line gets 12, losing 0.5; the fourth 9, losing 3.5, and the rest get average payoffs of 6.5, losing 6 each. Since the total gains from queuing are 8 and the losses 16, we can say that, in one fairly clear sense, queuing is inefficient.
We should note that it is in the power of the authority (the airline, in this case) to prevent this inefficiency by the simple expedient of not respecting the queue. If the clerks were to ignore the queue and, let us say, pass out lots for order of service at the time of their arrival, there would be no point for anybody to stand in line, and there would be no effort wasted by queuing (in an equilibrial information state).
Roger A. McCain 