Cooperative Games

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All of the examples so far have focused on non-cooperative solutions to "games." We recall that there is, in general, no unique answer to the question "what is the rational choice of strategies?" Instead there are at least two possible answers, two possible kinds of "rational" strategies, in non-constant sum games. Often there are more than two "rational solutions," based on different definitions of a "rational solution" to the game. But there are at least two: a "non-cooperative" solution in which each person maximizes his or her own rewards regardless of the results for others, and a "cooperative" solution in which the strategies of the participants are coordinated so as to attain the best result for the whole group. Of course, "best for the whole group" is a tricky concept -- that's one reason why there can be more than two solutions, corresponding to more than concept of "best for the whole group."

Without going into technical details, here is the problem: if people can arrive at a cooperative solution, any non-constant sum game can in principle be converted to a win-win game. How, then, can a non-cooperative outcome of a non-constant sum game be rational? The obvious answer seems to be that it cannot be rational: as Anatole Rapoport argued years ago, the cooperative solution is the only truly rational outcome in a non-constant sum game. Yet we do seem to observe non-cooperative interactions every day, and the "noncooperative solutions" to non-constant sum games often seem to be descriptive of real outcomes. Arms races, street congestion, environmental pollution, the overexploitation of fisheries, inflation, and many other social problems seem to be accurately described by the "noncooperative solutions" of rather simple nonconstant sum games. How can all this irrationality exist in a world of absolutely rational decision makers?

Credible Commitment

There is a neoclassical answer to that question. The answer has been made explicit mostly in the context of inflation. According to the neoclassical theory, inflation happens when the central bank increases the quantity of money in circulation too fast. Then, the solution to inflation is to slow down or stop increasing in the quantity of money. If the central bank were committed to stopping inflation, and businessmen in general knew that the central bank were committed, then (according to neoclassical economics) inflation could be stopped quickly and without disruption. But, in a political world, it is difficult for a central bank to make this commitment, and businessmen know this. Thus the businessmen have to be convinced that the central bank really is committed -- and that may require a long period of unemployment, sky-high interest rates, recession and business failures. Therefore, the cost of eliminating inflation can be very high -- which makes it all the more difficult for the central bank to make the commitment. The difficulty is that the central bank cannot make a credible commitment to a low-inflation strategy.

Evidently (as seen by neoclassical economics) the interaction between the central bank and businessmen is a non-constant sum game, and recessions are a result of a "noncooperative solution to the game." This can be extended to non-constant sum games in general: noncooperative solutions occur when participants in the game cannot make credible commitments to cooperative strategies. Evidently this is a very common difficulty in many human interactions.

Games in which the participants cannot make commitments to coordinate their strategies are "noncooperative games." The solution to a "noncooperative game" is a "noncooperative solution." In a noncooperative game, the rational person's problem is to answer the question "What is the rational choice of a strategy when other players will try to choose their best responses to my strategy?"

Conversely, games in which the participants can make commitments to coordinate their strategies are "cooperative games," and the solution to a "cooperative game" is a "cooperative solution." In a cooperative game, the rational person's problem is to answer the question, "What strategy choice will lead to the best outcome for all of us in this game?" If that seems excessively idealistic, we should keep in mind that cooperative games typically allow for "side payments," that is, bribes and quid pro quo arrangements so that every one is (might be?) better off. Thus the rational person's problem in the cooperative game is actually a little more complicated than that. The rational person must ask not only "What strategy choice will lead to the best outcome for all of us in this game?" but also "How large a bribe may I reasonably expect for choosing it?"

A Basic Cooperative Game

Cooperative games are particularly important in economics. Here is an example that may illustrate the reason why. We suppose that Joey has a bicycle. Joey would rather have a game machine than a bicycle, and he could buy a game machine for $80, but Joey doesn't have any money. We express this by saying that Joey values his bicycle at $80. Mikey has $100 and no bicycle, and would rather have a bicycle than anything else he can buy for $100. We express this by saying that Mikey values a bicycle at $100.

The strategies available to Joey and Mikey are to give or to keep. That is, Joey can give his bicycle to Mikey or keep it, and Mikey can give some of this money to Joey or keep it all. it is suggested that Mikey give Joey $90 and that Joey give Mikey the bicycle. This is what we call "exchange." Here are the payoffs:

Table 12-1

If we think of this as a noncooperative game, it is much like a Prisoners' Dilemma. To keep is a dominant strategy and keep, keep is a dominant strategy equilibrium. However, give, give makes both better off. Being children, they may distrust one another and fail to make the exchange that will make them better off. But market societies have a range of institutions that allow adults to commit themselves to mutually beneficial transactions. Thus, we would expect a cooperative solution, and we suspect that it would be the one in the upper left. But what cooperative "solution concept" may we use?

Pareto Optimum

We have observed that both participants in the bike-selling game are better off if they make the transaction. This is the basis for one solution concept in cooperative games.

First, we define a criterion to rank outcomes from the point of view of the group of players as a whole. We can say that one outcome is better than another (upper left better than lower right, e.g) if at least one person is better off and no-one is worse off. This is called the Pareto criterion, after the Italian economist and mechanical engineer, Vilfredo Pareto. If an outcome (such as the upper left) cannot be improved upon, in that sense -- in other words, if no-one can be made better off without making somebody else worse off -- then we say that the outcome is Pareto Optimal, that is, Optimal (cannot be improved upon) in terms of the Pareto Criterion.

If there were a unique Pareto optimal outcome for a cooperative game, that would seem to be a good solution concept. The problem is that there isn't -- in general, there are infinitely many Pareto Optima for any fairly complicated economic "game." In the bike-selling example, every cell in the table except the lower right is Pareto-optimal, and in fact any price between $80 and $100 would give yet another of the (infinitely many) Pareto-Optimal outcomes to this game. All the same, this was the solution criterion that von Neumann and Morgenstern used, and the set of all Pareto-Optimal outcomes is called the "solution set."

Alternative Solution Concepts

If we are to improve on this concept, we need to solve two problems. One is to narrow down the range of possible solutions to a particular price or, more generally, distribution of the benefits. This is called "the bargaining problem." Second, we still need to generalize cooperative games to more than two participants. There are a number of concepts, including several with interesting results; but here attention will be limited to one. It is the Core, and it builds on the Pareto Optimal solution set, allowing these two problems to solve one another via "competition."

Roger A. McCain