SOLUTION CONCEPTS

For a superadditive game in coalition function form, the only rational arrangement is the grand coalition. If the grand coalition is formed, nothing can be lost (since the grand coalition must have a value no less than those of any proper coalitions into which it can be decomposed) and something will usually be gained. All that remains is to determine how the value of the grand coalition will be divided among the decision makers. As we recall from Chapter 3, there are several such solution concepts.

The Core and Related Concepts

Probably the most widely discussed solution concept for games in coalition function form is the core. The simple idea behind the core of a cooperative game is that no group can be denied the value that they could obtain if they were to form a coalition and act independently of the rest. As an illustration, consider Game 2.5. A singleton coalition that would produce the public good would then have a value of at most 4, so will not produce the public good. The singleton coalition would face a unified opposition, a two- person coalition, capable of producing two units of the public good. The opposition coalition would refuse to produce the public good, presumably in order to increase its bargaining power, since producing the public good would raise the value of the singleton to 7 or 9. Therefore, v{a} = v{b} = v{c} = 5. A doubleton coalition that would produce the two units of public good would be worth 12, whereas if it does not produce its value is 10. Moreover there is nothing the opposition singleton coalition can do to reduce the doubleton’s payoff below 12, so the doubleton will choose to produce the public good1 and v{a,b} = v{b,c} = v{a,c} = 12. The grand coalition of all three agents will be worth 24 if it produces three units of the public good and less if not, so it will produce them and v{a, b, c} = 24.

The payoff to agent j, after side payments are made, is denoted by x. A set of payments x. for the n players in the game, consistent with the value of the grand coalition, is called an imputation.2 Accordingly, suppose xa = 5, xb = 5, xc = 10. Then xa and xb can instead form a doubleton coalition and earn 12, which they can divide among themselves. Thus we exclude the imputation 5,5,10 from the core. This process of coalition-shopping is called recontracting. In general, by the same reasoning, we exclude from the core any schedule of payoffs that does not satisfy:

  • 1.1. Xa $ 5
  • 1.2. xb $ 5
  • 1.3. xc $ 5
  • 1.4. Xa + Xb $ 12
  • 1.5. xa 1 xc $ 12
  • 1.6. Xb 1 xc $ 12
  • 1.7. xa 1 Xb 1 xc # 24

Adding inequalities 1.4-1.6 we obtain 2(Xa 1 Xb 1 Xc) $ 36, that is, Xa 1 Xb 1 Xc $ 18. Comparing this with inequality 1.7, we see that there are infinitely many imputations that satisfy the criteria for the core in this example. In particular 8,8,8; 8,6,10; and 12,6,6 all are members of the core.

Here is another example, Game 8.1. Once again it will be a three-person game and all singleton coalitions are worth 5 if no production takes place. There are two techniques of production, both of which have economies of scale so that they can be undertaken only by coalitions with two or more members.3 Technology 1 generates profits of 4 for those who undertake it but produces a polluting waste that has to be assigned to some individual agent (who need not be a member of the group that undertakes production with Technology 1) and reduces that person’s payoff by 5. Technology 2 generates a profit of 3 and no waste.

A singleton will face a united opposition that can reduce the singleton’s value to zero by producing with technology 1 and assigning the waste to the singleton. Therefore, v{a} = v{b} = v{c} = 0. A doubleton can achieve a value of 14 by producing using technology 1, and there is nothing the opposing singleton can do to reduce the doubleton’s payoff below 14. Therefore, v{a,b} = v{b,c} = v{a,c} = 14. The grand coalition has a value with no production of 15, with Technology 1 of 14, and with Technology 2 of 18. Therefore Technology 2 will be used and v{a,b,c} = 18.

For this game, in order to prevent any two-person group from dropping out and shifting to Technology 1, xa + xb + xc $ 21 is necessary. Since, however, xa + xb + xc # 18 is also necessary, there are no imputations that satisfy the criteria for the core of the game. That is, the core for this game comprises the null set. This is often expressed by saying “the core does not exist,” but strictly speaking, the core always exists, although (as in this case) it may be null.

Both of these games are symmetrical, but this need not be so. Of course, nonsymmetrical games, in which coalition values depend on the individual members of the coalitions, will be more complex, and in some cases very much so.

There are a number of properties that we might like a solution to have. Two of the most important are that it should never be null and should correspond to a unique imputation. Clearly the core satisfies neither of these. However, there are a number of desirable properties that it does have.

Suppose we have two games played by the same set of players, Г = (n,v(C) and 0 = (n,w(C)), and there are constants a and p such that for any coalition C w(C) = a + p v(C). The two games are said to be strategically equivalent. Suppose then that whenever x is a solution of Г, a + px is a solution of 0. Then the solution is covariant under strategic equivalence. In more ordinary terms, it says that the solution will be unchanged by a change in the scale of measurements of payoffs, and that is persuasively a good property for a solution to have. The core has this property4 (Peleg and Sudholter, 2003, p. 25).

The core also has a property of anonymity, which means that the solution does not depend on the identities of the players except so far as their contributions to the values of coalitions are concerned. To see how this might fail, consider a toy solution concept, which is meant only as a bad example. Assign xa = v{a}, xb = v{a,b} - v{a}, xc = v{a,b,c} - v{a,b}, and so on if there are more than three players. Now consider another game, 0, that is a permutation of Г; that is, we simply take a, b, and c in a different order, such as c, b, a. Nevertheless w{a, b} = v{a, b} and so on. But if we apply the toy solution concept using the new order we have xc = v{c}, xb = v{a,c} - v{c}, xa = v{a,b,c} - v{a,c}. For the toy solution concept, the solution depends on how the players are ordered. A solution concept that is independent of this ordering is symmetrical and if it is similarly independent of any identity the agents may have apart from what is expressed in the coalition function, it is anonymous. The core has these properties (Peleg and Sudholter, 2003, p. 26).

If a solution concept gives each agent a payoff no less than he could get acting as a singleton coalition, it is said to have individual rationality. The core has this property (ibid., p. 27).

Thus, despite its shortcomings, the core has some properties that we do want to find in a solution, and it captures the idea that a group of players will in general get at least what they can obtain by acting independently.

 
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