# THE PROBLEM OF APPLICABILITY

By comparison with noncooperative game theory, at least, there have been relatively few applications of cooperative game theory. In chapters to follow this book will argue that the simplifying assumptions lead to a theory that is simply too abstract to be useful in a wide range of applications. However, there are a few important applications in economics. We will review three: games of exchange, games of production, and applications to the allocation of cost in a multidivisional organization. The first two are applications of the core, while the last begins from the Shapley value.

## The Market as Implementation of the Core

Among the most important applications of cooperative game theory is the study of games of exchange. In a game of exchange, there are two or more types of players, who differ in their endowments of particular goods and services or in their preferences or both. Coalitions are formed for the purpose of making reciprocal transfers of the goods and services, that is, exchanges. The benefit of doing so is that each agent may at the end find himself with a collection of goods and services that he likes better than the endowment he began with.

Most contributions to this literature do not assume transferable utility, but instead adopt the non-transferable-utility approach due to Shapley and Shubik (1952). For simplicity, we will consider an example of trade in indivisible units of two goods.^{9} Suppose, then, that we have two traders interested in exchanging olive oil for wine. (This will be game 8.2.) Traders of type *a* are endowed, at the beginning of the game, with three

*Table 8.1 Preferences for a game of exchange*

Barrels of oil |
Barrels of wine |
A’s preferences |
B’s preferences |

0 |
0 |
16 |
16 |

0 |
1 |
15 |
15 |

0 |
2 |
14 |
14 |

0 |
3 |
12 |
12 |

1 |
0 |
13 |
13 |

1 |
1 |
11 |
11 |

1 |
2 |
7 |
8 |

1 |
3 |
5 |
5 |

2 |
0 |
10 |
10 |

2 |
1 |
8 |
7 |

2 |
2 |
6 |
4 |

2 |
3 |
3 |
2 |

3 |
0 |
9 |
9 |

3 |
1 |
4 |
6 |

3 |
2 |
2 |
3 |

3 |
3 |
1 |
1 |

barrels of olive oil, while traders of type *b* are initially endowed with three barrels of wine, and the barrels cannot be divided. With just two traders, then, an individual may find himself with 0, 1, 2, or 3 barrels of oil and 0, 1, 2, 3 barrels of wine. There are 16 such combinations and, for the purposes of our example, we need to know the preferences of both players with respect to all 16. These are shown in Table 8.1. The preferences are expressed as first, second, and so on, so smaller numbers are better. Thus, for example, the third column tells us that a player of type *a* prefers one barrel of oil and three of wine (fifth preference) to two of oil and one of wine (eighth preference), while the last column tells us that a player of type *b* prefers one of oil and three of wine to one of oil and two of wine.

Suppose, then, that it is proposed to exchange two barrels of oil for one of wine. This would please the type *b* trader, raising him from his twelfth to his fourth preference, but it would reduce the type *a* trader from his ninth to his eleventh preference; so the type *a* trader would veto the trade. Suppose, on the other hand, that it is proposed to trade one barrel of oil for two of wine. This would raise the type *a* trader to his sixth preference and the type *b* trader to his eleventh. Thus we may say that the allocation^{10} that results from the one-oil-for-two-wine trade, two oil and two wine for type *a* and one oil and one wine for a type b, *dominates* the initial allocation. In general, an allocation x will dominate an allocation y if there is a coalition at least one member of which is better off, and none worse off, with x than with y. With only two traders in the game, we need consider only the grand coalition and the singleton coalitions.

As before, the core will consist of all allocations that are undominated. As before, the core may not be unique. For this game, with just one trader of each type, we have

2, 2 for *a* and 1,1 for *b* resulting from a 1-for-2 trade 1, 2 for *a* and 2,1 for *b* resulting from a 2-for-2 trade 1, 3 for *a* and 2,0 for *b* resulting from a 2-for-3 trade

In each of these cases, neither person can do better as a singleton, that is, if no exchange takes place. Thinking in terms of prices, or rates of exchange, we see that the price of a barrel of wine can vary from Vi a barrel of oil to 1 barrel of oil within the core. (At a one-to-one exchange rate, the exchange of one for one is dominated by the exchange of two for two, as an exchange of one each leaves each with his eighth preference while the two-for-two exchange leaves each at his seventh. Put otherwise, at a price of 1, each person will offer two units for trade, and this is the market equilibrium).

Now suppose that we have two traders of each type, and suppose that the allocation proposed is that type a’s get 2,2 and b’s 1,1, corresponding to a price of V. Instead, consider a coalition of one *a* and two b’s, and of the *b’s,* suppose b_{1} transfers two of wine to *a, b _{2}* transfers one, and

*a*transfers one of oil to each. This leaves

*a*with 1,3, his fifth preference; b

_{1 }with 1,1, his eleventh, and b

_{2}with 1,2, his eighth.

*a*and b

_{2}are better off than they were in the proposed allocation, and b

_{1}is no worse off. (If we allowed wine to be divided, b

_{2}could offer b

_{1}a cup or two from b

_{2}’s second barrel, to make it worthwhile for

*b*

_{1}to join the coalition.) Thus, the three- person coalition permits an allocation that dominates the proposed 1-for-2 allocation, and the 1-for-2 allocation (and the price of Vi) is no longer in the core. What we see is that with more traders, we may have more complex coalitions, and these impose more constraints on the allocations that can belong to the core, so that the core is smaller in a larger game.

Now, suppose that we have three agents of each type, and the proposed allocation gives a’s 1,3 and b’s 2,0, as in the 2-for-3 exchange and the price of 2/3. Suppose instead that a coalition is formed of 2 a’s and 3 b’s with the nine barrels of each type allocated so that each *a* gets 2,1; two *b’s* get 1,2, and the third *b* gets 1,3. This means that the a’s are at their fourth preference, rather than their fifth as in the proposed allocation, and two b’s are at their eighth and one at his fifth preference rather than their tenth, as in the proposed allocation. We now have a 2*a* and 3*b* coalition dominating the 2-for-3 exchange, and the price of 2/3 is no longer in the core. Once again, the larger game allows for more complex coalitions that impose more constraints on the allocations in the core, resulting in a smaller core.

In the literature on market games, as in this example, the outcomes are vectors of quantities of different goods and coalitions are formed for reallocation of the initial endowments of goods. Usually goods and services are assumed to be divisible and the conventional neoclassical assumptions are made: the individual may be indifferent between two vectors of goods and services, but preferences are convex, meaning that a weighted average of two vectors of goods and services will be preferred to either of the two vectors that are averaged.^{11} It is then demonstrated that

- (1) the core of a market game is never null;
- (2) while the core is usually not unique, increasing the number of agents in the game tends to eliminate some allocations from the core, so that the size of the core is smaller for larger games, as in the example;
- (3) the supply-and-demand equilibrium allocation and ratio of exchange is always a member of the core.

This is a striking result. It says that, for games of exchange, the noncooperative game defined by competitive markets yields a result in the core, and that a great multilateral contract by way of the grand coalition could not improve on bilateral trade mediated through competitive markets. In the language of implementation theory, markets implement the core for this class of games.