Cooperative Games and Solution Concepts

Abstract

Let us consider the following situation. Five people, Anna, Bert, Carol, Dave, and Ellen decide to combine forces to start their own business. Each one of them has some skills and some capital to contribute to the venture. After careful analysis of the situation they conclude that they can achieve a yearly profit of 100 (in $10,000). Of course, this amount has to be divided among them. At first assigning 20 to each person seems a reasonable allocation. However, after some additional analysis Dave and Ellen figure out that if the two of them work together without the other three, they can make a yearly profit of 45. This is more than the 40 that they would receive according to the allocation above. Anna, Bert, and Carol also perform some additional analysis and realize that the three of them together can only make a profit of 25. So it is in their interest to keep Dave and Ellen in the group. Consequently, they decide to give 46 to Dave and Ellen, and to divide the remaining 54 equally among the three of them. This seems to settle the problem. To make sure that they cannot do better, Carol, Dave and Ellen decide to find out how much profit the three of them can make without the other two. It turns out that they can make a profit of 70, which is more than the 64 (46+18) that the second allocation described above gives to them. Anna and Bert do not have enough capital to start on their own, so they cannot make any profit with only the two of them. Considering this, they decide to give 71 to Carol, Dave and Ellen, and to divide the remaining 29 equally between the two of them. If Carol, Dave, and Ellen also decide to divide the 71 equally among the three of them some further analysis reveals the following problem. It turns out namely, that Bert, Dave, and Ellen can make a profit of 65 which is more than the last allocation assigns to them (65 > 2 × 71/3+29/2).