Endogenous Formation of Links Between Players and of Coalitions: An Application of the Shapley Value

Consider the coalitional game v on the player set (1,2,3) defined by (1) v(S) = backslashleftbackslash backslashbeginarrayl 0backslashquad ifbackslashkern 1pt backslashleft| S backslashright| = 1, backslashbackslash 60backslashquad ifbackslashkern 1pt backslashleft| S backslashright| = 2, backslashbackslash 72backslashquad ifbackslashkern 1pt backslashleft| S backslashright| = 3, backslashbackslash backslashendarray backslashright. were |S| denotes the number of players in S. Most cooperative solution concepts ``predict'' (or assume) that the all-player coalition 1, 2, 3 will form and divide the payoff 72 in some appropriate way. Now suppose that P1 (player 1) and P2 happen to meet each other in the absence of P3. There is little doubt that they would quickly seize the opportunity to form the coalition 1, 2 and collect a payoff of 30 each. This would happen in spite of its inefficiency. The reason is that if Pi and P2 were to invite P3 to join the negotiations, then the three players would find themselves in effectively symmetric roles, and the expected outcome would be 24,24,24. P1 and P2 would not want to risk offering, say, 4 to P3 (and dividing the remaining 68 among themselves), because they would realize that once P3 is invited to participate in the negotiations, the situation turns ``wide open'' --- anything can happen.