ABSTRACT
7.1 Example: Consider the game (N,v) where
N = {1,2,3}, v(i) = 0 for all i,
https://www.w3.org/1998/Math/MathML"> v ( 1 2 ) = v ( 1 3 ) = v ( 1 2 3 ) = 1 , a n d v ( 2 3 ) = 1 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429042034/4eff4487-a367-4138-a35f-94df68001ede/content/math112.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 75While 2 and 3 are in symmetric positions in this game, it appears that 1 is in a stronger position. Two problems can be considered: what coalitions will form?; and how will the members of the coalitions so formed divide their worth among themselves? There is no uniquely "correct" way of dealing with these problems, but the "Bargaining Set" represents one approach to the second problem, taking the coalitions which form as given. Consider the case where 1 and 2 get together in a coalition. Suppose that they are considering the payoff vector (2/3,1/3,0). Player 1 can say that this is not satisfactory since he could get together with player 3 and establish the payoff vector (5/6,0,1/6), which would benefit both himself and player 3. But player 2 can reply that he could also offer 1/6 to player 3, establishing the payoff vector (0,1/3,1/6), where he is as well off as he was before, and 3 is as well off as he would be in player l's proposed deviating payoff vector. However, player 1 could propose establishing (0.8,0,0.2) together with player 3, so that if player 2 were to give 3 as much as he gets in this payoff vector he would have to get less himself than he did in the original payoff vector which was being considered: v(23) = 1/2, so the most 2 could get if he gave 0.2 to player 3, would be 0.3, while originally he got 1/3. In this way the superior "strength" of player 1 is revealed, and he might suggest that (0.7,0.3,0) represents a reasonable split of the proceeds between himself and player 2. But exactly as above, 1 could threaten with (0.74,0,0.26), for example, which 2 could not match. So it appears that player 1 will receive an even larger payoff. Consider, then, the payoff vector (0.8,0.2,0) as a candidate for agreement between 1 and 2. In this case it is possible 76for player 2 to threaten with (0,1/4,/4), a threat which player 1 is unable to match, since if he gives 3 at least 1/4, at most 3/4 (< 0.8) will he left for himself. Consider, however, the payoff vector (3/4,1/4,0); if player 1 threatens to join with 3, at the same time increasing his payoff from 3/4, he will have to give 3 at most 1/4, while player 2 can always counter such a move by threatening to join with 3, giving him 1/4 while maintaining his own payoff. Similarly, if player 2 threatens to join with 3, and at the same time increases his payoff, then he can give 3 at most 1/4, while player 1 can always counter such a move by threatening to join with 3, giving him 1/4, and maintaining his own payoff at 3/k. In this way neither 1 nor 2 can object in a convincing way to the payoff vector (3/4,1/4,0), and the arguments above indicate that this is the only payoff vector for which this is so (for the grouping of players under consideration): it is, in fact, the unique member of the "Bargaining Set" in the case where players 1 and 2 get together in a coalition.