Comment on Shepsle's “On the Size of Winning Coalitions”

Professor Shepsle has contributed a lucid and stimulating discussion of coalitions and solutions, but his analysis cannot be used to resolve the problem of the size of winning coalitions. His conclusion that "there are good reasons to expect minimum winning coalitions in all but the most extreme instances" is therefore unwarranted, although his comment that "there are many theoretical subtleties yet to be explored" is fully substantiated. Shepsle begins by ably summarizing my critique of the size principle. In n-person, zero-sum games with perfect information, there are situations in which a player who does not belong to a minimum winning coalition will try to bribe his way into that coalition. He will find it rational to use side-payments to increase the payoffs of each member of that minimum winning coalition in return for his joining them. Since each member of the minimum winning coalition would gain by accepting that deal, they would agree to expand their group to a larger-than-minimum size. If we allow only one of the players who does not belong to the minimum winning coalition to engage in side-payments, that is the end of the story; a larger-than-minimum winning coalition results. With competitive bribery, however, the result is simply indeterminate. Maybe the winning coalition will be of minimum size; maybe it will be larger than minimum. All we can know for sure is that the number of players whose payoffs will be greater than zero will be at most the number of players required to form a minimum winning coalition. A quick word about that competitive bribery: Shepsle correctly notes that "there are very real incentives for e to bid up the price of bribery." Yes, there are, but they have limits; and those limits restrict e's bribe function to the same range that d's has. That range, in the game we keep using, is 20. It cannot be 25, as Shepsle would have it, because there are no amputations dominating those associated with a minimum winning coalition/maximum losing coalition situation that provide e with that many points to spend. Shepsle's suggested imputation (12, 12, 12, -20, -16) (see his Table 3, "A Bribe Response") would do so, if it indeed reflected e's only option. It does not do so, of course, since as Shepsle immediately points out it is strategically irrelevant to e. That is because the coalition { d, e } is effective for (10, 10, 10, -15, -15), which dominates the suggested payoff vector. The strategic analysis of d's situation must be, mutatis mutandis, the strategic analysis of e's. Shepsle's main thrust, however, is that neither Professor Riker nor myself ties our analyses explicitly to "a formal solution." Doing so would certainly improve our theoretical leverage on coalition behavior. Because he recognizes some critical difficulties with rigorous solution concepts, he cannot yet remedy this shortcoming, and so undertakes "a more modest inquiry" into an "heuristically interesting stable set" (p. 511). That inquiry, however, is too restricted to achieve his intentions. He begins by identifying a solution (in the sense of a von Neumann-Morgenstern stable set) to our five-person game, namely, "the set V of permutations of the vector (10, 10, 10, x, -30-x) with -20 <x <-10." He then asks whether there is "any limitation to the size of the winning coalition when the outcome of [the game] is an element of V," and concludes that the answer is No; "the size of the winning coalition cannot be determined by a stipulation of the von Neumann-Morgenstern stable set" (p. 512). That conclusion, he notes, rests upon "feasibility considerations." Does it change when some "motivational features" are specified? He concludes that "in a more general context, the answer in fact is ambiguous. For games like Butterworth's, however, the answer is unequivocally negative, and the size principle is sustained" (p. 512). But he can reach that conclusion only because his analysis misses a crucial point of the MPG principle. He first looks at a coalition structure's "acceptability." Since he confines his examination to V, he is concerned only with bargaining by players outside the minimum winning coalition when the payoffs to the positive gainers are fixed, and fixed a priori. No wonder, then, that he finds only one extreme case in which those gainers would shift from a situation in which the size of the winning coalition = the number of positive gainers = a minimum winning coalition: From the losers' point of view, the game is rigged. When the winnings of the positive gainers are specified, that is, so is the coalition structure, if we also specify the distribution of losses. If, as in Shepsle's example, we have not done the latter, then there is only one possible winning coalition that is larger than a minimum one-the case in which x (d's payoff) takes an extreme value. As Shepsle's discussion (equation (5.4)) shows clearly, this is because the value of a winning coalition will have to be achieved by add-