## Info

the complex possibilities facing each player i (i.e., world region) in game (N,U) by a single number (pi(N,U) representing the value to i of playing the game. Thus, the value of game (N,U) is an nX 1 vector in which each element represents the "expected value" to a player of playing the game, where (pi(N,U) represents the expected payoff to player i under a randomization scheme on all coalitions S she can join. For a simple two-player case, the egalitarian solution is represented by the expression (Pl{N,U) - (p2({ 1} ,U) = (p2{N,U) - <p1({2}, U), where (p^N, U) + (P2(N,U) = U(n). This relationship states that in the egalitarian Shapley value solution, player 1 gets the same utility out of the presence of player 2 as the latter gets out of the former. Extending this concept to a multiple-player game, the general formula is (see, e.g., Mas-Colell, Whinston, and Green, 1995):

ieN ieS

where s is the number of players in a coalition S . The basic principle (marginality principle) behind the share (pi(N,U) is that when a player joins a coalition, she receives the marginal amount [i/(5u{/})-i/(5')] . The probability that a random ordering of coalition SczN forms as the union of i and its predecessors equals the probability that / is in the sth place, which is J/ multiplied by the probability that S — {/} forms when we randomly select s-lmembers from AT-{/} , which is s!(n-s-l)!. For any given random ordering of players, we calculate the marginal contribution of every player / to its set of predecessors in this ordering.

Consider the simple game denoted in (2) between five world regions (a,b,c,d,e) whose population levels are, respectively, 5, 15, 20, 25, and 35.