The Shapley Value, The Core, and The Shapley - Shubik Power Index: A Simulation

The concept of Shapley values was first introduced into the game theory by Shapley [1953] in the framework of transferable payoffs, playing an important role in the game theory. According to a standard textbook on game theory it is interpreted as in what follows. “Suppose that all the players are arranged in some order, all orders being equally likely. Then player i’s Shapley value is the expected marginal contribution over all orders of player i to the set of players who precede him” (Osborne and Rubinstein [1994, p.291]). This concept has been applied to the parties’ power in the political election, as Shapley-Shubik Power Index (Shapley and Shubik [1954]). The Shapley-Shubik Index has been applied to actual election systems (Leech [2002] and Suzuki [1994]). The aim of this paper is, first, to provide the Mathematica programs to compute Shapley values and Shapley-Shubik Power Index. Utilizing them the relation between the core and Shapley values, and the paradox of Shapley-Shubik Power Index are examined. In Section I the relation between core and Shapley values is examined. It is known that the Shapley values may not belong to the core. Utilizing simulation approach, we compute the probability of “the Shapley values belong to the core”. In Section II, the paradox of ShapleyShubik Power Index is examined. It is often observed that even if a political party loses in an actual election, its Shapley-Shubik Power Index rises, and vice versa. Utilizing simulation approach, this paradox is examined theoretically.