Semivalues as power indices

Abstract

A restricted notion of semivalue as a power index, i.e. as a value on the lattice of simple games, is axiomatically introduced by using the symmetry, positivity and dummy player standard properties together with the transfer property. The main theorem, that parallels the existing statement for semivalues on general cooperative games, provides a combinatorial definition of each semivalue on simple games in terms of weighting coefficients, and shows the crucial role of the transfer property in this class of games. A similar characterization is also given that refers to unanimity coefficients, which describe the action of the semivalue on unanimity games. We then combine the notion of induced semivalue on lower cardinalities with regularity and obtain a series of characteristic properties of regular semivalues on simple games, that concern null and nonnull players, subgames, quotients, and weighted majority games.

Introduction

Shapley (1953) initiates the value theory, probably the most active branch within the cooperative game theory. The Shapley value applies to every cooperative game and gives a single payoff vector to the players for each game. If the characteristic function of the game is exclusively involved, many game theorists agree with this a priori solution concept. Shapley and Shubik (1954) suggest the first power index when applying the Shapley value to simple games. Shapley (1962) provides the foundations for an independent theory of simple games. As he says: “Much of the sophisticated analytical apparatus that has been devised to solve or otherwise cope with the more general numerical games is easier to appreciate and to apply in the context of simple games, and only rarely does the application reduce to a triviality. Finally, a surprising number of multiperson games found in practice are simple. Thus for several reasons––methodological, didactic, and practical––the theory of simple games invites a self-contained, independent treatment.”

Banzhaf (1965) introduces a new index of power on simple games (independently redefined by Coleman (1971)), that Owen (1975) extends to all cooperative games in a dummy-independent normalized form. Dubey (1975) uses a “transfer property” in order to get an axiomatic characterization of the Shapley–Shubik index of power as a function on simple games. Owen (1978) tries to axiomatize the Banzhaf value and finds the dictatorial and marginal indices. Dubey and Shapley (1979) characterize the Banzhaf–Coleman index of power.

Weber (1979) is the first to suggest the idea of semivalue, and precisely on simple games. Dubey et al. (1981) formally introduce in an axiomatic way, on all cooperative games, this new notion that encompasses both the Shapley and Banzhaf values. Gambarelli (1983) compares different power indices on simple games. Young (1985) gives a characterization of the Shapley value that, by using monotonicity, avoids additivity. Einy (1987) obtains a characterization by weighting coefficients of semivalues on simple games similar to that given by Dubey et al. (1981). Lehrer (1988) provides an axiomatization of the Banzhaf value on cooperative games. Straffin (1988) suggests a nice criterion to choose among the Shapley–Shubik and Banzhaf–Coleman power indices depending on the relationships between the agents. Weber (1988) gives an alternative axiomatic presentation of semivalues and introduces the more general notion of probabilistic value. Hart and Mas-Colell (1988) provide the idea of potential for the Shapley value. Haller (1994) obtains a “proxy” property for the Banzhaf value. Feltkamp (1995) offers very close axiomatic characterizations for the Shapley and Banzhaf values and also for the corresponding power indices.

In the second half of the nineties, several authors have been especially concerned with semivalues (and/or, in particular, with the Banzhaf value). Ruiz (1995) obtains new axiomatizations for the Banzhaf value. Dragan (1996) defines its potential. Dragan (1997) gives different interesting properties of semivalues. Calvo and Santos (1997) and Dragan (1999) independently get a potential for every semivalue. Carreras and Freixas (1999) introduce regular semivalues. Carreras and Freixas (2002) show the semivalue versatility as a very interesting tool for practical applications. Puente (2000) devotes most of her Ph.D. thesis to semivalues and, especially, to their action on simple games. Very recently, Laruelle and Valenciano (2001) give new parallel axiomatizations for both classical power indices.

This is a very brief overview of the history of semivalues and their ancestors. We have not mentioned extensions of the Shapley value as e.g. Aumann and Drèze (1974), Myerson (1977), Owen (1977), Kalai and Samet (1987), Carreras (1991) or Amer and Carreras (1995), all of them using additional information not supplied by the characteristic function of the game, because the comments would deserve several pages. Furthermore, we sincerely apologize for omitting, for obvious space reasons, many other interesting papers on value theory.

From our point of view, several interesting conclusions arise from this history. The first one is the existence of a strong interaction between the general cooperative game theory and the simple game theory: very often, concepts and results in one field are successfully translated to the other by either restriction or extension procedures. In fact, the class of simple games undergoes two almost opposite forces: on the one hand, it is, as Shapley (1962) pointed out, a “test-class” for many cooperative concepts, and its development cannot therefore be expected far from the general cooperative theory; but, on the other hand, the interpretations that can be attached to a simple game increase and the applications to very different fields are more and more exciting. This latter tendency strengthens the need of a self-contained treatment for simple games as suggested by Shapley (1962).

The Banzhaf value has raised a new status within the value theory, and semivalues give a sort of path between the two classical values. More and more properties first discovered for the Shapley value are found to hold also for the Banzhaf value and most of semivalues. Semivalues might well be “the values of the 21th century”. The inefficiency of all semivalues (the only exception is the Shapley value) does not seem to be at all any source of problems in the application to simple games, where the notion of power becomes essential and is well captured by the proportions between payoffs rather than by their absolute values.

In addition to the substantial theoretical work made during the last years on the semivalue notion, the possibilities of application increase seriously: it is our opinion that semivalues give the way to include, in the evaluation of games, additional information not stored in the characteristic function of the game and unable to be handled by means of previous concepts. To this end, the interesting semivalues do not differ very much from the two classical values (for details, we refer to Carreras and Freixas (2002)).

The aim of this paper is that of providing a self-contained theory for semivalues on simple games only, i.e. as power indices. We feel this goal well justified by the previous arguments. The organization of the paper is then as follows. In Section 2 we include a minimum of preliminaries that essentially refer to the idea of semivalue for general cooperative games. In Section 3, we specialize to semivalues on simple games, that we call power semiindices, and give the main theorem of the paper, that clearly reminds the characterization obtained by Dubey et al. (1981). We also provide a definition of semiindices in terms of unanimity coefficients and supply the necessary and suficient conditions that these coefficients must satisfy in order to define a semiindex. Finally, Section 4 deals with induced semiindices and, especially, with the properties derived from regularity. Some of these properties were studied by Felsenthal and Machover (1995) for the Shapley–Shubik, Banzhaf–Coleman, Johnston (1978) and Deegan and Packel (1978) power indices. Our results concern nonnull players, the balanced contributions property, the null player exclusion property for subgames, the null and nonnull player joining properties in passing to the quotient game, superadditivity and subadditivity (that yield a characterization of the Banzhaf–Coleman power index among semiindices), and donation properties in weighted majority games. A remark on multisemiindices and coherence close the paper.

Summing up, the paper tries to contribute to a better understanding of semiindices as a very consistent alternative or complement to the two clasical power indices, especially emphasizing on the sensibility of regular semiindices.