On the generalized dimension and codimension of simple games

Highlights

• We generalize the dimension and codimension concepts of simple games.

• We study this generalization for strict subclasses of weighted voting games.

• We characterize the games that are intersection/union of pure weighted voting games.

• We compute the generalized (co)dimension of every simple game up to 6 players.

Abstract

Weighted voting games are simple games that can be represented by a collection of integer weights for each player so that a coalition wins if the sum of the player weights matches or exceeds a given quota. It is known that a simple game can be expressed as the intersection or the union of weighted voting games. The dimension (codimension) of a simple game is the minimum number of weighted voting games such that their intersection (union) is the given game. In this work, we analyze some subclasses of weighted voting games and their closure under intersection or union. We introduce generalized notions of dimension and codimension regarding some subclasses of weighted voting games. In particular, we show that not all simple games can be expressed as intersection (union) of pure weighted voting games (those in which dummy players are not allowed) and we provide a characterization of such simple games. Finally, we experimentally study the generalized dimension (codimension) for some subclasses defined by establishing restrictions on the representations of weighted voting games.

Introduction

Cooperative games are decision models made up of players, individuals, or agents who can decide to associate in groups or coalitions to reach a common goal. Each coalition has an associated benefit, commonly quantified as a real number (von Neumann & Morgenstern, 1944). A relevant class of cooperative games is the one formed by the so-called simple games, where the benefit of each coalition is always binary: either 0 (losing) or 1 (winning). A simple game is defined by a set of winning coalitions that meet the monotony property. Simple games have applications in diverse settings such as voting theory, decision theory, social choice theory, logic and threshold logic, circuit complexity, artificial intelligence, geometry, linear programming, Sperner theory, order theory, among others (Eiter, Makino, Gottlob, 2008, Engel, 1997, Molinero, Riquelme, Serna, 2015a, Taylor, Zwicker, 1999).

A relevant subclass of simple games is formed by the weighted voting games, noted by $\mathcal{WVG}$. A weighted voting game with $n$ players is a simple game that can be represented by a vector of $n+1$ real numbers, one for a quota, and the remaining for the weights associated with each player so that a coalition wins if the sum of its weights meets or exceeds the quota. It is well-known that the weights and quota in these representations can be restricted to integer numbers (Taylor & Zwicker, 1999). Observe that each such representation defines a weighted voting game. However, a weighted voting game admits infinitely many representations. It is also well-known that some simple games are not weighted voting games. However, each simple game can be expressed as the intersection (or the union) of weighted voting games (Taylor & Zwicker, 1999). From this property, the concepts of dimension and codimension of a simple game arise. Recall that the dimension of a simple game is the minimum number of weighted voting games whose intersection coincides with the game. The codimension is defined similarly changing intersection by union. When analyzing the collective decision procedure for multi-agent systems or real-world scenarios, the dimension plays a relevant role. It usually requires less space to provide the game as an intersection of weighted voting games that list all its winning coalitions. The dimension aspects of real-world voting systems have been studied intensively within the field of computational social choice (Cheung, Ng, 2014, Freixas, 2004, Kilgour, 1983, Taylor, Zwicker, 1999). We want to point out that there exist real voting systems with a high dimension and codimension. One example is the voting rules stated for the EU Council with 28 countries, after the Treaty of Lisbon in November 2014, with at least dimension 8 (i.e., it cannot be represented as the intersection of seven or fewer weighted voting games) Kober & Weltge (2020) and at least codimension 2000 (Kurz & Napel, 2015). Since dimension and codimension are dual concepts (Kurz, Molinero, Olsen, & Serna, 2016), the dual of a game with high dimension has high codimension. Furthermore, there are simple games with a high dimension but low codimension, and vice versa.

Many theoretical results and examples of dimension (and a few of codimension) for simple games have been studied during the last two decades, e.g., Freixas & Molinero (2010), Taylor & Zwicker (1999), Taylor & Pacelli (2008). Some authors have focused on constructive procedures to find the dimension of a simple game (Freixas, Puente, 2001, Freixas, Puente, 2008, Olsen, Kurz, Molinero, 2016), and others in computational complexity results (Deĭneko, Woeginger, 2006, Freixas, Molinero, Olsen, Serna, 2011, Molinero, Olsen, Serna, 2016). In addition, some researchers are interested in finding the dimension of simple games representing voting systems in real life (Kober, Weltge, 2020, Kurz, Napel, 2015). Another line of research seeks simple games with high dimension (Kurz, 2021, Olsen, Kurz, Molinero, 2016, O’Dwyer, Slinko, 2017). Despite the above, no complete classification of dimension or codimension of simple games is known even for a small number of players.

As the number of weighted voting games grows rapidly, a natural direction to explore is to gather further knowledge on the capability of some restricted families of $\mathcal{WVG}$ to generate, through intersections or unions, simple games having or not similar properties. Once a class of $\mathcal{WVG}$ is fixed, we want to study the simple games that can be obtained as intersection (union) of elements in the class, i.e., the closure of the family under either union or intersection. In this way, by considering an initial subclass with a smaller number of games, we expect to make computationally doable listing all the games in the closure for small values of $n$. The above leads us to introduce a generalized definition of dimension (codimension) regarding a subclass of $\mathcal{WVG}$. Of course, we know that, for some subclasses, we will be unable to generate all simple games, and in those cases, it will be of interest to characterize the obtained families.

In this paper, we initiate the study of generalized dimension and codimension for subclasses of pure weighted voting games, namely, $p-\mathcal{WVG}$. A game is pure if it does not have dummy players. Restricting to games without dummy players is natural because those players do not belong to minimal winning coalitions. Moreover, in the context of threshold logic, pureness is known as non-degeneration. In fact, pure simple games are the so-called monotone non-degenerate Boolean functions (Wegner, 1985). Our contribution is divided into two parts. In the first part, we analyze theoretically generalized dimensions concerning $p-\mathcal{WVG}$. In the second part, we experimentally analyze other smaller subclasses of $p-\mathcal{WVG}$, developing a systematic method to achieve such a classification. We classify the generalized dimension and codimension of all games with up to 6 players with respect to these classes.

Our main theoretical result is a characterization of the simple games that can be expressed as intersection (union) of pure weighted voting games. Our characterization shows that all simple games except for a small set of non pure weighted voting games can be expressed as intersection of pure weighted voting games. The same is true with respect to union. In this case, the excluded games are those whose dual cannot be expressed by intersection of games in $p-\mathcal{WVG}$. We complement this characterization by showing that a pure weighted voting game can be expressed as an intersection (union) of non-self-dual pure weighted voting games. Furthermore, for simple games that can be expressed as intersection (union) of pure weighted voting games, we give an upper-bound on their generalized dimension with respect to $p-\mathcal{WVG}$ and non-self-dual-$p-\mathcal{WVG}$. Complementing this result, we show that the generalized dimension values, regarding $\mathcal{WVG}$ or $p-\mathcal{WVG}$, form a contiguous interval on the integers. This result is used later in the experimental part.

To define the subclasses of $p-\mathcal{WVG}$ to perform the experimental enumeration and subsequent computations, we start by considering a small set of integer representations, the so-called canonical minimum representations (Molinero, Serna, & Taberner-Ortiz, 2021). Briefly speaking (see definitions later) a canonical representation has the property that the player’s weights are sorted in decreasing order. In addition, a canonical minimum representation identifies a set of isomorphic games in a unique way, where a player is dummy if and only if it has weight 0. The uniqueness of the canonical minimum representations makes it possible to generate them quite efficiently. Furthermore, the number of such representations, although still high, is much smaller than the number of minimum sum representations of $\mathcal{WVG}$. Furthermore, canonical minimum representations allow us to identify easily self-dual games, thus providing a framework to study the subclasses of $\mathcal{WVG}$ mentioned in Table 1.

Notice that each subclass of canonical representations identifies a subclass of $\mathcal{WVG}$. Sometimes we will use only the associated game up to isomorphism, keeping one game for each class of isomorphic games. At other times, we consider all possible permutations, thus considering isomorphic games as different games.

This work will finish by conducting an intensive study on the generalized dimension and codimension of pure simple games with up to 6 players with respect to several subclasses of $p-\mathcal{WVG}$. The results allow us to establish the strict containment relationship among the closure under union (intersection) of subfamilies of $p-\mathcal{WVG}$ and subclasses of simple games. In particular, it is known that the intersections of pure weighted voting games defined by canonical minimum representations can generate only some games of the so-called complete simple games, but not all of them. However, the result is not constructive, and therefore particular examples of those games are not known. As a result of our experiments, we obtain complete simple games that cannot be obtained as intersections of canonical minimum sum representations.

The paper is organized as follows. In Section 2 we start stating the definitions of the main subclasses of simple games used in the paper together with some results on dimension and codimension. Section 3 is devoted to the definition of different kinds of representations for weighted voting games. We use these types of representations to define the associated subclasses of $\mathcal{WVG}$ and introduce the concept of generalized dimension. The theoretical result are given in Section 4 and the experimental ones in Section 5. We conclude in Section 6 posting some open problems.