Simple voting games are (simple) structures with various interesting properties, studied in game theory and combinatorics [11,57]. A simple voting game is a pair (P,W), where P
usually represents a finite set of voters, and W ⊆2P the set of winning coalitions, required to be closed under taking supersets. The game admits a quota representation whenever there is a quota q∈Rand a weight functionm:P →R+ such that
A∈ W ⇔X
a∈A
m(a)≥q,
which means that there is a way to assign a weight to each player, in such a way that a coalition is winning exactly when the collective weight of the players in the coalition reaches (or surpasses) the required quotaq.
Our representation problem bears a close connection to the theory of simple games. Consider a selection structure (Ω,A) with Ω ={ω1, ..., ωn}. Let Di:={X⊆Ω|ωiσX}, the collection of sets dominated by ωi. Each state ωi generates a simple voting game
Gi= (Ω,Dic), where the closure-under-superset condition is satisfied thanks to the (No-swap) property (see page 66: the propery ensures that if X∈ Dic, X⊆Y, then Y ∈ D
ic). Now supposeµ is a representation for the induced order <∗σ. Then we have, for eachωi, that
X∈ Dic iff µ(X)≥µ(ωi).
In other words, ifµis a probabilistic representation for <∗σ, it is also weight representation of each game Gi. More specifically, µ simultaneously represents all games Gi, where the quota for each Gi isµ(ωi). Conversely, each such simultaneous quota representation of the
associated system of games {Gi}i≤n gives rise to a probability distribution representing the order<∗σ (it suffices to normalise each weight by the weight of the grand coalition Ω). Thus, finding necessary and sufficient conditions for<∗σ to be representable is equivalent to finding the exact conditions for the collection of games{Gi}i≤n to be simultaneously representable in this sense (note that an important aspect of simultaneous representations is that the quotas themselves depend on the weight function).
We can give this problem a possible game-theoretic interpretation in terms of what we could call ‘coordinated blocking games’. We first identify each state ωi∈Ω with a player. Each game Gi tells us which coalitions can block any decision that playerωi supports. If the collection of games{Gi}i≤n is simultaneously representable as in the above, we can say that the voting system is at least minimally coherent, in the sense that one can attribute weights to all players in a uniform manner39.
In the context of simple games, the (Scott0) axiom above (p. 66) entails that one cannot transform a collectionA1, ...., An of winning coalitions into a collection of losing coalitions by a sequence of pairwise exchanges of players from one coalition to another. We know from classical results on simple games (see [57]) that axioms (No-swap) and (Scott0) suffice for each individual game{Gi}i≤n to be weight-representable. The method should be straightforward to those familiar with the comparative probability literature: we identify each coalition with its characteristic function, and the (Scott0) axiom guarantees that there is a hyperplane separating the winning from the losing coalitions. The normal vector to the separating hyperplane determines the desired weight function.
In our case, the full (Scott) axiom on <∗σ suffices to ensure that there is a way of constructing representations of all the games Gi that are consistent with each other; e.g., that no separating hyperplane for Gi lumps together some winning and losing coalitions from another gameGj.
We can also interpret strongest-stable-set operators in the context of voting games. Consider a weighted voting game (P,W) with weight representation m:P→R+. For
each player p∈P, the projected game Gp= (P \ {p}, Wp) is given by Wp:={X⊆P \
{p} | P
q∈Xm(q)≥m(p)}. The probability function obtained by normalisingm gives rise to a selection function σ:P(P)→ P(P). Given any subset of players A⊆P, the functionσ
outputs the minimalX ⊆A such thatA\X is a losing coalition in every reduced gameGp (withp∈X). The set σ(A) represents the minimal coalition of players such that individual
39
We can illustrate this with the following scenario. Imagine that you are a historian researching the voting protocols of an ancient civilization. You know that each province sent a delegate to vote in a national council, but you have no explicit information about how exactly the outcomes of the votes were determined. The only information available is records of some results of the votes which specify which coalitions were able to block which delegate (that is, you have a collection of games{Gi}i≤nlike the above). The working hypothesis is that the vote of each delegate was accorded a fixedweight – proportional, say, to the population of the delegate’s province. In order to check if this hypothesis is at least consistent with your data, you must check if the games{Gi}i≤nare simultaneously representable.
playerp∈σ(A) can block (or ‘veto’) the coalition A\σ(A). Suppose, for instance, that we play the following game. First, we restrict attention to a subsetA of players. Then we play a champion game onA: a championp is picked at random from some pre-selected subset
X⊆A. The champion then votes against the entire opposition A\X. Different choices of possible champion sets X yield different chances of winning against the opposition. A decisive team is a subset of playersX ⊆A such that, no matter who is chosen from X as a champion, the opposition A\X loses against the champion. We can think ofσ(A) as the minimal decisive team in the champion game onA.