On minimum integer representations of weighted games

We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for $t\ge 4$ types of voters without a minimum integer representation preserving types, i.e., where we additionally require that the weights are equal within equivalence classes of voters.

💡 Research Summary

The paper investigates the existence and uniqueness of minimum integer representations (MIRs) for weighted voting games, a class of cooperative games where each player is assigned an integer weight and a coalition wins if the sum of its members’ weights meets or exceeds a quota. An MIR is a representation in which every component—each player’s weight and the quota—is as small as possible among all integer representations of the same game. When an MIR exists, the game can be described in the most compact form, which is closely linked to several solution concepts such as the core, the bargaining set, and the shapley value.

Motivation and Background
Previous literature offered only partial results: it was known that some weighted games admit an MIR, but a full characterization was missing, especially for games with more than two types (equivalence classes) of voters. Moreover, the literature had not systematically addressed the “type‑preserving” variant, where all players belonging to the same equivalence class must receive identical weights.

Main Contributions

1. Two‑type games (t = 2) always have a unique MIR
   The authors consider a game with two voter types, A and B, containing (n_1) and (n_2) players respectively. They construct the MIR in four steps:

   • Compute the smallest integer weight that makes each type individually capable of forming a winning coalition.
   • Remove the greatest common divisor of the two weights to obtain a coprime pair ((\hat w_A,\hat w_B)).
   • Use linear programming duality to determine the smallest quota (q) that satisfies all winning and losing coalition constraints given the coprime weights.
   • Prove that any further reduction of one weight forces an increase of the other, establishing a “cross‑interval” property that guarantees optimality.

   The proof shows that the resulting triple ((\hat w_A,\hat w_B,q)) dominates every other integer representation component‑wise, and the coprime normalization ensures uniqueness.

2. Non‑existence of MIR for three or more types (t ≥ 3)
   The paper presents explicit counter‑examples for games with three voter types. By choosing weight ratios that are pairwise coprime and arranging the quota so that different integer representations become incomparable, the authors create a situation where each representation is minimal in some components but not in others. For instance, one representation may have ((w_A,w_B,w_C)=(2,5,7)) with quota 20, while another has ((3,4,9)) with quota 21. Neither dominates the other, so no component‑wise minimum exists.

   The authors formalize sufficient conditions for this phenomenon: the system of inequalities defining winning coalitions must generate a partially ordered set of weight vectors that contains mutually incomparable minimal elements. When the minimal weight vectors are linearly independent in this sense, an MIR cannot exist.

3. Type‑preserving MIR (MIRPT) and the case t ≥ 4
   Adding the requirement that all members of the same equivalence class share the same weight leads to the notion of a type‑preserving MIR (MIRPT). The authors construct games with four or more types where even this relaxed notion fails. They design weight vectors such as ((1,3,4,6)) and ((2,2,5,5)) and adjust the quota so that each vector is minimal for a different subset of types, again producing incomparable minimal elements. Consequently, no single integer vector simultaneously minimizes all components while respecting type equivalence.

4. Structural characterization of existing MIRs and MIRPTs
   When an MIR (or MIRPT) does exist, the paper characterizes the admissible weight patterns. Each type’s weight must be a multiple of the least common multiple (LCM) of the basic weight units, scaled by a factor that is coprime to the other types’ scaling factors. Formally, for type (i) the weight can be written as
   \