The Wikipedia page on hypergraphs says
In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory.
I have not found links that describe that directly, either on that page, or through web searching. Looking for "voting games" is useless, and "hypergraph voting games" brings up some obscure-looking papers.
Can you
- Explain the link more directly.
- Provide intro-level references on the web (i.e. not books)
From wikipedia:
A coalitional game v is considered simple if payoffs are either 1 or 0, i.e. coalitions are either "winning" or "losing".[7]
Equivalently, a simple game can be defined as a collection W of coalitions, where the members of W are called winning coalitions, and the others losing coalitions. It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set. However, in other areas of mathematics, simple games are also called hypergraphs or Boolean functions (logic functions).
However I found the following definition which is a bit different here
A simple game is a pair in which and is a collection of subsets $\mathcal W$ of that satisfies: $\varnothing \not \in \mathcal W$, $X\in \mathcal W$ and if $A\in \mathcal W$ and $A\subseteq B$ then $B\in \mathcal W$.
I also found a third definition here
A simplegame is a game $(N,f)$ with $f(N) =1$ and $f(S)$ either $0$ or $1$ for all coalitionsS.
So I guess only the first definition allows for all hypergraphs to be simple games. The second definition is a lot more restrictive, and the last one allows for all hypergraphs that contain the total hyperedge $N$.
I would like to conclude by saying this is probably not very important and just some definitions that aren't very widespread.