A Map of Approval Voting Equilibria Outcomes

It is commonly accepted that the multiplicity of equilibria is ubiquitous in preference aggregation games with any voting method. We prove that this multiplicity is greatly reduced under some mild restrictions over social preferences when each voter can vote for as many candidates as she wishes (the Approval voting method). For scenarios with three candidates, we can hence build a map that associates any preference profile to its set of equilibria outcomes; this map is very close to the most well-known Tournament solutions.