Equilibrium
The word equilibrium refers, among other things, to concepts of game theory. The most well-known equilibrium in game theory is the Nash equilibrium.
Nash equilibria are situations where each player has chosen a strategy and no single player could improve his situation by unilaterally changing strategy while all the other players keep their strategies.
Cabal equilibria are situations where each player has chosen a strategy and no subset of players could simultaneously change behaviour (while those outside that subset keep their strategies) in a way that no player in that subset is worse off and at least one in the subset is better off.
All cabal equilibria are Nash equilibria but not vice versa.
Notes
Note that just because a voting method can have an equilibrium on a particular candidate, doesn't mean it will always. For example, consider Approval voting having an equilibrium on the Condorcet winner:
Here's my reasoning: consider a standard chicken dilemma:
Number Ballots 1 A 34 A>B>C 25 B>A>C 40 C where, under honesty, the 59 voters ranking both A and B over C would approve of both A and B.
If B-top voters strategically bury/don't approve A, and A-top voters vote honestly, B of course wins; and B is clearly not the honest CW, nor the honest Approval winner (A is both). If B voters do this first, the strategic equilibrium for A voters (supposing here that C is indeed honestly unacceptable to A-top voters) is to simply continue to vote honestly, and thus B wins. So it seems incorrect to generally claim that strategic voting under Approval has an equilibrium where the CW wins all the time.
Furthermore, there seems to be (in my view) considerable similarity between representing this election as a game under Approval voting and as a game under, say, Ranked Pairs. What I mean is that each of the A-B player factions has two moves really available (honesty and disapproval/burial respectively in each game). So then, the result for Approval voting is:
- Honesty (A) Disapprove (A) Honesty (B) A wins, minor B utility loss A wins, minor B utility loss Disapprove (B) B wins, minor A utility loss C wins, major A,B utility loss For Ranked Pairs, the game matrix is exactly the same:
- Honesty (A) Burial (A) Honesty (B) A wins, minor B utility loss A wins, minor B utility loss Burial (B) B wins, minor A utility loss C wins, major A,B utility loss So, to suggest that Approval voting selects a CW under strategic equilibrium seems equivalent to saying that in such scenarios, Ranked Pairs and similar Condorcet methods also must do so in such scenarios; which is clearly incorrect.
(A brief comment: note that despite these scenarios being called "chicken dilemmas", they do not in fact share the same payoff matrices with the actual game of chicken. The RP/Approval payoff matrix is something like
- Honesty (A) Burial (A) Honesty (B) -1, 0 -1, 0 Burial (B) 1, -1 -10, -10 whereas in the actual standard game of chicken, the matrix is more like
- Swerve(A) Straight (A) Swerve (B) 0, 0 -1, 1 Straight (B) 1, -1 -10, -10 this is an important difference; namely, we see from the RP and Approval payoff matrix that regardless of which choice B makes, A is better off or at least not worse off for having chosen Honesty, whereas in standard chicken this is not true (what's best for A depends heavily on what B does).)[1]