Which voting method is the most fair? I have heard of


*

*Plurality

*Borda Count

*Instant Runoff

*Sequential Pairwise


voting methods, where it mathematically can be proved which is the most fair and in which situations.
From my understanding Sequential Pairwise does not seam fair at all, as the order in which you compare have a influence on the outcome. With Instant Runoff I would assume is not suited when there are few voting options and most of them have few votes.
Question
Does anyone know which is the most fair and when each of the methods should be used?
 A: I like to mention the following voting system: let each voter rank all candidates in order of preference (a total preorder will do, i.e., voters are allowed to rank two candidates equally).  Now tabulate, for each pair $(i,j)$ of candidates, the number of voters who placed $i$ before $j$ minus the number who placed $j$ before $i$.  This gives an antisymmetric matrix which can be considered as the payoff matrix of a symmetric zero-sum game.  By the von Neumann minimax theorem, this game has an optimal mixed strategy, and moreover, it is unique in general (barring generalized "ties"; for example, it can be made almost surely unique by adding a voter with infinitesimal weight who votes for a random permutation in the order of the candidates, or by some other kind of tie-breaking strategy).  Now choose the candidate according to the probabilities given by this optimal strategy: this candidate is considered elected.  (Note that this is not deterministic: there is a random choice at the end; however, if there is a Condorcet winner, which is exactly the same thing as an optimal pure strategy in the game, then there will be no random choice.)
This voting system is optimal exactly in the sense that the optimal strategy for the game is optimal.  That is, if I choose the election's winner according to the voting system described above, and you choose the election winner according to some other voting system, then the expected number of voters happier with my choice (i.e., having placed my choice higher in their ballot) is at least as great as the expected number of voters happier with yours, since the difference between them is exactly the outcome of the game and I am playing the optimal strategy for that game.  So in this sense, this voting system is the best when it comes to making the most electors happy.
(I don't know if this voting system has a name.  I like to call it the Condorcet-von-Neumann or Condorcet-Nash voting system, because it is a Condorcet voting system obtained by using the Nash equilibrium for the obvious game.  But the idea is so obvious that it has probably been described countless times in various different ways.  I just don't have a reference.  Edit: From this question, I learn that the (or at least "a") standard name for this voting system is, in fact, "maximal lotteries"; and as I expected, it has been rediscovered a number of times.)
Of course whether it is politically acceptable to have an element of randomness in an election is another problem, and since that one is non-mathematical, I won't comment any further.
