Least-bad Single-winner Ranking Method?



  • The question came up in the context of a new method proposed in the present forum.



  • @Jack-Waugh I would say either Minimax Condorcet or Borda



  • Objectively, the answer is Game Theory Voting. However, it is basically impossible to explain to, well, anyone.

    The idea is that, given any two ranked-ballot voting methods, we can compare how many voters prefer the winner of one over the winner of the other, and vice versa. Essentially, treating the ballots as giving head-to-head votes between the candidates who would win under each method.

    On average, in the long run, game theory voting will produce winners who are preferred by as many or more voters than the winner produced by any other method. In other words, there is no ranked-ballot voting system whose winners are preferred by more voters than the game-theory winner, over the long run.

    It is a Condorcet method, and indeed a Smith method, however the actual tiebreaking procedure is a linear optimization problem, which can be solved using, eg. the simplex method, to find each candidate’s optimal probability of winning. The winner is then selected with those probabilities.



  • @NevinBR said in Least-bad Single-winner Ranking Method?:

    On average, in the long run, game theory voting will produce winners who are preferred by as many or more voters than the winner produced by any other method. In other words, there is no ranked-ballot voting system whose winners are preferred by more voters than the game-theory winner, over the long run.

    So "optimal" means that the expected value of the margin between the winner selected by GTV and by some other system will always be non-negative?



  • @NevinBR Thank you for the Game Theory Voting link. I see that there is an assumption that when a Condorcet selection exists that it's the best choice. I think there are situations where that is NOT a correct assumption. I tried to modify the scenario that the authors, Rivest and Shen, used, but it wasn't working out very well, so I simplified one of the scenarios that I had previously come across, which was from using randomly generated numbers.

    Scenario: There are 4 candidates and 13, 138 voters. With FPTP, candidate B gets 3,900 votes, C 3,547, D 3,456 and A 2,325. B wins Condorcet buy 2 votes, 6570/6568, over each of the other candidates. At first blush, B seems to be clearly the best choice.

    I exaggerated the randomly generated numbers into a scenario where ALL of the voters that picked B as their first choice picked D as their second choice. So, in a head to head with B and D, B got 2 more votes than D, but D was the second choice of all 3,900 supporters of B. Conversely, only 639 of the 3,456 D supporters chose B as their second choice.

    It seems to me that candidate D is preferred by the most people to the greatest degree.

    I didn't try to understand all of the Game Theory details, but it seems to me that there is a way to modify it so that it considers this in the weighting and sets the probabilities accordingly.

    I'm wondering what people think of this. (Both of the assertion of D being the best choice, and the potential of Game Theory to account for this.)

    Here are the full numbers for the scenario:
    0:Option A
    819:Option A>Option B>Option C>Option D
    52:Option A>Option B>Option D>Option C
    141:Option A>Option C>Option B>Option D
    898:Option A>Option C>Option D>Option B
    116:Option A>Option D>Option B>Option C
    299:Option A>Option D>Option C>Option B
    0:Option B
    0:Option B>Option A>Option C>Option D
    0:Option B>Option A>Option D>Option C
    0:Option B>Option C>Option A>Option D
    0:Option B>Option C>Option D>Option A
    1800:Option B>Option D>Option A>Option C
    2100:Option B>Option D>Option C>Option A
    0:Option C
    988:Option C>Option A>Option B>Option D
    658:Option C>Option A>Option D>Option B
    3:Option C>Option B>Option A>Option D
    667:Option C>Option B>Option D>Option A
    473:Option C>Option D>Option A>Option B
    668:Option C>Option D>Option B>Option A
    0:Option D
    1044:Option D>Option A>Option B>Option C
    445:Option D>Option A>Option C>Option B
    382:Option D>Option B>Option A>Option C
    257:Option D>Option B>Option C>Option A
    635:Option D>Option C>Option A>Option B
    693:Option D>Option C>Option B>Option A


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