User:Silvio Gesell/Tom's method

From electowiki

Tom's method is a single-winner electoral system that finds the winner of repeated score balloting at the highest level of strategy capable of producing a stable winner.

The strategy level, $, is the power the previous round's totals are raised to to generate the values the candidates' perceived probabilities of victory (PPVs) are proportional to. The first round is identical to single-round score, and each subsequent round proceeds as follows:

1) An expected winner utility (EWU) is generated for each voter. A voter's EWU is equal to the sum of the products of his original scores for the candidates and their PPVs:


where N is the number of candidates and sk is the voter's original score for candidate k.

k's PPV, in turn, is equal to his total score in the previous round, to the $th power, divided by the sum of all candidates' total scores to the $th power:


where Sk is k's total score in the previous round.

2) Each voter approves (i.e. scores 1) each candidate he scored higher than this round's EWU on the original ballot and disapproves (i.e. scores -1) each candidate he scored lower than this round's EWU. Repeat until the results of one round are identical to the results of a previous round. If more than one candidate have won since that previous round, start over with a lower $.

At $=0, Tom's method is identical to single-round score.

At $>0, Tom's method can converge on a winner that is different from the single-round score winner.

If there is a Condorcet winner, Tom's method necessarily converges on a winner at any $ and the Condorcet winner at any sufficiently large $.

If there is no Condorcet winner, Tom's method, above a certain $, ceases to converge upon a winner. The winner at that $, the greatest stable $, the greatest $ for which Tom's method converges upon a winner, is the rightful winner.

How best to find the greatest stable $ is an open question. It is almost certainly economical to first check for a Condorcet winner. To do so by Tom's method, let $=∞. At $=∞, voters approve all candidates they prefer to last round's winner and disapprove all candidates they prefer last round's winner to (this will find the Condorcet winner or absence thereof in a maximum of N rounds without ties or 2N rounds with ties). In the absence of a Condorcet winner, the problem is potentially intractable, so it may be best to set a maximum search time (or number of rounds) in advance and, after ruling out a Condorcet winner, repeatedly eliminating any Condorcet loser (or candidate outside the Mutual Majority set, if one can easily be found), and finding the $=0 winner, "start the clock". First try $=1 and, thereafter:

If convergence succeeds, start over after doubling $ (if convergence succeeded last time) or setting it to the geomean of current and previous $.

If convergence fails, start over after halving $ (if convergence failed last time) or setting it to the geomean of current and previous $.

Repeat until the time limit, and declare the winner of the last successful convergence the winner. In the special case of a time limit of 0, Tom's method reduces to Condorcet/Score, in which the score winner wins if and only if there is no Condorcet winner. Condorcet/Score in turn reduces strategically to Condorcet/Approval given a sufficiently large range, suggesting a ranked/approval ballot may be more appropriate if the time limit is 0.