User:Silvio Gesell/Tom's method

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:

$$EWU=\textstyle \sum_{k=1}^N s_k*PPV_k\displaystyle$$,

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:

$$PPV_k=\frac{{S_k}^$}{\textstyle \sum_{k=1}^N {S_k}^$\displaystyle}$$,

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.