Maximal lotteries

Maximal lotteries were first proposed by the French mathematician and social scientist Germain Kreweras in 1965^[11] and popularized by Peter Fishburn.^[1] Since then, they have been rediscovered multiple times by economists,^[8] mathematicians,^[1][12] political scientists, philosophers,^[13] and computer scientists.^[14]

Several natural dynamics that converge to maximal lotteries have been observed in biology, physics, chemistry, and machine learning.^[15][16][17]

The input to this voting system consists of the agents' ordinal preferences over outcomes (not lotteries over alternatives), but a relation on the set of lotteries can be constructed in the following way: if ${\displaystyle p}$ and ${\displaystyle q}$ are lotteries over alternatives, ${\displaystyle p\succ q}$ if the expected value of the margin of victory of an outcome selected with distribution ${\displaystyle p}$ in a head-to-head vote against an outcome selected with distribution ${\displaystyle q}$ is positive. In other words, ${\displaystyle p\succ q}$ if it is more likely that a randomly selected voter will prefer the alternatives sampled from ${\displaystyle p}$ to the alternative sampled from ${\displaystyle q}$ than vice versa.^[4] While this relation is not necessarily transitive, it does always admit at least one maximal element.

It is possible that several such maximal lotteries exist, as a result of ties. However, the maximal lottery is unique whenever the number of voters is odd.^[18] By the same argument, the bipartisan set is uniquely defined by taking the support of the unique maximal lottery that solves a tournament game.^[8]

Maximal lotteries are equivalent to mixed maximin strategies (or Nash equilibria) of the symmetric zero-sum game given by the pairwise majority margins. As such, they have a natural interpretation in terms of electoral competition between two political parties^[19] and can be computed in polynomial time via linear programming.

Suppose there are five voters who have the following preferences over three alternatives:

• 2 voters: ${\displaystyle a\succ b\succ c}$
• 2 voters: ${\displaystyle b\succ c\succ a}$
• 1 voter: ${\displaystyle c\succ a\succ b}$

The pairwise preferences of the voters can be represented in the following skew-symmetric matrix, where the entry for row ${\displaystyle x}$ and column ${\displaystyle y}$ denotes the number of voters who prefer ${\displaystyle x}$ to ${\displaystyle y}$ minus the number of voters who prefer ${\displaystyle y}$ to ${\displaystyle x}$.

${\displaystyle {\begin{matrix}{\begin{matrix}&&a\quad &b\quad &c\quad \\\end{matrix}}\\{\begin{matrix}a\\b\\c\\\end{matrix}}{\begin{pmatrix}0&1&-1\\-1&0&3\\1&-3&0\\\end{pmatrix}}\end{matrix}}}$

This matrix can be interpreted as a zero-sum game and admits a unique Nash equilibrium (or minimax strategy) ${\displaystyle p}$ where ${\displaystyle p(a)=3/5}$, ${\displaystyle p(b)=1/5}$, ${\displaystyle p(c)=1/5}$. By definition, this is also the unique maximal lottery of the preference profile above. The example was carefully chosen not to have a Condorcet winner. Many preference profiles admit a Condorcet winner, in which case the unique maximal lottery will assign probability 1 to the Condorcet winner. If the last voter in the example above swaps alternatives ${\displaystyle a}$ and ${\displaystyle c}$ in his preference relation, ${\displaystyle a}$ becomes the Condorcet winner and will be selected with probability 1.