What is the strategy-proofness of maximal lottery?

Question

The maximal lottery system is a voting system based in game theory. It is based in game theory. The general idea is that it is better than all the other voting systems.

My question is, is it susceptible to tactical-voting? Is it strategy-proof?

Answer 1

Tactical voting can occur. Say the preferences of three voters for three candidates are $1\gt2\gt3$, $3\gt1\gt2$ and $2\gg1\gtrsim3$. Then honest voting would result in the payoff matrix

$$ \pmatrix{0&1&1\\-1&0&1\\-1&-1&0} $$

with a pure Nash equilibrium for candidate $1$, whereas if the third voter pretended to have preference $2\gt3\gt1$ the payoff matrix would be

$$ \pmatrix{0&1&-1\\-1&0&1\\1&-1&0} $$

with a mixed Nash equilibrium with probability $\frac13$ for each candidate, which is preferable to the third voter since her preference for $2$ over the other candidates is stronger than her preference for $1$ over $3$.

P.S.: My example relies on the strength of preferences. This article, which provides insight on a number of related questions, gives an example (on p. $6$) that works independent of the strength of preferences: One voter has preference $1\gt3\gt2$, one has $1\gt2\gt3$, two have $2\gt3\gt1$ and one has $3\gt1\gt2$. The payoff matrix is the second one above with equilibrium probability $\frac13$ for each candidate, whereas if the first voter pretends to have preference $1\gt2\gt3$, the payoff matrix becomes

$$ \pmatrix{0&1&-1\\1&0&3\\1&-3&0} $$

with equilibrium probabilities $(\frac35,\frac13,\frac13)$, a gain for this voter's preferred candidate at the cost of both less preferred candidates.