Metric voting distortion is the worst-case loss in social cost incurred when a voting rule chooses a candidate from voters' ordinal rankings without knowing the distances that induce those rankings.
In the metric social choice theory model, a finite set 
of voters and a finite set 
of candidates lie in a common metric space with metric 
. Each voter ranks the candidates by increasing distance.
The social cost of a candidate 
is the average
 |
(1)
|
A preference profile 
records the rankings of all voters. If a deterministic
voting rule 
assigns each profile to a candidate, its metric voting distortion is
 |
(2)
|
For a randomized voting rule, 
is a probability distribution
on 
,
and the numerator is replaced by the expected value 
(Anshelevich et al. 2018).
The optimal distortion of a deterministic voting rule is 3. Anshelevich et al. (2018) established the lower bound using two-candidate examples, and Gkatzelis et al. (2020) gave a polynomial-time voting rule attaining it.
The randomized problem behaves differently. The conjecture that randomized rules could attain distortion 2 was disproved by Charikar and Ramakrishnan (2022). Their
lower bounds depend on the number 
of candidates and tend to approximately 2.1126 as 
tends to infinity. The bound is optimal for 
, but the optimal randomized distortion remains unknown for
.
On the upper bound side, Charikar et al. (2024) obtained a distortion less than 2.753 using a mixture involving a maximal lottery, the Nash equilibrium of an associated zero-sum game. Frank (2026) and Ye (2026) independently improved the bound to 2.5 by mixing a maximal lottery with Integrated Veto. Shah (2026) subsequently obtained
 |
(3)
|
This leaves a gap between the lower bound approaching 2.1126 and the upper bound 2.3282.
Shah's rule mixes Integrated Veto with a random-size stable lottery. For a positive-integer-valued random variable 
, such a lottery guarantees that, for every candidate 
,
the probability that a random voter prefers 
to the voter's favorite of 
independent draws from it is at most the expected
value of 
,
where the probability also averages over 
. The proof combines infinite-dimensional conic linear
programming duality with exact rational verification of polynomial
nonnegativity in the Bernstein basis.
Shah (2026) reports that GPT-5.6 Sol derived the mathematical proofs from research directions, literature connections, and proof strategies supplied by Shah. Shah checked all final mathematical details, generalized a model-proposed stable-lottery construction to random-size stable lotteries, and rewrote and simplified the exposition with GPT-5.6 Sol and Claude Opus 5. Independent review of the 2.3282 bound has not yet been reported.
