Abstract: We consider a standard mechanism design problem in voting environments with $n$ voters, $n \geq 2$. Each voters' type is a linear ordering over a finite set of alternatives. We assume that a voter’s beliefs about the types of other voters is positive correlated with her own type. We propose definitions of positive correlation. One is based on the Kemeny distance between linear orders and referred to as K-correlation. Thus a voter with type $P_i$ considers the preference profile of other voters $P_{-i}$ more likely than the profile $P_{-i}'$ if the Kemeny distance between every pair $(P_i , P_j )$ is strictly less than the Kemeny distance $(P_i , P_j )$ for every $j = i$. We also define a different notion of positive correlation referred to as TS (or top-set) correlation based on the likelihood of agreement of the $k$ best alternatives (for any $k$) for any pair of types. We show that $K$-correlation implies TS-correlation.

A voting rule is a mapping from the set of preference profiles to the set of alternatives. It is Ordinally Bayesian Incentive-Compatible (OBIC) (d'Aspemont-Peleg (1988) if truth-telling is a Bayes-Nash equilibrium (given beliefs of every voter) for every possible cardinalization of voter types. We investigate the set of OBIC voting rules satisfying two alternative robustness hypotheses on voters' beliefs.

The first of these properties is local robustness. We identify a condition which is necessary and sufficient for a voting rule to be OBIC with respect to TS correlation and satisfying local robustness with respect to these beliefs. We demonstrate that there is a large class of voting rules satisfying this condition. We also consider global robustness where voting rules are required to be OBIC with respect to all TS or $K$-correlated beliefs. Interestingly, there are non-dictatorial rules which satisfy these properties, so that robustness of beliefs which are positively correlated is not equivalent to strategyproofness. We show however that if efficiency is also required, then global robusteness of OBIC with respect to either $K$ or TS-correlation leads to dictatorship (provided that there are at least three alternatives).

The generally positive results contrast sharply with the negative results obtained for the independent case by Majumdar-Sen (2004) and bear a similarity with results in the auction design model.

[This is a joint work with Mohit Bhargava and Dipjyoti Majumdar]