SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Daniel Halpern
dhalpern@g.harvard.edu
Rachel Li
rachelli@college.harvard.edu
Ariel D. Procaccia
arielpro@seas.harvard.edu
In voting theory, when voters have ranked preferences over candidates, the celebrated Gibbard-Satterthwaite Theorem essentially rules out the existence of reasonable strategyproof methods for picking a winner. What if we weaken strategyproofness to only hold for Bayesian voters with beliefs over others' preferences? When voters believe other participants' rankings are drawn independently from a fixed distribution, the impossibility persists. However, it is quite reasonable for a voter to believe that other votes are correlated, either to each other or to their own ranking. We consider such beliefs induced by classic probabilistic models in social choice such as the Mallows, Placket-Luce, and Thurstone-Mosteller models. We single out the plurality rule (choosing the candidate ranked first most often) as a particularly promising choice as it is strategyproof for a large class of beliefs containing the specific ones we introduce. Further, we show that plurality is unique among positional scoring rules in having this property: no other scoring rule is strategyproof for beliefs induced by the Mallows model when there are a sufficient number of voters. Finally, we give examples of prominent non-scoring voting rules failing to be strategyproof on beliefs in this class, further bolstering the case for plurality.
One of the most celebrated results in voting theory is the Gibbard-Satterthwaite Theorem [8, 22]. It states that when voters express ordinal preferences over at least 3 candidates, there is no "reasonable" aggregation rule that is strategy-proof: there will always exist instances where voters will be incentivized to manipulate and lie about their preferences to achieve a better outcome.
However, one caveat about this strong negative result is that, a priori, a voter may need perfect information about how others vote to manipulate successfully. Perhaps, if the voter is slightly uncertain, no manipulation helps consistently enough to be worthwhile. Majumdar and Sen [13] analyzed exactly this question when voters have independent beliefs. That is, when a voter is considering whether or not to manipulate, they assume all others have rankings drawn independently from a fixed distribution. The classic notion of strategyproofness no longer makes sense in this probabilistic Bayesian setting, so they instead use the natural extension known as ordinally Bayesian incentive compatible (OBIC), essentially that the rules are strategyproof in expectation no matter what underlying cardinal values voters have. Their results, unfortunately, are widely negative. They show that for a "large" set of distributions, Gibbard-Satterthwaite still holds. There do exist distributions where many rules are OBIC, e.g., the uniform distribution over all rankings. Still, these positive examples are extremely brittle: even a slight perturbation leads back to the impossibility.
But independent beliefs are quite restrictive. They cannot capture several kinds of beliefs that would likely occur in practice. For one, when the number of voters is large, the uncertainty essentially vanishes. Suppose a distribution places probability  $1/4$  on other voters having the ranking  $a \succ b \succ c$ . In that case, when the number of voters is large, it is extremely unlikely that the proportion of voters with this ranking is anything other than  $1/4 \pm \varepsilon$ . In a real presidential election, a voter may quite plausibly believe that a candidate will receive anywhere between  $45\%$  and  $55\%$  of the votes, but this situation simply cannot be captured by a single independent ranking distribution. Second, one's own ranking may influence the probability placed on others. Suppose a voter, after much research, discovers that they prefer one proposal to another; they may reasonably believe others are clever enough to have reached a similar conclusion. In terms of their beliefs, they may place a slightly higher probability on others voting more similarly to them than not, no matter what their realized preferences are.
This has led follow-up work to consider the same question under correlated beliefs [1, 16, 14, 2]. However, besides some impossibilities, the work so far has largely been of the following form: for any reasonable voting rule, there exists a set of beliefs where the rule is OBIC. But perhaps the more natural direction is the converse: under a natural set of beliefs, is there a reasonable voting rule that is consistently OBIC? Can this property help us distinguish between voting rules, showing that under some reasonable beliefs, certain rules are not OBIC, thereby bolstering the case for the provably incentive-compatible ones? These are the questions we tackle.
Our contributions. We begin by presenting various classes of beliefs induced by classic probabilistic social choice models such as the Mallows [15], Thurstone-Mosteller [23, 18], and Placket-Luce [20, 12] models. In essence, these are the beliefs a voter would have if they assume that voter preferences were generated by such a model. Inspired by these models, we present a novel class of mildly correlated beliefs that includes all of them. We show that, under this class of beliefs, the plurality rule is OBIC.
Next, we provide a negative result: Among positional scoring rules (where each voter assigns a fixed score to each position in their ranking), plurality is unique in being OBIC when voters have Mallows beliefs. All other rules will become not OBIC when there are three candidates, at least when there are a sufficient number of voters. In addition, we provide some robustness checks on this negative result. A popular positional scoring rule known as Borda Count fails for any number of voters. By contrast, we identify other positional scoring rules that are OBIC with two voters, meaning our result could not be strengthened by relaxing the sufficient number of voters requirement.
Finally, we complement this more sweeping classification with examples of other prominent rules, such as Copeland and maximin, which fail to be OBIC with specific Mallows beliefs and few voters. This further bolsters the case for plurality as an unusually attractive rule when viewed through the lens of ordinal Bayesian incentive compatibility under correlated beliefs.
Related work. As mentioned above, the analysis of OBIC voting rules began with Majumdar and Sen [13] essentially providing the final word on independent beliefs; their notion of OBIC dates back to work on committee selection [5].
Since then, there have been a few lines of work on correlated beliefs with slightly different goals. The most closely related is that of Majumdar and Sen [14]. They define a large class of positively correlated beliefs based on the Kemeny metric and then show in a similar fashion to the Gibbard-Satterthwaite Theorem that any voting rule that is OBIC with respect to these beliefs, along with being Pareto efficient, is necessarily dictatorial. They do present one voting rule that is both OBIC with respect to these beliefs and nondictatorial (while not being Pareto efficient), but it is a clearly impractical rule that is designed to make a technical point.<sup>1</sup> Note that all the rules we consider are Pareto efficient.
Another line of work considers local OBIC. A voting rule is locally OBIC with respect to a class of beliefs if there exists a belief in the class such that any belief in a neighborhood of the original is OBIC. This means the rule remains OBIC even after a slight perturbation to the underlying belief. Bhargava et al. [1] and Bose and Roy [2] attempt to classify the set of locally OBIC voting rules with
Their rule is called Unanimity with Status Quo. There is one default candidate  $x$ . If there is a candidate  $y$  which every single voter places as their top choice, then  $y$  is elected, but in any other case,  $x$  wins.
respect to a large class of correlated beliefs and show that under minimal conditions, this requirement can be satisfied.
Mandal and Parkes [16] consider a different notion of incentive compatibility which, rather than requiring that no manipulation can lead to a utility gain in expectation, bounds the probability under which there is a utility gain. They again do this with respect to several different classes of beliefs, including one that we consider based on the Mallows Model.
Further afield, there are extensive lines of research on circumventing the Gibbard-Satterthwaite Theorem. We provide examples of three here, although there are many others. One line considers the complexity, showing that some rules, while in principle susceptible to manipulation, have instances where it is hard (in the worst case) to find such a manipulation [7, 3]. Another considers strategyproofness under restricted domains, where a voter's set of possible rankings is limited [7, 3]. A third considers the likelihood of an individual arriving in an instance where they are able to manipulate at all [17, 25].
Finally, without considering strategyproofness, there has been much work on probabilistic social choice, making use of the models on which our results are based, especially in learning preferences from data [11, 24, 10, 19].
We begin by introducing the classic social model and then later describe relevant definitions for social choice under uncertainty.
Classic social choice model. Let  $N = \{1, \dots, n\}$  be a set of  $n$  voters, and let  $\mathcal{A} = \{a_1, \dots, a_m\}$  be a set of  $m$  alternatives. Let  $\mathcal{L}$  be the set of rankings over  $\mathcal{A}$ , where for  $\sigma \in \mathcal{L}$ ,  $\sigma(j)$  is the  $j$ 'th candidate in ranking  $\sigma$  and  $\sigma^{-1}(a)$  is the ranking index of candidate  $a$ . We use the notation  $a \succ_{\sigma} b$  to denote that  $\sigma^{-1}(a) < \sigma^{-1}(b)$  and  $a \succeq_{\sigma} b$  to denote  $\sigma^{-1}(a) \leq \sigma^{-1}(b)$ , i.e.,  $a$  is strictly (or weakly) preferred to  $b$  under  $\sigma$ . Additionally, instead of writing  $\sigma = a \succ b \succ c$ , when it is clear from context, we will sometimes shorten this to  $\sigma = abc$ . Each voter  $i$  has a ranking  $\sigma_i \in \mathcal{L}$  and the tuple of these rankings  $\sigma = (\sigma_1, \dots, \sigma_n) \in \mathcal{L}^n$  is called the preference profile. We let  $\sigma_{-i} \in \mathcal{L}^{n-1}$  denote the profile without voter  $i$ , and for a ranking  $\sigma_i' \in \mathcal{L}$ , we let  $(\sigma_{-i}, \sigma_i')$  be the profile with  $\sigma_i$  replaced with  $\sigma_i'$ .
A voting rule is a function  $f$  that, given a profile  $\sigma$ , outputs a distribution over winning alternatives. We define several voting rules of interest here. Our theoretical results will primarily focus on positional scoring rules [26]. A positional scoring rule  $f$  is parameterized by a vector of  $(s_1, \ldots, s_m)$  where each  $s_j \in \mathbb{Z}_{\geq 0}$  with  $s_1 \geq \dots \geq s_m$  and  $s_1 > s_m$ . On a profile  $\sigma$ , for each voter  $i$ , their  $j$ 'th candidate  $\sigma_i(j)$  is given  $s_j$  points. The points are added up over all voters, and the winning candidate is the one with the most points. More formally, for a ranking  $\sigma \in \mathcal{L}$  and candidate  $c \in \mathcal{A}$ , we write  $\mathrm{SC}_c^f(\sigma) = s_{\sigma^{-1}(c)}$  for the points (or score) given to  $c$  by  $\sigma$ . For a profile  $\sigma$ , we write  $\mathrm{SC}_c^f(\sigma) = \sum_i \mathrm{SC}_c^f(\sigma_i)$  to be the total points. When  $f$  is clear from context, we may drop it from the notation. In deterministic settings, when there is a tie, a tie-breaking rule needs to be given (i.e., tie-break in favor of lower index candidates). Since we will be working in a probabilistic setting, it will be more convenient to assume uniform random tie-breaking, so that if there is a tie among  $k$  candidates, each wins with probability  $1/k$ . However, our results would continue to hold even with arbitrary deterministic choices. Two rules of particular interest are plurality, parameterized by the vector  $(1, 0, \ldots, 0)$ , and Borda count, parameterized by the vector  $(m - 1, m - 2, \ldots, 1, 0)$ .
We consider two additional rules beyond positional scoring rules, Copeland and maximin. To define them, for a profile  $\sigma$  we define the pairwise margin for two candidates  $a$  and  $b$ ,  $N_{ab}(\sigma) = |\{i|a\succ_i b\}|$ , i.e., the number of voters that prefer candidate  $a$  to candidate  $b$ .
For Copeland, we define the Copeland score for a candidate  $a$  as  $\sum_{b \neq a} \mathbf{I}[N_{ab}(\sigma) > n/2] + (1/2)\mathbf{I}[N_{ab}(\sigma) = n/2]$ . In words, the candidate gets one point for every other candidate they pairwise beat and a half point for every other candidate they pairwise tie. The Copeland winners are those with the highest Copeland scores (with uniform tie-breaking). For maximin, we define the maximin score for a candidate  $a$  as  $\min_{b \neq a} N_{ab}(\sigma)$ , i.e., the smallest margin by which  $a$  beats another candidate. Again, the maximin winners are those with the highest maximin scores (with uniform tie-breaking).
A voting rule  $f$  is called strategy-proof if for all profiles  $\sigma$ , all voters  $i$ , and all alternative manipulations  $\sigma_i' \in \mathcal{L}$ ,  $f(\sigma) \succeq_{\sigma_i} f(\sigma_{-i}, \sigma_i')$ . That is, no voter can ever improve the outcome of the vote by misreporting. A rule is called onto if for all candidates  $a \in \mathcal{A}$ , there is a profile  $\sigma$  where  $f(\sigma) = a$ , and is called dictatorial if there is a voter  $i$  for which  $f(\sigma) = \sigma_i(1)$ , i.e., voter  $i$  always gets their top choice. The Gibbard-Satterthwaite Theorem states that any rule for  $m \geq 3$  candidates that is strategy-proof and onto is necessarily dictatorial. Since "reasonable" rules must be onto and nondictatorial, this eliminates the possibility of any being strategy-proof.
These notions can also be extended to randomized rules. A utility function  $u$  is a mapping from candidates to real numbers. We say that  $u$  is consistent with a ranking  $\sigma$  if  $u(x) > u(y) \Leftrightarrow x \succ_{\sigma} y$ . A (randomized) voting rule  $f$  is called SD-strategy-proof if for all profiles  $\pmb{\sigma}$ , all voters  $i$ , all manipulations  $\sigma_i'$ , and all utility functions  $u$  consistent with  $\sigma_i$ ,  $\mathbb{E}[u(f(\pmb{\sigma}))] \geq \mathbb{E}[u(f(\pmb{\sigma}_{-i}, \sigma_i'))]$ . This says that no matter what underlying utilities an agent has, as long as they are consistent with their ranking, they cannot improve their expected utility by manipulation. An equivalent definition can be given with respect to stochastic dominance (hence the SD in the name). For  $k \leq m$ , let  $B_k(\sigma) = \{\sigma(1), \dots, \sigma(k)\}$  be the set of the  $k$  best alternatives according to  $\sigma$ . Then, SD-strategy-proofness can be rephrased as requiring that for all profiles  $\pmb{\sigma}$ , all voters  $i$ , all manipulations  $\sigma_i'$ , and all  $k \leq m$ ,  $\operatorname*{Pr}[f(\pmb{\sigma}) \in B_k(\sigma_i)] \geq \operatorname*{Pr}[f(\pmb{\sigma}_{-i}, \sigma_i') \in B_k(\sigma_i)]$ .
We say a rule  $f$  is unilateral if it depends only on a single voter, i.e., there is a voter  $i$  such that for all  $\sigma$  and  $\sigma'$ , if  $\sigma_i = \sigma_i'$ , then  $f(\sigma) = f(\sigma')$ . A rule  $f$  is called a duple if its range is two candidates, i.e., there is a pair of candidates  $a$  and  $b$  such that for all  $\sigma$ ,  $f(\sigma) \in \{a, b\}$ . Gibbard [9] extended the Gibbard-Satterthwaite theorem to randomized rules as follows: Any randomized rule  $f$  that is strategyproof is a mixture over unilateral and duple rules. Since unilateral and duple rules are seen as undesirable, this implies that finding a reasonable, strategy-proof voting rule, even allowing randomization, is a hopeless endeavor.
Social choice under uncertainty. A belief for voter  $i$  is a probability measure  $\mathbb{P}_i$  over the set of profiles  $\mathcal{L}^n$  (when the  $i$  is clear from context we will drop it from the notation). This describes  $i$ 's prior probability over profiles before considering their own ranking. After observing their own ranking  $\hat{\sigma}_i$ , the voter can update their posterior using the conditional distribution  $\mathbb{P}[\cdot \mid \sigma_i = \hat{\sigma}_i]$ . For notational convenience, we will often shorten this to  $\mathbb{P}[\cdot \mid \hat{\sigma}_i]$ .
In this model, we need a slightly different notion of strategyproofness. A voting rule is called ordinally Bayesian incentive compatible (OBIC) with respect to a beliefs  $(\mathbb{P}_1,\ldots ,\mathbb{P}_n)$  if for all voters  $i$  ,all rankings  $\hat{\sigma}_i$  , all manipulations  $\sigma_i^\prime$  , and all utility functions  $u$  consistent with  $\hat{\sigma}_i$ $\mathbb{E}[u(f(\pmb {\sigma}_{-i},\hat{\sigma}_i))]$ $\hat{\sigma}_i]\geq \mathbb{E}[u(f(\pmb {\sigma}_{-i},\sigma_i^{\prime})|\hat{\sigma}_i]$  . This is the natural generalization of SD-strategyproofness to a Bayesian setting. Just as with SD-strategyproofness, an equivalent definition is for all voters  $i$  , all rankings  $\hat{\sigma}_i$  all manipulations  $\sigma_i^\prime$  , and all  $k\leq m$ $\mathbb{P}[f(\pmb {\sigma}_{-i},\hat{\sigma}_i)\in B_k(\hat{\sigma}_i)\mid \hat{\sigma}_i]\geq \mathbb{P}[f(\pmb {\sigma}_{-i},\sigma_i^{\prime})\in B_k(\hat{\sigma}_i)\mid \hat{\sigma}_i]$
We now present a few possible choices of "reasonable" priors based on well-known probabilistic models of social choice. The first is based off of a Mallows Model [15]. This model is parameterized by a ground truth ranking  $\tau \in \mathcal{L}$  and a dispersion quantity  $\varphi$ . We define the Kendall tau distance between rankings  $d(\sigma_1, \sigma_2) = |\{(a, b) \in \mathcal{A}^2 | a \succ_{\sigma_1} b \wedge b \succ_{\sigma_2} a\}|$ , i.e., the number of pairs of candidates on which  $\sigma_1$  and  $\sigma_2$  disagree. In a Mallows Model, each voter's ranking is drawn independently with probability proportional to  $\varphi^{d(\sigma, \tau)}$ . More formally, the probability that a specific ranking  $\sigma$  is drawn is equal to  $\frac{\varphi^{d(\sigma, \tau)}}{Z}$  where  $Z = \sum_{\sigma \in \mathcal{L}} \varphi^{d(\sigma, \tau)}$  is the normalizing constant. One can easily check that if we extend the notion of Kendall tau distance to operate on a profile and a ranking, with  $d(\sigma, \tau) = \sum_i d(\sigma_i, \tau)$ , then the probability of sampling a profile  $\sigma$  is proportional to  $\varphi^{d(\sigma, \tau)}$  (this time with a  $Z^n$  normalizing constant).
We convert this model into a prior in two ways. The first we call a confident Mallows prior parameterized by  $\varphi$ . The agent assumes a ground truth  $\tau$  is first drawn from some (arbitrary) distribution, then, given this ground truth,  $\sigma_{i} = \tau$  with probability 1 and the remainder of the profile  $\sigma_{-i}$  is drawn from a Mallows Model with a fixed  $\varphi$  using  $\tau$ . Essentially, the agent believes that they correctly know the ground truth, but all others only approximate this truth using a Mallows Model. The conditional distribution over the remainder of the profile  $\sigma_{-i}$  given  $\hat{\sigma}_{i}$  then follows a standard Mallows model with the ground truth equal to  $\hat{\sigma}_{i}$ , so  $\mathbb{P}[\sigma_{-i}|\hat{\sigma}_i]\propto \varphi^{d(\sigma_{-i},\hat{\sigma}_i)}$ .<sup>2</sup>
2This is equivalent to the Conditional Mallows Model of Mandal and Parkes [16].
The second we call an unconfident Mallows prior. Here, the agent believes that the ground truth  $\tau$  is drawn uniformly at random, and then the entire profile (including their own ranking) is drawn from a Mallows Model. Therefore,  $\mathbb{P}[\pmb {\sigma}] = \frac{1}{m!}\sum_{\tau \in \mathcal{L}}\frac{\varphi^{d_{kt}(\pmb{\sigma},\tau)}}{Z^n}$ . Since for any  $\hat{\sigma}_i$ , by symmetry  $\operatorname*{Pr}[\sigma_i = \hat{\sigma}_i] = \frac{1}{m!}$ , we can write the conditional probability as
$$
\mathbb {P} [ \pmb {\sigma} _ {- i} \mid \hat {\sigma} _ {i} ] = \sum_ {\tau \in \mathcal {L}} \frac {\varphi^ {d ((\pmb {\sigma} _ {- i} , \hat {\sigma} _ {i}) , \tau)}}{Z ^ {n}} = \sum_ {\tau \in \mathcal {L}} \frac {\varphi^ {d (\pmb {\sigma} _ {- i} , \tau)}}{Z ^ {n - 1}} \cdot \frac {\varphi^ {d (\hat {\sigma} _ {i} , \tau)}}{Z}.
$$
We will obtain notation slightly and write  $\mathbb{P}[\tau \mid \hat{\sigma}_i] = \frac{\varphi^{d(\hat{\sigma}_i,\tau)}}{Z}$  and  $\mathbb{P}[\pmb {\sigma}_{-i}\mid \tau ] = \frac{\varphi^{d(\pmb{\sigma}_{-i},\tau)}}{Z^{n - 1}}$  so that
$$
\mathbb {P} [ \boldsymbol {\sigma} _ {- i} \mid \hat {\sigma} _ {i} ] = \sum_ {\tau \in \mathcal {L}} \mathbb {P} [ \boldsymbol {\sigma} _ {- i} \mid \tau ] \cdot \mathbb {P} [ \tau \mid \hat {\sigma} _ {i} ].
$$
We can interpret this as saying the voter has a posteriori over ground truths,  $\mathbb{P}[\tau \mid \hat{\sigma}_i]$ , and is using this posterior to infer the probability of the rest of the profile. Intuitively, the agent is uncertain over ground truths but, due to the observation of their ranking, places higher weight on ground truths that are closer to their own ranking. This decomposition is possible because the rest of the profile  $\sigma_{-i}$  is conditionally independent of  $\hat{\sigma}_i$  given the ground truth  $\tau$ .
The Thurstone-Mosteller model is defined with respect to underlying means  $\mu_c$  for each candidate  $c \in \mathcal{A}$ . To sample a ranking, a value  $X_c \sim \mathcal{N}(\mu_c, 1)$  is drawn independently for each candidate  $c$  from a normal distribution with variance 1 around the mean. The resulting ranking is the order of the  $X_c$  values from highest to lowest. The Placket-Luce model is defined with respect to underlying weights  $w_c > 0$  for each candidate  $c \in \mathcal{A}$ . To sample a ranking, we iteratively select a candidate  $c$  from the remaining unchosen candidates  $P$  with probability  $\frac{w_c}{\sum_{c' \in P} w_c'}$ .
To convert these models to beliefs, we assume that the voter believes there are underlying distinct means  $\mu_1 > \dots > \mu_m$  (resp. weights  $w_1 > \dots > w_m$ ) but is uncertain about which candidate has which mean (resp. weight). To relate this to the Mallows belief, we will call this order  $\tau$ , the ground truth. In the confident version, the voter believes that their ranking is always equal to  $\tau$ , but all other votes are drawn from the corresponding model. In the unconfident version, the voter believes a priori that  $\tau$  is drawn uniformly at random, and then all voter rankings, including their own, are drawn from the corresponding model. As with the Mallows beliefs, the voter can do a Bayesian update to compute a posterior about which candidate was assigned to which weight. We can again decompose
$$
\mathbb {P} [ \pmb {\sigma} _ {- i} \mid \hat {\sigma} _ {i} ] = \sum_ {\tau} \mathbb {P} [ \pmb {\sigma} _ {- i} \mid \tau ] \cdot \mathbb {P} [ \tau \mid \hat {\sigma} _ {i} ],
$$
where, by Bayes' rule,  $\mathbb{P}[\tau \mid \hat{\sigma}_i] \propto \mathbb{P}[\hat{\sigma}_i|\tau]$ , the probability of generating  $\hat{\sigma}_i$  under the model with ground truth  $\tau$ .
Note that to make this more general, it would also make sense for the voter to believe there is a distribution over means or weights; our results continue to hold with this more general class; however, for ease of presentation, we focus on the more restricted form.
3 Plurality is OBIC
We start by defining a class of beliefs that we call top-choice correlated. The class is similar in spirit (although incomparable) to the class of top-set correlated beliefs introduced by Bhargava et al. [1].