SlowGuess/NeurIPS_2023_OCR huggingface dataset:
To define the class, given a profile  $\sigma$  and a candidate  $c\in \mathcal{A}$ , we let  $\mathrm{PLU}_c(\pmb {\sigma}) = |\{i\mid \sigma_i(1) = c\} |$  be the plurality score of  $c$ , i.e., the number of voters that rank  $c$  first. Further, we let  $\mathrm{PLU}(\pmb {\sigma})$  be the vector of plurality scores indexed by the candidates. A belief  $\mathbb{P}_i$  is top-choice correlated if the following holds. Fix a ranking  $\hat{\sigma}_i$  and let  $a = \hat{\sigma}_i(1)$ . Then, for all candidates  $b\neq a$  and all pairs of plurality vectors  $\mathbf{r}$  and  $\mathbf{r}'$  such that  $r_c = r_c'$  for all  $c\neq a,b$ ,  $r_a = r_b'$ ,  $r_b = r_a'$ , and  $r_a > r_b$ ,  $\mathbb{P}[\mathrm{PLU}(\pmb{\sigma}_{-i}) = \mathbf{r}\mid \hat{\sigma}_i]\geq \mathbb{P}[\mathrm{PLU}(\pmb{\sigma}_{-i}) = \mathbf{r}'\mid \hat{\sigma}_i]$ . This says that if the voter is told the remaining plurality scores of all other candidates except  $a$  and  $b$ , as well as possible scores for  $a$  and  $b$ , they would think it is more likely that  $a$  (their top choice) has the higher score. In other words, all else being equal, the voter's top choice is more likely to perform better than other candidates.
We now claim that all of the specific beliefs we have introduced are top-choice correlated, suggesting that this condition is quite weak.
Lemma 1. The confident and unconfident versions of Mallows, Thurstone-Mosteller, and Placket-Luce beliefs under any parameter settings are top-choice correlated.
The proof of Lemma 1 can be found in Appendix A. For confident versions of these beliefs, this is relatively straightforward as the models directly place higher mass on the voter's top choice being chosen. For the unconfident versions, slightly more intricate analysis is necessary to show that more mass is placed on ground truth rankings where the voter's top choice is higher, and from this, we can reach the same conclusion.
Despite the breadth of the class of top-choice correlated beliefs, it turns out that plurality is OBIC for all beliefs in this class.
Lemma 2. Under any top-choice correlated beliefs, plurality is OBIC.
Proof. Let  $f$  be the plurality voting rule,  $i$  be an agent, and  $\mathbb{P}$  be their top-choice correlated belief. Suppose  $i$  observes  $\hat{\sigma}_i$ , and let  $u$  be an arbitrary utility function that is consistent with  $\hat{\sigma}_i$ . Let  $a = \hat{\sigma}_i(1)$  be their top-ranked alternative.
Let  $\sigma_{i}^{\prime}$  be a possible manipulation for voter  $i$ , and let  $b = \sigma_{i}^{\prime}(1)$  be the top-ranked alternative. Notice that if  $a = b$ , then the outcome under plurality is identical, and this manipulation cannot be an improvement. Hence, from now on, we assume that  $b \neq a$ .
For  $\sigma_{-i}$ , let  $\mathrm{UG}(\sigma_{-i}) = \mathbb{E}[u(f(\sigma_{-i},\sigma_i'))] - \mathbb{E}[u(f(\sigma_{-i},\hat{\sigma}_i))]$  be the expected utility gain of switching from  $\hat{\sigma}_i$  to  $\sigma_i'$  when others report  $\sigma_{-i}$ . We wish to show that  $\mathbb{E}[\mathrm{UG}(\sigma_{-i})|\hat{\sigma}_i]\leq 0$ , where the expectation is over the belief  $\mathbb{P}$ . To simplify notation, we will allow the utility function  $u$  to operate on (nonempty) sets of candidates  $S$  as  $u(S) = \frac{1}{|S|}\sum_{c\in S}u(c)$ . Note that when the set of plurality winners on a profile is  $S$ , the expected utility is  $u(S)$ .
We now partition the possible  $\sigma_{-i}$  based on their utility gain. Let  $C \subseteq \mathcal{A} \setminus \{a, b\}$  be a (nonempty) set candidates not including  $a$  and  $b$ . For each set  $C$ , we define eight sets of profiles  $\sigma_{-i}$  depending on the winners under  $(\sigma_{-i}, \hat{\sigma}_i)$  and  $(\sigma_{-i}, \sigma_i')$ ,  $E_1(C)^a, E_1(C)^b, E_2(C)^a, E_2(C)^b, E_3(C)^a, E_3(C)^b, E_4(C), E_5(C)$ . In each  $E(C)$  set,  $C$  will be the set of candidates excluding  $a$  and  $b$  with the highest plurality score. We abuse notation slightly and write  $\mathrm{PLU}(C)$  for the (tied) plurality score of each of these candidates and  $\mathrm{PLU}(a)$  and  $\mathrm{PLU}(b)$  for the plurality scores of  $a$  and  $b$ , respectively. The sets are otherwise defined by the set of plurality winners in  $(\sigma_{-i}, \hat{\sigma}_i)$  and  $(\sigma_{-i}, \sigma_i')$ . The definitions can be found in Table 1. One can check that these (disjoint) sets collectively cover all possible  $\sigma_{-i}$  where  $\mathrm{UG}(\sigma_{-i})$  is nonzero.
We can now rewrite the expected utility gain in terms of these sets. For each set  $E(C)$ , we write  $\mathrm{UG}(E(C))$  for the expected utility gain for profiles  $\sigma_{-i} \in E(C)$  (which will always be the same for all  $\sigma_{-i} \in E(C)$ ). From this, we have
$$
\begin{array}{l} \mathbb{E}[\mathrm{UG}(\boldsymbol{\sigma}_{-i})|\hat{\sigma}_{i}] = \sum_{\substack{C\subseteq \mathcal{A}\setminus \{a,b\} \\ C\neq \emptyset}}\Bigg(\sum_{j = 1}^{3}\big(\mathbb{P}[E_{j}(C)^{a}|\hat{\sigma}_{i}]\mathrm{UG}(E_{j}(C)^{a}) + \mathbb{P}[E_{j}(C)^{b}|\hat{\sigma}_{i}]\mathrm{UG}(E_{j}(C)^{b})\big) \\ \left. + \mathbb {P} [ E _ {4} (C) | \hat {\sigma} _ {i} ] \mathrm {U G} (E _ {4} (C)) + \mathbb {P} [ E _ {5} (C) | \hat {\sigma} _ {i} ] \mathrm {U G} (E _ {5} (C))\right) \\ \end{array}
$$
Our goal, again, is to show that this expression is at most 0. Notice that for each  $C$ ,  $\mathrm{UG}(E_4(C)) = u(b) - u(a) < 0$  and  $\mathrm{UG}(E_5(C)) = \frac{1}{|C|} (u(b) - u(a)) < 0$  because  $u(a) > u(c)$  for all other candidates  $c$ . In what remains, we show that for all  $C$  and each  $j \leq 3$ ,
$$
\mathbb {P} \left[ E _ {j} (C) ^ {a} \right] \mathrm {U G} \left(E _ {j} (C) ^ {a}\right) + \mathbb {P} \left[ E _ {j} (C) ^ {b} \right] \mathrm {U G} \left(E _ {j} (C) ^ {b}\right) \leq 0. \tag {1}
$$
Fix an arbitrary  $C$ . To do this, we show that for each  $j$ ,  $\mathrm{UG}(E_j(C)^a) \leq 0$ ,  $-\mathrm{UG}(E_j(C)^a) \geq \mathrm{UG}(E_j(C)^b)$ , and  $\mathbb{P}[E_j(C)^a] \geq \mathbb{P}[E_j(C)^b]$ . Together, these imply (1).
We analyze the case of  $j = 1$ ; the arguments for  $j = 2$  and  $j = 3$  are very similar. Notice that
$$
\operatorname {U G} \left(E _ {1} (C) ^ {a}\right) = u (C \cup \{a \}) - u (a) = \frac {| C |}{| C | + 1} (u (C) - u (a)),
$$
Set	Condition	(σ−i, δi) Winners	(σ−i, σ′i) Winners
E1(C)a	PLU(a) = PLU(C) > PLU(b) + 1	a	C ∪ {a}
E1(C)b	PLU(b) = PLU(C) > PLU(a) + 1	C ∪ {b}	b
E2(C)a	PLU(a) = PLU(C) = PLU(b) + 1	a	C ∪ {a,b}
E2(C)b	PLU(b) = PLU(C) = PLU(a) + 1	C ∪ {a,b}	b
E3(C)a	PLU(C) = PLU(a) + 1 > PLU(b) + 1	C ∪ {a}	C
E3(C)b	PLU(C) = PLU(b) + 1 > PLU(a) + 1	C	C ∪ {b}
E4(C)	PLU(a) = PLU(b) ≥ PLU(C)	a	b
E5(C)	PLU(C) = PLU(a) + 1 = PLU(b) + 1	C ∪ {a}	C ∪ {b}
Table 1: Definition of the sets  $E_{1}(C)^{a}, E_{1}(C)^{b}, E_{2}(C)^{a}, E_{2}(C)^{b}, E_{3}(C)^{a}, E_{3}(C)^{b}, E_{4}(C)$ , and  $E_{5}(C)$ . They contain all  $\sigma_{-i}$  that satisfy the corresponding condition. In each set, the contained  $\sigma_{-i}$  all have the same winners in both  $(\sigma_{-i}, \hat{\sigma}_{i})$  and  $(\sigma_{-i}, \sigma_{i}')$  as seen in the corresponding columns.
and symmetrically
$$
\operatorname {U G} \left(E _ {1} (C) ^ {b}\right) = u (b) - u (C \cup \{b \}) = \frac {| C |}{| C | + 1} (u (b) - u (C)).
$$
Since  $u(a)$  is maximal,  $\mathrm{UG}(E_1(C)^a) \leq 0$ . Further,
$$
- \operatorname {U G} \left(E _ {1} (C) ^ {a}\right) - \operatorname {U G} \left(E _ {1} (C) ^ {b}\right) = \frac {| C |}{| C | + 1} (u (a) - u (b)) \geq 0.
$$
Finally, to show  $\mathbb{P}[E_1(C)^a] \geq \mathbb{P}[E_1(C)^b]$ , we consider the vectors of plurality scores (indexed by candidates)  $\mathbf{r}^a$  and  $\mathbf{r}^b$  that lead a profile  $\sigma_{-i}$  to end in up in  $E_1(C)^a$  and  $E_1(C)^b$ , respectively. Due to the symmetry, there is a natural bijection between these two sets of vectors obtained by swapping the  $a$  and  $b$  components. Further, by the definition of  $E_1(C)^a$  and  $E_1(C)^b$ , the  $a$  component of  $\mathbf{r}^a$  is always strictly larger than the  $b$  component. Since  $\mathbb{P}$  is top-choice correlated, for any two vectors that differ by swapping the  $a$  and  $b$  components,  $\mathbb{P}[\cdot |\hat{\sigma}_i]$  always places higher mass on the vector in  $\mathbf{r}^a$ . Therefore,  $\mathbb{P}[E_1(C)^a|\hat{\sigma}_i] \geq \mathbb{P}[E_1(C)^b|\hat{\sigma}_i]$ , as needed.
From these two lemmas, we immediately derive our main positive result.
Theorem 1. When voters have beliefs that are any of the confident or unconfident versions of Mallows, Thurstone-Mosteller, or Placket-Luce under any parameter settings, plurality is OBIC.
4 Other Voting Rules Are Not OBIC
From the positive result about plurality, one might wonder whether satisfying OBIC with respect to these beliefs is a relatively weak condition. If several rules satisfy it, this property is not useful for a mechanism designer who is comparing between rules to implement. In this section, we show this is not the case. Specifically, we focus on both the confident and unconfident variants of Mallows beliefs. Our main theoretical negative result is that plurality is uniquely OBIC among positional scoring rules in certain regimes of Mallows beliefs.
Theorem 2. Let  $f$  be a non-plurality positional scoring rule on three candidates. If a voter has unconfident or confident Mallows beliefs with  $\varphi \leq .988$ , for a sufficiently large  $n$ ,  $f$  is not OBIC.
Below we provide a detailed proof sketch. However, we shunt some unwieldy technical derivations into two lemmas relegated to the appendix.
Proof sketch of Theorem 2. Fix a non-plurality scoring vector  $(s_1, s_2, s_3)$ . Without loss of generality, we can translate and scale the vector such that  $s_3 = 0$  and  $s_1$  and  $s_2$  are relatively prime integers with  $s_2 > 0$ . Fix a voter  $i$  with unconfident Mallows beliefs  $\mathbb{P}$  with parameter  $\varphi \leq .988$ ; we will describe later how to extend it to a confident Mallow's belief. Suppose they observe ranking  $\hat{\sigma}_i = a \succ b \succ c$ . We will show that for sufficiently large  $n$ , it will be beneficial to switch to  $\sigma_i' = a \succ c \succ b$ . More specifically, we will show that this manipulation increases the probability that candidate  $a$  wins, which means the rule cannot be OBIC with respect to these beliefs. Formally, we will show that,
$$
\Pr [ f (\boldsymbol {\sigma} _ {- i}, \sigma_ {i} ^ {\prime}) = a \mid \hat {\sigma} _ {i} ] > \Pr [ f (\boldsymbol {\sigma} _ {- i}, \hat {\sigma} _ {i}) = a \mid \hat {\sigma} _ {i} ],
$$
$$
\Pr [ f (\sigma_ {- i}, \sigma_ {i} ^ {\prime}) = a \mid \hat {\sigma} _ {i} ] - \Pr [ f (\sigma_ {- i}, \hat {\sigma} _ {i}) = a \mid \hat {\sigma} _ {i} ] > 0.
$$
Notice that since we are looking at the difference in probabilities of two events, we can ignore their intersection when both reports lead to  $a$  as the winner. That is, the left-hand side is equal to<sup>3</sup>
$$
\Pr [ f (\sigma_ {- i}, \sigma_ {i} ^ {\prime}) = a \wedge f (\sigma_ {- i}, \hat {\sigma} _ {i}) \neq a \mid \hat {\sigma} _ {i} ] - \Pr [ f (\sigma_ {- i}, \hat {\sigma} _ {i}) = a \wedge f (\sigma_ {- i}, \sigma_ {i} ^ {\prime}) \neq a \mid \hat {\sigma} _ {i} ].
$$
To shorten notation, we will label the events of interest as  $\mathcal{E}^{cb} = \{f(\pmb{\sigma}_{-i},\pmb{\sigma}_i^{\prime}) = a\wedge f(\pmb{\sigma}_{-i},\hat{\sigma}_i)\neq a\mid \hat{\sigma}_i]\}$  (i.e., ranking  $c$  above  $b$  causes  $a$  to win) and  $\mathcal{E}^{bc} = \{f(\pmb{\sigma}_{-i},\hat{\pmb{\sigma}}_i) = a\wedge f(\pmb{\sigma}_{-i},\pmb{\sigma}_i^{\prime})\neq a\mid \hat{\sigma}_i]\}$  (i.e., ranking  $b$  above  $c$  causes  $a$  to win), so we wish to show that
$$
\mathbb {P} \left[ \mathcal {E} ^ {c b} | \hat {\sigma} _ {i} \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} | \hat {\sigma} _ {i} \right] > 0.
$$
Using the definition of the Mallows belief model, we can expand the left-hand side using ground truths to
$$
\sum_ {\tau \in \mathcal {L}} \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau \right]\right) \mathbb {P} [ \tau \mid \hat {\sigma} _ {i} ]. \tag {2}
$$
Consider the  $\tau = a\succ c\succ b$  term. Notice that, by symmetry  $\mathbb{P}[\mathcal{E}^{cb}\mid a\succ c\succ b] = \mathbb{P}[\mathcal{E}^{bc}\mid a\succ b\succ c]$  and  $\mathbb{P}[\mathcal{E}^{bc}\mid a\succ c\succ b] = \mathbb{P}[\mathcal{E}^{cb}\mid a\succ b\succ c]$ . Further  $\mathbb{P}[a\succ c\succ b\mid \hat{\sigma}_i] = \varphi \cdot \mathbb{P}[a\succ b\succ c\mid \hat{\sigma}_i]$  as  $d(a\succ c\succ b,\hat{\sigma}_i) = d(a\succ b\succ c,\hat{\sigma}_i) + 1$ . Hence,
$$
\begin{array}{l} \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = a \succ c \succ b \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = a \succ c \succ b \right]\right) \mathbb {P} \left[ a \succ c \succ b \mid \hat {\sigma} _ {i} \right] \\ = - \varphi \cdot \left(\left(\mathbb {P} [ \mathcal {E} ^ {c b} \mid \tau = a \succ b \succ c ] - \mathbb {P} [ \mathcal {E} ^ {b c} \mid \tau = a \succ b \succ c ]\right) \mathbb {P} [ a \succ b \succ c \mid \hat {\sigma} _ {i} ]\right), \\ \end{array}
$$
or in words, the  $\tau = a\succ c\succ b$  term is exactly equal to  $-\varphi$  times the  $\tau = a\succ b\succ c$  term. In fact, this same property holds for the other two pairs of ground truth rankings where  $a$  remains in the same position and  $b$  is swapped with  $c$ , so  $b\succ a\succ c$  with  $c\succ a\succ b$  and  $b\succ c\succ a$  with  $c\succ b\succ a$ . Hence, we can write the entire expression (2) as
$$
\begin{array}{l} (1 - \varphi) \cdot \left(\left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = a \succ b \succ c \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = a \succ b \succ c \right]\right) \mathbb {P} [ a \succ b \succ c \mid \hat {\sigma} _ {i} ] \right. \\ + \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = b \succ a \succ c \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = b \succ a \succ c \right]\right) \mathbb {P} \left[ b \succ a \succ c \mid \hat {\sigma} _ {i} \right] \\ \left. + \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = b \succ c \succ a \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = b \succ c \succ a \right]\right) \mathbb {P} \left[ b \succ c \succ a \mid \hat {\sigma} _ {i} \right]\right). \\ \end{array}
$$
Notice that since we wish to show this is strictly larger than 0 and  $1 - \varphi > 0$ , we show only that the sum of the probability terms is positive. Additionally, subbing in the values of  $\mathbb{P}[\tau \mid \hat{\sigma}_i]$  with the corresponding Kendall tau distances, the above simplifies to
$$
\begin{array}{l} \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = a \succ b \succ c \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = a \succ b \succ c \right]\right) \\ + \varphi \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = b \succ a \succ c \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = b \succ a \succ c \right]\right) \tag {3} \\ + \varphi^ {2} \left(\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = b \succ c \succ a \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = b \succ c \succ a \right]\right). \\ \end{array}
$$
We will now show that for some  $c_{1} > c_{2}$  to be chosen later, the first positive term  $\mathbb{P}[\mathcal{E}^{cb} \mid \tau = a \succ b \succ c] \in \Omega(c_{1}^{n})$  and each negative term  $\mathbb{P}[\mathcal{E}^{bc} \mid \tau] \in O(c_{2}^{n})$ , which implies that for sufficiently large  $n$ , the entire sum is positive, as needed. In addition, for the result to hold with confident Mallows rather than unconfident, it is only required that the first difference be positive, i.e.,
$$
\mathbb {P} \left[ \mathcal {E} ^ {c b} \mid \tau = a \succ b \succ c \right] - \mathbb {P} \left[ \mathcal {E} ^ {b c} \mid \tau = a \succ b \succ c \right] > 0.
$$
This is also directly implied by showing the above bounds.
We relegate these arguments to the following two lemmas, established in Appendices B and C, respectively.
Lemma 3.  $\mathbb{P}[\mathcal{E}^{bc}\mid \tau ]\in O(c_2^n)$  for  $c_{2} = e^{\frac{1 - \varphi^{2}}{2(1 + \varphi + \varphi^{2})}}\sqrt{\frac{\varphi(1 + 2\varphi)}{1 + \varphi + \varphi^{2}}}$ .
Note that since  $f$  is randomized, to make this precise, we would need to specify the joint distribution of its outputs on different inputs. However, the remainder of the proof will not rely on how this is done, so the joint distribution can be arbitrary.
Lemma 4.  $\mathbb{P}[\mathcal{E}^{cb}\mid \tau = a\succ b\succ c]\in \Omega (c_1^n)$  for  $c_{1} > e^{\frac{1 - \varphi^{2}}{2(1 + \varphi + \varphi^{2})}}\sqrt{\frac{\varphi(1 + 2\varphi)}{1 + \varphi + \varphi^{2}}}$ .
Together, the two lemmas imply the desired result.
We now complement this result with some additional robustness checks. First, even though the result is asymptotic, in special cases of interest, this is, in fact, not necessary.
Theorem 3. With three candidates, when a voter has unconfident or confident Mallows beliefs with  $\varphi < 1$ , Borda Count is not OBIC for any  $n \geq 2$ .
Notice that  $n = 1$  is a degenerate case with no other voters, so  $n \geq 2$  is the strongest we can hope for. The proof of Theorem 3 can be found in Appendix D. The beginning is nearly identical to the proof sketch of Theorem 2, but they diverge after this point. While Lemmas 3 and 4 are asymptotic in nature, the corresponding portion for Theorem 3 requires careful counting of the number of profiles satisfying different conditions to ensure that for any fixed  $n$ , the inequalities hold.
In light of Theorem 3, it may seem plausible that Theorem 2 could be strengthened to hold for all  $n$  rather than just asymptotically. However, we can give examples where this is not the case, suggesting that a "sufficiently large  $n$ " requirement may be necessary. In particular, there are scoring rules that, while not being plurality, are "close" to plurality in the sense that  $s_2$  is so tiny it only matters when there is a tie among the plurality winners. For example, say we have the scoring rule  $(4,1,0)$  with  $n = 3$  voters. If any candidate receives two first-place votes, they immediately have 8 out of the 15 available points, so they necessarily win. Only when each candidate is ranked first by one voter is there any difference. We show in Appendix E that such close-to-plurality rules, at least for  $n = 3$ , are OBIC for confident Mallows beliefs with any  $\varphi < 1$ .
Finally, we consider other prominent non-scoring rules, namely Copeland and maximin. Note that for explicitly-defined beliefs and a number of voters  $n$ , we can determine whether or not a rule is OBIC by computing the probabilities of winners under all possible manipulations. We do so for the aforementioned rules under both confident and unconfident Mallows beliefs with  $\varphi = 0.25, 0.5, 0.75, n = 2, \ldots, 50$ , and  $m = 3$ . The results can be found in Table 2. Although slightly mixed in the sense that in a few specific cases, OBIC holds, the key takeaway is that none of the rules considered are as consistently OBIC as plurality.
Copeland	maximin with φ = 0.25, 0.5	maximin with φ = 0.75
confident Mallows	even n	n ≠ 3	n ≠ 3
unconfident Mallows	all	all	n ≠ 6
Table 2: Scenarios where the Copeland and maximin rules fail to be OBIC with respect to Confident and Unconfident Mallows beliefs with  $\varphi = {0.25},{0.50},{0.75},n = 2,\ldots {50}$  ,and  $m = 3$  .
In summary, we have considered the problem of strategic voting when voters have certain correlated beliefs over others. We have singled out plurality as an auspicious choice, being incentive compatible for a large class of beliefs, and have complemented this with negative results showing other prominent voting rules do not satisfy this property. However, our work is certainly not the final word on this topic. The current negative results are only for three candidates, and although we believe they should extend to a larger number, the technical work in showing this seems to get quite messy. Further, although we have checked many prominent voting rules, we have not ruled out the existence of other "reasonable" rules that perform as well as plurality while simultaneously satisfying other desiderata.
Finally, taking a more practical viewpoint, although OBIC is a theoretically compelling condition, it is susceptible to common criticisms of models of voter behavior. As with many models of this form, the utility difference of misreporting, or of choosing any vote for that matter, can be very low, and it is debatable whether this is the driving force in how voters make decisions. It raises questions similar in spirit to the so-called Paradox of Voting [6]: why would any rational agent choose to vote if the cost almost certainly outweighs the expected benefits? Despite these challenges, we do believe that the exploration and refinement of models such as OBIC can lead to an improved understanding of voter behavior and, ultimately, to the development of more effective voting systems.
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