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# Anchor-proofness in Voting

Federico Fioravanti, Zoi Terzopoulou

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Federico Fioravanti, Zoi Terzopoulou. Anchor-proofness in Voting. 2026. hal-05494428

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# Anchor-proofness in Voting*

Federico Fioravanti¹ and Zoi Terzopoulou²

¹Université Jean Monnet Saint-Étienne, CNRS, Université Lyon 2, emlyon business school, GATE, 42023, Saint-Étienne, France.

²CNRS, Université Jean Monnet Saint-Étienne, Université Lyon 2, emlyon business school, GATE, 42023, Saint-Étienne, France.

## Abstract

This work contributes to a foundational question in economic theory: how do individual-level cognitive biases interact with collective-choice mechanisms? We study a setting where voters hold intrinsic preference rankings over a set of alternatives but cast approval ballots to determine the collective outcome. The ballots are shaped by an anchoring bias: alternatives are presented sequentially by a social planner, and a voter approves an alternative if and only if it is acceptable and strictly preferred to all alternatives previously encountered. We first analyze which approval-based voting rules are anchor-proof, in the sense that they always select the same winner regardless of the presentation order. We show that this requirement is extremely demanding: only very restrictive rules satisfy it. We then turn to the potential influence of the social planner. On the upside, when the planner has no information about the voters' intrinsic preferences, she cannot manipulate the outcome.

## 1 Introduction

A large literature documents systematic cognitive biases in individual decision making, including the *anchoring bias*, where early information serves as a reference point that affects subsequent evaluations (Tversky and Kahneman, 1974). The consequences of individual biases on welfare, learning, and market behavior are generally well understood within economic theory (see, for example, Thaler, 1980; Kahneman et al., 1986; Bordalo et al., 2012). In contrast, comparatively little is known about how such biases propagate in collective decision-making environments. This gap is important: many economic institutions, from committees and platforms to public-participation mechanisms, aggregate decisions

<footnote>*This research was funded by the French National Research Agency within the project ANR-24-EXMA-0001 PEPR MathsVivES CONDORCET.</footnote>
that are themselves shaped by individual cognitive processes. Thus, understanding how biases play a role in mapping preferences to outcomes is essential for the design of collective-decision rules. To illustrate, consider Example 1.

**Example 1.** In participatory budgeting platforms, citizens consider different projects and approve certain ones to receive funding in their neighborhood.<fnref n="1" /> An overlooked feature of this setting is that the order in which the projects are presented matters. Consider a simple scenario where a voter is presented with only two projects *a* and *b*, and she prefers project *a* over project *b*. If the voter encounters *a* first on the participatory budgeting platform, then *a* might turn into an *anchor* and make *b* seem less appealing by comparison—leading to an approval for *a* but a rejection for *b*. However, if the voter sees project *b* first, then she might approve both *a* and *b*, since her evaluations will be framed differently. At the aggregate level, this can reverse outcomes: For instance, take a group of five voters, two of whom prefer project *x* to *y* and three of whom prefer project *y* to *x*.

<table><caption>Table 1: Everyone sees their top alternative first.</caption><thead><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>x</td><td>x</td><td>y</td><td>y</td><td>y</td></tr></tbody></table><p>approval winner: y</p>

<table><caption>Table 2: Everyone sees first alternative *x* and then alternative *y*.</caption><thead><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead><tbody><tr><td>x</td><td>x</td><td>x, y</td><td>x, y</td><td>x, y</td></tr></tbody></table><p>approval winner: x</p>

If everyone sees their top alternative first when navigating the participatory budgeting platform, and hence only approves that alternative, then, *y* will be the project with most approval votes as shown in Table 1. However, in a non-personalized version of the platform where everyone sees alternatives in the same order (for example, first *x* and then *y*), then *x* will be the project with the most approvals as shown in Table 2. △

Sequential presentation of the alternatives is common both in institutional contexts of collective decision-making and in digital-choice architectures, where ordering is an implicit design feature.<fnref n="2" /> In such environments, anchoring can alter whether a voter approves an alternative by setting the reference point against which new alternatives are evaluated. Consequently, the induced approval ballots may reflect not only voters’ underlying preferences but also the given presentation order. This raises questions about the robustness of standard voting rules and the extent to which social planners can influence outcomes through choice architecture.

**Our Contribution.** We study a framework of approval voting, where voters possess intrinsic ordinal preferences over a finite set of alternatives and also

<footnote n="1">¹See for example the Paris platform www.paris.fr/budget-participatif.</footnote>

<footnote n="2">²In modern online platforms that rely on recommender systems, the ordering presented to users is personalized. On the other hand, electoral contexts usually present alternatives in a fixed order for all voters.</footnote>
hold an acceptability relation over the alternatives. A key feature of our model is that approval ballots are generated from preferences via a simple *anchoring* mechanism: a social planner specifies the order in which a voter encounters the alternatives, and the voter approves an alternative if and only if it is acceptable and strictly preferred to all previously encountered alternatives. Hence, the presentation order determines the mapping from preferences to approval ballots.

Our analysis addresses two central questions. First, we study the existence of approval-based voting rules that are robust to the anchoring bias, called *anchor-proof*, meaning that the set of selected winners is invariant with respect to the presentation order. Such rules guarantee that collective outcomes faithfully represent the voters’ underlying preferences, and are immune to the behavior of a social planner that may be driven by personal interests. We find that every non-constant rule violates anchor-proofness in its most demanding form; yet, there do exist reasonable rules that satisfy anchor-proofness for restricted preference structures or particular presentation orders. Second, for non-robust rules, we study the strategic role of the social planner in more depth. When the planner has her own preference over outcomes, we examine whether she can exploit the presentation order to induce a favorable collective decision. Our results bring to light cases in which such manipulation is possible and specify how it depends on the rules and the planner’s available information. Notably, we show that although the planner is often able to manipulate even with limited information about the voters’ preferences, complete lack of such information prohibits this endeavor.

**Related Literature.** A well-developed strand of the literature on social choice theory analyzes the relationship between voters’ underlying (ordinal) preferences and the approval ballots they cast. The distinction between a voter’s preference and her ballot, as well as the incentives that result to the latter, has been made explicit, e.g., by Endriss et al. (2009). Similarly to the seminal work on approval voting by Brams and Fishburn (1983), much of this literature assumes that a sincere voter with an intrinsic linear preference will form a so-called *consistent* approval ballot where all approved alternatives are preferred over all disapproved ones (Barrot et al., 2013; Endriss, 2013). In particular, this consistency assumption is central on results that compare outcomes of approval-based rules to outcomes of ranking-based rules (Gehrlein, 1981; Regenwetter and Grofman, 1998; Regenwetter and Tsetlin, 2004). Terzopoulou et al. (2025) perform this comparison exercise from a dual perspective, exploring how to generate ranking ballots from approval preferences, again assuming that approved alternatives are always preferred over disapproved ones.

Terzopoulou (2023) examines related settings where voters face costs of accessing their full intrinsic preferences, and the order in which they encounter the alternatives determines the subset over which the ballots are formed. The input of the aggregation rules in this work consists of partial orders rather than approval ballots, and the focus is on voters’ capacity limits.

Importantly, the behavioral construction proposed in our paper highlights
a distinct source of discrepancy between preferences and ballots that is neither strategic (incentive-driven) nor purely capacity-driven. Moreover, our approach generates approval ballots from underlying linear preferences but does not, in general, satisfy the common consistency requirement of prior models of the literature. Indeed, in the presence of anchoring, a voter may approve an alternative *y* and later reject an alternative *x* even though she prefers *x* to *y* (because for instance, she may encounter *y* early on but then, before seeing *x*, she may encounter a more preferred alternative *z*).

Empirical and experimental evidence support the plausibility of ordering ef-
fects in large-scale choice environments, suggesting that early options are more
likely to be selected. This resonates with our framework: early options may seem
satisfactory precisely because they have not yet been compared against poten-
tially better alternatives. Eye-tracking experiments demonstrate that users dis-
proportionately trust and select results that appear early in their Google search,
even when those results are objectively less relevant (Pan et al., 2007). Research
on product listings (notably a randomized variation in Expedia's rankings) also
indicates that higher-ranked products attract significantly more consumer at-
tention and clicks (Ursu, 2018). These findings motivate modeling presentation
order as a potentially designer-controlled feature that impacts expressed choices.

At first glance, our sequential anchoring mechanism may seem structurally
related to optimal-stopping selection settings, classically embodied by the sec-
retary problem. The secretary problem, tackled within a long theoretical litera-
ture, studies single-agent sequential choice when options arrive in random order
and decisions are irrevocable (for an overview, see Ferguson, 1989). Our model
differs in three key respects: (i) voters are influenced by the anchoring bias
when forming their ballots, rather than trying to optimize an objective over the
sequence of presented alternatives; (ii) the presentation order does not need to
be random and may be set by a planner who has incentives over the outcome;
and (iii) our focus is on the collective consequences given a group that engages
in approval voting rather than on individual decision making.

Our mechanism is also related to the literature on agenda manipulation,
originating by Farquharson (1969) and Black (1958), which studies how the or-
der of binary comparisons can affect the outcome of collective decision-making
processes. Under complete information, Miller (1977) and Banks (1985) char-
acterize the set of alternatives that an agenda-setter can induce a committee to
select, with subsequent extensions by Barberà and Gerber (2017) and Roessler
et al. (2018). More recently, settings with incomplete information have been
examined by Kleiner and Moldovanu (2017) and Gershkov et al. (2019). Our
framework differs from this line of work along several dimensions. First, the
sequence in which alternatives are presented may vary across voters. Second,
voters are shown individual alternatives sequentially rather than engaging in
a sequence of pairwise comparisons. Third, voting takes place only after all
approval ballots have been generated, rather than at each stage of the process.

Finally, there is growing interest in the new area of *behavioral social choice*,
which examines how cognitive constraints shape individual preferences and con-
sequently collective outcomes. Regenwetter et al. (2009) argue that standard
models of social choice presuppose stable and context-independent evaluations, whereas empirical evidence consistently exhibits context sensitivity and framing effects. They call for theoretical frameworks that incorporate behavioral regularities. Bonnefon (2010) shows that even preferences over aggregation procedures are malleable. Using controlled experimental settings of judgment aggregation, he demonstrates that a simple positive or negative framing of the judgment at hand can reverse individuals' procedural preferences, which can in turn shift the collective outcome. Baujard and Igersheim (2011) discuss how voting behavior depends on the particular electoral system in the context of French presidential elections. Binder et al. (2015) provide complementary field evidence using a natural experiment in California ballot initiatives. They show that framing effects meaningfully alter electoral support for low-salience propositions, while vigorous campaign activity substantially attenuates these effects.

**Roadmap.** Section 2 introduces our proposed voting model and relevant axiomatic properties. Section 3 presents an exhaustive analysis of the different variations of our central axiom — anchor-proofness — and demonstrates impossibility and possibility results for each one of them. Section 4 presents the model where a social planner can manipulate, and provides certain characterization results. Section 5 concludes and discusses future lines of research.

## 2 Model

Let $N = \{1, \dots, n\}$ be a finite set of voters and $X$ a finite set of alternatives, with $|X| = m > 2$. For an arbitrary set $Y$, we denote by $L(Y)$ the set of all linear orders (i.e., a complete and acyclic binary relations) over $Y$. We call a ranking over the set of alternatives $X$ a linear order $p \in L(X)$. If $(x, y) \in p$, we will say that alternative $x$ is preferred to $y$ in $p$. For simplicity, we often write $p = (p^1, p^2, \dots, p^m)$; that is, $p^k$ is the alternative that appears in the $k^{th}$ position of the ranking. The most preferred alternative in a ranking appears in position 1, and the least preferred alternative holds position $m$. Following the model introduced by Brams and Sanver (2009), each voter $i \in N$ is endowed with an intrinsic preference-approval over the set of alternatives:

$$p_i = (p_i^1, \dots, p_i^m)$$

A preference-approval $p_i \in P(X)$ is a ranking over the alternatives associated with an acceptability threshold $t_i \in \{1, \dots, m\}$ that captures the maximum position that voter $i$ considers acceptable for approving the corresponding alternative in the ranking. Formally, $P(X) = L(X) \times \{1, \dots, m\}$. Whenever the acceptability threshold is not relevant, we will treat $p_i$ simply as a linear order. So, an alternative $x$ with position $k$ in the ranking $p_i$ is acceptable if $k \le t_i$. Let $ACC(p_i) = \{p_i^1, \dots, p_i^{t_i}\} \subseteq X$ be the set of acceptable alternatives by voter $i$ with preference-approval $p_i$. Note that the most preferred alternative of a voter is always acceptable by that voter.
We denote by $\boldsymbol{p} = (p_1, \dots, p_n) \in P(X)^N$ the profile of preference-approvals of all voters in $N$. When $ACC(p_i) = X$ (equivalently $t_i = m$), we say that the preference-approval is *tolerant*. Moreover, if for all $i \in N$ it is the case that $ACC(p_i) = X$, we call the profile tolerant. We define $tol(p_i) \in P(X)$ to be the *tolerant* version of $p_i$: the associated ranking of $tol(p_i)$ is the same as of $p_i$ and moreover all alternatives are acceptable in $tol(p_i)$. The tolerant version of a preference-approval profile $\boldsymbol{p} = (p_1, \dots, p_n)$ is $\boldsymbol{tol}(\boldsymbol{p}) = (tol(p_1), \dots, tol(p_n)) \in P(X)^N$. When $|ACC(p_i)| = 1$ (equivalently $t_i = 1$), we say the preference-approval is *intolerant*. If for all $i \in N$ it is the case that $|ACC(p_i)| = 1$, we call the profile intolerant.

Given a profile $\boldsymbol{p}$ and an alternative $x \in X$, we define the plurality and the acceptability points of $x$ respectively, as follows:

$$ 
\begin{aligned}
\text{plur}_{\boldsymbol{p}}(x) &= |\{i \in N \mid x = p_i^1\}| \\
\text{acc}_{\boldsymbol{p}}(x) &= |\{i \in N \mid x \in ACC(p_i)\}|
\end{aligned}
$$

Thus, the plurality points of $x$ equals to the number of voters that rank $x$ first in $\boldsymbol{p}$; while the acceptability points of $x$ equals the number of voters that accept $x$ in $\boldsymbol{p}$. We also write $\text{PLUR}(\boldsymbol{p})$ for the set of alternatives that are ranked first by at least one voter in $\boldsymbol{p}$, and $\text{ACC}(\boldsymbol{p})$ for the set of alternatives that are acceptable by at least one voter in $\boldsymbol{p}$:

$$ 
\begin{aligned}
\text{PLUR}(\boldsymbol{p}) &= \{x \in X \mid x = p_i^1 \text{ for some } i \in N\} \\
\text{ACC}(\boldsymbol{p}) &= \{x \in X \mid x \in ACC(p_i) \text{ for some } i \in N\}
\end{aligned}
$$

Let $S(X)$ be the set of all permutations over the set of alternatives. Each voter $i \in N$ is shown the alternatives in steps, one at a time, according to the order prescribed by her corresponding permutation $\sigma_i$ in the vector $\boldsymbol{\sigma} = (\sigma_1, \dots, \sigma_n) \in S(X)^N$. Given a voter $i$ and an order $\sigma_i$, let $\sigma_i^k \in X$ be the alternative that is presented to her in step $k$. Let $A(X)$ be the set of all non-empty approval ballots over the set of alternatives—that is, $A(X) = 2^X \setminus \emptyset$. Voter $i$ with preference-approval $p_i$ creates an approval ballot $A_{p_i, \sigma_i} \in A(X)$ based on the order $\sigma_i$ via the following iterative procedure, where $k > 1$:

Step 1: $A_{p_i, \sigma_i}^1 = \{\sigma_i^1\} \cap ACC(p_i)$.

Step $k$: $A_{p_i, \sigma_i}^k = \begin{cases} (A_{p_i, \sigma_i}^{k-1} \cup \{\sigma_i^k\}) \cap ACC(p_i) & \text{if } (\sigma_i^k, x) \in p_i \text{ for all } x \in A_{p_i, \sigma_i}^{k-1} \\ A_{p_i, \sigma_i}^{k-1} & \text{otherwise} \end{cases}$

Finally: $A_{p_i, \sigma_i} = A_{p_i, \sigma_i}^m$

In words, a voter constructs her ballot by examining the alternatives sequentially, one at a time, and with commitment: once an alternative is approved, it cannot be revoked later. When the voter encounters the $k^{th}$ alternative in the presentation order, for every $k$, she exhibits an anchoring bias: she approves
the alternative if and only if she deems it acceptable and she has not previously encountered another more preferred alternative. Said differently, at every step k the anchor is the best alternative encountered so far.

Note that it is not the case that every approval ballot $A_{p_i, \sigma_i}$ formed by a preference-approval $p_i$ and an order $\sigma_i$ is consistent, although this is common in prior literature (for example, Brams and Sanver, 2009). Under this anchoring process, an alternative outside a voter's approval ballot may be more preferred by that voter to some alternative inside the ballot (see Example 2).

**Example 2.** Consider a tolerant preference-approval $p_i = (x, y, z)$ and an order $\sigma_i = (z, x, y)$. Then, $A_{p_i, \sigma_i}^1 = \{z\}$ (because $z \in ACC(p_i)$), $A_{p_i, \sigma_i}^2 = \{x, z\}$ (because $(x, z) \in p_i$), and $A_{p_i, \sigma_i}^3 = A_{p_i, \sigma_i}^2 = \{x, z\}$ (because $x \in A_{p_i, \sigma_i}^2$ but $(y, x) \notin p_i$). Thus, the voter approves $z$ but disapproves $y$ although she intrinsically prefers $y$ to $z$, generating the approval ballot $A_{p_i, \sigma_i} = \{x, z\}$. This happens because $z$ is acceptable and is examined at the very beginning, so it is approved since no better anchor exists at the time; but the voter has already seen the more preferred alternative $x$ when $y$ is examined, so $y$ is disapproved since it is compared to a better anchor. $\Delta$

We next introduce two particular orders that depend on a preference-approval $p_i$ of voter $i$:

* $\underline{\sigma_i}$ is the order that presents to voter $i$ the alternatives from least to most preferred in $p_i$. Formally, $\underline{\sigma_i} = (p_i^m, p_i^{m-1}, \dots, p_i^1)$. Thus:

$$A_{p_i, \underline{\sigma_i}} = ACC(p_i)$$

* $\overline{\sigma_i}$ is the order that presents to voter $i$ the alternatives from most to least preferred in $p_i$. Formally, $\overline{\sigma_i} = (p_i^1, p_i^2, \dots, p_i^m)$. Thus:

$$A_{p_i, \overline{\sigma_i}} = \{p_i^1\}$$

Given a profile $p$ and an order $\sigma$, we denote with $app_{p,\sigma}(x)$ the approval points of $x$ in $p$ under the order $\sigma$, that is:

$$app_{p,\sigma}(x) = |\{i \in N | x \in A_{p_i,\sigma_i}\}|$$

It is easy to see that $plur_p(x) \leq app_{p,\sigma}(x) \leq acc_p(x)$ for any profile $p$ and order $\sigma$. Moreover, Remark 1 shows that our setting allows certain liberties with respect to the approval ballot formed by a voter $i$. Specifically: first, any acceptable subset of alternatives that includes the voter's most preferred one can become the voter's approval ballot under some suitable presentation order; second, given a specific presentation order, any subset of alternatives can become the approval ballot of some voter with a suitable preference-approval; and third, given a specific presentation order, any subset of alternatives that includes the one appearing first in the order can become the approval ballot of some voter with a suitable tolerant preference-approval.
**Remark 1.** Let $A \subseteq X$ be a subset of alternatives. Then the following hold:

* for every preference-approval $p_i \in P(X)$ there exists an order $\sigma_i$ such that:

$$A_{p_i, \sigma_i} = (A \cap ACC(p_i)) \cup \{p_i^1\}$$

* for every order $\sigma_i$ there exists a (potentially non-tolerant) preference-approval $p_i \in P(X)$ such that:

$$A_{p_i, \sigma_i} = A$$

* for every order $\sigma_i$ there exists a tolerant preference-approval $p_i \in P(X)$ such that:

$$A_{p_i, \sigma_i} = A \cup \{\sigma_i^1\}$$

A profile of ballots $A_{p,\sigma}$ is a vector of approval ballots obtained for voters with intrinsic approval-preferences $p$ after the presentation of the alternatives following the order vector $\sigma$:

$$A_{p,\sigma} = (A_{p_1,\sigma_1}, \dots, A_{p_n,\sigma_n}) \in A(X)^N$$

An approval-based voting rule $F_a: A(X)^N \to 2^X$, takes a profile of approval ballots and outputs a set of winning alternatives. For the remainder of the paper, we will simply refer to a voting rule, assuming that its inputs are profiles of approval ballots.

Our main question is thus phrased as follows:

How does the order in which voters examine alternatives influence the outcome of a voting rule?

Intuitively, we examine how the voters' anchoring bias might influence the outcome, as different approval ballots can be elicited from the same preference, and if this can be exploited by a social planner. In some extreme cases, the formation of an approval ballot is not that influenced by the presentation order of the alternatives. For example, when the voter's preference-approval is intolerant and only her favorite alternative is acceptable and can be approved. On the other side, when $\sigma_i^1 = p_i^1$, the voter encounters her favorite alternative first, which becomes the anchor and no other alternative is approved, making the approval ballot heavily influenced by the presentation order.

This implies Remark 2.

**Remark 2.** The following hold:

* For a preference-approval profile $p$ and two orders $\sigma$ and $\pi$ such that $\sigma_i^1 = \pi_i^1 = p_i^1$ for all voters $i \in N$, it holds that $A_{p,\sigma} = A_{p,\pi}$.

* For every preference-approval profile $p$ such that $t_i = 1$ for all $i \in N$, it holds that $A_{p,\sigma} = A_{p,\pi}$ for all orders $\sigma$ and $\pi$.
We proceed with a central definition of this paper, that of *anchor-proofness*, i.e., robustness of outcomes with respect to the alternatives' presentation orders.

**Definition 1.** We call a voting rule $F_a$ **anchor-proof** for a profile $p$ if $F_a(\mathbf{A}_p, \sigma) = F_a(\mathbf{A}_p, \pi)$ for every two orders $\sigma, \pi \in S(X)$.

We call a voting rule $F$ **anchor-proof** if $F$ is anchor-proof for every profile $p$.

The most popular voting rule is *standard approval voting (SAV)*, which selects as winners the alternatives with the highest approval count in a profile $\mathbf{A} = (A_1, \dots, A_n)$. Concretely:

$$SAV(\mathbf{A}) = \arg\max_{x \in X} |\{i \in N \mid x \in A_i\}|$$

Another simple voting rule is what we call *nomination* (and can also be thought off as a *quota-1 rule*): An alternative is a winner if and only if it is ‘nominated’, i.e., approved by at least one voter. Formally:

$$Nom(\mathbf{A}) = \{x \in X \mid x \in A_i\}$$

Now we propose a number of axioms, which we consider desirable properties of general approval-based rules. The first one states that the names of the voters should not be considered important in selecting a winning alternative.

**Definition 2.** We say that $F_a$ is **anonymous** if for every permutation $\lambda: N \to N$ and approval profile $\mathbf{A} = (A_1, \dots, A_n) \in A(X)^N$, it holds that $F_a(A_{\lambda(1)}, \dots, A_{\lambda(n)}) = F_a(\mathbf{A})$.

On the other hand, the names of the alternatives should not be relevant for the rule.

**Definition 3.** We say that $F_a$ is **neutral** if for every permutation $\mu: X \to X$ and approval profile $\mathbf{A} = (A_1, \dots, A_n) \in A(X)^N$, it holds that $F_a(\mu(A_1), \dots, \mu(A_n)) = \mu(F_a(\mathbf{A}))$, where $\mu(A) = \{\mu(x) \mid x \in A\}$.

The next three axioms state different degrees of consensus that a voting rule should satisfy.

**Definition 4.** We say that $F_a$ satisfies **weak unanimity** if for every approval profile $\mathbf{A}$ such that $U = \{x \in X : x \in A_i \text{ for all } i \in N\} \neq \emptyset$, it holds that $F_a(\mathbf{A}) \subseteq U$.

**Definition 5.** We say that $F_a$ satisfies **total unanimity** if for every approval profile $\mathbf{A}$ where $A_i = X$ for all $i \in N$, it holds that $F_a(\mathbf{A}) = X$.

**Definition 6.** We say that $F_a$ satisfies **unanimity** if for every approval profile $\mathbf{A}$ such that $U = \{x \in X : x \in A_i \text{ for all } i \in N\} \neq \emptyset$, it holds that $F_a(\mathbf{A}) = U$.

Note that weak unanimity is a weaker axiom than the Pareto condition proposed by Brandl and Peters (2022) for approval voting. It requires all winners to be unanimously approved, but does not require selecting all unanimously approved alternatives. Standard approval voting rule (SAV) satisfies unanimity and thus also satisfies weak and total unanimity. The nomination rule (Nom) satisfies total unanimity but violates weak unanimity.
# 3 Anchor-proofness

This section investigates questions of *anchor-proofness*—specifically, conditions under which the outcome of a voting rule remains invariant to the presentation order of the alternatives. Our findings can be organized around six sub-questions (summarized in Figure 1). Fixing a voting rule, we ask:

1. Are the approval winners independent of the presentation order for all ranking profiles?

2. Does there exist a ranking profile for which the approval winners are in-dependent of the presentation order?

3. Does there exist a pair of distinct presentation orders that produce the same approval winners for an arbitrary ranking profile?

4. Given an arbitrary ranking profile, does there exist a pair of different presentation orders that produce the same approval winners?

5. Given an arbitrary pair of presentation orders, does there exist a ranking profile for which the approval winners coincide?

6. Does there exist some ranking profile and some pair of distinct presentation orders for which the approval winners coincide?

Together, these six questions form a complete landscape for assessing qualitative aspects of anchor-proofness: any meaningful inquiry into whether a profile's winners depend on the presentation order can be stated as one of them. Clearly, the first question is the strongest and most demanding (and indeed has a negative answer for all non-constant rules), while the last question is the weakest and has a trivial positive answer for all rules. The intermediate questions (2-5) are not logically related. Note also that questions (4-6) are included mostly for conceptual completeness and are of limited practical interest, since they require information about a concrete ranking profile (resp. a presentation order) to construct a suitable presentation order (resp. ranking profile). The topic of information needed to influence the voting outcome will be revisited in Section 4.

## 3.1 Anchor-Proofness for All Profiles

We start the presentation of our main results with Theorem 1, showing that anchor-proofness is impossible for non-constant rules within the restricted domain of tolerant profiles. Using the intermediate Lemma 1, we will then extend this impossibility to the general domain of all preference-approval profiles in Theorem 2. Note that a voting rule that is constant for all tolerant profiles is not necessarily constant for all, possibly non-tolerant, profiles, while the other direction holds. This means that Theorem 1 is implied by Theorem 2, but we demonstrate both since the former is used for the proof of the latter.
<figure><img src="image_3.png" /><figcaption>Figure 1: Summary of our results. The quantifiers in each frame show the conditions under which we demand that $F(\mathbf{A}_{p,\sigma}) = F(\mathbf{A}_{p,\pi})$, always assuming that $\sigma \neq \pi$. A ‘Yes’ (resp. ‘No’) answer means that the corresponding condition can (resp. cannot) be satisfied.</figcaption></figure>
**Theorem 1.** Let $F_a$ be a voting rule that is anchor-proof for all tolerant profiles $\mathbf{p}$. Then, $F_a$ is constant in the domain of approval profiles induced by tolerant preference-approval profiles. That is: $F_a(\mathbf{A}_{\mathbf{p},\sigma}) = F_a(\mathbf{A}_{\mathbf{q},\pi})$ for all tolerant profiles $\mathbf{p}$ and $\mathbf{q}$ and all orders $\sigma$ and $\pi$.

*Proof.* Assume that $F_a$ is not constant. Then, there exist two tolerant profiles $\mathbf{p}$ and $\mathbf{q}$ (not necessarily different) and two orders $\sigma$ and $\pi$ such that $F_a(\mathbf{A}_{\mathbf{p},\sigma}) \neq F_a(\mathbf{A}_{\mathbf{q},\pi})$.

Let us define two orders $\sigma^*$ and $\pi^*$, such that:

* $\sigma_i^{*1} = q_i^1$ and $\sigma_i^{*2} = p_i^1$

* $\pi_i^{*1} = p_i^1$ and $\pi_i^{*2} = q_i^1$

The following holds because $\mathbf{p}$ and $\mathbf{q}$ are tolerant and $F_a$ is anchor-proof:

$$F_a(\mathbf{A}_{\mathbf{p},\sigma^*}) = F_a(\mathbf{A}_{\mathbf{p},\sigma}) \neq F_a(\mathbf{A}_{\mathbf{q},\pi}) = F_a(\mathbf{A}_{\mathbf{q},\pi^*})$$

But by construction, we have that $A_{p_i, \sigma_i^*} = A_{q_i, \pi_i^*}$ for all $i \in N$. Thus we must have that $F_a(\mathbf{A}_{\mathbf{p},\sigma^*}) = F_a(\mathbf{A}_{\mathbf{q},\pi^*})$, a contradiction. $\square$

**Lemma 1.** Let $F_a$ be a voting rule that is anchor-proof for all preference-approval profiles $\mathbf{p}$. Then, $F_a(\mathbf{A}_{\mathbf{p},\sigma}) = F_a(\mathbf{A}_{\mathbf{tol}(\mathbf{p}),\sigma})$ for all profiles $\mathbf{p}$ and orders $\sigma$.

*Proof.* By anchor-proofness, we know that $F_a(\mathbf{A}_{\mathbf{p},\sigma}) = F_a(\mathbf{A}_{\mathbf{p},\pi})$ given a profile $\mathbf{p}$, for all orders $\sigma$ and $\pi$. Specifically:

$$F_a(\mathbf{A}_{\mathbf{p},\sigma}) = F_a(\mathbf{A}_{\mathbf{p},\bar{\sigma}}) = F_a(\{p_1^1\}, \dots, \{p_n^1\})$$

Analogously:

$$F_a(\mathbf{A}_{\mathbf{tol}(\mathbf{p}),\sigma}) = F_a(\mathbf{A}_{\mathbf{tol}(\mathbf{p}),\bar{\sigma}}) = F_a(\{p_1^1\}, \dots, \{p_n^1\})$$

We conclude that $F_a(\mathbf{A}_{\mathbf{p},\sigma}) = F_a(\mathbf{A}_{\mathbf{tol}(\mathbf{p}),\sigma})$. $\square$

**Theorem 2.** Let $F_a$ be a voting rule that is anchor-proof. Then, $F_a$ is constant. That is: $F_a(\mathbf{A}_{\mathbf{p},\sigma}) = F_a(\mathbf{A}_{\mathbf{q},\pi})$ for all preference-approval profiles $\mathbf{p}$ and $\mathbf{q}$ and all orders $\sigma$ and $\pi$.

*Proof.* By Lemma 1, it is sufficient to show that $F_a(\mathbf{A}_{\mathbf{tol}(\mathbf{p}),\sigma}) = F_a(\mathbf{A}_{\mathbf{tol}(\mathbf{q}),\pi})$ for all tolerant profiles $\mathbf{tol}(\mathbf{p})$ and $\mathbf{tol}(\mathbf{q})$. This is true from Theorem 1. $\square$

Our main observation so far involves an impossibility: There are no interesting rules that are indifferent to the alternatives' presentation order. Notably, this holds because the social planner might be able to carefully select an order, considering specific information about each voter's preference. We will see in Section 4 that possibilities arise when such information is not available.
## 3.2 Existence of an Anchor-Proof Profile

Having addressed the first question in our list about anchor-proofness in the general domain of preference-approvals, we move on to the second question that searches, more conservatively, the existence of some profile for which anchor-proofness is achievable. By the second point of Remark 2, we know that such profiles exist within all preference-approval ones: in particular, every intolerant profile does the job, independently of the voting rule. Theorem 3 shows that this ceases to be possible if we look only within the domain of tolerant profiles and apply a rule satisfying the reasonable axioms of weak and total unanimity. However, the impossibility for the tolerant domain is not generalizable to arbitrary voting rules: Example 3 illustrates that there exist non-neutral rules that are anchor-proof for tolerant profiles.

**Theorem 3.** Let $F_a$ be a voting rule that satisfies weak and total unanimity. Then, $F_a$ is not anchor-proof for any tolerant profile $p$.

*Proof.* Consider an arbitrary tolerant profile $p$ and suppose for contradiction that $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$ for all orders $\sigma, \pi$. Then, it must hold that

$$F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\sigma}) = F_a(X, \dots, X)$$

From total unanimity, this means that $F_a(\mathbf{A}_{p,\sigma}) = X$ for all orders $\sigma$. We distinguish two cases.

First, suppose that all preference-approvals in $p$ have the same top alternative $x$, i.e., $p_i^1 = x$ for all voters $i \in N$. Then, take the order $\sigma$ to be such that $\sigma_i^1 = x$ for all $i \in N$. This means that $F_a(\mathbf{A}_{p,\sigma}) = (\{x\}, \dots, \{x\})$, so weak unanimity applies for some alternative $y \neq x$ implying that $y \notin F_a(\mathbf{A}_{p,\sigma}) = X$. We have reached a contradiction.

Second, suppose that there exist two voters $r \neq j$ and two alternatives $x \neq y$ such that $p_r^1 = x \neq y = p_j^1$. Then, consider again the order $\sigma$ such that $\sigma_i^1 = x$ for all $i \in N$. This means that $x \in A_{p_i,\sigma_i}$ for all $i \in N$ and $y \notin A_{p_r,\sigma_r} = \{x\}$. By weak unanimity, we have that $y \notin F_a(\mathbf{A}_{p,\sigma}) = X$. We have reached a contradiction. □

**Example 3.** Consider the following voting rule $F_a$. Fix an alternative $x \in X$. Then, for an arbitrary approval profile $\mathbf{A} = (A_1, \dots, A_n)$ define:

$$F_a(\mathbf{A}) = \begin{cases} \{x\} & \text{if } x \in A_i \text{ for all } i \in N \\ X & \text{otherwise} \end{cases}$$

Now for the profile $p$ such that $p_i^1 = x$ for all $i \in N$ it holds that $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$ for all orders $\sigma$ and $\pi$. Note that this rule violates neutrality as well as weak and total unanimity.

As in the previous section, here as well, utilizing crucial information about the voters' preferences by the social planner makes natural voting rules fail anchor-proofness. However, there are some exceptions: Considering for instance
the setting of participatory budgeting, if all projects are so good such that voters accept them all, then there is no escape from the potential influence of a social planner. Moreover, even without such unanimity requirements, we are able to find rules which in certain situations are immune to anchor biases.

### 3.3 Order Pairs Preserving Outcomes for All Profiles

We continue with the third question in our list, specifying the existence of a pair of orders producing the same winners for all profiles. This is clearly a weaker requirement than anchor-proofness, which spans across all orders. Proposition 1 (utilizing Lemma 2) proves that such a pair of orders does not exist if we use a rule satisfying weak and total unanimity, even if we restrict attention to the domain of all tolerant profiles. Examples 4 and 5 show that the impossibility is circumvented in the restricted domain of tolerant profiles or, respectively, in the general domain of all preference-approval profiles, if we consider the nomination rule (which, recall, fails weak unanimity) or, respectively, a rule that violates neutrality. Remark 3 explains that the result for the nomination rule does not extend to the general domain.

**Lemma 2.** Consider two orders $\sigma_i$ and $\pi_i$ and two preferences $p_i$ and $p'_i$ such that the following three conditions hold:

(i) $A_{p_i, \sigma_i} = X$;

(ii) $A_{p_i, \pi_i} = A \neq X$;

(iii) $A_{p'_i, \sigma_i} = A' \supseteq A$.

Then, we have that $A_{p'_i, \pi_i} = A$.

*Proof.* Let us first show that $A_{p'_i, \pi_i} \subseteq A$. Aiming for a contradiction, suppose that there exists an alternative $x \in X$ such that $x \in A_{p'_i, \pi_i}$ but $x \notin A = A_{p_i, \pi_i}$. The fact that $x \notin A = A_{p_i, \pi_i}$ implies the existence of another alternative $a \in A$ such that:

$$ (a, x) \in p_i \quad \text{and} \quad (a, x) \in \pi_i $$

Now, $x \in A_{p'_i, \pi_i}$ and $(a, x) \in \pi_i$ together imply that:

$$ (x, a) \in p'_i $$

But by $(a, x) \in p_i$ and knowing from the hypothesis (i) that $A_{p_i, \sigma_i} = X$, we have that:

$$ (x, a) \in \sigma_i $$

This, together with the fact that $(x, a) \in p'_i$, implies that $a \notin A_{p'_i, \sigma_i} = A' \supseteq A$, which is a contradiction since $a \in A$.

Next, let us show that $A \subseteq A_{p'_i, \pi_i}$. Aiming for a contradiction, suppose that there exists an alternative $y \in X$ such that $y \in A \subseteq A'$, thus $y \in A' = A_{p'_i, \sigma_i}$, but $y \notin A_{p'_i, \pi_i}$. Now, from the hypotheses (i) and (ii), it must be the case that
any two alternatives $a, b \in A$ are ordered in the same way by $\sigma_i$ and by $\pi_i$: that is, $(a, b) \in \sigma_i$ if and only if $(a, b) \in \pi_i$ (otherwise, to have that $A_{p_i, \sigma_i} = X$, one of the alternatives $a, b$ would not be included in $A = A_{p_i, \pi_i}$). So, $y \in A_{p'_i, \sigma_i}$ means that:

for all $a \in A$ such that $(a, y) \in p'_i$ it holds that $(y, a) \in \sigma_i \cap \pi_i$

But since we have that $y \notin A_{p'_i, \pi_i}$, there must exist another alternative $x \notin A$ such that $(x, y) \in p'_i$ and $x \in A_{p'_i, \pi_i}$ (where $(x, y) \in \pi_i$). This is impossible, since in the first part of this proof we showed that $A_{p'_i, \pi_i} \subseteq A$. We have reached a contradiction.

We conclude that $A_{p'_i, \pi_i} \subseteq A$ and $A \subseteq A_{p'_i, \pi_i}$, and we are done. $\square$

**Proposition 1.** Let $F_a$ be a voting rule that satisfies weak and total unanimity. Then there is no pair of orders $\sigma \neq \pi$ such that $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$ for all (tolerant or not) preference-approval profiles $\mathbf{p}$.

*Proof.* Suppose for contradiction that there are two orders $\sigma$ and $\pi$ such that $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$ for all preference-approval profiles $\mathbf{p}$. We divide the proof into two cases, according to the number $k$ of voters for which the orders are different. That is, $k = |\{i \in N | \sigma_i \neq \pi_i\}|$.

**Case 1:** $k = 1$.

We can assume without loss of generality that $\sigma_1 \neq \pi_1$, that is, the orders are different for the first voter but the same for all other voters. Consider the profile $\mathbf{p}$ such that $\mathbf{A}_{p,\sigma} = (X, \dots, X)$ and $\mathbf{A}_{p,\pi} = (A_1, \dots, A)$. By weak unanimity, we have that $F_a(A_1, \dots, X) \subseteq A_1$, and by total unanimity $F_a(X, \dots, X) = X$. Thus, $F_a(A_1, \dots, X) \neq F_a(X, \dots, X) = X$, a contradiction with our hypothesis.

**Case 2:** $k > 1$.

We can assume without loss of generality that $\sigma_i \neq \pi_i$ for all $i \in \{1, \dots, k\}$, that is, the orders are different for the first $k$ voters and the same for all other voters. Consider the profile $\mathbf{p}$ such that $\mathbf{A}_{p,\sigma} = (X, \dots, X)$ and $\mathbf{A}_{p,\pi} = (A_1, \dots, A_k, X, \dots, X)$. By weak and total unanimity, $\cap_{i=1}^k A_i = \emptyset$, otherwise $F_a(A_1, \dots, A_k, X, \dots, X) \subseteq \cap_{i=1}^k A_i \neq X = F_a(X, \dots, X)$. By Lemma 2, we can create a profile $\mathbf{p}^i$, for $1 \le i \le k$, such that $\mathbf{A}_{p^i,\sigma} = (X, \dots, X, A_i, X, \dots, X)$ and $\mathbf{A}_{p^i,\pi} = (A_1, \dots, A_k, X, \dots, X)$. By weak unanimity we have that

$$F_a(A_1, \dots, A_k, X, \dots, X) = F_a(X, \dots, X, A_i, X, \dots, X) \subseteq A_i$$

for all $i = \{1, \dots, k\}$. Thus, we have that

$$F_a(A_1, \dots, A_k, X, \dots, X) \subseteq \cap_{i=1}^k A_i = \emptyset,$$

a contradiction. $\square$

**Example 4.** Let $F_a$ be Nom. Then, the two orders $\sigma$ and $\pi$ defined below are such that $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$ for all tolerant profiles $\mathbf{p}$.

Suppose that $n \ge m$ and define $\sigma \neq \pi$ such that for every alternative $x \in X$ it is the case that $x = \sigma_i^1 = \pi_i^1$ for some voter $i \in N$. Then, all alternatives are nominated in all profiles $\mathbf{p}$, so $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi}) = X$.
Second, suppose that $n < m$. Let $X'$ be the first $n$ alternatives according to some arbitrary order of $X$. Define $\sigma \neq \pi$ such that the three following conditions hold:

* for every alternative $x \in X'$ it is the case that $x = \sigma_i^1 = \pi_i^1$ for some voter $i \in N$;

* all alternatives in $X'$ appear before all alternatives in $X \setminus X'$ for every $i \in N$;

* all alternatives in $X \setminus X'$ appear in the same order for every $i \in N$.

Then, all alternatives in $X'$ are nominated in all profiles $p$, so $X' \subseteq F_a(A_{p,\sigma})$ and $X' \subseteq F_a(A_{p,\pi})$. Moreover, given $x \in X$, define the following sets:

$$A_x^\sigma = \{y \in X : (y, x) \in \sigma_i \text{ for some } i \in N\}$$

$$A_x^\pi = \{y \in X : (y, x) \in \pi_i \text{ for some } i \in N\}$$

Note that $A_x^\sigma = A_x^\pi$ for all $x \in X \setminus X'$. Now, an alternative $x \in X \setminus X'$ is a winner according to the nomination rule if and only if there exists a voter $i \in N$ such that $(x, y) \in p_i$ for all $y \in A_x^\sigma$ (that is, if and only if $x$ is more preferred over all alternatives that are previously presented to a voter). $\Delta$

Note that Example 4 does not work for the general domain (including all non-tolerant profiles). In fact, no pair of different orders can guarantee the same outcome for all profiles given the nomination rule (see Remark 3).

**Remark 3.** Consider arbitrary $\sigma \neq \pi$. Then, there exist $i \in N$ and $x, y \in X$ such that $(x, y) \in \sigma_i$ and $(y, x) \in \pi_i$. Take the profile $p$ such that $x$ is not acceptable for every $j \in N \setminus \{i\}$, while $x$ is acceptable for $i$ and $p_i^1 = y$ and $p_i^2 = x$. Then, $x \notin A_{p_j, \sigma_j}$ and $x \notin A_{p_j, \pi_j}$ for all $j \in N \setminus \{i\}$. Also, $x \in A_{p_i, \sigma_i}$ but $x \notin A_{p_i, \pi_i}$. This means that for $F$ being Nom it holds that $x \in F(A_{p,\sigma})$ but $x \notin F(A_{p,\pi})$.

**Example 5.** Consider a non-neutral voting rule according to which the nomination rule is followed except for two alternatives $x$ and $y$ that are excluded from the winning set for all profiles. Then, consider two orders $\sigma$ and $\pi$ that follow the construction of Example 4 with the exception that $x$ and $y$ are shown last for all voters. Then, the reasoning of Example 4 follows. $\Delta$

In contrast to the several impossibilities produced in previous sections, this section sheds light to certain positive observations, as we are able to find two orders for which a given voting rule is indifferent. This result becomes relevant for instance in a participatory budgeting scenario where all projects are acceptable by everyone. A regulatory entity that wants a fair process might then force the social planner to use only the presentation orders that make Nom anchor-proof. We can see this as a minimal guaranty that the regulatory institution has against bad-intended social planners.
## 3.4 Profiles Tailored to a Fixed Pair of Orders

Lastly, we examine whether for any pair of orders there exists a profile that produces the same winners for both orders. The answer is trivial if we allow for any profile: the outcomes of intolerant profiles are invariant for all orders (recall Remark 2). However, if we consider a voting rule satisfying unanimity, the answer becomes negative when we only allow for tolerant profiles (Proposition 2). Example 6 shows that a positive answer is achievable for other rules, such as a less decisive variation of SAV.

**Proposition 2.** Consider a voting rule $F_a$ that satisfies unanimity. Then, there exist two orders $\sigma$ and $\pi$ such that a $F_a(\mathbf{A}_{p,\sigma}) \neq F_a(\mathbf{A}_{p,\pi})$ for all tolerant profiles $p$.

*Proof.* Take two orders $\sigma$ and $\pi$ such that $\sigma_i^1 = x$ and $\pi_i^1 = y$, for alternatives $x \neq y$ and for all $i \in N$. Then, suppose for contradiction that $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$ for some tolerant profile $p$. By construction, we know that $x \in A_{p_i,\sigma_i}$ and $y \in A_{p_i,\pi_i}$ for all $i \in N$. Thus, unanimity implies that $\{x,y\} \subseteq F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi})$. Without loss of generality, suppose that $(x,y) \in p_i$ for some $i \in N$ (the case where $(y,x) \in p_i$ is completely symmetric). Then, since $\pi_i^1 = y$, we have that $x \notin A_{p_i,\pi_i}$. So by unanimity, $x \notin F_a(\mathbf{A}_{p,\pi})$. Contradiction. $\square$

**Example 6.** Consider the voting rule $F_a$ that returns all alternatives when at least one voter approves at least two alternatives and behaves as SAV otherwise (when each voter approves only one alternative). We can find a profile $p$ that is anchor-proof for $F_a$. Indeed, for arbitrary orders $\sigma \neq \pi$ consider a preference-approval profile such that $p_1 = (\sigma_1^2, \sigma_1^1, \dots)$ and $p_2 = (\pi_1^2, \pi_1^1, \dots)$. It will be the case that $A_{p_1,\sigma_1} = \{\sigma_1^1, \sigma_1^2\}$ and $A_{p_2,\pi_2} = \{\pi_2^1, \pi_2^2\}$. Thus, $F_a(\mathbf{A}_{p,\sigma}) = F_a(\mathbf{A}_{p,\pi}) = X$ according to the definition of the rule. $\triangle$

This section speaks about scenarios where the presentation orders are fixed and limited, for instance within electoral contexts that are supposed to exhibit the alternatives to voters in a pre-determined way. We ask whether there are specific preference structures that would induce robust outcomes for at least two different fixed presentation orders within groups of tolerant voters, thus restricting the potential influence of the social planner. We find that this — rather weak — requirement is not satisfied as long as the voting rule adheres to unanimity. However, there do exist situations where the social planner has no influence over the outcome. One of them concerns the tolerance of voters: In particular, consider scenarios where projects in participatory budgeting are extreme, in the sense that each project clearly benefits one part of the society while harming all the rest; then, voters can be assumed to accept only those projects from which they benefit and to reject all others, yielding an intolerant preference profile in which the outcome is independent of the presentation order. The other situation that sets constraints on the influence of the social planner does not necessarily demand intolerant voters, but employs a voting rule that is only meaningful in situations of indirect intolerance: unless each voter only
approves a single alternative where simple SAV may be applied, that rule returns a tie between all alternatives and is thus completely undecisive.

# 4 Strategies for the Social Planner

Previously, we studied anchor-proofness as a robustness property of voting rules with respect to the order in which alternatives are presented to voters. We now delve more deeply into this notion from a strategic perspective.

We consider that a social planner may have her own preferred outcome of a voting process. Then, choosing a presentation order is a strategic decision aimed at influencing that outcome. In this angle, a preference-approval profile is anchor-proof if and only if it is immune to manipulation by any planner: no matter which order is chosen, the resulting outcome remains unchanged. We can thus view anchor-proofness as a benchmark for worst-case robustness against strategic manipulation by the planner. This equivalence specifically assumes that the planner has complete access to voters' preferences, and is thus able to tailor individual presentation orders. This underlies our characterizations of anchor-proof profiles under SAV and Nom (Theorems 4 and 5), as well as our general result for all weakly unanimous rules (Theorem 6).

We then relax the assumption of full information and study precisely how the planner's ability to manipulate the outcome depends on the information she possesses about voters' preferences. We first show that, under zero information, manipulation is impossible for any rule satisfying weak and total unanimity, SAV being one of them (Theorem 7), and also for the nomination rule (Theorem 10). We then demonstrate that even limited aggregate information — such as acceptability points or plurality information — can restore manipulability for broad classes of rules (Theorems 8, 9, 11, and 12).

## 4.1 Anchor-proof Profiles under Full Information

Let us start with our results regarding anchor-proof profiles under SAV. Loosely speaking, homogeneity seems to help with anchor-proofness, but to an extent, since profiles with more than one unanimously accepted alternative are not anchor-proof.

**Proposition 3.** SAV is anchor-proof for a profile $p = (p_1, \dots, p_n)$ only if $p$ does not have more than one unanimously accepted alternative; that is, only if $|\{x \in X \mid x \in ACC(p_i) \text{ for all } i \in N\}| \le 1$.

*Proof.* Suppose that the profile $p$ has at least two unanimously accepted alternatives $x$ and $y$. Without loss of generality, suppose that there exists a voter $i \in N$ such that $(x, y) \in p_i$ — that is, voter $i$ prefers $x$ to $y$. Using the order $\sigma$ such that $\sigma_i = x$ for every $i \in N$, the social planner can make sure that $x$ gets $n$ approval points while all other alternatives (including $y$) get fewer approval points, and thus $x$ is the unique SAV winner. Alternatively, using $\pi$ such that
$\pi_i^1 = y$ for every $i \in N$, the social planner can ensure that $y$ gets $n$ approval points and is thus an SAV winner. So $\boldsymbol{p}$ is not anchor-proof. $\square$

**Theorem 4.** SAV is anchor-proof for a profile $\boldsymbol{p}$ if and only if:

for all $x \in \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$ it holds that $\operatorname{plur}_{\boldsymbol{p}}(x) \ge \operatorname{acc}_{\boldsymbol{p}}(y)$ for all $y \in X$,
where the equality only holds exactly when $y \in \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$.

*Proof.* Let $F_a$ be SAV.

(only if) Suppose that there is an alternative $y \notin \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$ such that for some $x \in \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$ it is the case that $\operatorname{plur}_{\boldsymbol{p}}(x) \le \operatorname{acc}_{\boldsymbol{p}}(y)$. We know that $F_a(\mathbf{A}_{\boldsymbol{p},\bar{\sigma}}) = \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$. By Remark 1, there exists an order $\boldsymbol{\mu}$ such that for every voter $i$ who finds $y$ acceptable, it holds that $A_{p_i, \mu_i} = p_i^1) \cup \{y\}$; and $A_{p_k, \mu_k} = p_k^1$ otherwise. Therefore $y \in F_a(\mathbf{A}_{\boldsymbol{p},\boldsymbol{\mu}}) \ne \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$, which means that $\boldsymbol{p}$ is not anchor-proof. If $y \in \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$, the proof is similar.

(if) For any presentation order, each alternative $x \in \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$ gets an amount of approval points equal to its plurality points (recall that every alternative at the top of a preference ranking will be approved by that voter, independently of the presentation order). Every other alternative gets an amount of approval points at most equal to its acceptability score. Thus for any order $\boldsymbol{\sigma}$ we have that $F_a(\mathbf{A}_{\boldsymbol{p},\boldsymbol{\sigma}}) = \operatorname{argmax}_{x \in X} \operatorname{plur}_{\boldsymbol{p}}(x)$. $\square$

Theorem 5 answers the same question for *Nom*.

**Theorem 5.** *Nom* is anchor-proof for a profile $\boldsymbol{p}$ if and only if $\mathrm{PLUR}(\boldsymbol{p}) = \mathrm{ACC}(\boldsymbol{p})$.

*Proof.* Let $F_a$ be Nom.

(only if) Suppose that $\mathrm{PLUR}(\boldsymbol{p}) \ne \mathrm{ACC}(\boldsymbol{p})$. Then, there exist an alternative $x \in X$ and a voter $i \in N$ such that $x \in \mathrm{ACC}(p_i)$ and $x \ne p_i^1$. Consider the order $\bar{\sigma}$ and the different order $\pi$ such that $\pi_j^1 = p_j^1$ for all $j \in N \setminus \{i\}$ and $\pi_i^1 = x$ and $\pi_i^2 = p_i^1$. Then, by definition of the nomination rule we have that $F_a(\mathbf{A}_{\boldsymbol{p},\bar{\sigma}}) = \mathrm{PLUR}(\boldsymbol{p}) \ne \mathrm{PLUR}(\boldsymbol{p}) \cup \{x\} = F_a(\mathbf{A}_{\boldsymbol{p},\pi})$.

(if) Suppose that $\mathrm{PLUR}(\boldsymbol{p}) = \mathrm{ACC}(\boldsymbol{p})$. For every $x \in \mathrm{PLUR}(\boldsymbol{p})$ and order $\boldsymbol{\sigma}$ we know that $x \in A_{p_i, \sigma_i}$ for some voter $i$. Then, by definition of the nomination rule we have that $\mathrm{PLUR}(\boldsymbol{p}) \subseteq F_a(\mathbf{A}_{\boldsymbol{p},\sigma}) \subseteq \mathrm{ACC}(\boldsymbol{p}) = \mathrm{PLUR}(\boldsymbol{p})$ for every order $\boldsymbol{\sigma}$, which means that $F_a(\mathbf{A}_{\boldsymbol{p},\sigma}) = \mathrm{PLUR}(\boldsymbol{p})$ for every order $\boldsymbol{\sigma}$. $\square$

Theorems 4 and 5 characterize the anchor-proof profiles for two particular rules, namely SAV and Nom. Next, Theorem 6 characterizes the profiles that are anchor-proof for all weakly unanimous rules.<fnref n="3" /> Recall that an alternative $x \in X$ is called *unanimously accepted* in a profile $\boldsymbol{p} = (p_1, \dots, p_n)$ if $x \in \mathrm{ACC}(p_i)$ for every $i \in N$.

<footnote n="3">³We may attempt to characterize the profiles for which an arbitrary weakly unanimous rule is anchor-proof. But since any rule can be transformed into a weakly unanimous rule by adding the condition to select the set of unanimously approved alternatives when this set is non-empty, anchor-proofness is strongly rule-dependent.</footnote>
**Theorem 6.** It holds that every weakly unanimous voting rule is anchor-proof for a profile **p** if and only if **p** is intolerant or it has a unique unanimously accepted alternative that is ranked first by every voter.

*Proof.* Given an approval profile $\mathbf{A} = (A_1, \dots, A_n)$, let us write $U(\mathbf{A}) = \{x \in X \mid x \in A_i \text{ for every } i \in N\}$ for the set of unanimously approved alternatives. We now proceed to the proof.

(only if) Let **p** be a profile for which every weakly unanimous rule is anchor-
proof, and suppose for contradiction that **p** is not intolerant and it does not
have a unique unanimously accepted alternative that is ranked first by every
voter. We will consider three cases, and on each of these three cases we will
present a weakly unanimous voting rule $F_a$ that is not anchor-proof for **p**.

**Case 1:** **p** is a profile with more than one unanimously accepted alternative.

By Proposition 3, we know that SAV is not anchor-proof for **p**.

**Case 2:** **p** is a profile with a unique unanimously accepted alternative *x*, but *x* is not ranked first by every voter.

We define the weakly unanimous rule $F_a$ such that $F_a(\mathbf{A}) = U(\mathbf{A})$ if $U(\mathbf{A}) \neq \emptyset$, and $F_a(\mathbf{A}) = X$, otherwise. Let $\boldsymbol{\sigma}$ be an order such that $\sigma_i^1 = x$ for all $i \in N$. Then we have that $F_a(\mathbf{A}_{p,\sigma}) = \{x\} \neq X = F_a(\mathbf{A}_{p,\sigma})$.

**Case 3:** **p** is a profile without a unanimously accepted alternative.

We define the weakly unanimous rule $F_a$ such that $F_a(\mathbf{A}) = U(\mathbf{A})$ if $U(\mathbf{A}) \neq \emptyset$, and otherwise $F_a(\mathbf{A})$ selects the largest approval ballot in $\mathbf{A}$ (in case of a tie, it selects the approval ballot of the voter with the minimal index). Then for the profile $\mathbf{p}$ we know that $U(\mathbf{A}_{p,\sigma}) = \emptyset$ for all orders $\sigma$. So we have that $|F_a(\mathbf{A}_{p,\bar{\sigma}})| = 1$ (because under $\bar{\sigma}$ all approval ballots are of size 1) and $|F_a(\mathbf{A}_{p,\sigma})| > 1$ (because the profile is not intolerant, so under $\sigma$ there exists at least one approval ballot of size larger than 1).

(if) Let $\mathbf{p}$ be an intolerant profile. Then by Remark 2, we have that every approval-based rule is anchor-proof for $\mathbf{p}$

Now let $\mathbf{p}$ be a profile with a unique unanimously accepted alternative $x$ that is ranked first by every voter. Then, for all $i \in N$ and all $\sigma_i \in S(X)$, it is the case that $x \in A_{p_i, \sigma_i}$. Thus we have that $U = \{x\}$, and by weak unanimity it holds that $F_a(\mathbf{A}_{p,\sigma}) = \{x\}$ for all orders $\sigma$, which means that $F_a$ is anchor-proof for $\mathbf{p}$. $\square$

It should be noted that when adding total unanimity to the requirements of
Theorem 6, the result remains the same. This is because total unanimity only
plays a role in the particular profile where every approval ballot is exactly $X$.
The profiles employed in the characterisation already omit this profile.

## 4.2 Anchor-proofness under Partial Information

So far, we have been able to find a small number of profiles for which particular
rules are anchor-proof, suggesting that voters wield only limited influence. Cru-
cially however, these results rest on a strong assumption about the information
available to the planner: anchor-proofness fails precisely because the planner
might be able to select a specific presentation order relying on the intrinsic preferences of the voters.

Next, we turn to settings where the planner might not have full information about voters' preferences. Using the model of Reijngoud and Endriss (2012), we define a notion called 'information function' that maps each preference profile $p$ to a type of information. Specific such functions info that we will consider are the following:

* **Zero:** $info_0(p)$ returns a constant value, independently of $p$; that is, it provides no useful information.

* **Acceptability points:** $info_{acc}(p)$ returns, for each alternative, the number of voters that consider it acceptable in $p$.

* **Acceptability sets:** $info_{\{acc\}}(p)$ returns, for each alternative, the set of voters that consider it acceptable in $p$.

* **Plurality points:** $info_{pl}(p)$ returns, for each alternative, the number of voters that rank it on top in $p$.

* **Plurality sets:** $info_{\{pl\}}(p)$ returns, for each alternative, the set of voters that rank it on top in $p$.

* **Full:** $info_1(p)$ returns precisely the profile $p$.

Each information function gives rise to a set of profiles that the social planner considers possible, consistently with the provided information. Formally, the set of profiles that are considered possible given a certain profile $p \in P(X)^N$ is defined as follows:

$$W_{info}(p) = \{p' \in P(X)^N | info(p) = info(p')\}$$

For example, $W_{info_0}(p) = P(X)^N$ and $W_{info_1}(p) = \{p\}$ for any profile $p$. Next, Definition 7 formalizes the comparison between information functions in terms of the degree of information they provide.

**Definition 7.** Let info and info' be two information functions. We call info at least as informative as info' if for every profile $p$ it holds that $W_{info}(p) \subseteq W_{info'}(p)$.

Considering the information functions that we presented, $info_0$ (respectively, $info_1$) is the least (respectively, the most) informative of all; $info_{acc}$ is less informative than $info_{\{acc\}}$; and $info_{pl}$ is less informative than $info_{\{pl\}}$. Note that $info_{pl}$ may at first appear at least as informative as $info_{acc}$, but technically this is not true since $info_{pl}$ restricts the alternatives that can be on the top of the profile, while $info_{acc}$ restricts the acceptance thresholds of the voters.

Now, we assume that the social planner has a preference $>_{sp}$ over all the possible outcomes of the aggregation process (that is, $>_{sp} \in L(2^X)$). Thus, she might have the incentive to show the alternatives in a specific order (which we call a *strategy*) to obtain a more preferred outcome.
We ask whether, under partial information, the social planner can manip-
ulate the voting outcome to her advantage. Given a set of possible profiles
$W \subseteq P(X)^N$ consistent with the information of the social planner and a voting
rule, we call a strategy *optimal* for the social planner if (i) for every possible
profile in $W$ it induces an outcome that is at least as good as the outcome of ev-
ery other strategy, and (ii) in some possible profile in $W$ it induces an outcome
that is strictly better than the outcome of some other strategy.<fnref n="4" /> If the social
planner has an optimal strategy given her information, then she can manipulate
the voting rule. The following definition formalizes this idea.

**Definition 8.** Let *info* be an information function and $F_a$ be a voting rule. We say that $F_a$ is **manipulable under info on a profile p by a social planner with preference $\succ_{sp}$** if there exists a strategy $\sigma^* \in S(X)$ for which the following two conditions hold:

* for every profile $p' \in W_{info}(p)$, and for every order $\sigma \neq \sigma^*$, it is the case that $F_a(\mathbf{A}_{p',\sigma^*}) \succeq_{sp} F_a(\mathbf{A}_{p',\sigma})$;

* for some profile $p' \in W_{info}(p)$, and for some order $\sigma \neq \sigma^*$, it is the case that $F_a(\mathbf{A}_{p',\sigma^*}) \succ_{sp} F_a(\mathbf{A}_{p',\sigma})$.

We next define the general notion of manipulability for a voting rule $F_a$.

**Definition 9.** Let *info* be an information function and $F_a$ be a voting rule. We say that $F_a$ is **manipulable under info** if it is manipulable on some profile **p** by a social planner with some preference $\succ_{sp} \in L(2^X)$.

Definition 9 is a direct analog of the definition of manipulability in voting under partial information by Reijingoud and Endriss (2012), which has been adopted in numerous other works since its original conception (see, for example, Endriss et al., 2016; Veselova, 2020; Gori, 2021). It is a rather demanding notion, according to which a strategy will only be used if it is ‘safe’, meaning that the planner will never risk obtaining a worse outcome in some possible scenario. The strength of this condition increases the relevance of the manipulability results that we will soon present. We start with an intuitive observation: an increase on the degree of the planner’s information benefits her ability to manipulate.

**Lemma 3.** Take an information function *info* that is at least as informative as an information function *info'*. Then, every voting rule $F_a$ that is manipulable under *info'* is also manipulable under *info*.

*Proof.* Since $F_a$ is manipulable under *info'* for a planner's preference $\succ_{sp}$, an optimal order $\sigma^*$, and a profile $p$, we know the following from the second condition of manipulability:

$$F_a(\mathbf{A}_{p',\sigma^*}) \succ_{sp} F_a(\mathbf{A}_{p',\sigma}) \text{ for some order } \sigma \neq \sigma^* \text{ and profile } p' \in W_{info'}(p)$$

<footnote n="4">⁴Case (ii) is important to avoid the scenario where all strategies give the same outcome in all possible profiles, meaning that none of them can be considered optimal.</footnote>
We will show that $F_a$ is manipulable under info for the same planner's preference $>_{sp}$ and optimal order $\sigma^*$, and for the profile $p'$. Note that $p' \in W_{info}(p')$ by definition. So the second condition of manipulability is also satisfied for info because of the following:

$$F_a(\mathbf{A}_{p',\sigma^*}) >_{sp} F_a(\mathbf{A}_{p',\sigma}) \text{ for some order } \sigma \neq \sigma^* \text{ and profile } p' \in W_{info}(p')$$

From the first condition of manipulability of info', the following holds:

$$F_a(\mathbf{A}_{p'',\sigma^*}) \succeq_{sp} F_a(\mathbf{A}_{p'',\sigma}) \text{ for all orders } \sigma \neq \sigma^* \text{ and profiles } p'' \in W_{info'}(p)$$

In addition, since info is at least as informative as info', we know that $W_{info}(p) \subseteq W_{info'}(p)$ for all profiles $p$. Specifically, it holds that $W_{info}(p') \subseteq W_{info'}(p')$. Moreover, the fact that $p' \in W_{info'}(p)$ implies that $W_{info'}(p) = W_{info'}(p')$ by definition. So the first condition of manipulability is also satisfied for info because of the following:

$$F_a(\mathbf{A}_{p'',\sigma^*}) \succeq_{sp} F_a(\mathbf{A}_{p'',\sigma}) \text{ for all orders } \sigma \neq \sigma^* \text{ and profiles } p'' \in W_{info}(p')$$

Our proof is concluded. □

<figure><img src="image_4.png" /></figure>

Theorem 7 establishes a strong positive result: under complete lack of information, any voting rule satisfying weak and total unanimity is not manipulable, irrespective of the social planner's preferences.

**Theorem 7.** Let $F_a$ be a voting rule that satisfies weak and total unanimity. Then, $F_a$ is not manipulable under $info_0$.

*Proof.* Aiming for a contradiction, suppose that $\sigma^*$ is an optimal strategy for a social planner with preference $>_{sp}$, who has zero information. Now consider the tolerant profile $p$ such that $A_{p_i, \sigma_i^*} = X$ for all $i \in N$ (this can be done by Remark 1). By total unanimity we have that $F_a(\mathbf{A}_{p, \sigma^*}) = X$. Fix a voter $j$, and let $\sigma \neq \sigma^*$ be an order such that $\sigma_i = \sigma_i^*$ for all $i \in N \setminus \{j\}$, where $\sigma_j$ and $\sigma_j^*$ differ only in the last two showed alternatives. Thus, we have that $A_{p_i, \sigma_i} = X$ for all $i \in N \setminus \{j\}$, and $A_{p_j, \sigma_j} = X \setminus \{x\}$ for some alternative $x$. By weak unanimity, we have that $F_a(\mathbf{A}_{p, \sigma}) = A \subseteq X \setminus \{x\}$. By optimality of $\sigma^*$, it is the case that $X >_{sp} A$.

Since $\sigma_i^* \in A_{p_i, \sigma_i}$ for all $i \in N$ and $m > 2$, it is possible, by Remark 1, to find a profile $p'$ such that $\mathbf{A}_{p', \sigma^*} = \mathbf{A}_{p, \sigma}$. Thus, $F_a(\mathbf{A}_{p', \sigma^*}) = A$. Again, by Remark 1, we can find an order $\sigma'$ such that $A_{p'_i, \sigma'_i} = X$ for all $i \in N$, and by total unanimity, $F_a(\mathbf{A}_{p', \sigma'}) = X$. By optimality of $\sigma^*$, we have that $A >_{sp} X$. Thus, we reach a contradiction. □

<figure><img src="image_5.png" /></figure>

Since SAV satisfies weak and total unanimity, the following holds:

**Corollary 1.** SAV is not manipulable under $info_0$.

The next two results show that manipulability of reasonable voting rules is recovered if the planner holds partial information about the voters' preferences,
such as about the relevant acceptability or plurality points of the alternatives. Theorem 8 specifically concerns all weakly unanimous rules that are also anonymous and neutral. Note that if we do not consider anonymity and neutrality, we need to include some ad-hoc condition ensuring that the rule produces at least two different outcomes within some interesting sets of possible profiles.

**Theorem 8.** Let $F_a$ be a voting rule that satisfies weak unanimity, anonymity, and neutrality. Then, $F_a$ is manipulable under $info_{acc}$.

*Proof.* Consider a profile $p$ where a unique alternative $a$ has $n$ acceptability points and all other alternatives have $n-1$ acceptability points. Take a social planner with preference $\succ_{sp}$ that ranks $\{a\}$ first. Then, the order $\sigma$ such that $\sigma_i^1 = a$ for all $i \in N$ is an optimal strategy for the social planner. Indeed, the first condition of manipulability is satisfied because $F_a(\mathbf{A}_{p,\sigma}) = \{a\} \supseteq_{sp} X'$, for every $X' \subseteq X$ that can be the outcome of the that profile under another order, as $a$ is selected by weak unanimity. For the second condition of manipulability, we need to find a profile with the same acceptability points as $p$ for all alternatives, in which some order produces an outcome different from $\{a\}$. For this purpose, consider a profile $p'$ together with an order $\sigma'$ such that each alternative is excluded by exactly one approval ballot. Then, anonymity and neutrality imply that $F_a(\mathbf{A}_{p',\sigma'}) = X$, and we are done. $\square$

**Theorem 9.** Let $F_a$ be a voting rule that satisfies weak and total unanimity. Then, $F_a$ is manipulable under $info_{pl}$.

*Proof.* Consider a profile $p$ where a unique alternative $a$ has $n$ plurality points. Take a social planner with preference $\succ_{sp}$ that ranks $\{a\}$ first. Then, the order $\sigma$ such that $\sigma_i^1 = a$ for all $i \in N$ is an optimal strategy for the social planner. Indeed, the first condition of manipulability is satisfied because $F_a(\mathbf{A}_{p,\sigma}) = \{a\} \supseteq_{sp} X'$, for every $X' \subseteq X$ that can be the outcome of that profile under another order, as $a$ is selected by weak unanimity. For the second condition of manipulability, we need to find a profile with the same plurality points as $p$ for all alternatives, in which some order produces an outcome different from $\{a\}$. For this purpose, consider a tolerant profile $p'$ where all voters rank $a$ first and consider the order $\sigma$ such that $b$ is shown first and $a$ second for every voter. Then, by total unanimity, $F_a(\mathbf{A}_{p',\sigma'}) = X$, and we are done. $\square$

Corollary 2 holds because SAV satisfies the conditions of Theorems 8 and 9, and Lemma 3 implies that manipulability pursues as long as the information of the planner increases.

**Corollary 2.** SAV is manipulable under $info_{acc}$ and $info_{pl}$ (as well as under $info_{\{acc\}}$ and $info_{\{pl\}}$).

We continue with exploring the degree of information that deems the nomination rule manipulable. First, we show that when no information is available, Nom cannot be manipulated.

**Theorem 10.** Nom is not manipulable under $info_0$.
**Proof.** Aiming for a contradiction, suppose that $\sigma^*$ is an optimal strategy for a social planner with preference $\succ_{sp}$, who has zero information. Wlog, suppose that $\sigma_i^* = (x, y, z, \dots)$. Consider the profile $p$ such that all the voters but $i$ accept only alternative $x$, while voter $i$ ranks $z$ first, $y$ second, and $x$ third; and accepts only those three alternatives. Then $\text{Nom}(A_{p,\sigma^*}) = \{x, y, z\}$, and by optimality, $\{x, y\} \succeq_{sp} X'$ for every $X' \subseteq X$. In particular, for $X' = \{x, z\}$, that is the outcome obtained when $\sigma_i = (x, z, \dots)$ is used. Now consider $p'$ such that all the voters but $i$ accept only alternative $x$, while voter $i$ ranks $z$ first, $x$ second, and $y$ third, and accepts only those three alternatives. Then $\text{Nom}(A_{p,\sigma^*}) = \{x, y\}$, and by optimality, $\{x, z\} \succeq_{sp} \{x, y, z\}$, the outcome obtained when $\sigma_i = (y, x, z, \dots)$ is used. Thus, we reach a contradiction, showing that $\text{Nom}$ is not manipulable under $\text{info}_0$.

□

**Theorem 11.** $\text{Nom}$ is manipulable under $\text{info}_{acc}$ (as well as under $\text{info}_{\{acc\}}$).

**Proof.** Consider a profile $p$ where alternative $a$ has $n$ acceptability points, alternative $b$ has 1 acceptability point, and no other alternative has acceptability points. Take a social planner with preference $\succ_{sp}$ that ranks $\{a, b\}$ first. Then, the order $\sigma$ such that $\sigma_i = (b, a, \dots)$ for all $i \in N$ is an optimal strategy for the social planner. Indeed, the first condition of manipulability is satisfied because $\text{Nom}(A_{p,\sigma}) = \{a, b\} \succeq_{sp} X'$, for every $X' \subseteq X$ that can be the outcome of that profile under another order. For the second condition of manipulability, we need to find a profile with the same acceptability points as $p$ for all alternatives, in which some order produces an outcome different from $\{a, b\}$. For this purpose, consider a profile $p'$ where $a$ is ranked first by every voter together with an order $\bar{\sigma}$. Then $\text{Nom}(A_{p',\bar{\sigma}}) = \{a\}$, and we are done.

□

**Theorem 12.** $\text{Nom}$ is manipulable under $\text{info}_{pl}$ (as well as under $\text{info}_{\{pl\}}$).

**Proof.** The proof is similar to the proof of Theorem 9, without needing to use weak and total unanimity to justify the outcomes.

□

<table><caption>Table 3 summarizes the results of this section.</caption><thead><tr><td>Planner's Info</td><td>Manipulation Possible?</td></tr></thead><tbody><tr><td>info<sub>1</sub></td><td>Yes, unless anchor-proof profile (Thms 4, 5, 6)</td></tr><tr><td>info<sub>0</sub></td><td>No (for W.unan. &amp; T.unan., Thm 7; for Nom, Thm 10)</td></tr><tr><td>info<sub>acc</sub></td><td>Yes (for W.unan., Anon., Neut., Thm 8; for Nom, Thm 11)</td></tr><tr><td>info<sub>pl</sub></td><td>Yes (for W.unan. &amp; T.unan., Thm 9; for Nom, Thm 12)</td></tr></tbody></table>

<caption>Table 3: Manipulability across informational assumptions.</caption>

Note that except for the case of zero information, all information functions described in this section allow us to distinguish the status of the different alternatives in a profile (which is sensible in several contexts of collective decisions,
including polls of political elections and preliminary results of participatory budgeting contests). But in other settings, we can imagine that the provided information does not allow the planner to separate the different alternatives. The reader might suspect that for reasonable rules such as SAV, manipulability is then avoidable, since the planner has to fix a presentation order of the alternatives in her strategy without being able to exclude any permutation of the alternatives in the profiles she considers possible. However, this conjecture does not hold in general as Example 7 shows.

**Example 7.** Consider a specific information function called $info_X$ such that $W_{info_X}(p) = \{\mu(p) | \mu \in S(X)\}$ for every profile $p$. Conceptually, under this information function all names of the alternatives are indistinguishable for the planner, but the abstract structure of the profile (i.e., the specific positions in which each given (un-named) alternative appears) is known.

* **Alternative-structure:** $info_X(p)$ returns the acceptability thresholds of all voters' preference-approvals in $p$ and the family $\{(i, k) \in N \times \{1, \dots, m\} | p_i^k = x\}_{x \in X}$.

Let $X = \{a, b, c\}$ and $n = 4$. We will show that SAV is manipulable under $info_X$. Suppose that the preference of the social planner is lexicographic:

$$ \{a\} \succ_{sp} \{a, b\} \succ_{sp} \{a, c\} \succ_{sp} \{a, b, c\} \succ_{sp} \{b\} \succ_{sp} \{b, c\} \succ_{sp} \{c\} $$

Consider a profile $p$ that induces a class of possible profiles under $info_X$ for which the following conditions hold:

* the preference-approval of voter 1 has acceptability threshold 2, while those of voters 2, 3, and 4 have acceptability threshold 1.

* voters 1 and 2 rank the same alternative first, and voters 3 and 4 rank the same alternative first.

* the alternative that voter 1 ranks second is the same as the alternative that voters 3 and 4 rank first.

Then, with an exhaustive case-analysis over all six possible profiles under relating of the three alternatives, we can see that a strategy $\sigma^*$ of a social planner with $\sigma_1^* = (a, b, c)$ is optimal. $\Delta$

We wrap up this section by highlighting an open question: is SAV manipulable when the information function reveals precisely the acceptability threshold of every voter (without further details about the profile structure)?

# 5 Conclusion

Our analysis of anchor-proof voting rules demonstrates how axiomatic characterizations in social choice can be extended to accommodate systematic deviations
from rational preference formation. We have started by considering voters with
intrinsic preference rankings who are subject to an anchoring-bias, influencing
the formation of their approval ballots via the presentation order of the alterna-
tives. Our central characterization result shows that no approval-based voting
rule (except constant ones) is robust to the impact of such an anchoring-bias.
On the positive side, we also find that a social planner who has no information
about the voters' underlying preferences cannot choose the presentation order
to safely steer the voting outcome to her advantage.

A natural extension of our framework is to consider ballots with richer struc-
tures, particularly ranked ballots. In such settings, the derived ballots under an
anchoring bias would typically correspond to incomplete preference orders over
the set of alternatives. A preliminary conjecture is that every tops-only rule
— that only takes into account the most preferred alternatives of each voter to
determine the outcome — is anchor-proof, since the most preferred alternative
always appears in the corresponding derived ballot regardless of the presenta-
tion order. This conjecture naturally connects the present work to the broader
literature on collective decision-making under incomplete preferences, where the
interplay between informational incompleteness and behavioral robustness may
yield novel characterization results.

A further direction involves exploring alternative mechanisms for deriving
anchor-biased ballots. One possibility is to consider trichotomous ballots, where
alternatives are classified as approved, disapproved, or neutral. Under this
formulation, an alternative that is accepted by a voter but appears after a more
preferred one could be treated as neutral rather than disapproved, offering a
more fine-grained representation of cognitive constraints.

Beyond anchoring, developing a systematic taxonomy of cognitive biases
in voting could give rise to novel characterizations across different rules, anal-
ogously to classical results like Arrow's theorem but incorporating bounded
rationality. In general, different biases may require distinct formal representa-
tions of how reported ballots deviate from underlying preferences, suggesting a
rich space of models to explore. Framing effects, loss aversion, and social confor-
mity biases represent particularly promising candidates, as they admit tractable
mathematical representations while exhibiting substantial empirical prevalence.

The negative results established in our paper motivate yet another research
direction: designing voting rules with quantitatively optimal *degrees* of immu-
nity to specific biases. From a computational social choice perspective, un-
derstanding which biases can be efficiently detected or corrected through algo-
rithmic pre-processing of preference data, versus which require fundamentally
different aggregation mechanisms, has important practical implications for both
human and multi-agent collective-decision contexts.

Finally, some negative results might be circumvented through randomization
mechanisms, for instance where alternatives are displayed in random orders with
all orderings equally likely. However, when certain alternatives are objectively
superior or inferior (as may be determined through quality assessments or ex-
pert evaluations) one might assign greater probability to orders that position
such alternatives earlier or later respectively. The formal development of such
probabilistic models, potentially incorporating techniques from online learning and mechanism design, may yield further insights into the existence and characterization of bias-robust social choice rules.

All these directions suggest that behavioral social choice, incorporating bounded rationality as a fundamental consideration rather than implementation noise, can give rise to both theoretically richer frameworks and more reliable mechanisms for collective decision-making. Bias-robust rules extend naturally beyond voting, to participatory budgeting, committee selection, peer review, recommendation systems, and multi-agent resource allocation, where systematic cognitive biases similarly threaten the normative properties of aggregation procedures.

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