Anchor-proofness in Voting

1. Problem Statement

The paper addresses a foundational question in economic theory and social choice: How do individual-level cognitive biases interact with collective-choice mechanisms? Specifically, it investigates the impact of anchoring bias on approval voting.

• Context: In many collective decision-making settings (e.g., participatory budgeting, online platforms), alternatives are presented sequentially to voters.
• The Bias: Voters exhibit an anchoring bias where early information serves as a reference point. A voter approves an alternative if and only if it is acceptable and strictly preferred to all previously encountered alternatives.
• The Core Issue: The presentation order (controlled by a "social planner") determines the mapping from a voter's intrinsic preferences to their approval ballot. This raises two critical questions:
  1. Are there voting rules that are anchor-proof (i.e., the set of winners is invariant regardless of the presentation order)?
  2. To what extent can a social planner manipulate the outcome by strategically ordering alternatives, given varying levels of information about voter preferences?

2. Methodology and Model

A. The Voting Model

• Voters & Alternatives: A finite set of voters $N$ and alternatives $X$ ($|X|=m > 2$).
• Intrinsic Preferences: Each voter $i$ has a preference-approval $p_i = (p_i^1, \dots, p_i^m, t_i)$, consisting of:
  • A linear ranking of alternatives.
  • An acceptability threshold $t_i$ (alternatives ranked $1$ to $t_i$ are acceptable).
• Ballot Generation (Anchoring Mechanism):
  • A social planner specifies a presentation order $\sigma_i$ for each voter.
  • The voter iterates through the order. At step $k$, if the current alternative $\sigma_i^k$ is acceptable and strictly preferred to the current "anchor" (the best alternative seen so far), it is added to the ballot.
  • Key Feature: Once an alternative is approved, it stays; if a better alternative is seen later, it becomes the new anchor, potentially causing subsequent acceptable alternatives to be rejected. This can lead to inconsistent ballots (e.g., approving $z$ and rejecting $y$ even if $y \succ z$).

B. Definitions

• Anchor-proofness: A voting rule $F$ is anchor-proof for a profile $p$ if $F(A_{p,\sigma}) = F(A_{p,\pi})$ for all presentation orders $\sigma, \pi$.
• Manipulability: A rule is manipulable if a planner with specific information and preferences can choose an order $\sigma^*$ that yields a strictly better outcome than any other order, without risking a worse outcome in any possible profile consistent with their information.

C. Information Functions

The paper analyzes the planner's ability to manipulate under different information sets:

• Zero Info ($info_0$): No information about preferences.
• Partial Info: Acceptability points, Plurality points, or sets of voters supporting specific alternatives.
• Full Info ($info_1$): Complete knowledge of the preference profile.

3. Key Contributions and Results

I. Impossibility of Anchor-Proofness (General Case)

The authors establish a strong negative result regarding the existence of robust rules:

• Theorem 1 & 2: No non-constant voting rule is anchor-proof for all profiles (even within the restricted domain of "tolerant" profiles where all alternatives are acceptable).
• Implication: If a social planner has full information, they can almost always manipulate the outcome of any reasonable rule by exploiting the presentation order. The only rules that are truly order-invariant are constant rules (always selecting the same winner regardless of input).

II. Existence of Anchor-Proof Profiles

While no rule is universally anchor-proof, specific profiles may be robust:

• Intolerant Profiles: If every voter only accepts their top choice ($t_i=1$), the outcome is invariant to order (Remark 2).
• Tolerant Profiles: For rules satisfying Weak Unanimity and Total Unanimity, no tolerant profile is anchor-proof (Theorem 3).
• Exceptions: Non-neutral rules (e.g., a rule that always selects a specific alternative if unanimously approved) can be anchor-proof for specific profiles.

III. Strategic Manipulation under Partial Information

The paper characterizes when a planner can manipulate the outcome based on their information level:

• Zero Information ($info_0$):
  • Theorem 7: Any rule satisfying Weak and Total Unanimity (including Standard Approval Voting, SAV) is not manipulable if the planner has zero information.
  • Theorem 10: The Nomination rule is also not manipulable under zero information.
  • Significance: Lack of information acts as a shield against manipulation.
• Partial Information:
  • Theorem 8 & 9: If the planner knows Acceptability Points or Plurality Points, manipulation becomes possible for a broad class of rules (including SAV and Nomination).
  • Corollary 2: SAV is manipulable under partial information regarding acceptability or plurality.
• Full Information: Manipulation is possible unless the profile is inherently anchor-proof (e.g., intolerant profiles).

IV. Extension to Ranked Ballots

The authors extend the model to top-truncated ranked ballots (voters submit a ranking of a subset of alternatives).

• Theorem 13: A rank-based voting rule is anchor-proof if and only if it is tops-only (the outcome depends solely on the top-ranked alternative of each voter).
• Trade-off: Rules that use less information (like Plurality) are more robust to anchoring bias. Rules requiring more detailed preference data are more vulnerable to manipulation via ordering.

4. Significance and Implications

1. Behavioral Social Choice: The paper bridges the gap between behavioral economics (anchoring bias) and social choice theory. It demonstrates that standard voting rules, often assumed to be robust, are highly sensitive to the "choice architecture" (presentation order) when voters are boundedly rational.
2. Design of Collective Decision Mechanisms:
   • Robustness: To ensure outcomes reflect true preferences rather than presentation order, mechanisms should either restrict the information available to the agenda-setter (planner) or use "tops-only" rules (like Plurality) in ranked settings.
   • Participatory Budgeting: The findings suggest that in platforms where projects are presented sequentially, the order can reverse election outcomes (as shown in Example 1). To ensure fairness, platforms might need to randomize order or limit the planner's ability to tailor orders based on voter data.
3. Informational Trade-offs: The paper highlights a fundamental tension: Informational richness increases manipulability. Rules that demand more detailed voter input (beyond just the top choice) provide the planner with more leverage to exploit anchoring biases. Conversely, "informationally austere" rules are inherently more robust.
4. Theoretical Contribution: It provides a new axiom (anchor-proofness) and characterizes the landscape of voting rules under cognitive biases, showing that "perfect" robustness is impossible for non-trivial rules in the general approval setting, but achievable in specific restricted domains or with ranked ballots.

Conclusion

The paper concludes that while no approval-based rule is universally immune to anchoring bias, the threat of manipulation can be mitigated by limiting the planner's information or by adopting voting rules that rely on minimal information (tops-only). Future work should explore other cognitive biases (framing, loss aversion) and probabilistic mechanisms (randomized ordering) to further enhance the robustness of collective decision-making.