Parameter Consistency Veto

• Parameter consistency veto is a framework in social choice and learning-augmented algorithms that uses (p, q)-veto rules to select candidates based on ordinal inputs.
• It establishes a trade-off between consistency, when predictions are accurate, and robustness in worst-case scenarios, achieving optimal distortion bounds.
• The BoostedSV rule incorporates voter veto weights and a prediction boost parameter to finely balance social cost minimization with risk control.

The parameter consistency veto is a concept in social choice and learning-augmented algorithm design, central to the study of metric distortion for ordinal voting. The framework considers a collection of voters and candidates situated in a metric space with unknown inter-point distances. The challenge is to select a candidate that minimizes the sum of voter-to-candidate distances (“social cost”) using only ordinal (ranked) input. The (p, q)-veto core rule generalizes the “Simultaneous Veto” mechanism by assigning normalized weight vectors $p$ to the voters (veto power) and $q$ to the candidates (support). In the learning-augmented context, the parameter consistency veto designates the trade-off frontier between robustness (worst-case distortion given arbitrary predictions) and consistency (distortion under perfect prediction), instantiated through a parameterized algorithm that leverages a machine-learned prediction via the q distribution (Berger et al., 2023).

1. Metric Distortion and Ordinal Social Choice

The metric distortion problem formalizes the use of ordinal (ranked) ballots for candidate selection within a metric space $(X,d)$, where only the orderings but not the numeric values $d(v, c)$ are accessible. Formally, the social cost of candidate $c$ is:

$SC(c; d) = \sum_{v \in V} d(v, c) .$

An algorithm receives rankings $\{\sigma_v\}_{v\in V}$, where each $\sigma_v$ orders $C$ in a manner consistent with their metric distances. The distortion metric is defined as the worst-case ratio (over all consistent metrics and ranking profiles) between the social cost of the output and the optimal cost:

$$\operatorname{dist}(\mathrm{ALG}) = \sup_\sigma\sup_{d:\ d\trianglelefteq \sigma} \frac{SC(\mathrm{ALG}(\sigma); d)}{SC(c^*(d); d)} .$$

This abstraction captures the loss in social welfare due to the restriction to ordinal information.

2. The (p, q)-Veto Core Rule

The (p, q)-veto core is a class of deterministic social choice mechanisms parameterized by weight distributions $q$0 and $q$1. The process operates as follows:

• Each candidate $q$2 receives an initial score proportional to $q$3.
• Each voter $q$4 “vetoes” (down-votes) her current least-preferred candidate at a rate $q$5, reducing the candidate’s score continuously.
• Candidates whose scores reach zero last form the veto core.

A practical instantiation is the Boosted Simultaneous Veto (BoostedSV) rule where $q$6 is uniform and $q$7 is guided by a prediction, typically the output of a learning algorithm. In the absence of predictions or with uniform $q$8, the rule reduces to the deterministic Simultaneous Veto and achieves a known worst-case distortion bound of 3 (Berger et al., 2023).

3. Learning-Augmented Framework: Consistency and Robustness

The learning-augmented setting introduces an external prediction $q$9 believed (with unknown accuracy) to be the optimal candidate. The algorithm can boost the initial score $(X,d)$0 of this predicted candidate by an adjustable parameter $(X,d)$1, calibrated via a “confidence” parameter $(X,d)$2. The trade-off between robustness and consistency is then shaped by the choice of $(X,d)$3:

• Consistency quantifies distortion in the case where the prediction is correct ($(X,d)$4).
• Robustness quantifies distortion in the worst case, over all possible (possibly adversarial) predictions.

A refined error parameter $(X,d)$5 further measures the normalized prediction error. The algorithm aims to achieve low distortion both when the prediction is accurate and when it is unreliable, interpolating between the classical voting rule (unboosted Simultaneous Veto) and the fully prediction-trusting approach.

4. Algorithmic Details and Distortion Bounds

The concrete BoostedSV rule applied in (Berger et al., 2023) operates as follows:

1. For $(X,d)$6 voters and $(X,d)$7 candidates, let $(X,d)$8 denote the number of voters ranking $(X,d)$9 first.
2. Scores are initialized as $d(v, c)$0 for $d(v, c)$1, and $d(v, c)$2 where $d(v, c)$3.
3. Down-voting is executed at total rate $d(v, c)$4 until all scores reach zero.
4. The set of last remaining candidates forms the output; if $d(v, c)$5 is among them, $d(v, c)$6 is selected.

Key results are as follows:

Parameter	Formula	Interpretation
Consistency	$d(v, c)$7	Distortion if $d(v, c)$8
Robustness	$d(v, c)$9	Worst-case distortion
Refined error-based bound	$c$0	Interpolates with prediction error

The trade-off curve between consistency and robustness is tight; no deterministic rule can uniformly outperform these ratios for all $c$1. As $c$2, both bounds converge to 3 (Simultaneous Veto); as $c$3, consistency tends to 1 but robustness diverges (Berger et al., 2023).

5. Theoretical Guarantees and Optimality

The main theorems establish that the BoostedSV algorithm realizes the optimal trade-off frontier for deterministic learning-augmented algorithms in this framework. Proofs combine triangle-inequality arguments over the down-voting process with “veto map” pairings to relate the accumulated scores and distances. In particular, the additional score (boost) for the prediction can increase social cost, but the increase is carefully bounded. The result generalizes directly to arbitrary (p, q)-veto core rules but receives its sharpest, most interpretable form when $c$4 is uniform and $c$5 allocates extra mass to the predicted optimum.

6. Parameter Choices and Extensions

Adjusting the boost parameter $c$6 allows fine-grained control of the robustness-consistency trade-off. Any Pareto-optimal distortion pair on the frontier can be achieved by solving for the corresponding $c$7 in the formulas provided. Further, the bound can be improved in special metric geometries; for example, in the $c$8-decisive case (where each voter’s second choice is at least $c$9 times farther than her first), the 3-distortion is replaced by $SC(c; d) = \sum_{v \in V} d(v, c) .$0. A plausible implication is that structure in the metric space can further tighten parameter consistency veto guarantees.

7. Significance and Open Problems

The parameter consistency veto and the (p, q)-veto core provide a rigorous framework for incorporating predictions in the design of social choice rules, establishing robust guarantees even under adversarial information. The full characterization of distortion for arbitrary $SC(c; d) = \sum_{v \in V} d(v, c) .$1 remains open, suggesting avenues for further generalization and potential for leveraging more complex or distributed prediction guidance (Berger et al., 2023). The methodology connects algorithmic mechanism design, voting theory, and machine learning under rigorous worst-case and consistency-based analysis.