Ranking with Intensities Ballot Format

• The ranking with intensities ballot format is a voting approach where agents rank alternatives and indicate strength differences using normal (≻) and intense (≻≻) preferences.
• Positional Scoring Matching (PSM) rules leverage this format to achieve a worst-case metric distortion below 3, overcoming limitations of standard ordinal methods.
• Theoretical analysis of distortion bounds and the price of ignoring intensities demonstrates how minimal intensity information enhances social cost optimization in metric aggregation.

A ranking with intensities ballot format is an extension of classical ordinal voting in which each agent ranks alternatives and explicitly annotates the strength of their preferences between adjacent alternatives. This is realized by using two types of comparisons: a normal preference (≻) and an intensive preference (≻≻), allowing voters to indicate not only order but also "gaps" in their rankings. This augmentation provides additional expressiveness, which, under certain frameworks, can significantly improve the worst-case performance of voting rules, especially in metric social choice and similar aggregation settings (Abbaszadeh et al., 5 Jan 2026).

1. Formal Specification

Let $A$ denote a set of $m$ alternatives and $N$ a set of $n$ agents. A ranking-with-intensities ballot for agent $i$ is a pair $(\pi_i, \iota_i)$:

• $\pi_i : [m] \rightarrow A$ is a bijection encoding a strict total order $$\pi_i(1) \succ \pi_i(2) \succ \dotsb \succ \pi_i(m)$$.
• $$\iota_i : [m-1] \rightarrow \{ \succ, \succ\!\!\succ \}$$ assigns either a normal or intensive preference mark between each adjacent pair in their ranking.

A collective profile is $$\mathcal{P} = ((\pi_1,\iota_1), \dotsc, (\pi_n,\iota_n)) \in S(A)^n$$, where $S(A)$ is the collection of all such annotated rankings over $A$.

Intensity gaps are parameterized by $\alpha \in [0,1]$, intended to model gaps in the underlying (unknown) agent-to-alternative cost metric $d$. Specifically, if an agent $i$ reports $\pi_i(j) \succ\!\!\succ \pi_i(j+1)$, the intended constraint is $$d(i,\, \pi_i(j)) \leq \alpha \cdot d(i,\, \pi_i(j+1))$$; for $\succ$, no such constraint is imposed (Abbaszadeh et al., 5 Jan 2026).

2. Incorporation into the Metric-Distortion Framework

The metric social choice paradigm assumes a hidden metric $d$ over $N \cup A$, with $d$ satisfying metric axioms. Each agent $i$ faces a cost $d(i,a)$ for alternative $a$, and the aggregate cost (social cost) of selecting $a$ is $SC_d(a) = \sum_{i} d(i,a)$. The goal is to choose an $a$ minimizing $SC_d(a)$ despite not knowing $d$.

A profile $(\pi,\iota)$ is $\alpha$-consistent with $d$ if:

• For mandatory elicitation: for all $i$ and $j$,
  • If $\iota_i(j) = \succ\!\!\succ$, then $d(i,\pi_i(j)) \leq \alpha \cdot d(i,\pi_i(j+1))$,
  • If $\iota_i(j) = \succ$, then $d(i,\pi_i(j)) > \alpha \cdot d(i,\pi_i(j+1))$.

Under voluntary elicitation, only the inequality for $\succ\!\!\succ$ is enforced.

The $\alpha$-distortion of a voting rule $f : S(A)^n \to A$ on an election $E$ is

$$\mathrm{dist}_\alpha(f,E) = \sup_{d : (\pi, \iota) \models d} \frac{SC_d(f(\pi,\iota))}{SC_d(\operatorname{opt}_d)}$$

where the supremum is over all $d$ consistent with the reported intensities, and $\operatorname{opt}_d$ is the social cost minimizer (Abbaszadeh et al., 5 Jan 2026).

3. Positional Scoring Matching Rules

A central contribution is the definition and analysis of the Positional Scoring Matching (PSM) rules.

• Fix a unit-sum scoring vector $s = (s_1,\ldots,s_m)$ with $\sum_j s_j = 1$.
• For a profile $(\pi, \iota)$, assign each agent $i$ a uniform weight $p_i = 1/n$, and let $q(a) = (1/n) \sum_i s_{\pi_i^{-1}(a)}$ for each alternative $a$.
• Construct, for each $a \in A$, a bipartite domination graph $G_{p,q}^E(a)$ with edge $(i, c)$ iff $a \succ_i c$, assigning weights $p_i$ to agents and $q(c)$ to alternatives.
• By the Ranking-Matching Lemma (Gkatzelis–Halpern–Shah 2020), for any profile there exists an alternative $a$ such that $G_{p,q}^E(a)$ admits a fractional perfect matching.
• The rule $PSM_s$ selects any such $a$.

For the family of "moderate-up-to-$k$" profiles, where agents specify their first strong gap at position $k+1$, the $\alpha$-distortion of a PSM rule $PSM_r$ is controlled by a linear optimization over the scoring vector $r$'s first $k+1$ entries, expressible as a zero-sum game and solved explicitly by a recurrence for $(r^k, t_k)$ (Abbaszadeh et al., 5 Jan 2026).

4. Theoretical Guarantees and Bounds

Distortion below 3

A key result is that by using ranking with intensities ballots and appropriate PSM rules, the deterministic worst-case metric distortion is strictly less than $3$ for all finite $m$, breaking a longstanding barrier in the field.

• Lower bound: Any deterministic rule with $\alpha$-consistent intensities must have distortion at least

$$1 + 2 \max\left(\alpha, \frac{1-\alpha^{\lfloor m/2 \rfloor}}{1+\alpha^{\lfloor m/2 \rfloor}} \right)$$

• Upper bound (moderate-up-to-$k$): For each $k$, there exists a PSM member with distortion at most $2 + \max(\alpha, t_k)$, where $t_k$ is computed from the equilibrium of the zero-sum game described above.
• General upper bound (heterogeneous agents): For $\ell_{max}$ the maximum first "intensive" mark over the agents, achievable distortion is $\leq 2+\max(\alpha, t_{\ell_{max}})$.
• If a fraction $\beta$ of agents use a position $>\ell$ for their first strong gap, the resulting distortion increases to at most $D_\ell + \frac{\beta}{1-\beta}(1+D_\ell)$ (Abbaszadeh et al., 5 Jan 2026).

Price of Ignoring Intensities (POII)

The price of ignoring intensities quantifies the possible degradation in distortion when the intensities in ballots are disregarded. For mandatory elicitation:

$$\mathrm{POII}(\alpha) = \frac{3 \left(\alpha^{\lfloor m \rfloor_{\text{even}} + 1} + 2\alpha^{\lfloor m/2 \rfloor} - 1 \right)}{1+\alpha^{\lfloor m/2 \rfloor}}$$

For voluntary elicitation:

$$\mathrm{POII}^v(\alpha) = \frac{3}{2\alpha^{\lfloor m/2 \rfloor}+1}$$

Explicit constructions demonstrate that approximating the true optimum while being intensity-blind can lead to distortions up to these ratios (Abbaszadeh et al., 5 Jan 2026).

5. Example Formats and Comparative View

A "moderate-up-to-$k$" ballot is one in which the agent reports a unique strong (intensive) gap at position $k+1$, expressing that only their top $k+1$ alternatives are separated by a strong preference from the remainder. This format enables further refinement of cardinal-like information in an otherwise ordinal framework.

Alternative formats incorporating grades ("ranking with intensities" via cardinal grades or abstentions, as in (Laraki et al., 2023)), also allow agents to express variable intensities but operate under a different set of axiomatic properties, such as strategy-proofness under single-peaked preferences. The Phantom-Proxy approach, for instance, aggregates grades and proxies, tolerates abstention and ineligibility, and employs lexicographic tie-breaking to yield a strict ranking, but it is not optimized for metric distortion—the focus of the Ballot-with-Intensities approach (Laraki et al., 2023).

6. Connections, Variants, and Special Cases

Connections

• The ranking-with-intensities format strictly generalizes standard ordinal voting and relates to partially cardinal formats.
• The matching-based social choice rules originate from metric distortion theory, with optimality proofs deeply connected to concepts from linear programming duality and fractional matchings.

Special Cases and Insights

• In one-dimensional (line) metric spaces with two alternatives, an explicit distortion bound can be computed: $\max((3-\alpha)/(1+\alpha), 2\alpha+1)$. A simple two-alternative rule $f^{tal_\alpha}$ is conjectured to attain this bound exactly (Abbaszadeh et al., 5 Jan 2026).
• Empirical evaluations show that modest increases in reported intensity granularity (i.e., small $\ell_{max}$) lead to significant improvements in worst-case distortion. The improvement vanishes (returns to $=3$) when the intensity information is omitted or when all preferences are only "normal" (Abbaszadeh et al., 5 Jan 2026).

7. Significance and Implications

The introduction of the ranking-with-intensities format addresses expressiveness limitations of standard rank-based systems, enabling voting rules to exploit minimal intensity information with theoretically significant consequences. In metric aggregation scenarios, the format allows for deterministic rules to breach the classical distortion barrier of $3$, a feat unachievable with standard ordinal input. The price-of-ignorance bounds confirm that this extra information, often only a single "bit" per adjacent pair, yields nontrivial control over social cost guarantees. The formal analysis paves the way for further exploration of minimally enriched ballot formats and their practical potential in both voting and large-scale grading contexts (Abbaszadeh et al., 5 Jan 2026, Laraki et al., 2023).