Distortion — the formal price of a ranked ballot¶

A 301 theory page. Distortion is how mainstream computational social choice formalized the question this library keeps circling: if a ballot records only order and never degree, how much is that costing you? The literature's answer is a ratio with theorems attached — and it cuts both ways. It openly adopts the cardinal premise (the ground truth is voter utility; a ranked ballot is explicitly modeled as lossy compression of it), which sounds like a slam-dunk for scored ballots — until you read the headline theorem: under the spatial assumption, the best purely ordinal rules land within 3× of optimal, and 3 is provably the best possible. Rankings are lossy and the loss is capped at a small constant. Both halves are proved.

Level: 301 · deep dive Builds on the spatial model (201→301) and pairs with VSE — the average-case cousin of this worst-case metric.

Companions: Cardinal utility — what the "utilities" assumed below actually are, and whether they can be added up · What makes a good winner? · Simulate utilities, not ballots — the same "utility is the ground truth" move, in code · Scores vs. ranks — the ballot-level distinction this measures · Does Arrow apply to STAR? — the other place the ordinal/cardinal line is load-bearing · Grading as a rival primitive — the assault on ordinal ballots from the economics side.

What distortion measures¶

Assume — as social choice has since von Neumann and Morgenstern — that voters really do have cardinal utilities: numbers saying how good each candidate would be for them. A candidate's social welfare is the sum of those numbers (equivalently, in the metric setting, their social cost is the sum of distances to all voters, and the optimal candidate minimizes it — the utilitarian, least-total-unhappiness pick that Tideman & Plassmann's center and VSE's 100% mark both point at).

If we could see the utilities, picking a winner would be trivial arithmetic. We can't. A voting rule is handed only the rankings those utilities induce. So:

Distortion of a rule = the worst-case ratio between the welfare of the best possible candidate and the welfare of the candidate the rule actually elects — over every possible electorate and every set of utilities consistent with the ballots the rule saw.

                            welfare of the BEST candidate
distortion(f)  =   max     ────────────────────────────────────
                 over all   welfare of the candidate f ELECTS
                 profiles

Distortion 1 = the rule always finds the welfare-maximizing winner. Distortion 3 = never worse than 3× optimal, even against an adversary placing voters as maliciously as the ballots allow. Distortion "unbounded" = there is no worst case; the rule can be arbitrarily bad.

Read what that construction assumes, because it is the whole ranked-vs-rated argument made formal: the rule is penalized for only seeing rankings. The survey's own framing — the loss is "due to having access to preferences of limited expressiveness, particularly ordinal rankings."

That premise sits at the center of mainstream computational social choice, not at its advocacy fringe — Procaccia & Rosenschein (2006) defined it, and it has fifteen-plus years of AAAI/IJCAI/FOCS results behind it. If you want a citation for "academics increasingly treat utility as the thing elections are trying to find, and rankings as a compression of it," this is the one to reach for.

The founding paper, in four results¶

Worth knowing on its own terms, because most citations of it stop at the definition — and because two of the four are small enough to run.

Nobody is perfect, and it takes three voters to prove it. Proposition 1: every social choice function has distortion greater than 1 at 3 voters and 2 candidates. The proof is a matched pair of electorates with identical rankings and opposite welfare-optimal winners — reproduced here as two runnable ballot files, where you can watch STAR's scoring round report the difference and its runoff decline to act on it. This is the whole ranked-vs-rated argument at minimum scale, and it also collides productively with May's theorem.
Unit-sum normalization is one-person-one-vote. Proposition 3: constraining every voter's utilities to the same total is equivalent, for distortion purposes, to leaving them unconstrained but weighting each voter by their own total. Read backwards, that's the formal case for a bounded ballot: refusing to normalize is identical to giving louder voters more votes. (Drop the constraint entirely and distortion is unbounded at 3 voters and 2 candidates — the paper's Remark 1.) The repo argues this from principle on one person, one vote and the equally weighted vote; this is the theorem underneath.
Distortion is NP-complete to compute. Proposition 4, by reduction from Knapsack. Which is why this field proves bounds by hand instead of measuring them — and why the average-case instrument (VSE) is simulated rather than solved.
The escape hatch, and its price. The paper's second half restricts utilities to rank positions — Monroe's misrepresentation — and the impossibilities lift: every method gets a finite bound, Copeland (Ranked Robin) beats STV again, and Borda scores a perfect 1 because the measure is the Borda count. That's a page of its own: misrepresentation.

And a caveat this page owes you, because the paper is explicit about it. Distortion was defined for multiagent systems — software agents that genuinely compute exact utilities — not for human elections. The authors' own motivation concedes that a human would "probably find it impossible to evaluate each candidate precisely" in utility terms. So the founding paper does not establish that human voters have the utilities the framework assumes; it assumes them for agents and proves things about the assumption. What carries the framework over to real elections is the metric branch below, which replaces introspection with geometry. Anyone quoting distortion as proof that voters "really have" utilities is quoting past the source — see cardinal utility for that fight.

Two models — and the model decides the verdict¶

This is the part most people quoting distortion get wrong, and it is the same discipline this library already imposes on simulation results (never quote a rate without the model). There are two standard settings, and they give wildly different answers.

1. Normalized / unit-sum. Each voter's utilities sum to 1, but are otherwise arbitrary. No structure: a voter may love one candidate and despise every other, or be nearly indifferent across the field. This is the adversarial model — the analogue of impartial culture on the simulation ladder.

2. Metric. Voters and candidates are points in a metric space (an "ideological distance"), and cost = distance. Preferences are consistent with a geometry. This is the spatial model taken literally — far more realistic, and far more constrained.

The unit-sum model says ordinal ballots are catastrophic. The metric model says they are pretty good. Both are true statements about their models, and any argument that quotes one without naming it is doing the thing this repo tells you not to do.

Model 1 — unit-sum: rankings are catastrophic¶

Result	Bound	Source
Borda	unbounded	Procaccia & Rosenschein 2006
Plurality	Θ(m²)	Caragiannis & Procaccia 2011
every deterministic ordinal rule	Ω(m²) — Plurality's bound is optimal	Caragiannis et al. 2017
best randomized rule	O(√m · log* m), with an Ω(√m) floor	Boutilier et al. 2015

Read the third row slowly. It is not "Plurality is bad." It is: no deterministic rule reading only rankings can do better than quadratic-in-the-candidate-count loss. Rearranging the ballots doesn't help; inventing a cleverer ranked method doesn't help. The loss is in the information, not the tabulation.

Model 2 — metric: rankings are surprisingly cheap¶

Rule	Metric distortion	Source
any Condorcet winner (when one exists)	≤ 3	Anshelevich et al. 2015
any deterministic rule reading only rankings	≥ 3 (matching lower bound)	Anshelevich et al. 2015
Plurality Matching	= 3 — optimal, conjecture resolved	Gkatzelis, Halpern & Shah 2020 (FOCS)
Plurality Veto	= 3 — same bound, far simpler rule	Kizilkaya & Kempe 2022
Copeland — i.e. Ranked Robin	≤ 5	Anshelevich et al. 2015
Munagala & Wang's rule	4.236 — first break below 5	2019
Ranked Pairs	Θ(√m) — not constant	Goel et al. 2017; Kempe 2020
STV (⇒ RCV-IRV single-winner)	between Ω(√log m) and O(log m) — not constant	Skowron & Elkind 2017
Plurality, Borda	2m − 1 (linear in the field)	Anshelevich et al. 2015
k-approval, Veto	2n − 1 (linear in the electorate)	Anshelevich et al. 2015
approval-style input	unbounded	Pierczyński & Skowron 2019

Whether the 3 could actually be attained by a rule (Condorcet winners don't always exist) was open for years — the optimal metric distortion conjecture — until Gkatzelis, Halpern & Shah resolved it, and Kizilkaya & Kempe then found an almost embarrassingly simple optimal rule: Plurality Veto — give each candidate their first-place count as a score, then let voters in turn each decrement their least favorite remaining candidate; last one standing. Distortion exactly 3.

The survey states the moral bluntly, and it is a counter to the strong pro-score claim: as long as voters and candidates live in a metric space, "it is always possible to choose a close-to-optimal alternative based only on the ordinal rankings of the agents, without even the need to know their actual locations in the space."

How is a bound of 3 even possible? The triangle inequality is smuggled-in cardinal information¶

A ranking hides intensity — it reports A > B identically whether the voter adores A and despises B or can barely tell them apart. In the unstructured model that really is fatal, per Model 1's table.

The metric assumption changes the game because distances can't lie independently — they must satisfy the triangle inequality. If a voter ranks W > X, they're at least as close to W as to X; and W and X can't be far from each other unless that voter is also far from X. Geometry converts ordinal counts into cardinal bounds: a majority saying "W over X" doesn't tell you how much better W is, but it caps how much worse W can possibly be. The intensity information the ballot dropped isn't recovered — it's bounded by the assumed shape of the space.

Here is the whole proof that a Condorcet winner W has social cost ≤ 3× any rival X. Let S be the majority of voters preferring W to X (so |S| ≥ n/2), and write cost(·) for the sum of distances to all n voters:

Voters in S: each has d(v, W) ≤ d(v, X) — that's just what their ballot says.
W and X are near each other: for any u ∈ S, d(W, X) ≤ d(u, W) + d(u, X) ≤ 2·d(u, X). Averaging over S (at least n/2 voters): d(W, X) ≤ 4·cost(X)/n.
Voters outside S: each has d(v, W) ≤ d(v, X) + d(W, X). Summing all n voters: cost(W) ≤ cost(X) + (n/2)·d(W, X) ≤ cost(X) + 2·cost(X) = 3·cost(X). ∎

Three lines, and every line is triangle inequality plus "a majority said so." That's the entire mechanism: majority preference + geometry = a utility guarantee, extracted from ballots that never mentioned utility.

Why you can't do better than 3 — near-indifference is invisible¶

The matching lower bound is one picture. Two candidates, A and B, distance 1 apart on a line; half the voters sit exactly at B, the other half sit at the exact midpoint (and tie-break their ranking toward A):

A ●-----------○-----------● B
              ↑            ↑
         n/2 voters    n/2 voters
        (midpoint:      (at B:
         ½ from each)   0 from B, 1 from A)

The ballots are a perfect 50/50 split — A>B from the midpoint voters, B>A from B's — so a deterministic rule must pick someone, and by symmetry the adversary can arrange the geometry against whichever it picks. Say it picks A: cost(A) = (n/2)(½) + (n/2)(1) = 3n/4, while cost(B) = (n/2)(½) + 0 = n/4. Ratio: exactly 3.

Look where the loss lives: the midpoint voters are essentially indifferent — but a ranking has no way to say "barely." Their forced-strict A>B reads identically to B-voters' heartfelt B>A, and that is precisely the intensity information a scored ballot exists to carry (preference vs. support). The theory quantifies the score-ballot argument and caps it in the same breath: what rankings hide is real, it costs up to 3× in the worst case, and it can cost no more — because geometry won't let indifference hide anywhere worse. (Randomized rules dip just below 3 — random dictatorship already achieves 3 − 2/n, and recent work pushes the frontier toward ~2.7 — but nothing reaches 1; some price is information-theoretically forced.)

What this says about the methods in this library¶

Ranked Robin has a constant metric-distortion bound; RCV-IRV does not. Copeland — which is exactly what Ranked Robin tabulates — stays within a factor of 5 of the welfare optimum no matter how many candidates run. STV/IRV's bound grows with the field (O(log m), with a super-constant floor, so it isn't an artifact of loose analysis). Elimination leaks: the center squeeze is exactly a distortion event — the low-cost central candidate discarded for lacking first choices. And Plurality's 2m − 1 prices first-choice-only as the most lossy compression of all.

That is a mainstream-academic argument for Condorcet over instant runoff, on the same ballot — from a literature with no stake in the American reform fight. It fits this library's standing preference for singing the virtues rather than attacking RCV: the ballot is fine, the count is where the welfare goes.

Condorcet-consistency is not what earns the 5. Careful here. A Condorcet winner, when one exists, is within 3 — that's the lemma above. But a Condorcet-consistent rule must also do something sensible when no Condorcet winner exists, and that's where rules diverge: Copeland's bound runs through the uncovered set (it always elects from it), while Ranked Pairs is equally Condorcet-consistent and still lands at Θ(√m). "It's a Condorcet method, so it has low distortion" is a claim the literature does not support; don't make it.

Approval-only input is unbounded. A cardinal ballot with too few levels is not automatically better than a ranking — it is, in the metric setting, worse than every ranked rule in the table. This is the cleanest available refutation of "cardinal ⇒ more information ⇒ better outcomes" as a blanket claim, and it belongs in any honest STAR-vs-Approval discussion (see how often STAR and Approval disagree, where the same fragility shows up empirically as the cutoff problem).

Read all of it the repo's way (severity × frequency): these are worst cases. On realistic electorates every serious method sits far below its bound most of the time — which is what the average-case metric measures. Distortion is the guarantee; VSE is the expectation. With one large caveat, proved in 2024: "average case" is not automatically the same thing as "realistic," and on the standard random model it rescues nothing at all — below.

Districts multiply all of it by k. Everything above assumes one electorate counted once. Slice the voters into k districts that each elect a representative and then choose among the representatives, and every bound in both tables gains a factor of k — a cost that survives even when each district counts perfect cardinal utilities, because it is a property of the architecture rather than the ballot. That has its own page: distributed voting — the measured price of counting by district.

Does STAR's runoff change the math? (Yes — it's where the lemma plugs in)¶

STAR isn't in the tables because its ballot is cardinal — the framework analyzes rules that see only rankings, so it doesn't directly score STAR at all. No unconditional worst-case distortion number for STAR exists in this literature; don't quote one. The nearest published relative is The Distortion of Approval Voting with Runoff (Ebadian et al., AAMAS 2023) — same score-then-runoff shape, cruder ballot. That's a real, citable open gap. That paper now has a runnable companion in this library, and its headline is the unit-sum mirror image of the metric insurance worked below: there, adding a majority runoff makes approval voting's distortion worse — Θ(m) → Θ(m²) — because a majority check structurally blocks the valuable Condorcet loser, the beloved-of-a-minority candidate with the highest welfare in the race (their Example 5.1, counted on nine ballots in the companion). Majority runoff = liability in the unit-sum model, insurance in the metric one; both are theorems, and the model decides.

What can be said rigorously is conditional, and the machinery maps onto STAR's two rounds surprisingly cleanly:

The runoff is pairwise majority — the exact fact the three-line proof feeds on. STAR's automatic runoff elects whichever finalist a majority prefers, so the winner W pairwise-beats the runner-up X, and the proof applies: cost(W) ≤ 3·cost(X). The runoff is a built-in distortion clamp between the finalists — whatever the scoring round surfaced, the majority check guarantees the elected finalist isn't more than 3× costlier than the one it passed over. In distortion language, this is precisely the "utilitarian pick, then majority check" hybrid design: the scoring round chases the low-cost candidate, the runoff bolts on the triangle-inequality insurance.
One honest wrinkle: step 2 of the proof needs |S| ≥ n/2 — a majority of all voters. STAR's runoff uses the decided-voters denominator, so with heavy Equal Support the winner's preferring-set can be under half the electorate, and the constant degrades accordingly. The clamp is real; it just isn't literally 3 in the equal-support-heavy corner.
When the Condorcet winner reaches the runoff, STAR elects them (a Condorcet winner beats any opponent head-to-head, including the other finalist) — and then inherits the full ≤ 3-versus-everyone guarantee. Since STAR elects the Condorcet winner in the overwhelming majority of simulated and real profiles, the ≤ 3 clamp is the typical regime, not a corner case.
What the runoff can't fix: a finalist round the optimum never reached. If the utility-optimal candidate misses the top two, no majority check can recover them. And the scoring round's worst-case behavior isn't covered by this theory at all, because how a voter maps distances onto 0–5 scores is a modeling step outside the ordinal framework (the same step that drives every simulation's results). With honest, well-calibrated scores the scoring round is hunting the utilitarian optimum directly — better than any ordinal rule can promise; with adversarial or badly quantized scores it can miss. So the theory gives STAR a conditional guarantee and hands the rest to average-case evidence, which is VSE's department — where STAR's scoring round is exactly why it tops the charts.

The strongest result for the pro-score case¶

If you take one row out of the whole literature, take this one. Amanatidis, Birmpas, Filos-Ratsikas & Voudouris studied rules that read rankings plus a few numeric queries — "on a scale, how good is this one?" — asked of each voter:

Cardinal information per voter	Worst-case unit-sum distortion	Source
none (rankings only)	Θ(m²)	Caragiannis et al. 2017
one query	O(m) — and no deterministic 1-query rule beats Ω(√m), even when values are 0/1	Amanatidis et al. 2021; Caragiannis & Fehrs 2024 (Thm 18)
λ queries, λ constant	Θ(min{n, m}^1/λ) — optimal; each extra query takes another root off the bound	Ebadian & Shah 2025
O(log m) queries	O(log m), randomized	Caragiannis & Fehrs 2024 (Thm 15)
to reach a constant	Ω(log m) queries are necessary; O(log² m) suffice	Caragiannis & Fehrs 2024 (Thm 17); Amanatidis et al. 2021

And — the part that matters most — these bounds hold without any normalization assumptions.

The middle row is the one to memorize, and it is newer than the rest: each additional query takes another root off the distortion. One query buys you m, two buys √m, three buys ∛m, and the bound is now known to be optimal — Ebadian & Shah's AAAI-25 outstanding paper settled it against a matching lower bound. Reading the top and bottom rows together: the first scrap of intensity information is worth more than every clever thing you could do with the rankings.

That is the precise, defensible form of "scores are more powerful expression." Not "cardinal ballots are better" (Approval's unbounded row kills that), and not "rankings are hopeless" (the metric factor of 3 kills that). The defensible claim is: a small amount of intensity information buys a large, provable reduction in worst-case welfare loss.

Don't over-map this onto the STAR ballot, though. A "query" here is the mechanism asking one voter for one candidate's exact value; a 0–5 ballot instead collects a coarsely quantized value for every candidate. That is a different object — richer in coverage, poorer in precision — so these rows are not a distortion bound for STAR (there still isn't one). What they establish is the direction and the steepness: intensity information is cheap to collect and pays out fast.

Does averaging rescue rankings? Not by itself¶

The natural hope, once you've read the Θ(m²) row, is that it's a pathology — a construction an adversary builds, not a thing that happens. Take the average instead of the worst case and surely the ordinal rules look fine. Caragiannis & Fehrs (WINE 2024) went and checked, and the answer is no.

Their model is the obvious one: each voter draws a value for each candidate independently from a common distribution, and ranks accordingly. That is exactly an impartial culture electorate. They then define average distortion as the ratio of the expected optimal welfare to the expected welfare of what the rule elects — a ratio of expectations, not an expectation of ratios, which is the choice that keeps rare events from swallowing the metric.

Theorem 1. For every voting rule — deterministic or randomized — average distortion on binary value distributions is Ω(m).

And the flip side, which is what makes it bite: the trivial rules — return a fixed candidate, or draw one uniformly at random while ignoring every ballot — have average distortion at most m for any distribution. Put those together:

On an impartial-culture electorate, picking the winner out of a hat is within a constant factor of the best that any voting rule can do on average. Not Plurality-is-bad; every rule, including ones nobody has invented yet.

This is the theorem underneath a caution this library was already printing from the empirical side. The simulation ladder warns that impartial culture manufactures near-ties so that "methods never get to show their skill at ferreting out the best candidate," and simulate utilities, not ballots files IC as adversarial, not realistic. Caragiannis & Fehrs prove the strong form: under IC there is no skill to show. Anyone quoting a method-comparison statistic computed on impartial culture is quoting a number the theory says is nearly information-free — which is why this repo's rule is never quote a rate without the model.

Read the negative result carefully, in both directions:

It does not say real elections are hopeless. IC is the adversarial end of the modelling spectrum, and the escape route for ordinal rules was never averaging — it is the metric assumption, which is a statement about structure, not about typicality. Structure is what saves ranked ballots; randomness doesn't.
It does kill the loose move of treating "worst case" as a synonym for "exotic corner." Averaging over a standard random model changes the exponent (m² → m) and nothing else that matters.

And then the same escape hatch appears, one section early. Their positive results are queries again: a deterministic mechanism (Mean) making a single value query per voter achieves constant average distortion on binary distributions, and a randomized one-query mechanism (RtMean) achieves O(log m + log(σ²/μ²)) for a general distribution with mean μ and variance σ² — logarithmic in m for most distributions you'd actually write down. Worst case and average case, from opposite directions, arrive at the same sentence: one question per voter is the difference between "no better than a coin flip" and "constant."

Distortion vs. VSE — same question, opposite instruments¶

	Distortion	VSE / Bayesian Regret
Case analyzed	worst case	average case
Method	proved (theorems, bounds)	simulated (thousands of elections)
Output	a ratio, often in m or n	a percentage, 0–100%
Optimum is	ratio 1	100%
Depends on	which model (unit-sum / metric)	which voter model + strategy mix
Home turf	academic CS/econ (AAAI, IJCAI, FOCS)	reform advocacy (advocacy-adjacent — disclose it)
Weak spot	worst cases may be exotic	assumptions are contestable, not proved

They are the same underlying commitment — utility is the target, a ballot is a lossy channel — measured from opposite ends. That's the useful thing to know if someone dismisses VSE as reform-movement math: the peer-reviewed literature made the identical modeling choice, and simply proved bounds instead of running simulations. VSE's premise is mainstream; only its methodology is advocacy-side.

The table's empty corner is no longer empty, and that is the news from the average-distortion section. Read the first two rows as a grid: worst-case-and-proved is distortion, average-case-and-simulated is VSE, and average-case-and-proved was, until recently, nobody's department. Caragiannis & Fehrs occupy it — and the first thing found there is that the two instruments do not simply meet in the middle. VSE's optimism comes from its voter model (spatial, structured, correlated), not from the mere act of averaging. Swap in an unstructured random electorate and the average-case verdict is nearly as bleak as the worst-case one. That's a caution for VSE too: the percentage you get out is a fact about the electorate you simulated.

The fair reading, both directions¶

Concede to the cardinal camp: the ground truth in this entire literature is utility. Rankings are formally modeled as lossy compression of cardinal preferences — a premise at the center of mainstream computational social choice, not a fringe scored-ballot talking point. Anyone claiming intensity is meaningless or unmeasurable is arguing against the field's own foundations. And the query results show that a little intensity buys a lot.

Concede to the ordinal camp: the loss is bounded, small, and worst-case. Distortion 3 for the best rules — and ≤ 3 for any Condorcet winner — means "rankings are lossy" cannot be inflated into "ranked methods are utility-blind." The Condorcet ideal and the utilitarian ideal usually agree; this theorem is the formal reason why: majority preference geometrically implies near-optimal total utility. A debater who says "only scored ballots can find the utilitarian winner" is overclaiming by exactly this theorem.

Two more that cut against us, because the reading-these-fairly rule applies to results we like:

Truthful mechanisms are essentially ordinal. Bhaskar, Dani & Ghosh (2018), and earlier Filos-Ratsikas et al. (2014) in matching: the best strategyproof mechanisms are ordinal, and asymptotically as good as the best ordinal non-truthful ones. Once you demand honesty-compatibility, cardinal input stops paying. This is the distortion literature independently rediscovering the objection every score-voting critic raises: the extra information isn't stable under strategy.
Ballots aren't utilities. Every voter min-max normalizes onto 0–5. That rendering step is itself a distortion the theory usually assumes away, and it is the same crux as the approval cutoff.

And the load-bearing assumption, disclosed: the 3 depends on the metric premise — the spatial model with its triangle inequality. That model explains most real voting behavior, but it is a lens, not reality: drop it and ordinal distortion blows up with the field size. So the honest one-sentence summary is: to the extent electorates are spatial, rankings lose at most a small constant; the case for scored ballots then rests on average-case performance (VSE), expressiveness, and strategy resistance — not on a worst-case rescue. That's a narrower claim than either camp's slogan, which is usually the sign you've got it right.

Using it in an argument¶

Say this:

Modern social choice treats voter utility as the thing an election is trying to find, and a ranked ballot as a compressed encoding of it — that's what "distortion" measures. In the adversarial model, no deterministic ranked rule beats quadratic loss. In the realistic metric model, good ranked rules get within a factor of 3–5, so rankings are not hopeless. But asking voters for even a little intensity provably collapses the worst case from quadratic to constant. That's the real case for score ballots — a bounded, proven improvement, not a claim that rankings don't work.

Don't say "academics have shown scores are better" — Approval's unbounded distortion and the ordinal-truthfulness results are both sitting right there, and a reader who knows the literature will produce them.

Sources¶

Academic — the right tier for theorems.

Anshelevich, Filos-Ratsikas, Shah & Voudouris, Distortion in Social Choice Problems: The First 15 Years and Beyond (IJCAI 2021 survey; arXiv:2103.00911) — the single best entry point; every bound quoted above is in it.
Procaccia & Rosenschein, The Distortion of Cardinal Preferences in Voting (CIA 2006, LNAI 4149, pp. 317–331; Springer) — the founding paper: §3 is summarized above and its Proposition 1 is runnable; §4 is the misrepresentation page.
Boutilier, Caragiannis, Haber, Lu, Procaccia & Sheffet, Optimal Social Choice Functions: A Utilitarian View (AIJ 2015) — the unit-sum randomized bounds.
Anshelevich, Bhardwaj, Elkind, Postl & Skowron, Approximating Optimal Social Choice under Metric Preferences (AIJ 2018; arXiv:1512.07590) — Copeland ≤ 5, the ≤ 3 Condorcet lemma, the ≥ 3 lower bound, the scoring rules.
Skowron & Elkind, Social Choice under Metric Preferences: Scoring Rules and STV (AAAI 2017; arXiv:1611.08549) — STV's O(log m).
Gkatzelis, Halpern & Shah, Resolving the Optimal Metric Distortion Conjecture (FOCS 2020) — distortion 3, achieved.
Kizilkaya & Kempe, Plurality Veto: A Simple Voting Rule Achieving Optimal Metric Distortion (IJCAI 2022).
Amanatidis, Birmpas, Filos-Ratsikas & Voudouris, Peeking Behind the Ordinal Curtain: Improving Distortion via Cardinal Queries (AIJ 2021) — the query results.
Ebadian et al., The Distortion of Approval Voting with Runoff (AAMAS 2023) — the closest published analogue to STAR's shape; runnable companion: the valuable Condorcet loser, counted.
Caragiannis & Fehrs, Beyond the Worst Case: Distortion in Impartial Culture Electorates (WINE 2024) — the average-distortion definition, the Ω(m) impossibility, the one-query Mean/RtMean mechanisms, and the Ω(log m) query lower bound.
Ebadian & Shah, Every Bit Helps: Achieving the Optimal Distortion with a Few Queries (AAAI 2025, outstanding paper) — Θ(min{n, m}^1/λ) with λ queries; settles the fixed-query question.
Filos-Ratsikas & Voudouris, Revisiting the Distortion of Distributed Voting (Theory of Computing Systems 2024, open access) — what districts cost; see distributed voting.

Lean disclosure: this is peer-reviewed CS/economics with no stake in the US reform fight — the most neutral tier available on this question, and unusually so for voting-method material. Its blind spot is the opposite of advocacy's: it optimizes worst cases over models chosen for tractability, and says nothing about ballot usability, spoilage, or whether voters can actually fill the thing in.