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Costly Voting in the Hotelling-Downs Model

Guy Wolf, Reshef Meir

arXiv:2609.29869v1cs.GTcs.MA

TL;DR

The paper asks which feature of voter participation helps explain candidate divergence where the full-participation Hotelling–Downs model predicts median convergence. It models turnout as costly and endogenous, and finds that several forces affect polarization while turnout level alone does not.

  • Problem

    The full-participation Hotelling–Downs model predicts median convergence, but the introduction points to candidate divergence and leaves open which feature of the participation margin matters.

    Source: §1 · p. 1

  • Method

    The paper extends Hotelling–Downs by making turnout depend on whether the relative utility gain from a preferred candidate exceeds each voter's cost, with alienation and candidate competitiveness as additional parameters.

    Source: §1 · p. 2

  • Results

    Voter dispersion, cost responsiveness in the reverse-hazard-rate sense, and alienation raise polarization; competitiveness lowers it, while turnout level does nothing.

    Source: §6 · pp. 45–46

  • Takeaways & Limitations

    The voter-side polarizing forces do not interfere with each other and persist at every level of competitiveness.

    Source: §6 · p. 46

    Without additional conditions, a solution to the equilibrium conditions need not be an equilibrium, and whether some such solution is an equilibrium remains open.

    Source: §3.1 · p. 19

Abstract

from arXiv · show

We study a partial-participation variation of the Hotelling-Downs model. Voters each have a cost to vote, and only vote when the comparative gain from their preferred candidate exceeds the cost. Under this model the median voter theorem breaks, and we study the extent of polarization under equilibria in different voters and cost distributions. We find that the main predictor of polarization is the reverse-hazard-rate of the cost distribution, indicating that the driver of polarization under our model is the willingness of voters to respond to changes in positions of candidates. We then extend the model by adding parameters governing alienation and candidate competitiveness, showing that our results are robust even when taking into account other realistic factors.

1 Introduction

The introduction connects candidate divergence and voter abstention to a gap in the Hotelling–Downs model: it does not explain which feature of participation drives divergence. The paper introduces costly voting and examines how voter and cost distributions, alienation, and candidate objectives shape polarization.

1 Introduction

  • The canonical full-participation model predicts median convergence, while the introduction sets that prediction against observed candidate divergence and declining turnout.Source: §1 · pp. 1–2
  • The model makes turnout depend on whether the relative utility gain from a preferred candidate exceeds each voter's cost, with arbitrary voter and cost distributions and two additional participation and candidate-objective dials.Source: §1 · p. 2

1.1 Our Contribution

  • Costly voting breaks median convergence: polarization rises with voter dispersion, reverse-hazard-rate cost ordering, and alienation, while competitiveness lowers it and turnout level does nothing.Source: §1.1 · pp. 2–3
  • A Lipschitz payoff bound, exact grid quadrature, and adaptive refinement turn finite computation into a certificate ruling out equilibria in a continuous-strategy game.Source: §1.1 · p. 3

1.2 Related Work

  • The model's distinctive combination is stakes-based turnout, a full cost distribution, no pivot-probability discount, and analysis for arbitrary voter and cost distributions.Source: §1.2 · p. 6

2 Model

The model replaces full participation with voter-by-voter costly voting, then examines how this changes candidate competition and adds alienation and competitiveness as extensions.

2 Model

  • The model’s baseline departure is voter-by-voter cost–benefit participation, with alienation and candidate competitiveness added later as separate extensions.Source: §2 · p. 7

2.1 The Hotelling–Downs Benchmark

  • With full participation, the classical Hotelling–Downs game places both candidates at the median; the model changes the voting decision, not the utility comparison.Source: §2.1 · pp. 7–8

2.2 Voters, Costs, and Participation

  • Independent voting costs gate participation on the preferred candidate’s distance advantage, making turnout platform-sensitive; candidate votes come from their flank and half of the contested middle.Source: §2.2 · pp. 8–10

2.3 Candidates and Objectives

  • Candidates choose platforms simultaneously to maximize their own vote shares, while total turnout can be below one.Source: §2.3 · p. 10

2.4 Equilibrium and a Running Example

  • In the doubly-uniform example, candidate 1’s best response to c2 = 0.8 is c1 = c2/3, and the best-response curves intersect at two equilibria.The trade-off is between proximity to more voters and turnout gained from greater separation.Source: §2.4 · p. 12, Figure 2 · p. 12

2.5 Extensions: Alienation and Competitiveness

  • Alienation makes voting benefits depend on distance from one’s own candidate, while competitiveness lets candidates weight opponents’ votes negatively as β rises.At β = 0 candidates maximize vote share; at β = 1 they maximize margin.Source: §2.5 · pp. 13–14

3 Costly Voting

This section characterizes costly-voting equilibria in the baseline model, examining how electorate and cost-distribution shape and voters’ responsiveness to candidate moves relate to equilibrium separation and turnout. It also considers reverse-hazard-rate comparisons and power-cost examples.

3.1 Equilibrium Characterization

  • At every interior stationary profile, inner and outer marginal masses balance for each candidate; in aggregate, exactly half of total marginal mass lies on the inner arms.This balance holds for every voter and cost distribution in the baseline model.Source: §3.1 · pp. 16–17
  • Pure-equilibrium existence is established for several benchmark settings, but remains open for uniform costs when the electorate’s density ratio exceeds 4.The paper conjectures baseline pure existence more generally and notes that the high-ratio case has equilibria in one example but no general proof or counterexample.Source: §3.1 · p. 17
  • With uniform costs, equilibria trap exactly half the electorate; when ρ_v ≤ 4, symmetric quartile profiles are equilibria.Under the density-ratio condition, each candidate’s payoff is concave on her own side of the opponent, subject to cross-over checks.Source: §3.1 · p. 18
  • With uniform voters, every equilibrium is symmetric, and regular costs yield a unique equilibrium gap and a unique equilibrium up to relabeling.Without regularity, the scalar equilibrium equation can have multiple roots, not all of which are equilibria.Source: §3.1 · p. 19

3.2 Polarization

  • Nearly twice the equilibrium separation is supported by a hollow rather than peaked electorate under uniform costs, because the quartile profile transmits the electorate’s interquartile range one for one.The comparison holds for symmetric electorates with the same density ratio, ρ_v = 3.Source: §3.2 · pp. 20–21, Figure 4 · p. 21
  • Equilibrium depends on cost shape through the reverse hazard rate r = k/K, not turnout level: scaling costs scales vote shares and turnout but leaves best responses and equilibria unchanged.The reverse hazard rate measures how proportionally turnout responds to changes in the stakes.Source: §3.2 · pp. 21–22
  • A higher reverse-hazard rate implies a wider equilibrium gap; the comparison depends only on responsiveness at the incumbent gap, though a pointwise order is a sufficient condition.Equilibrium can be represented as a crossing of required and supplied responsiveness; super-regularity makes the equilibrium unique.Source: §3.2 · pp. 22–23
  • Polarization comes from nearly indifferent flank voters responding to outward moves, not from a small flank; in the baseline, a wider gap raises turnout, while alienation can restore the reverse channel.The account identifies responsive participation as the mechanism and says alienation does not close that channel for α > 0.Source: §3.2 · pp. 23, 26
  • Turnout falls from 0.375 to 0.269 while the equilibrium gap shrinks from 1/2 to about 0.4226, showing that lower turnout does not imply more polarization.Adding never-voters can drive turnout arbitrarily low without changing platforms; the model attributes turnout and polarization to cost-distribution shape rather than one causing the other.Source: §3.2 · pp. 23–25
  • For power costs K(t)=t^p, equilibrium separation equals the mean voting cost, p/(1+p)=E[κ], so increasing p raises responsiveness and polarization.The example shows that one cost-distribution shift can produce rising polarization alongside falling participation, without either outcome causing the other.Source: §3.2 · p. 25
  • Every interior symmetric candidate profile is the unique equilibrium for exactly one power-cost distribution, while quartile equilibrium for all symmetric electorates is special to uniform costs.Rescaling costs cannot move the equilibrium quantile; moving it requires changing the reverse hazard rate.Source: §3.2 · p. 25

4 Alienated Voters

This section examines how alienation and cost distributions shape turnout and equilibrium polarization under costly voting, including how polarization relates to voter dispersion. It shows that stronger alienation can increase candidate separation toward full separation while changing who participates, with turnout responses to polarization that need not be monotone.

4 Alienated Voters

  • Alienation changes turnout rather than candidate comparisons alone: with costly voting, its distance penalty bites hardest at the midpoint and pushes candidates apart.Without costly voting, alienation cancels from the comparison between candidates and the median equilibrium survives.Source: §4 · p. 26

4.1 Equilibrium at a Fixed α

  • Alienation creates an inner abstention zone at every profile and can create outer dead zones; unlike uniform voters, some non-uniform electorates can sustain dead-zone equilibria.Alienation also shrinks the range of density ratios for which the first-order conditions are known to characterize equilibria.Source: §4.1 · pp. 27–29
  • For uniform voters and super-regular costs, the equilibrium is unique and symmetric at every α > 0.The section also gives a master first-order condition for arbitrary electorates and cost distributions, including cases with outer dead zones.Source: §4.1 · pp. 28–29
  • With alienation, the equilibrium voter quantile depends on the electorate rather than remaining distribution-free, as it does in the baseline.Across the examined electorates, the quantile spans [0.194, 0.207] at α = 1 and [0.125, 0.153] at α = 4.Source: §4 · p. 26, §4.1 · p. 30

4.2 Costs and Polarization at a Fixed α

  • At every fixed α, a higher reverse-hazard-rate cost distribution produces a larger equilibrium gap; stronger alienation also polarizes at fixed costs.The cost comparison is also one of stochastically higher costs when the higher reverse hazard rate distribution has eK(1) ≤ K(1).Source: §4.2 · p. 30
  • For affine stretches of a fixed electorate, power costs transmit mass polarization proportionally; with a uniform electorate, rising cost elasticity amplifies it and falling elasticity damps it.The transmission result holds at every α, including α = 0.Source: §4.2 · p. 31
  • Polarization raises turnout locally at equilibrium, but for α > 0 turnout eventually falls as the gap grows, so outside pressure can push candidates into a turnout-decreasing range.In the doubly-uniform benchmark, turnout is single-peaked; at α = 1 its peak is 9/11 ≈ 0.818, above the equilibrium gap of 0.6.Source: §4.2 · pp. 32–33

4.3 Higher α Polarizes

  • For every super-regular cost distribution, the equilibrium gap rises strictly with alienation and tends to full separation, even at a fixed level of participation.The fixed-participation result distinguishes alienation-driven polarization from the cost channel.Source: §4 · p. 26, §4.3 · p. 33
  • For uniform voters and every super-regular cost distribution, equilibrium separation increases with alienation and tends to 1 as α → ∞.Fig. 8 shows the increasing separation curves for four cost distributions; stronger alienation also depresses turnout in the doubly-uniform benchmark.Source: §4.3 · p. 33
  • Alienation pushes candidates outward because their most extreme flank voters have the least at stake and abstain first, giving candidates an added incentive to reach their own flanks.The boundary voter’s turnout fraction is K(b)/K(∆) = 2/(2 + α), which falls as alienation grows and can be restored only by moving outward.Source: §4.3 · pp. 33–34
  • Alienation increases polarization even at unchanged turnout: admissible cost-distribution scalings can match turnout across α = 0 and α > 0 while the alienated equilibrium gap remains larger.The scalings preserve each game's equilibrium, so the gap difference follows from the alienation result rather than a turnout difference.Source: §4.3 · pp. 34–35
  • Alienation and cost shifts have distinct participation signatures: alienation creates two active-electorate humps around the candidates and an inner dead zone, while a cost shift changes participation without changing its shape.With α = 0, participation on each flank tracks voter density because the gain is flat at ∆; alienation makes the active electorate bimodal.Source: §4.3 · p. 35

5 Competitive Candidates

This section examines how candidate competitiveness and alienation affect equilibrium polarization, characterization, stability, and turnout. It also considers how these effects depend on the margin-game endpoint and electorate shape, including cases where pure equilibria are eliminated.

5.1 Equilibrium at a Fixed β

  • At fixed β < 1, the inner mass receives weight 1+α+β and the outer mass 1+α−β; this balance holds across scale transformations for all α ≥ 0 and β ∈ [0, 1].As β rises toward 1, the inner share falls from 1/2 toward α/[2(1+α)], making competition the model’s de-polarizing force.Source: §5.1 · pp. 36–37, §5 · p. 36
  • Under super-regularity, the β < 1 equilibrium is unique and symmetric; with uniform costs and arbitrary electorates, the mass-gap law traps mass 1−β and quantile profiles move toward the median.For non-uniform electorates, characterization is partial; symmetric stationary profiles exist, and competition makes alienation dead zones easier to sustain.Source: §5.1 · pp. 38–39

5.2 Costs and Polarization at a Fixed β

  • More responsive cost distributions sustain weakly more polarization at every β < 1, while mass polarization scales elite polarization exactly under power costs and otherwise depends on cost elasticity.Under uniform costs, balance conditions characterize equilibria within specified density-ratio thresholds; beyond uniform costs, the concavity argument has an unresolved obstruction.Source: §5.1 · p. 37, §5.2 · p. 39, Proposition 40 (Mass polarization… · pp. 39–40

Proposition 40 (Mass polarization scales elite polarization, general

  • Competition lowers equilibrium turnout even though the turnout curve itself is unchanged across β; more polarized candidates raise turnout locally at every competition level.Competition changes the equilibrium’s location on that curve by narrowing the candidate gap.Source: Proposition 40 (Mass polarization… · p. 40

5.3 Higher β De-polarizes

  • For β < 1, the equilibrium gap falls strictly as competitiveness β rises at every alienation level, and it still falls when turnout is held fixed.This monotonicity relies on the super-regularity conditions supporting uniqueness; with a non-monotone K/k, an equilibrium branch can instead polarize as β rises and disappear in a fold.Source: §5.3 · pp. 40–41

5.4 The Margin Game (β = 1)

  • At β = 1, the margin-game equilibria form a band of zero-margin profiles, and with uniform voters some tied profiles leave almost nobody voting.Without alienation, the median is the unique equilibrium; for a hollow-centre electorate, the median fails, and some electorates have no pure equilibrium.Source: §5 · p. 36, §5.4 · pp. 41–42
  • As β approaches 1, the selected profile is maximally polarized within the margin-game band when α > 0, whereas with α = 0 it converges to the median.For α > 0, the limit gap is α/(2+α), the band’s upper endpoint; at α = 0, the gap is (1−β)/2.Source: §5.4 · pp. 42–43, Figure 9 · p. 43
  • With α > 0, a centrally peaked electorate makes the median a strict local equilibrium, while a hollow center makes the median fail.For uniform voters, v′′ ≡ 0 is the knife-edge associated with the band E(α).Source: §5.4 · p. 43
  • For margin maximizers, local curvature at the median—not overall voter dispersion—determines whether the center holds against small candidate deviations.A highly dispersed electorate can sustain the median if peaked there, while a barely dispersed electorate can lose it if locally hollow.Source: §5.4 · p. 44
  • At α = 1 with uniform costs, computer-assisted certificates show no pure equilibrium for arcsine or Beta(0.8, 0.8) voters.The proof converts finite-grid security-value calculations into a continuum result using a Lipschitz bound; the authors expect broader non-existence for sufficiently hollow electorates but do not prove it.Source: §5.4 · pp. 44–45
  • For arcsine voters at α = 1, no pure equilibrium exists for β ≥ 0.9168, whereas one is found numerically at β = 0.90; at α = 0, one persists for every β < 1.This contrast indicates that non-existence arises from alienation and competitiveness together, not competitiveness alone.Source: §5.4 · p. 45

6 Discussion and Further Research

The discussion frames polarization as the balance of voter-side forces and candidate competitiveness, while turnout levels themselves do not determine equilibrium polarization. It also identifies empirical and theoretical boundaries on these conclusions.

What drives polarization

  • A more dispersed electorate, costlier voting in the reverse-hazard-rate sense, and stronger alienation raise equilibrium polarization; competition lowers it, while turnout level has no causal role.The discussion presents these directions as the model’s comparative statics; the table collects the forces and their effects.Source: §6 · pp. 45–46, Table 3 · p. 47

The margin game

  • For an arcsine electorate, the margin game has no pure equilibrium: its maximum security value is negative at c = 0.829, unlike the median for centrally peaked voters or every point of the band for uniform voters.Endpoint-aware quadrature is needed; a naive grid makes the arcsine maximum 47 times too negative.Source: Figure 11 · p. 46

Benchmarks and extensions

  • The doubly-uniform benchmarks organize costly voting, alienation, and competitiveness as distinct forces with explicit comparative statics, but their reported bounds are sufficient rather than tight.The discussion cautions that the joint-extension bounds do not reduce continuously to the preceding row as α approaches zero.Source: §6 · pp. 46, 48, Table 4 · p. 48

Open questions

  • The discussion leaves open sharp density-ratio thresholds, extensions of the alienation result beyond uniform electorates, and a general pure-equilibrium existence theory.It also identifies an unresolved certified gap and uncharacterized mixed equilibria in the margin game.Source: §6 · pp. 46–47

Empirical implications

  • The model’s empirical prediction concerns the composition of abstainers rather than participation rates, distinguishing uniform changes in voting costs from abstention concentrated among ideologically distant voters.The discussion cites Dutch municipal evidence consistent with centrist voters being more likely to abstain, while noting the test has a confound the model assumes away.Source: §6 · p. 47

Scope and further research

  • The analysis leaves out costs correlated with voter position; if ideologically committed voters are cheaper to mobilize, the model may understate polarization.The discussion distinguishes this fixed voter-side cost variation from alienation, which changes with candidate positions.Source: §6 · p. 48

Further research

  • Possible extensions include nonlinear or probabilistic voter utilities, repeated elections, more than two candidates, endogenous entry, asymmetric candidates, and finite electorates.These are identified as directions for extending the costly-turnout framework.Source: §6 · pp. 48–49

A Deferred Proofs

The appendix supplies deferred proofs and supplementary results, including arguments about equilibrium deviations, comparative statics, and uniqueness of equilibrium gaps.

Uniqueness

  • The appendix establishes uniqueness of the equilibrium-gap root by showing that its defining function decreases from positive to negative over the feasible interval.As the gap grows, the relevant interval shifts right and shrinks; the non-increasing reverse hazard rate makes the function strictly decreasing.Source: A Deferred Proofs · p. 49

Root equilibrium

  • A root profile is shown to be a global mutual best response by establishing single-peaked own-side payoffs and showing that far-side deviations are dominated by their mirror images.The proof also rules out maximizers in the dead-zone region.Source: A Deferred Proofs · pp. 49–50

Support of best responses

  • Candidates outside the electorate’s support cannot be best responders, since moving to the support boundary weakly improves their payoff and generally improves it strictly.The argument uses pointwise benefit changes and monotonicity of the cost distribution.Source: A Deferred Proofs · p. 50

Rescaling costs

  • Rescaling the cost distribution preserves the conditions for the reverse-hazard-rate comparison, and the equilibrium gap increases with the rescaling parameter when alienation is positive.The proof obtains the gap’s increase from the equilibrium equation in voter-position space.Source: A Deferred Proofs · p. 50

Alienation comparative statics

  • With alienation, the equilibrium gap increases in α and approaches full separation, as follows from the sign of the implicit derivative and the bound ∆* > α/λ.The proof uses that the function defining the equilibrium gap is strictly decreasing in the gap.Source: A Deferred Proofs · p. 51

Symmetric stationary profiles

  • For uniform voters, an untied symmetric stationary profile cannot have a dead zone: its payoff derivative is negative there, forcing positive benefits at a stationary profile.The first-order condition then yields the stated equation for the symmetric profile.Source: A Deferred Proofs · p. 51

B Omitted Proofs: Baseline and Margin Game

This appendix gathers deferred proofs and supplementary material for the paper’s main results.

Appendix scope

  • The appendix collects proofs deferred from Sections 3 to 5, with some remaining proofs located in Appendix A and Appendices I to Q.It also contains supplementary results, worked examples, and certificates.Source: B Omitted Proofs: Baseline and Margin… · p. 52

Proof organization

  • This appendix proves results from Sections 3 and 5 stated without proof in the main text; proofs of remaining results appear in Appendix A and Appendices I to Q.For the reader’s convenience, each result is restated before its proof.Source: B Omitted Proofs: Baseline and Margin… · p. 52

B.1 Preliminaries: Payoffs and Their Derivatives

This section characterizes equilibrium positions across uniform-cost and uniform-voter cases, including the effects of voter and cost distributions and alienation. It establishes median-position uniqueness in the zero-alienation margin game and shows that the only equilibrium places both candidates at q, as deviations from q admit a profitable deviation.

B.1 Preliminaries: Payoffs and Their Derivatives

  • Under uniform costs, equilibria are interior with c1<c2 and satisfy V(m)=2V(c1)=2V(c2)−1.If ρv≤4, candidates' payoffs are concave on each side of the opponent, and any quartile profile V(c1)=1/4, V(c2)=3/4 is an equilibrium; with symmetric voters, it is symmetric.Source: §B.1 · pp. 53–55
  • With uniform voters, every equilibrium is symmetric and has gap Δ∈(0,1) solving K(Δ)=(1−Δ)k(Δ); if ψ(t)=t+K(t)/k(t) is strictly increasing, it is unique up to relabeling.Here ψ(Δ)=1.Source: §B.1 · p. 55
  • If two uniform-voter cost distributions satisfy k̃/K̃≥k/K pointwise, the equilibrium gap for the former is at least as large; if also K̃(1)≤K(1), then K̃≤K pointwise.Source: §B.1 · p. 56
  • With uniform voters and alienation α≥0, the margin-game equilibria are exactly the profiles in [1/(2+α),(1+α)/(2+α)]², each with margin zero, independently of K.Source: §B.1 · p. 56
  • With α=0 and voters and costs satisfying Assumption 2, the median profile (cmed,cmed) is the unique margin-game equilibrium, and every unilateral deviation strictly lowers the deviator’s margin.Source: §B.1 · p. 58
  • Every equilibrium has c1 = c2 = q, because any equilibrium with one candidate away from q permits a profitable deviation.A candidate can deviate to q to improve its margin; applying the argument symmetrically establishes the result for both candidates.Source: §B.1 · p. 58

C Existence and Uniqueness of Equilibria

The benchmark balance conditions have solutions, but solving them does not generally establish a pure Nash equilibrium. The section distinguishes existence of candidate roots from equilibrium existence and identifies conditions that yield uniqueness or guarantee a mixed equilibrium.

C Existence and Uniqueness of Equilibria

  • Under uniform voters, solutions to the necessary conditions exist, but they may fail to be equilibria; pure-equilibrium existence in general remains open.The symmetric-case hypotheses of Corollary 11 guarantee equilibrium existence.Source: C Existence and Uniqueness of Equilibria · pp. 59–60
  • When the monotonicity condition holds, the interior root is unique and yields the unique equilibrium; without it, roots can fail to be equilibria.Non-increasing or log-concave cost densities provide sufficient conditions for this monotonicity.Source: C Existence and Uniqueness of Equilibria · p. 60
  • For one uniform-voter cost density, three symmetric profiles satisfy the conditions, but the middle profile is not a Nash equilibrium.Candidate 1 can profit by moving from c1 = 1/2 to approximately 0.473.Source: C Existence and Uniqueness of Equilibria · p. 60
  • Uniqueness can also fail with non-uniform voters: a symmetric voter density admits three equilibria, including two asymmetric mirror images.The section notes that the uniform-cost sufficiency result identifies some equilibria, not necessarily all of them.Source: C Existence and Uniqueness of Equilibria · p. 60
  • A possibly mixed Nash equilibrium always exists because payoffs are continuous on the compact strategy space.Continuity at tied profiles follows because vote shares vanish as candidates approach each other.Source: C Existence and Uniqueness of Equilibria · pp. 60–61

D Power Costs against a Non-Uniform Electorate

The appendix examines power costs with non-uniform electorates and a separate uniform-cost example. Numerical checks find equilibria across the tested power-cost cases, while the uniform-cost example has three equilibria, two asymmetric.

D Power Costs against a Non-Uniform Electorate

  • Standard sufficiency criteria miss this power-cost family except at p = 1, and the theorem for non-increasing costs requires uniform voters.For p < 1, the cost density is non-increasing but its endpoint density ratio is infinite.Source: D Power Costs against a Non-Uniform… · p. 61
  • In all 20 tested combinations of five symmetric electorates and p ∈ {1/2, 1, 2, 3}, the unique symmetric stationary profile is a Nash equilibrium.The largest profitable deviation found was 1.1 × 10^-16.Source: D Power Costs against a Non-Uniform… · p. 61
  • For p ≠ 1, the equilibrium quantile shifts by less than 0.03 across the tested range, with the direction depending on whether the electorate is peaked or hollow.At p = 1, the quantile is 1/4 for every electorate; departures from the uniform-voter value grow with |p−1| and distance from uniformity.Source: D Power Costs against a Non-Uniform… · p. 61

E Example with Multiple Equilibria (Uniform Costs)

  • Under uniform costs, three distinct solutions are all numerically verified as Nash equilibria, with two asymmetric mirror-image profiles despite a symmetric electorate.Their payoffs are approximately (0.075, 0.212), (0.106, 0.106), and (0.212, 0.075).Source: E Example with Multiple Equilibria… · p. 62

F Stationary Profiles Need Not Be Equilibria (Uniform Costs)

The appendix separates stationary conditions from global equilibrium verification, illustrates turnout comparative statics, and extends preliminary equilibrium results across alienation and competitiveness parameters. Its examples show that stationary profiles can fail global optimality, while the stated interiority result applies only to specified parameter cases.

F Stationary Profiles Need Not Be Equilibria (Uniform Costs)

  • Among three stationary profiles for a four-cluster electorate under uniform costs, the middle one is not a Nash equilibrium because candidate 2 can profitably relocate to another local maximum.Her payoff rises from approximately 0.1353 to 0.1365; the two outer profiles are Nash equilibria.Source: F Stationary Profiles Need Not Be… · pp. 62–63

G Turnout under Linear Costs

  • Under uniform voters and linear costs, the appendix computes vote shares at symmetric profiles and derives total turnout using payoff symmetry.The supplied passages state the calculation's scope but do not give a readable turnout result.Source: G Turnout under Linear Costs · p. 63

H Turnout Comparative Statics: A Worked Example

  • At equilibrium, turnout falls from approximately 0.54 at ϑ = −2 to 20/81 ≈ 0.25 at ϑ = 2, less than the decline with fixed platforms.The equilibrium gap widens as costs rise, moderating the turnout decline in this example.Source: H Turnout Comparative Statics: A… · pp. 63–64

I Preliminaries for all (α, β)

  • Every pure equilibrium is untied and interior in the listed cases: β = 0, α = 0, or uniform voters; the result does not extend to β = 1.The section also extends the baseline derivative and balance conditions across the parameter square.Source: I Preliminaries for all (α, β) · p. 65

J Results for the baseline (α = β = 0): a reflection lemma

For symmetric electorates with β = 0, a candidate’s own-side optimality at a mirror profile suffices for Nash equilibrium. Reflection and monotonicity rule out profitable cross-over deviations, but this argument does not extend to β > 0.

J Results for the baseline (α = β = 0): a reflection lemma

  • With β = 0 and a symmetric electorate, if c1 maximizes its vote share against the mirrored opponent, the mirror profile is a Nash equilibrium.The result holds for arbitrary cost distributions under the stated assumption.Source: J Results for the baseline (α = β =… · p. 67
  • Reflection makes cross-over deviations payoff-equivalent to own-side deviations, while monotonicity in the opponent’s position completes the argument.The proof uses symmetry of voter density and non-decreasing K.Source: J Results for the baseline (α = β =… · p. 68
  • For β > 0, the reflection lemma fails because changing the opponent’s position also changes the opponent’s share, so cross-over deviations require explicit checks.Source: J Results for the baseline (α = β =… · p. 68

K.1 A sufficiency threshold under uniform costs

With uniform costs, a bound on voter-density variation yields single-peaked payoffs and an equilibrium characterization; the analysis extends the sufficiency result to non-uniform costs, identifies conditions for dead zones, and characterizes equilibria with competitive candidates and uniform voters.

K.1 A sufficiency threshold under uniform costs

  • F(α) falls from 4 at α = 0 through 3 at α = 1 toward 1, while equilibrium existence above the sufficiency threshold remains open.The authors attribute the shrinking sufficiency range to the dead-zone portion of the payoff.Source: §K.1 · pp. 69–70
  • When ρv ≤ F(α), own-side payoffs are single-peaked, with maximizers in the no-dead-zone region, and equilibrium requires balance conditions plus no profitable crossover deviation.For symmetric electorates, the bound also guarantees a symmetric equilibrium.Source: §K.1 · p. 69

K.2 Beyond uniform costs

  • For non-increasing cost densities, ρvρk ≤ F(α) preserves single-peakedness and the equilibrium characterization; at α = 0, the bound becomes ρvρk ≤ 4.For symmetric electorates, the result also gives existence; the stated proposition assumes differentiable costs, with the conclusion extending to continuous non-increasing costs.Source: §K.2 · p. 71

K.3 Dead zones

  • For v = Beta(8, 8) and α = 2, the unique symmetric stationary profile (0.381, 0.619) is a genuine equilibrium with xout 1 = 0.26.The passage reports that the equilibrium was verified numerically as mutual global best responses.Source: §K.3 · p. 73
  • Dead zones require sufficiently high alienation and concentrated electorates, whereas uniform electorates have no dead-zone equilibrium.The necessity result also rules out dead zones when its alienation condition is not met.Source: §K.3 · pp. 72–73

L.1 Uniform voters, α = 0: complete characterization

  • With uniform voters and α = 0, every equilibrium is symmetric and has a gap satisfying ψβ(∆) = 1; if K/k is non-decreasing, the equilibrium is unique.The characterization also describes symmetric equilibrium profiles through maximization of F over the stated interval.Source: §L.1 · p. 73

L.2 Uniform voters, α > 0: complete characterization

This section characterizes equilibrium existence and platform positions under uniform voters with α > 0, including how competitiveness and alienation shape the symmetric equilibrium gap and how stationarity and uniform-cost single-peakedness thresholds behave inside and outside the dead zone.

L.2 Uniform voters, α > 0: complete characterization

  • The unique Nash equilibrium for uniform voters with α > 0 is symmetric, has no dead zones, and has gap ∆∗>∆0, the symmetric margin-game gap.The gap is the unique root of J(∆)=Lβ in (∆0,1).Source: §L.2 · p. 75
  • As α ↓ 0, the α > 0 theory deforms continuously onto Theorem 63’s characterization.For doubly-uniform and polynomial K, the reported closed forms match the roots of the respective equations to 10^-6 across the tested α and β values.Source: §L.2 · p. 76

L.3 Comparative statics and their limits

  • Under the stated reverse-hazard-rate conditions, the equilibrium gap decreases strictly with β, increases strictly with α when α > 0, and is weakly larger for more eager electorates at every β.The reverse-hazard-rate comparison holds uniformly in competitiveness.Source: §L.3 · pp. 76–77
  • As β ↑ 1, symmetric stationary gaps converge to ∆0; for α > 0, the approach is linear in 1−β when k(0+) is finite and positive, but slows to order (1−β)^(1/p) when K(t)∼C t^p, p>1.At α=0, if K(t)/k(t)∼ct, the rate is c(1−β); the passage notes c=1 gives a universal 1−β rate.Source: §L.3 · p. 77
  • Without the monotone-reverse-hazard-rate hypothesis, greater competitiveness can increase polarization, and an explicit cost distribution yields three equilibria with a middle branch that increases in β before a fold leaves one equilibrium.The middle and upper branches collide at β=3/47 and ∆=1/2; the equilibrium count falls from three to one through a non-smooth fold.Source: §L.3 · p. 78
  • At a symmetric stationary profile, the slope of K/k at the gap determines whether it is an equilibrium; in the three-root example, the middle root is a local payoff minimum and not an equilibrium.The reported values of χ′β at the three roots are +4.5, −0.5, and +0.94; the negative middle-root value identifies the local minimum.Source: §L.3 · p. 79

L.4 The β ↑1 limit for general electorates

  • For general electorates, any limit of Nash equilibria as β ↑ 1 is a margin-game equilibrium; with α=0 the limit is the median profile, while under Theorem 64’s hypotheses the full limit is the maximally polarized symmetric profile.For uniform voters with α > 0, every limit point lies in the margin-game equilibrium set.Source: §L.4 · p. 79
  • For the arcsine and Beta(0.8,0.8) hollow electorates with α=1 and uniform voting costs, grid certificates show that the margin game has no pure equilibrium.The reported certificates give ε′=6.97×10^-3 for the arcsine electorate and ε′=2.40×10^-3 for Beta(0.8,0.8).Source: §L.4 · pp. 80–81
  • For v=Beta(1/2, 1/2) and α=1, equilibria are numerically found through β=0.90 but not for β≥0.91; certified non-existence holds for β≥0.9168.The numerical threshold is an observation; the cited theorem-grade non-existence result is sharper than the corollary’s conservative bound.Source: §L.4 · pp. 81–82

L.5 Quantile structure at α = 0 under uniform costs

  • With α=0 and uniform costs, every equilibrium has voter mass (1−β)/2 between the platforms, independently of the voter distribution.For symmetric electorates, the quantile equilibrium moves from the quartiles at β=0 toward the median as β approaches 1; the concavity threshold rises with β, and multiplicity is not excluded.Source: §L.5 · pp. 82–83

M.1 Two distribution-free laws of stationarity

  • At every interior stationary profile, (1+α+β)I=(1+α−β)O, while (1+α−β)(I1−I2)=(1+α+β)(O1−O2).These distribution-free inner–outer and asymmetry laws generalize the α=0, uniform-cost mass-gap law.Source: §M.1 · p. 85

M.2 The joint single-peakedness threshold

  • For uniform costs, α>0 and β∈[0,1), the joint threshold Fβ(α) is neither a β-dependent rescaling of F(α) nor shifted by β in one direction; competitiveness stabilizes concavity on the no-dead-zone piece.Source: §M.2 · pp. 86–87
  • For general non-increasing cost densities, no joint single-peakedness threshold follows from voter- and cost-density ratios alone; an additional bound on |k′|/inf k is needed.Uniform costs avoid this obstruction because k′=0.Source: §M.2 · p. 87

M.3 Stationary profiles and a continuation theorem

The section establishes when symmetric stationary profiles and Nash equilibria exist as competitiveness changes, then characterizes when alienation can sustain dead zones. The dead-zone threshold falls with competitiveness, while uniform voters remain immune to dead zones.

M.3 Stationary profiles and a continuation theorem

  • For every β < 1, a symmetric stationary profile exists; if ρv ≤ gβ(α), every such profile lies on the no-dead-zone branch.These are stationary-profile claims, not general Nash-equilibrium existence claims.Source: §M.3 · p. 88
  • For sufficiently small β, a unique symmetric β=0 stationary root continues to a symmetric Nash equilibrium whose positions converge to the β=0 profile.This continuation requires the strict β=0 umbrella condition and the unique-root assumption.Source: §M.3 · pp. 88–89
  • Verified numerical examples extend symmetric equilibria beyond small β, including cases outside the sufficient umbrella and one with unbounded ρv.The examples show that the umbrella is sufficient but not necessary.Source: §M.3 · p. 89

M.4 Dead zones and competitiveness

  • A dead-zone stationary profile requires ρv ≥ gβ(α), without requiring voter or profile symmetry; for symmetric electorates, the corresponding threshold is sufficient for existence.The supplied propositions state the necessity and the symmetric-electorate sufficiency conditions.Source: §M.4 · p. 89
  • As β rises, the dead-zone threshold falls, making alienation dead zones easier to sustain at stationarity; uniform voters have none for any β ∈ [0, 1).The threshold falls from the β=0 bound ρv ≥ λ/α to ρv ≥ gβ(α).Source: §M.4 · p. 90
  • Numerical minima at β = 0, 0.3, 0.6, and 0.9 closely match gβ(2), with differences attributed to the ramp width.The reported minima are 2.0152, 1.5447, 1.2524, and 1.0531; the corresponding bounds are 2.0000, 1.5385, 1.2500, and 1.0526.Source: §M.4 · p. 90

N Existence at α = 0 for every β < 1

At α = 0, quantile profiles provide pure equilibria for broad classes of symmetric electorates and every β < 1, including hollow electorates. With α = 1 and arcsine voters, a certificate instead rules out pure equilibria for β ≥ 0.9168.

N Existence at α = 0 for every β < 1

  • For α = 0, the quantile equilibrium result makes median selection non-vacuous for hollow electorates, whereas hollowness alone does not rule out equilibrium.The cited result covers convex symmetric electorates satisfying its conditions, including the arcsine electorate for all β < 1.Source: N Existence at α = 0 for every β < 1 · pp. 90, 92
  • For α = 0 and uniform costs, the quantile profile is a Nash equilibrium for every β < 1 when the symmetric electorate satisfies the stated conditions; this includes convex electorates and the arcsine density.For convex symmetric densities, the sufficient condition is η(β) ≤ 4/(1−β); the arcsine case satisfies it for every β < 1.Source: N Existence at α = 0 for every β < 1 · pp. 90–92
  • The arcsine electorate has a pure equilibrium for every β < 1 at α = 0, but has none for β ≥ 0.9168 at α = 1.At α = 0, the quantile equilibria converge to the median profile as β approaches 1.Source: N Existence at α = 0 for every β < 1 · p. 92, O The certified non-existence interval · pp. 92–93

O The certified non-existence interval

  • For α = 1, κ ∼ U(0,1), and arcsine voters, a computer-assisted certificate proves no pure equilibrium for β ≥ 0.9168.A pure equilibrium was found numerically at β = 0.90, placing the existence threshold in (0.90, 0.9168]; the lower bound is not certified.Source: O The certified non-existence interval · pp. 92–93
  • The certificate’s numerical boundary is near β = 0.908, where shrinking regret approaches the refinement slack and the certificate stops.The reported pure equilibrium at β = 0.908 and non-existence at β = 0.909 are numerical observations, not certified results.Source: O The certified non-existence interval · p. 93

P Independent verification of the Sections 3–5 results

An independent implementation recomputes numerical claims from Sections 3–5, while exact quadrature is important for security values with arcsine voters. The paper leaves general pure-equilibrium existence at α = 0 conjectural because the arbitrary-(v, k) case remains open.

P Independent verification of the Sections 3–5 results

  • The independent implementation passes all 14 checks of the numerical claims from Sections 3–5, including the scale-invariance proposition.Table 5 describes re-verification using closed forms for uniform-voter payoffs and incomplete-beta integrals.Source: P Independent verification of the… · p. 93, Table 5 · p. 94
  • At δ = 0.02 and α = 1, the Beta(2, 2) quartic check value and prediction are both −9.2444 · 10−7.For the arcsine case, the value is +1.9666 · 10−7 against a +1.9617 · 10−7 prediction; the residual shrinks at smaller δ.Source: P Independent verification of the… · p. 94
  • For arcsine voters, the exact security value is −8.2244 · 10−3, compared with −3.0238 · 10−3 for Beta(0.8, 0.8).The corresponding certified Lipschitz bounds are ≤−6.97·10−3 and ≤−2.40·10−3, respectively.Source: P Independent verification of the… · p. 94
  • Naive quadrature can produce security values wrong by orders of magnitude for the arcsine density’s endpoint singularity.For w(0.999, 0.5), the exact value is −0.0284, while a 103-cell trapezoid rule gives +79.2.Source: P Independent verification of the… · pp. 94–95

Q Conjecture and open items

  • Existence for every β < 1 at α = 0 and arbitrary (v, k) remains a conjecture, since the β = 0 case includes the open existence problem.The paper cites positive evidence for restricted classes but notes that the general case is unresolved.Source: Q Conjecture and open items · p. 95
  • The finiteness proviso of Lemma 53 and the C1 approximation step in Proposition 59 remain open, but neither affects the main-text results.Source: Q Conjecture and open items · p. 95
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