Costly Voting in the Hotelling-Downs Model

Guy Wolf; Reshef Meir

cs.GTSep 24, 2026

Costly Voting in the Hotelling-Downs Model

Authors: Guy Wolf, Reshef Meir

Organizations: Technion—Israel Institute of Technology

Abstract

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.

Figures & tables

Primitives and parameters
$V,v$ ; $K,k$	CDF/density of voter positions; of voting costs $\kappa$ (on $[0,1]$ )
$\rho_{v}$	density ratio $\sup v/\inf v$ (electorate evenness)
Profiles and induced behavior
$c_{i}$ ; $d_{i}(x):=\|x-c_{i}\|$	platform of candidate $i\in\{1,2\}$ ; distance to voter $x$
$c_{1}\leq c_{2}$ ; $\Delta$ ; $m$	labeling; separation $c_{2}-c_{1}$ ; midpoint $\tfrac{c_{1}+c_{2}}{2}$
$B_{i}(x)$ ; $p_{i}(x)$	benefit of voting for $i$ : $d_{-i}-d_{i}$ , Eq. 1 ; share of the voters at $x$ who vote for $i$ , Eq. 2

Table 1: Notation of the model. The last block belongs to the extensions of Section 2.5 ; the model of Sections 2.2 and 2.3 sets

\alpha=\beta=0

. Shorthands introduced later, in the analysis, are collected separately in Table 2 .

Analysis shorthands
$r:=k/K$ ; $G(t):=\int_{0}^{t}K$	reverse hazard rate of $K$ ( Corollary 4 ); integrated CDF
$\psi$ ; $\psi_{\beta}$	virtual cost $t+K/k$ , Eq. 6 ; its $\beta$ -weighting $t+\tfrac{1+\beta}{1-\beta}K/k$ , Theorem 5.1
$I_{i},O_{i}$	marginal voter mass on $i$ ’s inner/outer arm, Lemma 2
$\sigma$ ; $\eta(u)=u\,k(u)/K(u)$	electorate spread; elasticity of $K$ , Proposition 5
$\bar{\Delta}(\alpha)$	turnout-maximizing separation, Proposition 6
$c_{\mathrm{med}}$ ; $w(c,c^{\prime})$	median $V^{-1}(\tfrac{1}{2})$ ; margin of a candidate at $c$ against one at $c^{\prime}$ , Theorems 5.2 and 15

Table 2: Shorthands introduced in the analysis, collected for reference; each is defined where it first appears. The second block is used only from Section 4 on. The symbols of the model itself are in Table 1 .

Increase in…	$\Delta^{*}$	Holds when	Where
electorate spread (IQR)	$\uparrow$	unif. costs, $v$ symmetric, $\rho_{v}\leq 4$	Corollary 1
electorate scale $\sigma$	$\uparrow$	unif. voters, $K$ super-reg.; exact for power costs (any $v$ )	Proposition 5
voting costs (rhr order)	$\uparrow$	unif. voters, $K$ super-reg., any $(\alpha,\beta)$	Propositions 3 , 4.2 and 10
alienation $\alpha$	$\uparrow$	unif. voters, $K$ super-reg., $\alpha>0$ ; $\Delta^{*}\to 1$	Theorem 4.3
competitiveness $\beta$	$\downarrow$	unif. voters, monotone rhr only	Propositions 13 and 2
hollow center, $v^{\prime\prime}(\tfrac{1}{2})>0$	breaks median	margin game, unif. costs, $\alpha>0$ , $v$ symmetric	Theorem 5.4

Table 3: Every force in the model, and which way it moves equilibrium polarization

\Delta^{*}

. The last row is not a gap in the analysis but a theorem: the level of participation is an output of the model, never an input to it.

Setting	$\Delta^{*}$ (doubly unif.)	Umbrella $\rho_{v}\leq$	Where
baseline $(0,0)$	$\tfrac{1}{2}$ , at $(\tfrac{1}{4},\tfrac{3}{4})$	$4$	Corollaries 2 and 1
alienation $(\alpha,0)$	$\tfrac{2+\alpha}{4+\alpha}$	$\min\{\tfrac{2\lambda}{1+\alpha},\tfrac{\lambda}{\alpha}\}$	Corollaries 5 and 0.K.1
competition $(0,\beta)$	$\tfrac{1-\beta}{2}$ (quantile profile)	$\tfrac{4}{1-\beta}$	Corollaries 7 and 9
both $(\alpha,\beta)$	Eq. 14	$\min\{f_{\beta},g_{\beta}\}$	Corollaries 7 and 0.M.1
margin limit $(\alpha,1)$	band: $\Delta\in\big[0,\tfrac{\alpha}{\lambda}\big]$	knife-edge	Propositions 14 and 5.4

Table 4: The baseline and the two dials at a glance. Second column: the equilibrium in the doubly-uniform benchmark (

x,\kappa\sim U(0,1)

), with

\lambda=2+\alpha

. Third column: sufficient density-ratio bound on

v

for the first-order conditions to characterize equilibria under uniform costs, where

f_{\beta}=\tfrac{2\lambda(1+\alpha)+\beta}{(1+\alpha)^{2}}

(concavity) and

g_{\beta}=\tfrac{(1+\alpha)\lambda}{\alpha(1+\alpha+\beta)+\beta\lambda}

(dead zone), from Theorem 0.M.1 ; the bounds come from different arguments and are not claimed tight. The row ‘both’ is stated for

\alpha>0

and does not reduce to the row above it as

\alpha\downarrow 0

f_{\beta}\to 4+\beta\leq\tfrac{4}{1-\beta}

, because the

\alpha=0

bound uses an exact cancellation at the midpoint that the worst-case argument of Theorem 0.M.1 forgoes ( Remark 20 ), and

g_{\beta}\to 1/\beta

bounds a dead-zone piece of the strategy space that vanishes at

\alpha=0

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Result	Verified statement	Outcome
Proposition 7	$zK$ : equilibria fixed, turnout $\times z$ ( $\alpha\in\{0,1\}$ , $z\in\{1,\frac{1}{2},\frac{1}{10}\}$ )	exact
Theorems 4.1 and 4.3	$\Delta^{*}(\alpha)$ solves $\int_{b}^{\Delta}r=\log\frac{\lambda}{2}$ ; unique; increasing in $\alpha$ ; doubly-uniform value $\frac{2+\alpha}{4+\alpha}$	$\leq 2{\cdot}10^{-5}$
Theorem 4.2	$K=t^{2}\succeq_{\mathrm{rhr}}K=t$ polarizes at $\alpha\in\{0,1,2\}$	exact
Proposition 6	$T^{\prime}$ formula ( $10^{-10}$ ); peak $\bar{\Delta}=\frac{\lambda^{2}}{\alpha^{2}+6\alpha+4}$ , $\bar{\Delta}(1)=\frac{9}{11}$ ; $\Delta^{}<\bar{\Delta}$ ; $T^{}=\frac{2(3+\alpha)}{(4+\alpha)^{2}}$	exact
Proposition 5	isomorphism $u_{(V_{\sigma},K)}=u_{(V_{0},K_{\sigma})}$ pointwise ( $5{\cdot}10^{-9}$ ); power-law costs exactly proportional; exponential cost super-proportional	exact
Theorem 5.4	$w_{1}/\delta^{4}\to C(\alpha)v^{\prime\prime}(\frac{1}{2})$ with $C(1)=\frac{13}{27}$ ; $\mathrm{Beta}(2,2)$ and arcsine match the prediction at $\delta=0.02$ (values below); uniform: $0$ to machine precision	confirmed

Appendix

Table 5: Re-verification of results from Sections 3 , 4 and 5 by an independent implementation of the model. “exact” means closed forms (uniform-voter payoffs, incomplete-beta integrals) reproduced to the stated precision.

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Democracy on Rugged Landscapes: Phase Transitions in Optimal Voting Rules

Joshua Nunley

Indiana University, Bloomington, IN, USA

Majority Voting Democracy Landscapes Phase Transitions Terrain

Open paper AskPDF arXiv

Laws and institutions shape individual outcomes through complex interactions with citizens' diverse circumstances, yet how different voting methods navigate this coupled landscape remains poorly understood. We model collective governance as optimization on NK fitness landscapes, where shared bits (laws) are updated by voting while individual bits (personal traits) remain fixed. A cross-dependency parameter

α

controls how legislation's effects depend on individual circumstances. We compare eight standard voting methods and a generalized scoring family across landscape ruggedness

K \in \{1,\ldots,20\}

and

α\in [0,1]

with 1000 runs per configuration. Under direct democracy, the optimal voting method undergoes sharp phase transitions as a function of landscape complexity: cardinal score voting dominates on smooth landscapes, ordinal scoring with

p=0.35

at low-to-moderate ruggedness, Borda count across a wide middle range, and STAR voting at the highest complexity. A two-parameter empirical formula reduces the

(K, α)

plane to a single complexity axis for visualization. Borda count achieves the highest mean fitness and lowest variance across most of the parameter space. We further introduce a representative democracy model parameterized by identity weight

β

and candidate self-interest

p_{\mathrm{self}}

. Representation reshapes the complexity-dependent structure even under favorable conditions: cardinal score voting dominates across most regimes, with plurality emerging as the top method at high

β

and low-to-moderate

p_{\mathrm{self}}

Abstract

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