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Democracy on Rugged Landscapes: Phase Transitions in Optimal Voting Rules
Joshua Nunley
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper models collective governance as optimization on NK fitness landscapes, comparing eight standard voting methods and a generalized scoring family under direct and representative democracy. It demonstrates that optimal voting rules undergo sharp phase transitions based on landscape ruggedness (K) and cross-dependency (α). Cardinal score voting dominates smooth landscapes, ordinal scoring (p=0.35) and Borda count excel at low-to-moderate complexity, while STAR voting performs best at high complexity. Representative democracy reshapes these regimes based on identity weight (β) and candidate self-interest (p_self), with cardinal score generally dominating and plurality emerging under specific high-identity/low-self-interest conditions.
Entities (11)
Relation Signals (7)
NK Fitness Landscape → models → Collective Governance
confidence 98% · We model collective governance as optimization on NK fitness landscapes
Borda Count → dominates → Moderate Complexity
confidence 95% · Borda count across a wide middle range
Cardinal Score Voting → dominates → Smooth Landscapes
confidence 95% · cardinal score voting dominates on smooth landscapes
STAR Voting → dominates → High Complexity
confidence 95% · STAR voting at the highest complexity
Ordinal Scoring (p=0.35) → dominates → Low-to-moderate Ruggedness
confidence 90% · ordinal scoring with p=0.35 at low-to-moderate ruggedness
Representative Democracy → parameterizedby → Identity Weight (β)
confidence 90% · representative democracy model parameterized by identity weight β and candidate self-interest p_self
Representative Democracy → parameterizedby → Candidate Self-interest (p_self)
confidence 90% · representative democracy model parameterized by identity weight β and candidate self-interest p_self
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Abstract
Abstract: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 $\alpha$ 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 $\alpha \in [0,1]$ 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 $(K, \alpha)$ 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 $\beta$ 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 $\beta$ and low-to-moderate $p_{\mathrm{self}}$.
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- Source: https://arxiv.org/abs/2606.02813v1
- Canonical: https://arxiv.org/abs/2606.02813v1
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Democracy on Rugged Landscapes: Phase Transitions in Optimal Voting Rules Josh Nunley1 1Indiana University, Bloomington, IN, USA joshnunl@iu.edu Abstract 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∈1,…,20K∈\1,…,20\ and α∈[0,1]α∈[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=0.35p=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)(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 pselfp_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 pselfp_self. Introduction Democratic governance requires a population of individuals with diverse needs and circumstances to make binding collective decisions (laws) that affect everyone. A tax policy, zoning regulation, or healthcare mandate interacts differently with each citizen depending on their occupation, health, location, and other personal attributes. This interaction between shared institutions and individual heterogeneity is a defining feature of law as a complex adaptive system (Ruhl, 1996), yet formal models of voting rarely account for it. We propose a framework that captures this structure directly. Each individual is represented as a binary string on an NK fitness landscape (Kauffman, 1993), partitioned into a shared portion (bits modified collectively through voting, representing laws) and an individual portion (bits that are fixed and unique to each person, representing personal traits). The NK epistatic structure couples these portions: the fitness contribution of each law-bit depends on nearby trait-bits, and vice versa. A cross-dependency parameter α controls the fraction of each bit’s dependencies that cross between portions. At α=0α=0, dependencies are purely internal: laws interact only with other laws, so legislation affects all citizens identically. At α=1α=1, dependencies are purely cross-portion: every law’s effect depends on individual traits, but traits lack complex internal interactions. Intermediate values produce the richest landscapes. At α≈0.5α≈ 0.5, both internal and cross-dependencies are present, creating the highest effective complexity. This parameter captures a simple spectrum: from purely public goods (α=0α=0) to policies whose effect is entirely mediated by personal circumstances (α=1α=1). Within this framework, each round of voting is a step of collective optimization on a rugged landscape. Eight standard voting methods (plurality, approval, score, Borda, IRV, STAR, minimax, and random dictator) aggregate individual preferences differently, and thus navigate the landscape differently over many rounds. All voting methods receive the same input: a utility matrix U∈ℝM×2VU ^M× 2^V whose (i,p)(i,p) entry is the fitness voter i would obtain under proposal p. A voting rule can therefore be viewed as a transformation of the same utility field: different methods preserve, compress, or discard different aspects of U before selecting a winner. For example, plurality keeps only each voter’s top choice, score voting sums raw cardinal values, and Borda discards magnitudes but preserves full within-voter ordinal structure. From this viewpoint, voting methods differ not only in their standard social-choice properties but also in the kind of information processing they perform. Beyond direct democracy, we develop a unified model of representative democracy motivated by two simple distinctions in representation. The first is whether a candidate’s platform is delegate-like, reflecting constituency welfare, or trustee-like, reflecting the candidate’s own fitness in the model. We parameterize this as pself∈[0,1]p_self∈[0,1], the probability a candidate’s platform reflects their own interest rather than their constituency’s collective welfare. The second is identity versus policy voting: do citizens elect representatives who share their background and circumstances (descriptive representation (Mansbridge, 1999)) or those whose platforms promise the best outcomes? We parameterize this as β∈[0,1]β∈[0,1], the weight on identity-based versus policy-based candidate evaluation. Candidates’ constituencies are defined by nearest-neighbor Hamming distance in individual traits. The cross-dependency α then plays a natural role: when α=0α=0, individual traits are irrelevant to policy outcomes and identity voting carries no information. When α=1α=1, traits fully mediate policy effects, making identity a strong proxy for policy alignment. We study two questions. First, how does voting method choice affect long-run collective welfare (mean fitness, variance, and distributional outcomes) across varying landscape ruggedness K and law-trait coupling α? Second, how does the introduction of representation, and the choice of β and pselfp_self, alter these dynamics, and in what parameter regimes does representation help or hurt? We use the term phase transition in an empirical sense: as K and α vary, the identity of the best-performing voting rule changes abruptly across narrow regions of parameter space, producing stable performance regimes separated by sharp boundaries. These are not thermodynamic phase transitions, but regime transitions in the optimizer induced by changes in landscape structure. The central interpretation is that voting rules act as information transforms: they preserve some features of the voter-utility matrix while discarding others, and landscape complexity determines which preserved features are useful. Our main finding is that voting methods undergo sharp phase transitions as a function of landscape complexity: cardinal score voting dominates on smooth landscapes, ordinal scoring with p=0.35p=0.35 at low-to-moderate ruggedness, Borda count across a wide middle range, and STAR voting at the highest complexity. These transitions are well-described by a two-parameter empirical formula that reduces the (K,α)(K,α) plane to a single complexity axis for visualization. Borda count achieves both the highest mean fitness and lowest variance across the majority of the parameter space. Under representative democracy, the complexity-dependent regime structure is reshaped 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 pselfp_self. These results also suggest a new research question: whether voting rules can be ordered by computable properties of the utility transformation they implement, and whether such properties predict performance across environments. Figure 1: Mean fitness rank (1=best, 9=worst) for each voting method across the full (K,α)(K,α) grid. Green indicates top performance; red indicates poor performance. Regime transitions are visible as sharp boundaries between green and red regions. Related Work NK landscapes and collective search. The NK landscape model (Kauffman, 1993) provides a tunable framework for studying optimization on rugged fitness landscapes, parameterized by the number of components N and the degree of epistasis K. It has been widely applied to organizational decision-making (Levinthal, 1997; Rivkin, 2000), where groups search for high-fitness configurations under varying degrees of interdependence. In these models, organizations navigate rugged landscapes through hierarchical decomposition, imitation, or parallel search. Our work extends this tradition by modeling the search process as democratic voting rather than managerial or evolutionary search, and by introducing a partition between shared (law-like) and individual (trait-like) components that creates voter heterogeneity absent from prior NK collective search models. Computational social choice. The formal study of voting methods has a long history (Arrow, 1951; Gibbard, 1973; Satterthwaite, 1975). Arrow’s impossibility theorem established that no rank-order voting system can satisfy a small set of reasonable criteria simultaneously, while the Gibbard–Satterthwaite theorem showed that deterministic, non-dictatorial methods are susceptible to strategic manipulation. Computational approaches have since analyzed the complexity, manipulability, and axiomatic properties of various methods (Brandt et al., 2016). However, most analyses focus on single-shot elections or equilibrium outcomes rather than iterated collective optimization over many rounds, which is the regime most relevant to ongoing governance. Our framework bridges this gap by evaluating voting methods as long-run optimizers on a landscape where the quality of collective decisions accumulates over time. Law as a complex adaptive system. Legal scholars have increasingly applied complexity theory to law (Ruhl, 1996, 2008), arguing that legal systems exhibit emergent behavior, path dependence, and co-evolution with the populations they govern. Our model operationalizes this perspective: laws (shared bits) and individual traits co-determine fitness through epistatic interactions, and the voting process drives adaptation of the legal component. The cross-dependency parameter α formalizes the degree to which legislation interacts with individual circumstances, a central concern in regulatory design that complexity-theoretic analyses of law have identified but not previously modeled at the level of voting mechanisms. Model We model a population of M individuals making collective decisions on an NK fitness landscape. Each individual i is represented by a binary string i∈0,1Nx_i∈\0,1\^N, partitioned into a shared portion ∈0,1Nss∈\0,1\^N_s (identical across all individuals, representing laws) and an individual portion i∈0,1Ntt_i∈\0,1\^N_t (unique and fixed, representing personal traits), where Ns+Nt=N_s+N_t=N. Fitness Landscape Fitness is computed according to the standard NK model (Kauffman, 1993). Each bit position j contributes a fitness value fjf_j that depends on the bit at position j and K other positions determined by a dependency matrix D: F(i)=∑j=1Nfj(i[j],i[dj,1],…,i[dj,K]),F(x_i)= _j=1^Nf_j\! (x_i[j],\;x_i[d_j,1],…,x_i[d_j,K] ), (1) where dj,1,…,dj,Kd_j,1,…,d_j,K are the K positions on which position j depends. Each fjf_j maps a (K+1)(K\!+\!1)-bit substring to a fitness value drawn uniformly from [−1,1][-1,1]. Increasing K increases the ruggedness of the landscape, creating more local optima and making optimization harder. Cross-Dependency Parameter α We control the coupling between shared and individual portions through a cross-dependency fraction α∈[0,1]α∈[0,1]. For each bit position j, a fraction α of its K dependencies are drawn from the other portion, while the remaining 1−α1-α are drawn from the same portion. At α=0α=0, dependencies are entirely internal: laws affect all citizens identically. At α=1α=1, dependencies are entirely cross-portion: individual traits mediate all policy effects but lack complex internal structure. At α≈0.5α≈ 0.5, both are present, creating the highest effective complexity. Figure 2: Mean fitness rank (top) and variance rank (bottom) vs. fitted complexity K0/20K_0/20. The progression Cardinal score → Ordinal scoring (p=0.35p\!=\!0.35) → Borda (p=1p\!=\!1) → STAR is visible for efficiency; both Borda and ordinal scoring (p=0.35p\!=\!0.35) rank best for equity. Voting Process At each time step, the system selects V bit positions uniformly at random from the shared portion (we use V=2V=2 in direct democracy unless otherwise noted). This defines 2V2^V possible proposals (all binary configurations of those positions). Each individual i evaluates all proposals by computing the fitness they would obtain under each one, producing a utility matrix U∈ℝM×2VU ^M× 2^V. A voting method V then acts on this matrix to select a winning proposal, which is applied to all individuals’ shared portions. Equivalently, we may view each method as a function :ℝM×2V→1,…,2V,V:R^M× 2^V→\1,…,2^V\, (2) or as a two-stage map that first transforms U into proposal scores and then selects the maximiser. Under this interpretation, all methods solve the same task from the same input, but they differ in which information from U they retain. Some use only each voter’s top choice, others use full ordinal rankings, others use cardinal magnitudes, and pairwise methods reprocess the same profile through head-to-head comparisons. We study eight standard methods plus one member of a generalized scoring family: • Plurality: Each voter casts one vote for their highest-utility proposal; the proposal with the most votes wins. • Approval: Each voter approves all proposals that improve their fitness over the status quo (or all tied-for-best if none improve); the most approved proposal wins. • Cardinal score: Sum each voter’s raw utility for each proposal; highest total wins. • Borda: Each voter ranks proposals; points are assigned by rank and summed. • Instant-Runoff (IRV): Iteratively eliminate the proposal with the fewest first-place votes, redistributing those votes, until a majority winner emerges. • STAR: Sum scores to find the top two proposals, then elect the one preferred head-to-head by more voters. • Minimax (Condorcet): Select the proposal whose worst pairwise margin of defeat is minimized. • Random Dictator: A uniformly random voter’s top choice is selected. Serves as a baseline. • Ordinal scoring (p=0.35p\!=\!0.35): A positional scoring rule where rank k (out of n) receives score ((n−1−k)/(n−1))0.35((n\!-\!1\!-\!k)/(n\!-\!1))^0.35. This is a member of the family of positional scoring rules (Saari, 1995) that interpolates between antiplurality (p→0p→ 0) and plurality (p→∞p→∞), with Borda at p=1p=1. At p=0.35p=0.35, the rule is flatter than Borda, assigning substantial weight across the ranking while remaining purely ordinal. This exponent was identified from a sweep over the scoring family as the value that dominates at low-to-moderate complexity. Representative Democracy The representative model adds a second information bottleneck before aggregation occurs. Voters no longer vote directly over policy proposals; instead, they choose among candidates whose platforms may reflect either constituency welfare or personal self-interest. The parameter β controls how much voters choose candidates by identity similarity rather than policy consequences, while pselfp_self controls how often candidates choose platforms for themselves rather than their constituents. In the representative variant, C candidates are drawn uniformly at random from the population. Each candidate c represents a constituency: voters assigned by nearest-neighbor Hamming distance on individual traits. Platform formation (pselfp_self). With probability pselfp_self, candidate c follows a trustee-like rule and selects the proposal maximizing their own fitness; with probability 1−pself1-p_self, they follow a delegate-like rule and maximize mean constituency welfare. This setup interpolates between constituency-focused (pself=0p_self=0) and candidate-focused (pself=1p_self=1) platform formation. Candidate election (β). Voters evaluate candidates by a convex combination of policy utility upolicy(i,c)=Fi(πc)u_policy(i,c)=F_i( _c) and identity utility uid(i,c)=1−dH(i,c)/Ntu_id(i,c)=1-d_H(t_i,t_c)/N_t: u(i,c)=(1−β)u~policy(i,c)+βuid(i,c),u(i,c)=(1-β)\, u_policy(i,c)+β\,u_id(i,c), (3) where β∈[0,1]β∈[0,1] is the identity weight and both components are normalized to [0,1][0,1]. The same voting method used in direct democracy then elects a winning candidate whose platform is implemented. The (β,pself)(β,p_self) plane nests four limiting cases. Rows: election type (policy vs. identity) × platform type (constituency vs. self-interested): Model β pselfp_self Trustee-like 0 1 Delegate-like 0 0 Identity–trustee-like 1 1 Identity–delegate-like 1 0 The cross-dependency α interacts with β: when α=0α=0, identity utility carries no policy information and β>0β>0 adds variation unrelated to policy outcomes; when α=1α=1, traits fully mediate policy effects, making identity a strong proxy for policy alignment. Experiments All experiments use N=50N=50 bit positions with a voting portion of 0.50.5 (Ns=Nt=25N_s=N_t=25), a population of M=100M=100 individuals, V=2V=2 bits voted on per round, and all individuals sharing identical initial voting bits with unique, randomly generated individual bits. Experiment 1: K×αK×α Sweep (Direct Democracy) We sweep landscape ruggedness K∈1,2,…,20K∈\1,2,…,20\ and cross-dependency α∈0.00,0.05,0.10,…,1.00α∈\0.00,0.05,0.10,…,1.00\ (21 levels) under direct democracy for all eight standard voting methods. Each configuration is run for 150+50K150+50K iterations with 1000 independent runs, providing high-resolution coverage of the full (K,α)(K,α) plane. For each run we record mean fitness, fitness variance, minimum and maximum fitness at every iteration. In addition, we sweep a generalized positional scoring family parameterized by exponent p: rank k (out of n) receives score ((n−1−k)/(n−1))p((n\!-\!1\!-\!k)/(n\!-\!1))^p. This family includes antiplurality (p→0p→ 0), Borda (p=1p=1), and plurality (p→∞p→∞) as special cases. We evaluate 28 exponents spanning p∈[0.05,10]p∈[0.05,10] across the full (K,α)(K,α) grid with 1000 runs per configuration, identifying p=0.35p=0.35 as the exponent that dominates at low-to-moderate complexity. This ninth method is included in all subsequent analyses. Experiment 2: Representative Democracy Using the representative democracy model described in Section 3, we sweep β∈0, 0.5, 1β∈\0,\,0.5,\,1\ and pself∈0, 0.5, 1p_self∈\0,\,0.5,\,1\ (9 configurations) with C=5C=5 candidates selected uniformly at random, V=4V=4 bits voted on per round (16 proposals), across the full landscape grid: K∈1,…,20K∈\1,…,20\ and α∈0.0,0.1,…,1.0α∈\0.0,0.1,…,1.0\. All nine voting methods are tested, yielding 17,820 configurations with 500 independent runs each. The larger proposal space (V=4V=4 versus V=2V=2 in Experiment 1) ensures that candidates can hold distinct platforms and constituency welfare can meaningfully diverge from population welfare. Using the same (K,α)(K,α) grid as Experiment 1 allows direct comparison of the fitted complexity axis under representation. Results Direct Democracy: K×αK×α Sweep We present three results: (1) method performance is regime-dependent, (2) the (K,α)(K,α) plane reduces to a single complexity axis via a fitted formula, and (3) efficiency (mean fitness) and equity (fitness variance) dissociate. Mean Fitness Rank Phase Diagram Figure 1 shows the rank of each voting method (1=best, 9=worst) by terminal mean fitness at every (K,α)(K,α) configuration. No single method dominates. The (K,α)(K,α) plane is partitioned into regimes where different methods are optimal. At the lowest complexity (K≤3K≤ 3, α>0α>0), cardinal score voting ranks first: a simple utilitarian sum suffices when the landscape is nearly additive. Among the eight classical methods, as K increases to 44–99, plurality takes over in the mid-α range. However, ordinal scoring with p=0.35p=0.35 (identified from a sweep over exponents in [0.05,10][0.05,10]) dominates this entire low-to-moderate regime, outperforming plurality while maintaining Borda’s favorable equity properties (Figure 2). At moderate complexity (K≈8K≈ 8–1414), Borda count dominates: full ordinal information becomes necessary for top performance on a rugged landscape. At the highest complexity (K≥15K≥ 15), STAR voting emerges as rank 1 in the mid-α region, with its two-stage mechanism (scoring followed by pairwise runoff) suited to highly rugged landscapes. Approval voting performs poorly across much of the mid-α region. It exhibits a U-shaped pattern: it ranks first at extreme α values (α≈0α≈ 0 and α≈1α≈ 1) but drops to rank 7 across the entire mid-α band (0.25≤α≤0.800.25≤α≤ 0.80) for K≥5K≥ 5. This boundary is sharp, not gradual. One plausible mechanism is that approval is thresholded against the status quo: at intermediate cross-dependency, the status quo becomes a weak proxy for the collective optimum, so the approval profile discards useful preference information. Random dictator is rank 9 except at α=0α=0. This confirms its role as a coordination-free baseline. A Fitted Complexity Parameter To visualize regime transitions on a single axis, we fit parabolic iso-complexity curves K=a(K0−1)2(α−12−bK0−1)2+K0,K=a\,(K_0-1)^2 (α- 12- bK_0-1 )^\!2+K_0, (4) where K0∈[1,20]K_0∈[1,20] indexes complexity and a, b are fitted by maximizing rank-vector uniformity along iso-K0K_0 contours (Gaussian-kernel-weighted rank variance, bandwidth ε=0.5 =0.5). The fitted values are a=2.35a=2.35, b=0.29b=0.29. The center μ(K0)=12+b/(K0−1)μ(K_0)= 12+b/(K_0-1) converges to 12 12 at high complexity, consistent with α=12α= 12 producing the richest dependency structure. For K0=1K_0=1, where the parameterization is singular, we treat the degenerate iso-complexity contour separately as the flat line K=1K=1. Efficiency and Equity vs. Complexity Figure 2 shows mean fitness rank (top) and variance rank (bottom) for all nine methods plotted against normalized complexity K0/20K_0/20. The regime transitions collapse cleanly onto this single axis for both measures. Efficiency. Cardinal score dominates at low complexity, then ordinal scoring (p=0.35p\!=\!0.35) takes over through the low-to-moderate range, Borda dominates the mid-range, and STAR emerges at the highest complexity. Among classical methods alone, plurality briefly leads in the low-to-moderate range, but p=0.35p\!=\!0.35 strictly dominates it there. Equity. Both Borda and ordinal scoring (p=0.35p\!=\!0.35) achieve rank 1–2 for variance (most equitable) across nearly the entire complexity range. Ordinal scoring (p=0.35p\!=\!0.35) achieves the best or near-best equity while simultaneously dominating efficiency at low-to-moderate complexity, a combination no classical method achieves. Plurality and random dictator consistently produce high variance; IRV shows poor equity at high complexity. Figure 3: Mean rank vs. complexity for all nine methods across the 3×33× 3 (β,pself)(β,p_self) grid. Rows vary β (identity weight: 0, 0.5, 1); columns vary pselfp_self (candidate self-interest: 0, 0.5, 1). Representation reshapes the regime structure: high pselfp_self produces chaotic rankings; high β favors simpler methods. Representative Democracy We sweep β∈0,0.5,1β∈\0,0.5,1\ and pself∈0,0.5,1p_self∈\0,0.5,1\ across the full (K,α)(K,α) grid (K∈1,…,20K∈\1,…,20\, α∈0.0,0.1,…,1.0α∈\0.0,0.1,…,1.0\) with V=4V=4 bits, C=5C=5 candidates, and 500 runs per configuration. The (β,pself)(β,p_self) Grid Figure 3 presents a 3×33× 3 grid of rank-versus-complexity curves, one panel for each (β,pself)(β,p_self) configuration, using the same fitted complexity axis (Eq. 4) for comparability with direct democracy. Cardinal score voting ranks first or near-first in most panels of the grid. The exception is the high-β, low-to-moderate-pselfp_self regime (bottom-left two panels): plurality emerges as the top-ranked method here, since its concentrated signal resists the noise introduced by identity voting better than methods that require richer preference information. Regime structure persists under ideal representation. At (β=0,pself=0)(β=0,\,p_self=0) (policy-based voting with delegate-like platform formation) the rank-versus-complexity curves most closely resemble direct democracy. Multiple methods compete across the complexity axis, and the progression from simple to sophisticated methods is visible. This configuration represents ideal representation in the model: voters evaluate candidates on policy merit, and candidates choose platforms for their constituencies. Rank ordering becomes chaotic as pselfp_self increases. Moving rightward in the grid, the rank curves grow increasingly chaotic. At pself=1p_self=1, no method clearly dominates at any complexity level; the rank curves cross repeatedly and erratically, with no stable ordering. Self-interested representatives introduce noise that overwhelms the complexity-dependent signal: the proposals implemented reflect individual self-interest rather than any aggregation of collective preferences, rendering the choice of voting method effectively random. Method differentiation decreases as β increases. The effect is less severe than under high pselfp_self. At β=1β=1, voters elect representatives based on shared traits rather than policy outcomes. The resulting representatives may share their constituents’ circumstances but do not necessarily champion their preferred policies, reducing the signal available to the voting method. Voting method has little systematic effect at (β=1,pself=1)(β=1,\,p_self=1). This corner (identity voting with candidate-focused, trustee-like platform formation) produces the least differentiated rank curves. Representatives are elected for who they are, not what they propose, and then implement whatever benefits themselves. No voting method, in this model, recovers a strong aggregation signal when the representation layer discards both policy information (high β) and constituency alignment (high pselfp_self). Delegate-like Advantage The delegate-like endpoint (pself=0p_self=0) consistently outperforms the candidate-focused, trustee-like endpoint (pself=1p_self=1) in mean fitness when α>0α>0, with the largest gaps at mid-α and high K. At K=10K=10, α=0.5α=0.5, the delegate-like advantage is 1.39 mean fitness units (best-method fitness of 3.45 vs. 2.04). At K=20K=20, α=0.5α=0.5, the delegate-like endpoint more than doubles collective welfare (1.82 vs. 0.78). At α=0α=0, the advantage vanishes entirely; on landscapes with no cross-dependency, self-interest and constituency welfare coincide. The β–α Interaction Identity-based voting (β=1β=1) underperforms policy-based voting (β=0β=0) most severely at mid-α. At K=10K=10, α=0.5α=0.5, the penalty is 1.57 fitness units (best-method fitness of 3.45 under β=0β=0 vs. 1.88 under β=1β=1). At α=0α=0, the three β values converge exactly: individual traits carry no policy information, so identity voting introduces noise but does not systematically bias outcomes. At α=1α=1, the β values nearly converge: traits fully mediate policy effects, making identity similarity a reasonable proxy for policy alignment. Discussion Direct Democracy: Complexity-Dependent Optimization Under direct democracy, no voting method universally dominates: the optimal method depends on the complexity of the underlying decision landscape. Our framework asks a different question than the classical impossibility theorems of Arrow and Gibbard-Satterthwaite, which characterize methods by axiomatic properties on fixed preference profiles. We find that the empirical performance ordering of methods is a function of landscape complexity and undergoes sharp phase transitions. The progression Cardinal score → Ordinal scoring (p=0.35p\!=\!0.35) → Borda (p=1p\!=\!1) → STAR can be interpreted in terms of information aggregation depth. On smooth landscapes, preferences are nearly aligned and a simple sum of utilities suffices. As ruggedness increases, preferences fragment: a flatter ordinal scoring rule (p=0.35p\!=\!0.35), which spreads weight across the ranking, outperforms both the raw sum and plurality’s hard cutoff. At moderate complexity, full ordinal information becomes necessary for top performance (Borda, p=1p\!=\!1), and at the highest complexity, even ordinal information must be combined with pairwise verification (STAR). Each successive method in the progression extracts richer information from voters’ preferences, at increasing computational and communicative cost. Among classical methods, plurality occupies part of the regime where p=0.35p\!=\!0.35 dominates, suggesting that plurality captures a real low-complexity signal but that its hard cutoff is too coarse to exploit it fully. Voting Rules as Utility Transforms In our model, all voting methods operate on the same utility matrix U∈ℝM×nU ^M× n and can be understood as different transformations of this common input: plurality compresses each row to an argmax, Borda replaces magnitudes with ordinal ranks, score voting sums cardinal values directly, and pairwise methods reprocess the profile through head-to-head comparisons. Each transformation preserves different features of U—top-choice information, ordinal structure, cardinal intensity, or pairwise margins—while discarding others. This framing suggests a research direction. Given assumptions about the fitness landscape (ruggedness, cross-dependency structure), one might prove which properties of the utility transform lead to optimal collective optimization, connecting computational social choice to optimization theory. For instance, our results suggest that on smooth landscapes, preserving cardinal magnitudes is optimal (score voting), while on rugged landscapes, preserving full ordinal structure (Borda) or pairwise comparisons (STAR) becomes necessary. A formal theory characterizing this relationship could yield optimality results for voting methods as a function of landscape properties, rather than relying on axiomatic criteria alone. Borda’s Robustness and Equity Under direct democracy, Borda count is the most robust classical method across the middle complexity range, combining high mean fitness with consistently low variance. Its worst observed rank is approximately 4. This complements classical axiomatic work that identifies Borda by distinctive structural properties (Young, 1974), but in a dynamic, multi-round optimization setting far from the one-shot environments usually studied in social choice theory. Representation as an Information Bottleneck Subject to the model’s simplifications (sincere voters, binary candidate platforms, random NK landscapes; see Limitations), representation reshapes the regime structure even under favorable conditions. At (β=0,pself=0)(β=0,\,p_self=0), the direct democracy ordering is partially visible but already altered: representation acts as a lossy compression, and even candidates using the delegate-like rule cannot fully reconstruct the population-level signal that direct democracy exploits. The two representation parameters introduce qualitatively different distortions: high pselfp_self produces chaotic rank curves with no stable method ordering, while high β favors simpler methods. Plurality performs comparatively well when voters select on identity rather than policy, since its concentrated signal is more resistant to noise than methods requiring richer preference information. Voting method choice matters most under delegate-like representation. As either β or pselfp_self increases, the binding constraint shifts from aggregation quality to representation quality. Within these limits, the model identifies two ways the representation layer can weaken the signal available to a voting rule. High β captures selection on identity rather than policy. High pselfp_self captures candidates choosing platforms for themselves rather than constituents. In both cases, voting rules matter most when voters receive useful policy signals and candidates remain aligned with constituents (Mill, 1861; Achen and Bartels, 2016). Limitations Several simplifications limit generalizability. All agents vote sincerely; strategic voting could qualitatively change method comparisons, particularly for IRV and Borda (Gibbard, 1973). Individual portions are fixed; in reality, personal circumstances respond to collective decisions. The NK landscape is random; results are averages over realizations, not predictions for specific policy domains. The proposal space (V=2V=2 for direct, V=4V=4 for representative) is small. Our constituency assignment uses Hamming distance, a simplification of real representational geography. Finally, we study only three discrete values of β and pselfp_self; finer resolution might reveal additional structure. Conclusion We have presented an NK landscape framework for studying voting methods as collective optimizers. The main findings are: 1. Complexity-dependent method ordering. Under direct democracy, method performance undergoes sharp phase transitions: cardinal score → ordinal scoring (p=0.35p\!=\!0.35) → Borda (p=1p\!=\!1) → STAR as complexity increases. No method universally dominates. 2. Borda’s robustness. Under direct democracy, Borda count is the most robust classical method across the middle complexity range, combining high mean fitness with consistently low variance. 3. A flatter ordinal scoring rule. Under direct democracy, a sweep over the generalized scoring family reveals that p=0.35p=0.35 (flatter than Borda) dominates plurality in the low-to-moderate regime while matching Borda’s favorable variance properties. 4. Voting rules as utility transforms. The model frames voting methods as transformations of a common utility matrix, suggesting that formal connections between transform properties and landscape-dependent optimality may be possible. 5. Representation as information bottleneck. Under representative democracy, the regime structure is reshaped by candidate self-interest and identity voting, with simpler methods gaining advantage as representation degrades. Future work could explore: • strategic voting; • formal connections between utility transform properties and landscape-dependent optimality; • fitness-biased candidate selection to model incumbency advantage; • a third, socially transmitted “belief” portion of the bitstring, so that opinion dynamics on networks can interact with collective optimization. Within the limits of the model, the broader implication is that voting reform cannot be evaluated independently of assumptions about the structure of the policy environment. If policy effects are smooth and broadly aligned, cardinal aggregation performs well. If effects are rugged and heterogeneous, ordinal and pairwise structure become more valuable. Voting rules are therefore not only procedures for choosing winners; in this model, they are adaptive information-processing mechanisms whose performance depends on the landscape they are asked to navigate. Acknowledgments This work used the Big Red 200 supercomputer at Indiana University, supported by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute. AI tools were used to assist with editing the manuscript and assisting with code development. Multiple independent AI systems were employed, with outputs cross-referenced to identify inconsistencies. The author reviewed all AI-generated and AI-edited content, taking full responsibility for the final work. All code was further validated through systematic testing and visualization. References C. H. Achen and L. M. Bartels (2016) Democracy for realists: why elections do not produce responsive government. Princeton University Press. Cited by: Representation as an Information Bottleneck. K. 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