A Replication and Analysis of Tiebout Competition Using an Agent-Based Computational Model

Abstract

Replication is a critical element of the scientific process. This article is an effort to contribute to the slowly growing literature concerning the replication of agent-based computational models. We present a replication of Kollman, Miller, and Page’s model of Tiebout sorting. In that model, individual agents with heterogeneous preferences for government policies select among jurisdictions that offer the most satisfactory package of government services. This project makes three contributions to the literature. First, our successful replication provides the research community with a modernized version of that seminal model. Second, we confirm that earlier results with respect to the single jurisdiction setting are highly robust with respect to voter preferences, while the results for multiple jurisdiction settings are more sensitive. Finally, we demonstrate a technique for conducting sensitivity analyses that leverages a high-dimensional experimental design.

Keywords

tiebout competition agent-based modeling voting behavior local government

Introduction

Replication is a critical element of the scientific process. It confirms earlier findings and identifies exceptions to those findings. Although agent-based computational models have grown in popularity in recent years, the effort to replicate the results of earlier researchers has grown only slowly. Axtell, Axelrod, Epstein, and Cohen (1996) are generally credited with the first attempt to dock two different simulation models, while Hales, Rouchier, and Edmunds (2003) report the results of a model-to-model workshop, the intent of which was to encourage the replication of computational models. Wilensky and Rand (2007) argue that replication is even more important for computational models because it “increases our confidence in the model verification, leads to a reexamination of the original validation of the model, and at the same time, facilitates a common language and understanding among modelers.” See Sansores and Pavon (2005), North and Macal (2009), and Zhong and Kim (2010) for relatively recent examples of efforts to replicate computational models. Miodownik, Cartrite, and Bhavnani (2010) is an example of an attempt to extend and build upon a previously developed agent-based model that examines civic traditions in Italy.

This article is an effort to contribute to this growing literature. We present a replication of Kollman, Miller, and Page’s (1997) model of Tiebout sorting (hereafter KMP97). In that model, individual agents with heterogeneous preferences for government policies select among jurisdictions that offer the most satisfactory package of government services. We replicate the model in Java using the Recursive Porous Agent Simulation Toolkit (REPAST).¹ We then extend the analysis along a number of margins. First, we determine how sensitive the results in Kollman, Miller, and Page (1997) are to a wider array of types of voter preferences. Finally, we borrow from the operations research literature on data farming and employ a high-dimensional experimental design in support of a thorough sensitivity analysis.

This project makes three contributions to the literature. First, we build a replication of KMP97 that achieves relational equivalence overall, and distributional equivalence in certain aspects, thus providing the research community with a modernized version of that seminal model. Second, we find that KMP’s results with respect to the single jurisdiction setting are highly robust with respect to voter preferences, while their results for multiple jurisdiction settings are sensitive to variations. Finally, we demonstrate a technique for conducting sensitivity analyses that leverages a high-dimensional experimental design.

The Tiebout Hypothesis

A vast literature exists that examines the ability of local governments to provide public goods efficiently. However, most of the literature overlooks the differential effects of the collective choice institutions. In addition, the distribution of voter preferences may enhance or hinder the performance of political institutions. The dynamics between political institutions and voter preferences may greatly affect a local government’s ability to leverage its monopoly position in order to extract rents from its constituents.

In his seminal paper, Tiebout (1956) argues that, given certain assumptions, local governments provide a supply of public goods that approaches the competitive level. “If consumer-voters are fully mobile, the appropriate local governments, whose revenue-expenditure patterns are set, are adopted by the consumer-voters” (p. 424). Because local governments offer a variety of services and service levels, and constituents “vote with their feet”, local governments’ production of public goods will approach the efficient level. Local governments that succeed in providing those public goods that consumer voters demand tend to be rewarded with new constituents (taxpayers) and those who fail tend to be faced with declining populations and shrinking tax revenue. Some of the assumptions upon which Tiebout (1956, p. 419) relies include (1) consumer voters are fully mobile and can move freely; (2) consumer voters have full knowledge; (3) large number of communities; (4) restrictions due to employment are neglected; and (5) public services exhibit no externalities upon other jurisdictions.

In short, the Tiebout hypothesis states that voters will move to localities that offer that package of government services and taxation that fits their preferences and that voter’s ability to relocate places an effective limit on local government’s monopoly power. The present article examines the extent to which Tiebout’s conclusion holds, given a somewhat richer model that relaxes several of his assumptions—namely, those pertaining to the knowledge levels of local politicians and their abilities and incentives to compete politically for the favor of constituents.

The KMP97 Model

In Kollman, Miller, & Page (1997), the authors develop a computational model to measure the ability of different institutions to sort voters who possess preferences over the allocation of public goods into different jurisdictions. Heterogeneous agents reside in one of a fixed number of jurisdictions and are endowed with fixed, linearly separable preferences over a specified number of binary issues. During each time step, the jurisdictions determine their platform of government services to offer according to the voting rule in effect and the preferences of the constituents. The agents are then free to move to the jurisdiction that offers them the highest expected utility. The collective choice problem is resolved using one of the following institutions: democratic referendum, direct competition, or proportional representation. Agents are unable to forecast the effects that their presence will have on the platforms selected in jurisdictions to which they may move, which is a key aspect of their bounded rationality.

One of the authors’ overarching conclusions is the process through which the agents seek their optimal jurisdiction, and through which the imperfectly informed parties pursue the election-winning platform and is significantly more stable than McKelvey’s (1976) findings would suggest. In addition, they find that the order of how well political institutions perform relative to one another (in terms of increasing aggregate utility) in a single jurisdiction setting is generally opposite of how they perform in a multiple jurisdiction setting. Namely, democratic referendum, an institution that provides highly stable outcomes is found to be superior to the other collective choice mechanisms in a single jurisdiction model, followed by direct competition and proportional representation. When the number of jurisdictions is increased, proportional representation is found to outperform the others. In fact, as the number of jurisdictions increase, so too does the relative ability of proportional representation to sort voters according to their preferences (Kollman, Miller, & Page, 1997). This relationship is due to the propensity for unstable political outcomes under proportional representation and direct competition, thus allowing jurisdictions to “break out” of suboptimal equilibria, with greater regularity than democratic referendum.

Institutions and Hill-Climbing Parties

KMP consider the following voting institutions:

Democratic Referendum: A democratic referendum involves a simple majority rule vote on each issue. The outcome is the median level for each issue in the jurisdiction. This is the only institution examined which does not involve party participation.

Direct competition: Direct competition is a plurality contest between parties. Agents vote for the party proposing the platform that yields them the highest utility and the jurisdiction wholly adopts the winning party’s platform.

Proportional Representation: Under a proportional representation regime, each voter votes for the party proposing the platform that yields his or her the highest utility. Each party is then allocated “seats” according to the proportion of votes received.² The parties then vote separately and sincerely on each issue in a democratic referendum.

For political institutions that involve political parties, such as direct competition and proportional representation, a given number of parties compete within each jurisdiction during the electoral process. They stand separate and distinct from the voting populace and lack complete information regarding the preferences of voters within the constituency. They gradually adapt their platform by employing various search mechanisms in an attempt to find the platform that will garner the most votes in the election. The parties seek only reelection and have no policy preferences of their own. The adaptive technique primarily used by the parties in the present article is known as the hill-climbing heuristic procedure. Hill climbing is intended to simulate a party that “fine-tunes the policy positions of its candidate using polling and focus groups” (Kollman, Miller, & Page, 1998, p. 144).³ See Kollman, Miller, and Page (1992) for another model that implements adaptive political parties.

Replicating the hill-climbing algorithm is by far the most critical and challenging aspect of this project. In this section, we parse the exact text in order to highlight the assumptions we make concerning our interpretation of their description. Steps S1 through S3 are three consecutive sentences that outline the algorithm and are quoted directly from Kollman, Miller, and Page (1997, p. 983). We emphasize those elements that require an assumption or some amount of clarification.

S1: “A randomly generated current platform is given to each party prior to any elections.”

We assume that platforms are randomly assigned to parties at time t = 0 and that the starting point for each party during subsequent time steps is the platform from the previous time step. We experiment with randomly assigning new platforms to parties at the beginning of each time step though do not present the results here.

S2: “When a party is given a chance to adapt, it first generates a new platform by randomly perturbing its current platform on up to three issues.”

Because Kollman, Miller, and Page do not explicitly state that incumbent parties refrain from adapting their platforms, we assume that all parties adapt their platforms. We also experiment with a variation in which incumbent parties do not adapt their platforms, as in Kollman, Miller, and Page (1998).

Although the choice of words “up to” leaves some ambiguity, our algorithm perturbs exactly three issues during this stage. Earlier versions of our replication randomly drew the number of issues to perturb (one, two, or three) from a uniform distribution but suffered substantial alignment problems, especially with respect to the number of agent relocations at various design points.

S3: “If the new platform yields a higher vote total, it becomes the party's current platform; otherwise the current platform remains unchanged.”

For this step, we assume that parties compare the performance of their current (adapted) platform and “new” candidate platform separately in an informal poll of all voters against the adapted platforms of all other parties, as opposed to the platforms the parties ran in the previous election.

S4: A party continues its adaptive search for eight iterations (S1–S3), after which the next party is given a chance to adapt. A single round in which every party in the jurisdiction adapts its platform is a campaign cycle. For purposes of replication, each jurisdiction conducts five such cycles, with eight adaptive iterations for each party. At the conclusion of the “campaign,” the parties compete in an election in which their newly adapted platforms are offered.

Notation and Measures of Effectiveness

The following is a formal specification of the KMP97 model, the notation for which is drawn from Kollman, Miller, and Page (1997, 1998). Let N_a agents select among N_j different jurisdictions. In each jurisdiction, a total of N_i issues are resolved collectively. Let p_ji ∊ {Yes, No} give the position of jurisdiction j on issue i. When political parties are considered, let p_rji ∊ {Yes, No} give the position of party r in jurisdiction j on issue i. Let Platform P _j ∊ {Yes, No} ^Ni denote the vector of decisions across all N_i issues in jurisdiction j. Jurisdictions take binary positions on each issue, thus we think of such positions as the presence or absence of a particular public good, policy, or regulation.

Let ν_ai ∼ Uniform (−400/N_i , 400/N_i ) and give agent a’s utility for issue i. Furthermore, agent a’s utility from platform P _j is given by:

u_{a} (P_{j}) = \sum n_{a i} \cdot δ (p_{j i}),

where δ(Yes) = 1, and δ(No) = 0. An agent’s ideal platform will yield an expected value of 100, while the expected utility of a random platform is zero (Kollman, Miller, and Page, 1997).

For the replication process, we consider three outputs as measures of effectiveness. The first is per-capita utility, which is simply the mean utility of all agents in the population. In addition to the influence of political institution, per-capita utility tends to increase with the number of jurisdictions and increase with the number of parties.

The second measure we examine is the number of agent relocations during a simulation run. An agent executes a relocation whenever it changes jurisdictions. If a particular voting institution tends to encourage a large number of voters to relocate to different jurisdictions, this could be beneficial for a number of reasons. A relatively high number of moves can be indicative of instability, which, if it happens early in the simulation, could facilitate the sorting of voters into homogeneous constituencies that achieve high average utility. However, if the instability persists, it might be indicative of an inability to produce platforms that generate high utility for constituents.

We also look at a measure of interjurisdiction distance. The interjurisdiction distance between the platforms of two jurisdictions is equal to the number of issues for which policies differ divided by the total number of issues (Kollman, Miller, & Page, 1997, p. 989). We examine the mean interjurisdiction distance, which is the mean of all pairwise combinations.

A Successful Replication of KMP97

Wilensky and Rand (2007) describe six margins along which an original model and subsequent model may differ. They are time, hardware, languages, toolkits, algorithms, and authors. Approximately 16 years separate the creation of the two models by different authors. That fact, coupled with the fact that the current model is implemented in Java using the REPAST libraries, tools that did not exist when Kollman, Miller, and Page developed their model, means that the two models differ in hardware, languages, and toolkits. The algorithms were recreated as faithfully as possible from their descriptions in the referenced works, as the current author had no access to the original source code or output. Table 1 outlines additional details of the replication effort.⁴

Table 1.

Replication Summary.

Replication Standard	Partial Distributional Equivalence Relational Alignment
Focal measures	Per-capita utility Interjurisdiction distance Agent relocations
Level of communication with previous authors	None
Familiarity with language of original model	None
Examination of source code	None
Exploration of parameter space	None

We select per-capita utility, mean interjurisdiction distance, and agent relocations as focal measures because they are the only response variables Kollman, Miller, and Page discuss in the 1997 article. We place special emphasis on per-capita utility for purposes of the replication assessment because the primary focus of their analysis, and our subsequent analysis, is the effect of voting rules on voter welfare.

Axtell et al. (1996) describe three possible replication standards: numerical identity, distributional equivalence, and relational equivalence. Numerical identity achieves exact output for the same input parameters, an exceedingly difficult feat for models with stochastic elements. This level of replication is neither necessary nor feasible for the present project.

When possible, we strive to achieve distributional equivalence, which is replication such that the two models produce “distributions of measurements that are statistically indistinguishable” from each other (Axtell, Axelrod, Epstein, & Cohen, 1996, p. 127). In general, the process to determine whether the output distributions from two models are statistically similar depends on the distributions under consideration. Per-capita utility, as presented in Kollman, Miller, and Page (1997), is the average of the utility of the 1,000 agents during a particular run. Kollman, Miller, and Page report the mean per-capita utility for each of their design points after 200 replications. Thus, these 200 observations of mean per-capita utility comprise a sampling distribution. The Central Limit Theorem gives us a strong reason to believe that this sampling distribution is Normal $(\hat{μ} = \overset{ˉ}{X}, σ = s . e . (\overset{ˉ}{X}) \cdot \sqrt{200}$ ) (Devore, 2000, p. 233). For each design point under consideration, we employ a one-sample Kolmogorov goodness-of-fit test with the null hypothesis that there is no difference between our output distribution (200 observations) of per-capita utility and a corresponding normal distribution with KMP’s mean and standard deviation (Conover, 1999, pg 428).

Similarly, interjurisdiction distance is the average pairwise difference between the platforms of each jurisdiction and has a similarly constructed sampling distribution. Although, the per-run samples are small in some cases (i.e., for three jurisdictions there exist only two pairwise differences), at 200 replications the Central Limit Theorem ought to offer some protection here. We also employ a one-sample Kolmogorov goodness-of-fit test to assess equivalence in this case as well.

There are two limitations to assessing distributional equivalence in the manner we have chosen. The first is that, admittedly, we only test distributional equivalence at the level of the experiment. In order to attain a better picture of distributional equivalence, it would be necessary to compare the two models and measure how utility is distributed across each of the 1,000 agents at the end of each simulation run. Second, our Kolmogorov goodness-of-fit test compares the output distribution of the replication, with that of a theoretical normal distribution, rather than KMP’s experimental output. Although we believe we have made the best use of all available information, ultimately, the success of the replication hinges on whether we achieve relational equivalence.

Finally, relational equivalence between models achieves the “same internal relationship among their results” (Axtell et al., 1996, p. 135). If the replicated output generates the same qualitative conclusions, such as treatment A is superior to treatment B, then relational equivalence has been achieved. We do not have sufficient information regarding the distribution of agent relocations in order to confirm distributional equivalence. However, we employ comparison of means tests as an assessment of potential for distributional equivalence since similar distributions would share similar measures of centrality. Ultimately, we only pursue relational equivalence for this measure.

Although we only consider the parameter space that Kollman, Miller, and Page examine in their 1997 paper, in subsequent sections, we expand the parameter space and conduct a thorough sensitivity analysis of their results. Finally, it is a testament to KMP’s scholarship and clarity of exposition that the current author is able to develop a replication of their model with no contact with the authors or with the original model in any form.

Replication Assessment

In this section, we compare the results of experiments with the present model to KMP’s findings regarding similar experiments. As in Kollman, Miller, and Page (1997), all scenarios presented involve 1,000 agents with preferences on 11 binary issues and 10 election cycles per run. We replicate each design point 200 times. In addition, we employ common random numbers in each scenario in a further effort to reduce the variance between the estimates and enhance comparative ability⁵ (Law & Kelton, 2000). The random number generators that produce the agent’s preferences and initial locations are “restarted” at the same point in the pseudo-random number sequence for each scenario.

Single jurisdiction per-capita utility

The results for the single jurisdictional model are shown in Table 2 as are KMP’s findings for the same scenarios. In general, the model performs remarkably similar to the KMP model. First, as in the KMP model, all estimates of per-capita utility differ significantly from zero (the expected value of a random platform). More importantly, the relative ranks of the decision rules are identical, thus, we sufficiently achieve relational alignment.

Table 2.

Comparison of Results of Single Jurisdictional Models (200 Replications Each).

Single Jurisdiction	KMP		Seagren		One-Sample Kolmogorov Test
Single Jurisdiction	Per-Capita Utility	SE	Per-Capita Utility	SE	Test Stat	p Value
Democratic referenda	2.69	0.12	2.32	0.10	0.174	.00**
Direct competition (two parties)	1.45	0.13	1.74	0.11	0.113	.01*
Direct competition (three parties)	0.67	0.13	0.61	0.12	0.061	.46
Direct competition (seven parties)	0.33	0.13	0.27	0.11	0.054	.60
Proportional representation (three parties)	1.33	0.13	1.21	0.12	0.074	.22
Proportional representation (seven parties)	1.36	0.13	1.23	0.11	0.111	.01*

Note. KMP = Kollman, Miller, and Page’s model; SE = standard error.

*p < .1, **p < .001.

The replication successfully achieves distributional equivalence for certain design points. For each institution considered, we use a one-sample Kolmogorov goodness-of-fit test to test the null hypothesis that the distribution of mean per-capita utility estimates for the current model is that of a normal distribution, with KMP’s mean and standard deviation. We find that the present model’s output differs significantly from that of KMP97 for democratic referenda, two-party direct competition, and seven-party proportional representation.

Multiple jurisdiction per-capita utility

The results of the multiple jurisdiction scenarios are shown in Table 3, along with KMP’s corresponding findings. As with the single jurisdictional configuration, the model output matches the KMP output well. The output for each decision rule differs significantly from zero, and the relative rank of each rule is nearly identical, thereby achieving (near) relational alignment. The only discrepancy occurs in the case of three jurisdictions, where direct competition achieves greater per-capita utility than democratic referenda (p < .05). Interestingly, in every jurisdiction for the replication the voting rules are ranked proportional representation, direct competition, and democratic referenda in order of decreasing per-capita utility. So, the replication actually behaves in a slightly more uniform manner than the original.

Table 3.

Comparison of Results of Multiple Jurisdictional Models (200 Replications Each).

Multiple Jurisdictions	KMP		Seagren		One-Sample Kolmogorov Test
	Per-Capita Utility	SE	Per-Capita Utility	SE
	Per-Capita Utility	SE	Per-Capita Utility	SE	Test Stat	p Value
Three Jurisdictions
Democratic referenda	34.39	0.15	33.78	0.16	0.105	.02*
Direct competition (two parties)	34.15	0.14	34.56	0.11	0.188	.00**
Proportional representation (three parties)	35.56	0.11	34.85	0.11	0.220	.00**
Seven jurisdictions
Democratic referenda	48.29	0.13	48.18	0.12	0.062	.42
Direct competition (two parties)	49.90	0.13	49.13	0.12	0.218	.00**
Proportional representation (three parties)	51.80	0.12	50.60	0.10	0.325	.00**
Eleven Jurisdictions
Democratic referenda	55.46	0.12	55.34	0.11	0.059	.48
Direct competition (two parties)	57.03	0.13	56.14	0.12	0.235	.00**
Proportional representation (three parties)	58.93	0.12	57.61	0.10	0.357	.00**

Note. KMP = Kollman, Miller amd Page’s model; SE = standard error.

*p < .1, **p < .001.

For each decision rule, we again use a one-sample Kolmogorov goodness-of-fit test to test the null hypothesis that the distribution of mean per-capita utility estimates for the current model is that of a normal distribution, with KMP’s mean and standard deviation. The present model achieves distributional equivalence with the KMP’s model for democratic referenda in the 7 and 11 jurisdiction settings. However, due to behavior at the other design points, we only claim to achieve relational equivalence.

Interjurisdiction distance

The results for interjurisdiction distance also suggest that the replication has achieved relational equivalence, as Table 4 illustrates. The relative ranks for each of the voting rules for every jurisdiction are identical to that of the KMP model. The results of the one-sample Kolmogorov goodness-of-fit tests reveal that distributional equivalence is achieved at three design points. We are satisfied with relational equivalence in this case because the estimated means are relatively close and the replication matches the relative ranks of the voting rules.

Table 4.

Comparison of Results of Multiple Jurisdiction Models.

Multiple Jurisdictions	KMP		Seagren		One-Sample Kolmogorov Test
Multiple Jurisdictions	Interjuris Distance	SE	Interjuris Distance	SE	Test Stat	p Value
Three Jurisdictions
Democratic referenda	61.58	0.84	59.39	0.49	0.31	.00**
Direct competition (two parties)	65.21	0.41	65.67	0.21	0.63	.00**
Proportional representation (three parties)	66.67	0.36	66.58	0.05	0.50	.00**
Seven Jurisdictions
Democratic referenda	51.64	0.32	50.68	0.19	0.22	.02*
Direct competition (two parties)	53.96	0.23	53.73	0.14	0.17	.12
Proportional representation (three parties)	56.12	0.15	55.65	0.08	0.32	.00**
Eleven Jurisdictions
Democratic referenda	50.76	0.23	50.12	0.13	0.20	.03*
Direct competition (two parties)	52.26	0.13	51.94	0.08	0.17	.10
Proportional representation (three parties)	53.32	0.09	53.22	0.05	0.17	.11

Note. KMP = Kollman, Miller amd Page’s model; SE = standard error.

*p < .1, **p < .001.

Agent relocations

The final response variable we consider is the aggregate number of agent relocations during a simulation run, as indicated in Table 5. In this case, we receive more evidence of relational equivalence. In addition to the relative proximity of the estimated means, the rank order of the voting rules is, again, replicated. Although not a true test of distributional equivalence, we conduct a comparison of means test with unequal variances and report the results in Table 5.

Table 5.

Comparison of Results of Multiple Jurisdictional Models.

Multiple Jurisdictions	KMP		Seagren		Comparison of Means
	Agent Relocations	SE	Agent Relocations	SE
	Agent Relocations	SE	Agent Relocations	SE	Difference	p Value
Three Jurisdictions
Democratic referenda	864.2	17.24	830.2	21.79	−33.96	.01*
Direct competition (two parties)	915.7	14.67	1042.2	25.07	126.48	.00**
Proportional representation (three parties)	1277.6	34.41	1230.8	28.60	−46.80	.04*
Seven Jurisdictions
Democratic referenda	863.2	10.41	877.2	12.91	14.00	0.09
Direct competition (two parties)	1162.5	8.63	1196.4	29.39	33.90	0.03*
Proportional representation (two parties)	1371.1	25.75	1402.0	25.15	30.98	.09
Eleven Jurisdictions
Democratic referenda	887.3	5.26	901.4	7.07	14.06	.00**
Direct competition (two parties)	1293.7	7.13	1253.2	20.27	−40.50	.00**
Proportional representation (three parties)	420.3	21.74	1525.6	26.70	105.32	.00**

Note. KMP = Kollman, Miller amd Page’s model; SE = standard error. Null hypothesis is that there is no difference between the model outputs’ average relocations.

*p < .1, **p < .001.

Replication Summary

In summary, we succeed in the effort to replicate the KMP97 model. With regard to per-capita utility, the replication achieves near-complete relational alignment and even achieves distributional equivalence in many instances. The replication convincingly achieves relational equivalence for interjurisdiction distance and agent relocations as well.

Development and Analysis of the Extended Model

Although replication is a valuable part of the scientific process, we gain several additional benefits by replicating early computer models. First, doing so provides the research community with a new version of a seminal model. More importantly, we can leverage faster and more powerful computers that are capable of implementing a richer and more complex version of the original model.

In Kollman, Miller, and Page (1998), the authors examine the behavior of adaptive parties who seek electoral victory by evolutionarily changing their platforms in the hopes of increasing their success at the ballot box. However, they compare party performance, given constituencies with significantly more complex preferences. The voters in this model possess an ideal desired level of government for each issue as well as a separate strength of that preference. The authors test a number of different combinations of voter ideologies and types of strength distributions.

Model Formulation and Description

The notation for this model is identical to that in the Notation and Measures of Effectiveness Section earlier, with the exception of the policy space and the manner in which agents’ preferences regarding various policy levels are modeled. Let p_ji ∊ {0, R} give the position of jurisdiction j on issue i. The positions may be integers between 0 and R. When political parties are considered, let p_rji ∊ {0, R} give the position of party r in jurisdiction j on issue i. Let Platform P _j ∊ {0, R} ^Ni denote the vector of decisions across all N_i issues in jurisdiction j.

An agent possesses ideal point for issue i, d _ai, selected from the closed set {0, R}, where the parameter R is the maximum range for ideal points. Each ideal point has an associated intensity level, s_ai that is selected from the closed set {0,S}. Agents possess ideal points and intensities for each of the N_i issues under consideration. The results presented later in the section on Results for the Extended Model are for scenarios involving 10 issues under consideration, agents’ ideal points are in the set [0, 8], and agent’s intensities are drawn from the set [0, 2].

An agent’s utility is the negative of the squared weighted Euclidean distance between the agent’s ideal platform and the given platform, where the weights are the agent’s strengths on each issue (Kollman, Miller, & Page, 1998, p. 143).

U_{a} (P_{j}) = - Σ s_{a i} {(p_{j i} - d_{a i})}^{2} .

Agents possess complete information regarding the level of government services offered in all jurisdictions. They move costlessly to the jurisdiction whose platform offers them the highest utility, without any expectation regarding the effect they may have on political outcomes in their prospective jurisdiction. External effects between issues are neglected so voters are assumed to vote sincerely.

Preference Landscapes

As in KMP97, each agent has linearly separable preferences over each issue. In this case, agents possess a set of ideal points that characterize its desired levels of government service for each issue. Agents also possess a corresponding set of strengths that characterize the relative importance of each issue to the voter (Kollman, Miller, & Page, 1998, p. 143). The agents’ preferences remain constant throughout the simulation; however, they may be correlated in two ways. Individual voter ideal points may be correlated on different issues, and individuals’ strengths may be correlated to their ideal points.

Ideology is a term used to describe the manner in which voters’ ideal points are correlated (Kollman, Miller, & Page 1998, p. 145). An individual voter may have consistent or uniform ideology. A voter with consistent ideology has ideal points that are correlated across issues, with a bias toward a particular part of the issue spectrum. These ideal points are generated such that a bias point is selected which constrains all ideal points to within plus or minus one unit. For instance, if the bias point generated for a given agent is 2, then his ideal points for each issue must be an integer from the closed set [1, 3]. The ideal points for agents with uniform ideology are drawn from a uniform random distribution with no such biases.

An individual agent’s ideal points and strengths may be correlated in one of three manners. Centrist strengths are higher for ideal points that are closer to the middle of the distribution. Suppose ideal points are distributed between 0 and 6. Then for centrists, the maximum strength for an issue will be given to ideal points of 3. The strength for an ideal point of 2 will be higher than the strength associated with an ideal point of 1 and so on. Extremist strengths are higher for ideal points that are closer to the extremes. So, in the previous example, ideal points of 3 would have 0 strength, while ideal points of 0 or 6 would be assigned the highest strengths. Independent strengths are randomly and independently distributed so that on average, they are uncorrelated with the underlying ideal points.

Combining ideologies and strength distributions yields six potential electoral landscapes of preferences. These categories are neither exhaustive nor intended to fit actual particular distributions of preferences. They represent polar cases on which to test the performance of political institutions. They do, however, tend to generally correspond to ways in which preferences may be distributed once people decide how to vote. Individuals often feel strongly about divisive issues with relatively polar alternatives such as abortion or gun control. Although for other issues, voters may prefer positions far outside the mainstream but not attach much weight to them.

Kollman, Miller, and Page (1998) show that the electoral landscape formed by voter preferences with respect to the incumbent party’s platform varies in terms of ruggedness and slope. Ruggedness is a measure of the relative number of local extrema, while slope is a measure of the magnitude of those extremes. Voters with uniform preferences (i.e., randomly drawn and unbiased) with independent intensities (i.e., uncorrelated strengths of opinion) tend to form landscapes that are the most rugged (p. 153). Rugged landscapes hinder the ability of the challenger to locate the global optimum that may defeat the incumbent. Consistent preferences with centrist intensities (i.e. biased towards the middle of the spectrum) seem to result in less rugged landscapes that enable parties to quickly converge toward the neighborhood of the median. In this section, we test the robustness of KMP’s Tiebout sorting against a number of varied electoral landscapes.

Results for the Extended Model

The purpose of this section is to gain insight into the sensitivity of KMP97’s results with respect to voter preference landscapes. A finding that the qualitative relationships hold, even while testing over a variety of landscapes, lends additional credibility to their conclusions. However, to discover exceptions to their findings provides a more nuanced understanding of the way in which local voting rules affect the efficiency with which individuals collectively acquire public goods.

We vary institution, jurisdictions, voter ideology, and voter intensity in a full-factorial experimental design with levels outlined in Table 6. The experiment consists of a 336 design points, which we replicate 50 times each for a total of 16,800 runs. In the replication section, we employ common random numbers to the greatest extent possible in order to reduce the variance between design points and improve statistical power.

Table 6.

Full-Factorial Experimental Design.

Institution	Jurisdictions	Ideology	Intensity
Referenda	1	Consistent	Random
Direct competition (two parties)	3	Uniform	Centrist
Direct competition (three parties)	7		Extremist
Direct competition (seven parties)	11
Proportional representation (three parties)
Proportional Representation (seven parties)

The parameters held constant during this experiment are shown in Table 7. Since issue levels are integers, they can take on values between 0 and 8. Similarly, intensity can take on integer values on the closed set [0,2]. The factor levels in Table 7 were selected due to their proximity to the factor levels in Kollman, Miller, and Page (1998). The duration of each simulation run is 30 time steps, in contrast to 10 in Kollman, Miller, and Page (1997) because the dynamics of the model, given the richer electoral landscapes of preferences, were such that equilibrium is not always attained in 10 time steps. For the results presented, the response variables are all at time t = 30.

Table 7.

Parameters Held Constant in Experiment.

Static Factors
Description	Factor	Level
Number of issues under consideration	Number issues	10
Maximum preference level for issue	Range intensity	8
Maximum intensity strength for issue	Range variance	2
Maximum distance from ideal point	Range campaign	1
Number of cycles for adaptive parties	Length campaign	8
Number of adaptive iterations per cycle	Iterations perterb	5
Issues to perterb in campaign cycle	Issues	3
Proportion of voters polled	Polling sample	0.1

Single jurisdiction

The results for the single jurisdiction model are shown in Table 8. Recall that utility is the negative weighted Euclidean distance between an agent’s ideal platform and the platform offered in her jurisdiction. The means shown are the overall mean per-capita utilities achieved by the particular institution. We conduct a one-way analysis of variance on the effect of the institution while blocking on the effects of the six different landscapes. We use the Tukey–Kramer multiple comparison technique (Kramer, 1956; Tukey, 1953) to then determine which, if any, institutions generate statistically significant differences at a .05 level of significance. In the tables, the levels that do not share a letter are significantly different. For example, that fact that democratic referenda and two-party direct competition share an “A” indicates that the two levels are not significantly different. However, seven-party proportional representation is assigned a “B,” which indicates it is significantly different than democratic referenda.

Table 8.

Single Jurisdiction; Per-Capita Utility After 30 Elections.

One Jurisdiction
Institution		Mean
Democratic referenda	A	−73.30
Direct competition (2 parties)	A B	−76.34
Proportional representation (7 parties)	B	−78.30
Rroportional representation (3 parties)	C	−86.10
Direct competition (3 parties)	C	−88.89
Direct competition (7 parties)	D	−121.74

Note. Levels not connected by same letters are significantly different.

Table 8 confirms that the order between the institutions for the single jurisdiction case is nearly identical to that of Kollman, Miller, and Page (1997). The only difference is that seven-party proportional representation is slightly higher than the three-party proportional representation. This order is based on the overall means for each treatment (institution). It is possible that the particular performance of each institution relative to the others may depend on the landscape of voter preferences. Table 9 outlines the per-capita utility (at time 30) for each institution and its relative ranking by landscape.

Table 9.

Multiple Jurisdictions; Per-Capita Utility After 30 Elections.

Institution	Three Jurisdictions		Seven Jurisdictions		Eleven Jurisdictions
Democratic referenda	B	−59.67	A	−31.02	A	−26.17
Proportional representation (seven parties)	A	−39.58	A	−31.01	B	−27.54
Proportional representation (three parties)	A	−39.41	A	−31.38	C	−28.71
Direct competition (two parties)	A	−39.29	B	−32.63	D	−29.85

Note. Levels not connected by same letters are significantly different.

Referenda has the highest per-capita utility for each landscape, while seven-party direct competition has the lowest for each. Most other deviations from the order presented in Table 10 are slight. For example, two-party direct competition is ranked second, as it is in Table 10, in all but one landscape. Also of note, landscapes with centrist intensities tend to yield higher per-capita utility levels, while extremist intensities result in substantially lower levels.

Table 10.

Single Jurisdiction; Per-Capita Utility After 30 Elections by Preference Landscape.

One Jurisdiction	Uniform Ideology			Consistent Ideology
Institution	Random Strengths	Centrist Strengths	Extremist Strengths	Random Strengths	Centrist Strengths	Extremist Strengths
Democratic referenda	−66.69 (1)	−33.33 (1)	−122.42 (1)	−63.91 (1)	−36.66 (1)	−116.79 (1)
Direct competition (two parties)	−69.51 (2)	−34.65 (4)	−126.69 (2)	−66.37 (2)	−39.47 (2)	−121.34 (2)
Proportional representation (seven parties)	−70.90 (4)	−34.35 (2)	−128.79 (3)	−68.48 (3)	−40.10 (3)	−127.21 (3)
Rroportional representation (three parties)	−70.56 (3)	−34.37 (3)	−128.99 (4)	−81.79 (4)	−54.80 (4)	−146.06 (4)
Direct competition (three parties)	−72.18 (5)	−35.15 (5)	−133.31 (5)	−85.90 (5)	−57.67 (5)	−149.11 (5)
Direct competition (seven parties)	−81.62 (6)	−37.14 (6)	−150.50 (6)	−141.54 (6)	−100.16 (6)	−219.45 (6)

Note. Within-block rank in parentheses.

In the single jurisdiction scenario, the rank order of the aggregate performance of the institutions closely resembles that of Kollman, Miller, and Page (1997). Closer inspection of the relative performance of the institutions over each landscape reveals that this ordering holds with only minor exceptions. Thus, we firmly conclude that the original findings with respect to the single jurisdiction case are robust across a wide range of possible voter preference landscapes.

Multiple jurisdictions

In this section, we consider additional jurisdictions. As in KMP, we observe that the availability of additional jurisdictions tends to increase aggregate utility due to the ability of the voters to sort themselves into more homogeneous constituencies. The results for the multiple jurisdiction scenarios are displayed in Table 9.

Recall that KMP finds proportional representation to be superior to the other voting rules in all multiple jurisdiction cases. We find that two-party direct competition is superior for three jurisdictions, but democratic referenda returns to the top in the 7- and 11-jurisdiction cases. Although the difference between the top voting rules and three-party proportional representation is not statistically significant in the three- and seven-jurisdiction cases, as we increase the number of jurisdictions to 11 and beyond, the differences among each institution becomes statistically significant.

Table 11 shows the performance by landscape for the three-jurisdiction scenario. We find that three-party proportional representation offers highest per-capita utility for two landscapes, while the two-party direct competition is superior for four landscapes. Democratic referenda have the lowest per-capita utility for five of the six landscapes. The primary conclusion to draw from Table 11 is that with small numbers of jurisdictions, the qualitative rankings of institutions by per-capita utility are highly dependent on preference landscape. Although the relationship between preference landscape and the voting institution that achieves the highest utility is unclear at three jurisdictions, as the number of jurisdictions increases to 11, a more robust pattern emerges. This pattern is consistent with the relative ranking depicted in Table 9.

Table 11.

Three Jurisdictions; Per-Capita Utility After 30 Elections, by Preference Landscape.

Three Jurisdictions	Uniform Ideology			Consistent Ideology
Institution	Random Strengths	Centrist Strengths	Extremist Strengths	Random Strengths	Centrist Strengths	Extremist Strengths
Direct competition (two parties)	−57.60 (2)	−31.38 (4)	−107.29 (1)	−12.22 (2)	−13.15 (2)	−14.09 (1)
Proportional representation (three parties)	−57.24 (1)	−30.99 (2)	−107.34 (2)	−12.40 (3)	−13.30 (3)	−15.17 (3)
Proportional representation (seven parties)	−58.11 (3)	−30.78 (1)	−109.18 (3)	−11.96 (1)	−13.12 (1)	−14.30 (2)
Democratic referenda	−59.75 (4)	−31.01 (3)	−109.76 (4)	−45.94 (4)	−29.34 (4)	−82.25 (4)

Note. Within-block rank in parentheses.

For multiple jurisdictions (namely, greater than three), we find that the superior voting rule with respect to achieving the highest per-capita utility is democratic referenda. In contrast, KMP find that proportional representation is superior. One possible symptom of proportional representation’s relatively poorer performance is illustrated in Figure 1.

Figure 1.

Average agent relocations; three jurisdictions.

Figure 1 shows the average number of agent relocations tallied in each 10 time step segment of the simulation run in the three-jurisdiction scenario. All voting institutions experience a large number of relocations in the first 10 time steps of the simulation. However, by the 30th time step, the average number of moves is very low for democratic referenda and the two-party direct competition, while the three- and seven-party proportional show little sign of convergence after even 60 time steps. Although a large number of relocations are necessary early in a simulation run in order to facilitate sorting, proportional representation, as an institution, appears to have trouble coping with the complexity of the preference landscapes.

Kollman, Miller, and Page’s (1997) overarching conclusion concerning the stability of the electoral process involving boundedly rational agents is essentially confirmed. However, more particular conclusions regarding the relative performance of collective decision-making rules is clearly not robust over various electoral landscapes. Although it is ultimately an empirical question as to whether the voters’ preferences in a particular set of jurisdictions conform to one or more of the electoral landscapes considered in this model, further investigation of the link between electoral landscape will likely yield interesting results.

The Extended Model Exhibits Low Sensitivity to Arbitrary Parameter Values

In the years since Kollman, Miller, and Page developed and analyzed their models, computing power has grown exponentially. In addition, the state of the art of simulation analysis pushed forward by researchers working the operations research field of data farming has also improved substantially.⁶ We leverage both in this section in order to gain further insight into the robustness of our conclusions.

We examine a total of 13 factors. The factors include important qualitative variables such as voting institution and preference landscape that we examined earlier as well as arbitrary simulation parameters such as the number of issues or the number of campaign iterations. A full-factorial analysis of the factors listed in Table 12 requires 97,977,600 design points. Alternatively, a Nearly Orthogonal and Balanced Latin Hypercube design accommodates both continuous and categorical variables and only requires 512 design points (Viera, Sanchez, Kienitz, Belderrain, 2011). We replicate each design point 20 times for a total of 10,240 runs. The design achieves near orthogonality of the factors, which improves the power of the statistical analysis while requiring only a fraction of the design points of a full-factorial analysis.⁷

Table 12.

Experimental Design.

Nearly Orthogonal and Balanced Latin Hypercube Design
Description	Factor	Type	Levels	Values
Number of voters in polity	Voters	Continuous		(1,000, 5,000)
Number of jurisdictions in the system	jurisdictions	Continuous		(1, 20)
Number of political parties per jurisdiction	parties	Categorical	9	(2, 3, 4, 5, 6, 7, 8, 9, 10)
Number of issues under consideration	numberIssues	Continuous		(8, 24)
Maximum distance from ideal point	varianceRange	Categorical	3	(0, 1, 2)
Maximum preference level for issue	issueRange	Categorical	5	(4, 6, 8, 10, 12)
Maximum intensity strength for issue	intensityRange	Categorical	4	(1, 2, 3, 4)
Number of cycles for adaptive parties	campaignLength	Categorical	9	(4, 5, 6, 7, 8, 9, 10, 11, 12)
Number of adaptive iterations per cycle	campaignIterations	Categorical	10	(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)
Proportion of voters probed	pollProp	Categorical	3	(0.1, 0.7,1.0)
Issues to perterb in campaign cycle	perterbIssuses	Categorical	2	(0.2, 0.5)
Preference landscapes	PREF	Categorical	6	(0, 1, 2, 3, 4, 5)
Voting rule employed in system	INST	Categorical	3	(0, 1, 2)

For each of the response variables, namely, per-capita utility and agent relocations, we conduct a forward stepwise regression analysis with all main effects and first-order interactions in the candidate pool and use minimum Schwarz’s Bayesian Criterion as our stopping rule (Schwarz, 1978). The resultant meta-models are shown in Table 13. For clarity, only interactions containing voting rules are presented in the table, but interactions among other factors are present in the models. We are particularly interested in such interactions because their existence indicates that the performance of some voting rules depends upon simulation parameters and may affect qualitative conclusions.

Table 13.

Sensitivity Meta-Models.

Response	E (Per-Capita Utility)	E (Agent Relocations)
Adj. R ²	.823	.584
Observations	512	512
Degrees of freedom (model)	37	23

Factor	Estimate	p Value	Estimate	p Value
Intercept	160.67	<.0001	−11175.37	<.0001
Jurisdictions	2.52	0.007	226.31	<.0001
Parties	1.35	0.130	1051.91	<.0001
Voters	−0.01	0.003	2.84	<.0001
INST_direct_comp			2163.30	<.0001
INST_proportional_rep	8.35	0.006	3272.84	<.0001
PREF_indep−consist	38.25	<.006
PREF_cent-unif	23.23	<.0001
PREF_cent-consist	39.32	<.0001	−673.76	.017
PREF_ext-uniform	−48.95	<.0001	1188.01	<.0001
PREF_ext-consist	19.75	<.0001	−1182.52	<.0001
numberIssues	−10.29	<.0001
intensityRange	−2.32	0.805
varianceRange			2808.38	<.0001
issueRange	−8.95	0.000	18.47	0.845
campaignLength	3.16	0.008
perterbIssues			−62.08	0.001
pollProp			44.47	<.0001
INST_direct_comp × parties			259.76	0.002
INST_direct_comp × voters			0.93	<.0001
INST_direct_comp × pollProp			32.96	0.000
INST_proportional_rep × PREF_ext-consist[1]	9.23	0.002
INST_proportional_rep × jurisdictions	−2.31	<.0001
INST_proportional_rep × voters	0.01	0.003	1.19	<.0001

The meta-models confirm that statistically significant relationships exist between the response variables and factors such as voting institution, preference landscape, and jurisdictions. The models also confirm that relatively few statistically significant relationships exist between basic model parameters (the second to lowest group of factors in Table 13) and the response variables. For example, only numberIssues, issueRange, and campaignLength are statistically significant factors in the per-capita utility model. Similarly, only varianceRange, perturbIssues, and pollProp are statistically significant factors for the agent relocations model. However, even these relationships are simply additive and do not affect the relative performance of the voting rules.

The most important finding from the meta-models is that the only one-way interaction that includes a voting institution and an otherwise arbitrary simulation parameter is the direct_competition*pollProp term in the agent relocations meta-model. We already knew from the previous sections that the behavior of certain voting institutions depends on the number of parties and voters. However, the presence of the interaction means that as the proportion of the electorate that is polled during the adaptive party campaigns increases, so too does the propensity for agents to change locations under the direct competition voting rule.

Thus, we conclude the qualitative findings, namely the relative performance of the voting rules, with respect to per-capita utility and agent relocations are remarkably robust to choices of arbitrary simulation parameters.

Conclusion

Agent-based simulation is a relatively new tool in the social sciences, and it was more recently introduced to the field of political economy. The model we present in this article applies this modeling technique in an attempt to gain insight into the efficiencies with which political institutions are able to sort voters by preferences. The technique shows potential for more rigorous analysis of political, fiscal, and economic competition on the part of local governments in the future. Our successful replication provides interested researchers with the means to pursue such interests.

The most convincing finding in our analysis of the extended model is that the conclusion drawn by Kollman, Miller, and Page (1997) regarding the relative efficiencies of voting institutions in the single-jurisdiction scenario is highly robust. However, when we consider multiple jurisdictions, the results are far less clear. As the number of jurisdictions increase a stable pattern emerges, but it is one where democratic referenda is the most efficient, rather than proportional representation as in Kollman, Miller, and Page (1997).

Finally, we demonstrate a technique for conducting sensitivity analyses that leverages a high-dimensional experimental design. Our design requires only 512 design points in contrast to the 99 million required for a similarly rigorous full-factorial design. The ordinary least squares meta-models constructed from the response surfaces reveal that our conclusions are relatively robust with respect to arbitrary simulation parameters.

Footnotes

Author’s Note

For discussion and useful suggestions, I would like to thank Alexander Fink, Charles Rowley, Roger Congleton, and Richard Wagner, as well as two anonymous referees. I remain solely responsible for any errors still present. The opinions expressed in this article are the author’s alone and do not necessarily those of the Department of Defense or the United States Government.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

Notes

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