Three Stages of Decision Making
Buchanan and Tullock categorize democratic decision-making into two main stages: constitutional decision-making and legislative decision-making However, it is essential to recognize a third stage—elections—which play a crucial role in democracy and fundamentally differ from the other two stages.
Buchanan and Tullock conceptualize constitutional decisions as social contracts that bind individuals, emphasizing the importance of establishing voting rules and institutions for future governance They argue that the choice of how to decide is fundamental, advocating for the unanimity rule as the most basic principle in this process This rule holds a significant role in constitutional decision-making, ensuring that rational individuals reach mutually beneficial agreements similar to economic contracts According to their theory, individuals will only consent to a social contract if they agree to its terms, and unanimity ensures that all members must support collectivization before it can be enacted, allowing individuals the right to reject it if they do not agree.
While our research does not encompass vote trading due to the need for further foundational development, it provides valuable insights for future studies on this topic Specifically, the mathematical analysis of forming coalitions can illuminate the concessions required to enact legislation involving vote trading We urge scholars to explore these extensions in their work.
The decision-making process can be divided into three stages, with significant differences between constitutional and legislative decisions While unanimous agreement is often required for constitutional matters to prevent coercion, legislative decisions can operate under less-inclusive voting rules, like majority rule, to enhance efficiency This approach allows individuals to streamline the decision-making process for the numerous policy choices they face daily Buchanan and Tullock suggest that rational individuals may prefer majority rule for these frequent decisions, provided that citizens consent to such arrangements, ensuring that no one is forced into acceptance against their will.
Electoral decisions primarily focus on electing public officials, presenting unique challenges due to the large electorate and the difficulty of vote trading among citizens Unlike legislative voting, where the status quo significantly influences decision-making, elections typically lack a status quo alternative unless an incumbent seeks re-election Consequently, constitutional designers aim to treat all candidates equally, diverging from legislative voting rules Additionally, the high costs associated with organizing votes make it impractical for citizens to engage in multiple rounds of voting on various alternatives Instead, elections generally involve considering all candidates simultaneously, necessitating a different set of voting rules that are more suitable for the electoral phase compared to those proposed by Buchanan and Tullock for constitutional and legislative contexts.
The current book utilizes user-friendly computer-based simulations and robust analytical tools to explore the connections between voting rules and democratic outcomes This approach attracts scholars in comparative politics focused on institutional roles in democratic transitions, democratic theorists aiming to apply political philosophy, and computer scientists and constitutional political economists examining the practical application of computer models in social science Additionally, it offers a thoughtful reexamination of a classic work.
In Chapter 2, we examine the foundational arguments presented by Buchanan and Tullock in their seminal work, The Calculus of Consent They, along with Mueller, propose that government decision-making should be categorized into two distinct phases: the constitutional phase, which establishes the rules and framework, and the parliamentary phase, where decisions are made within that established framework.
2 Nevertheless, Buchanan and Tullock (1962) make an interesting argument about different candi- dates representing implicit bundles of vote trades See their pages 135–36.
6 1 Introduction phases correspond to the constitutional and legislative phases described in our book.
A key aspect of earlier discussions is that the institutions established during the constitutional phase should improve the well-being of certain individuals without negatively impacting others In the parliamentary phase, decision-makers must navigate these considerations while also aiming to streamline the process, minimizing the time and effort required to make multiple decisions efficiently.
In Chapter 3, we clarify key concepts used by Buchanan and Tullock, revealing that the connection between unanimity rule and various Pareto principles may not be as strong as they propose This analysis lays the groundwork for exploring the three phases of decision-making, highlighting how slight variations in definitions can significantly impact applications, especially in medium- and large-sized populations These insights are crucial for understanding the implications of Pareto concepts during the electoral phase of decision-making, suggesting that alternative voting rules might achieve Pareto optimal results more effectively than unanimity rule.
Chapter 4 explores the role of voting during the constitutional phase, focusing on the assertion that decision-making costs are negligible Through computer simulations and deductive analysis, we investigate whether the unanimity rule yields better Pareto superior and Pareto optimal outcomes compared to alternative voting methods Our findings encompass three key results, all pertaining to the concept of Pareto optimality, across scenarios involving random, sincere, or strategic proposals.
When individuals propose randomly, majority rule is generally more effective than unanimity rule in achieving a Pareto optimal outcome In cases where individuals propose sincerely, majority rule is at least as effective as unanimity rule in selecting a Pareto optimal result However, when individuals engage in strategic proposing and voting, unanimity rule consistently leads to a Pareto optimal outcome Additionally, other k-majority rules can also produce Pareto optimal results and typically yield outcomes that are very close to being Pareto optimal A k-majority rule requires a specific threshold of affirmative votes, which can include majority, supermajority, or unanimity rules for a proposal to be approved.
Unanimity rule, unlike majority rule, is often as effective in achieving outcomes that are both Pareto superior and Pareto optimal, particularly when efficiency demands that everyone is at least as well off as under the status quo This conclusion is supported by laboratory experiments and historical data from the adoption of the U.S Constitution Chapter 5 focuses on legislative voting, specifically analyzing the optimal k-majority rule concerning decision and external costs According to Buchanan and Tullock (1962) and Mueller (1996), a k-majority rule close to half of the voting body is preferred as it minimizes the combined costs in legislative settings.
We examine external costs and decision costs over a sequence of votes The in- troduction of multiple alternatives affects external costs and decision making costs
3 For example, the U.S House of Representatives requires 218 of its 435 members to sign a suc- cessful discharge petition In this case, k = 218 More precise definitions are offered in Chapter 3.
The decision-making process involves three key stages, emphasizing the importance of evaluating multiple alternatives This approach prompts a re-examination of the external cost function and enhances our understanding of decision-making costs By analyzing various alternatives, we can identify conditions under which specific majority rules effectively minimize total costs The optimal majority is influenced by how decision makers weigh the likelihood of passing proposals and the speed at which favorable options are identified However, majority rule tends to be optimal only under specific conditions, particularly when there is a jump discontinuity in the decision cost function, as suggested by Mueller (2003).
Chapter 6 analyzes four voting rules, focusing on three commonly used in elections: plurality rule, majority rule with a runoff, and instant runoff voting, while also considering the Borda count due to its recent relevance Given the vast number of voters, k-majority rules are seldom used, as most candidates are Pareto optimal, rendering the Pareto criterion ineffective The evaluation of these rules is based on six normative criteria: the Condorcet winner and loser criteria, majority criterion, consistency, reversal symmetry, and independence of eliminated alternatives Utilizing computer simulations of single-dimensional voting in single-member districts, we discover that the Borda count excels in independence of eliminated alternatives and performs comparably on other criteria Majority rule with a runoff consistently meets the majority criterion and avoids Condorcet losers, showing strong performance in consistency and reversal symmetry Ultimately, the effectiveness of each voting rule on the Condorcet winner criterion varies based on specific conditions, indicating that the ideal voting rule may differ according to societal values.
The book concludes with a few comments about the significance of our research for social contract theory and the creation of constitutions more broadly.
By analyzing impartial standards, researchers can identify which institutional frameworks are most likely to meet these criteria, enabling them to recommend fairer voting systems across various contexts This approach is beneficial for countries drafting new constitutions, such as Afghanistan and Iraq, as well as for policymakers designing local government institutions and legislatures reassessing their voting procedures, like the U.S Senate's review of the filibuster Additionally, the findings can assist smaller voting bodies, including boards of directors and university senates, in establishing more equitable and efficient decision-making rules.
Original Theories and Current Studies
Legislative Decision Making
Buchanan and Tullock's study on legislative decision-making focuses on various k-majority rules, where a proposal requires k "yea" votes to be approved This range can vary from a single vote to the unanimous agreement of all N individuals in a population Their analysis identifies the optimal k-majority rule, which aims to minimize the combined external costs and decision-making costs associated with the voting process.
External costs refer to the expenses individuals incur due to the actions of others, as defined by Buchanan and Tullock (1962) They suggest that these costs decrease as the number of individuals needed to reach a group decision increases When fewer individuals can make decisions, external costs rise, as those in the decisive coalition may prioritize their own interests over those outside it Conversely, when unanimous agreement is required, external costs tend to be minimal or even zero, since each member can veto decisions that negatively impact them.
Buchanan and Tullock (1962) illustrate the impact of decision-making on municipal street repairs, highlighting that if a single individual decides on repairs based on personal benefits, they are likely to prioritize roads they use, neglecting others and imposing external costs on those not in the decision-making group Conversely, if all residents must approve repairs, each will only agree if it benefits them, leading to zero external costs Thus, allowing one person to dictate repairs can result in significant negative externalities, while requiring unanimous approval minimizes these costs.
Buchanan and Tullock emphasize the concept of external costs, focusing on the expected costs incurred by individuals Individuals cannot predict whether they will be part of the decisive coalition or outside it, leading them to choose the most suitable k-majority rule based on this uncertainty Consequently, as the number of individuals needed for decision-making rises, the expected external costs are likely to diminish, although actual external costs for each individual may vary.
In contrast, decision making costs are the costs resulting from the time and effort needed to reach an agreement Buchanan and Tullock argue that such costs are an
1 See Heckelman and Dougherty (2010a) for a crude test of whether larger k-majority rules have negative effects on various tax increases.
Legislative decision-making becomes increasingly complex as the number of individuals involved in the process rises When a single person makes a decision, it requires minimal time and effort, as there is no need for negotiation However, as the size of the coalition necessary for agreement grows, more time is needed to reach a consensus, primarily because coalition members have fewer alternative options if disagreements arise within their group.
In the context of street repairs, having a single decision-maker can expedite the process, allowing for swift resolutions However, involving additional stakeholders in the decision-making coalition can prolong the planning phase, necessitating more time and effort When unanimous approval is required, the process becomes significantly more complex, demanding extensive coordination to ensure all parties are content and to prevent disputes over resource allocation.
Fig 2.1 Traditional External Costs and Decision Costs
Buchanan and Tullock (1962) illustrate decision-making costs using a graph where the vertical axis represents expected costs and the horizontal axis indicates the number of individuals involved in the decision The graph features a decreasing line for external costs and an increasing line for decision costs Notably, when one individual makes decisions for the group, it can lead to significant external costs while incurring minimal decision costs.
The voting rule ensures no delays in decision-making; however, the unanimity rule, while reducing external costs, significantly increases decision-making costs Buchanan and Tullock propose that the ideal decision-making rule should minimize the combined costs of these two factors, represented by the dashed line, which is achieved at point kI.
In the analysis of decision-making theories, the focus is on minimizing the combined expected costs of imposed decisions and individual decision-making When legislatures are faced with selecting a voting rule from a range of k-majority options, adopting a k-majority rule is deemed optimal for efficient outcomes.
Buchanan and Tullock, in their work *The Calculus of Consent* (1962), advocate for the adoption of unique institutional frameworks tailored to the values and preferences of each society, rather than prescribing a universal set of institutions They highlight the distinction between positive observations of rational decision-making in selecting institutions and normative recommendations for constitutional choices This approach allows for diverse societies to implement varying institutional structures while remaining fair, as different individuals prioritize different values Additionally, societies may opt for a unanimity rule during the constitutional phase, transitioning to a less inclusive majority rule for legislative decisions, demonstrating the flexibility in institutional design that Buchanan and Tullock propose.
The adoption of simple majority rule for routine group decisions regarding explicitly collectivized activities does not conflict with the requirement for unanimous consensus when it comes to changes in fundamental organizational rules, as noted by Buchanan and Tullock (1962).
When selecting the optimal k-majority rule, individuals should consider several key factors Decision costs tend to increase in environments with diverse opinions and limited information, compared to those with uniform opinions and accessible information Additionally, larger communities often face higher decision costs than smaller ones due to their complexity Lastly, communities with well-defined bills of rights are likely to incur lower external costs than those lacking such protections.
Buchanan and Tullock's analysis indicates that when both external costs and decision costs are considered, the optimal k-majority rule falls between 1 and N While majority rule is a potential candidate, there is no inherent justification for it being the k-majority rule that effectively minimizes total costs.
Others have adjusted Buchanan and Tullock’s argument in a way that makes majority rule much more likely to be optimal For example, Mueller (2003, pp 76-
Research suggests a potential "kink" in the decision cost function at N/2, indicating that for any k ≤ n/2, both policy A and its alternative can succeed For instance, if k is set at 35 out of 100 voters, a proposal to increase school expenditures may initially secure a majority with 40 votes Once this measure is approved, a subsequent counterproposal to reduce school expenditures by the same margin could emerge.
2 By “kink” Mueller meant a jump discontinuity.
Legislative decision-making can become problematic under a k-majority rule where k is less than or equal to 50, as it permits the formation of winning coalitions on both sides of an issue This scenario can lead to a deadlock, characterized by an ongoing cycle of opposing proposals Consequently, the costs associated with decision-making can rise significantly, making effective governance more challenging.
Constitutional Decision Making
Buchanan and Tullock argue that constitutional decisions differ significantly from legislative decisions, as they set the foundational rules that govern the legislative process for the long term In constitutional contexts, individuals face greater uncertainty regarding their future circumstances and interests, prompting them to adopt a more objective perspective and act as if they represent the average person within that framework.
The argument parallels Kant’s categorical imperative, advocating that individuals should make choices applicable to all in similar situations While there may be disagreements on the best institutions, Buchanan and Tullock argue that constitutional decisions are inherently more impartial than policy decisions, leading to greater homogeneity among decision-makers and reduced decision costs Rational individuals recognize the significance of constitutional choices, understanding that one such decision can influence numerous policy outcomes Consequently, they may prioritize external costs over decision costs Buchanan and Tullock assert that in the absence of decision costs, the unanimity rule emerges as the optimal voting mechanism.
The principle of unanimous consent holds a significant role in decision-making analysis, indicating that if the costs associated with making decisions are minimized, a rational individual would consistently advocate for unanimous agreement.
Buchanan and Tullock highlight that decision-making costs are significantly positive during the constitutional phase, suggesting that the process of establishing rules incurs substantial external costs These costs are so considerable that they overshadow the relative importance of the decision-making costs themselves.
4 Rae assumes voters are equally likely to support or oppose a proposal and that everyone votes.
If different assumptions are made, then his model does not necessarily predict that majority rule is optimal.
Requiring a threshold lower than unanimity creates uncertainty for rational individuals about their role in a coalition, potentially leading to worse outcomes due to coercive powers In contrast, the unanimity rule ensures that all members are included in negotiations, promoting Pareto improvements—changes that benefit at least one individual without harming others A situation where no further such improvements can be achieved is termed Pareto optimal.
Pareto optimality is a key measure of efficiency in welfare economics, linking consensual decision-making to the concepts of unanimity and Pareto improvements Buchanan and Tullock assert that a choice is considered Pareto optimal only when all parties agree, emphasizing that any decision-making process falling short of unanimity is likely to result in nonoptimal outcomes They position unanimity as the "ideal" voting rule, while recognizing that deviations from this principle are often necessary expedients.
Key constitutional decisions include determining whether the legislature will be unicameral or bicameral, as highlighted by Buchanan and Tullock (1962) Additionally, it is crucial to establish the proportion of the population that will serve as representatives, which significantly impacts the governance structure.
In "The Calculus of Consent," the authors discuss the method of election, emphasizing the significance of individual votes and randomization devices (Buchanan and Tullock, 1962, pp 205-8, 217–20) A key focus is the selection of the k-majority rule employed in the legislature, which recurs as a central theme throughout the work.
In the legislative process, individuals will select the most effective k-majority rule by weighing both external costs and decision-making costs incurred during the constitutional phase, as each legislative decision generates its own associated decision costs.
During the constitutional phase, individuals face the choice of opting into or out of collectivization Buchanan and Tullock argue that k-majority rules, which require less than unanimity, can lead to coercion among individuals and may result in an inefficient allocation of resources to the public sector While this does not imply that collectivization will extend into inappropriate areas, it suggests that excessive resources may be devoted to community-collectivized activities.
According to Buchanan and Tullock, traditional objections to the unanimity rule as impractical are rooted in binary decision-making scenarios, where only two mutually exclusive options are considered In contrast, they propose a constitutional decision-making framework that facilitates bargaining across a spectrum of alternatives.
5 Such concepts are defined more carefully in the next chapter.
6 Also see Buchanan and Tullock (1962, pp 94, 110)
In their 1962 study, Buchanan and Tullock explore the complexities of collective decision-making through a scenario involving three individuals contemplating the collectivization of fishing One person may oppose this idea due to personal preferences, such as a dislike for fish, making unanimous agreement difficult However, by expanding the decision to include the collectivization of coconut gathering, the individuals might engage in logrolling, allowing them to negotiate a compromise that benefits all parties involved.
The principle of mutual agreement applies to various activities, suggesting that if consensus cannot be reached on potential collectivization, it would not serve everyone's interests Imposing a coercive collective agreement on all parties would be unjust As noted by Buchanan and Tullock (1962), when trade is possible, it aligns with the concept of economic or market exchange.
255) Vote trading and bargaining help social contracts become more like economic contracts.
Representative Democracy
In Chapter 15, Buchanan and Tullock explore the application of their ideas to representative democracy, focusing on elections as a distinct decision-making phase They identify four key constitutional choice variables that individuals must evaluate simultaneously: the voting rule for selecting representatives, the basis of representation in the assembly, the degree of representation, and the k-majority rule for decision-making in the assembly Building on their previous discussion of the costs related to the fourth variable, they extend their analysis to the other three variables, highlighting the complexity of these choices in the democratic process.
When considering the balance of representation in decision-making, there exists a spectrum from a single representative making choices for all voters to direct democracy, where each eligible voter acts as a representative Constitutions establish a minimum number of representatives, reflecting a decision on the desired level of representation within a population Buchanan and Tullock suggest that determining the optimal number of representatives parallels the process of selecting an optimal majority rule; here, the critical factor is the ratio of representatives to the overall population As the proportion of representatives increases, decision-making costs rise due to the increased time required for consensus, while external costs tend to decrease.
The basis of representation is a crucial variable that influences how representatives are selected On one end of the spectrum, representatives are elected based on individual votes, while on the opposite end, representatives may be appointed through alternative methods.
Vote Trading and Other Themes
In ancient Athens, random selection played a significant role in appointing political officials, which can still be seen as a form of representation, albeit in a different context While most modern democracies tend to limit random elements in their political processes, this approach remains a valid constitutional choice, as discussed by Buchanan and Tullock (1962).
Buchanan and Tullock argue that individuals aim to minimize combined costs during the constitutional stage of decision-making If proposals to raise the k-majority rule in elections are successful, decision costs will rise while external costs will fall, prompting citizens to consider lowering the k-majority thresholds in specific variables Conversely, a shift from functional to random representation is likely to increase expected decision costs and decrease external costs, leading individuals to potentially lower thresholds in other variables This insight indicates that the analytical model for voting thresholds may have broader applications and highlights the intricate nature of decision-making that Buchanan and Tullock believe individuals can manage.
2.4 Vote Trading and Other Themes
Buchanan and Tullock's work explores several key themes, with vote trading being a central focus They conclude that vote trading is more advantageous when individuals receive different benefits from a collective good, as members of minority groups with strong interests are incentivized to trade votes more effectively Additionally, they argue that vote trading under majority rule enhances the efficiency of winning outcomes Through a simple three-person game, they demonstrate that the composition of the winning coalition significantly affects the solution set When side-payments are introduced, efficiency improves dramatically, as the winning coalition allocates resources to those who benefit most, regardless of their coalition membership, and redistributes payments to maximize returns for the coalition.
Buchanan and Tullock (1962) illustrate a scenario involving a township of 100 farmers who must make decisions on road repairs In this context, the initial group of farmers stands to gain $10 for every dollar invested in the repairs, highlighting the economic implications of collective decision-making in community resource management.
In a study of coalition dynamics regarding road repairs, three distinct groups emerge: the first coalition, which generates the highest return on investment; the second coalition, benefiting $5 for every dollar spent; and the third coalition, which sees a return of $1 for each dollar invested With a $33 grant available for road repairs, a coalition formed by all members of the third coalition and eighteen from the second coalition will strategically prioritize road repairs for the first coalition, as it yields the most productive outcome This coalition will require the first coalition to transfer a portion of the benefits, ensuring an efficient allocation of resources.
Buchanan and Tullock (1962) argue that side-payments of $10 per member to the winning coalition ensure that funds are allocated efficiently to those who value them most, leading to more equitable sharing of benefits They assert that this holds true even when the distribution of road repairs is uneven.
Buchanan and Tullock (1962) suggest that vote trading results in an outcome that lies between the extremes of no side-payments and complete side-payments They view vote trading as an indirect method of facilitating side-payments in decision-making processes.
Buchanan and Tullock examine bicameralism, suggesting it can effectively minimize anticipated external costs while keeping decision-making costs manageable (1962, p 236) Their analysis highlights how bicameral institutions enhance the depth of their study, although they also introduce greater complexity in the decision-making processes at the constitutional level.
Conclusion
"The Calculus of Consent" is a pioneering economic analysis of constitutional formation, introducing key concepts such as decision costs, external costs, and Pareto efficiency, which have influenced subsequent research Since its publication, scholars have explored various questions regarding the motivations behind a nation's pursuit of a new constitution, the processes involved in constitution-making, and the factors contributing to corruption in constitutional decision-making Additionally, researchers have revisited the historical efficiency of governmental institutions, highlighting the ongoing relevance of these themes in social science discourse.
In 1989, researchers began examining the factors that contribute to the success of democracies, drawing insights from notable studies by Przeworski (2005), Lijphart (1999), and Lipset (1963) These findings have been integrated with traditional public choice themes, particularly focusing on the implications of logrolling, as discussed by Riker and Brams.
1973), bicameralism (Riker, 1992; Diermeier and Myerson, 1999), agenda setting
7 For a criticism of the generality of these claims see Riker and Brams (1973).
(Koford, 1982), and legislative size (Crain and Tollison, 1977; Dougherty and Ed- ward, 2009).
The broad impact of "The Calculus of Consent" on various research questions prompts a revisitation of its central themes, which may now appear commonplace due to the contributions of Buchanan and Tullock Before delving into these themes, it is essential to define key terms and explore the connection between unanimity rule and several Pareto concepts This analysis reveals that the seemingly close relationship between unanimity rule and Pareto improvements may not be as straightforward as it appears.
Definitions
The unanimity rule and the Pareto criterion are often viewed as similar concepts, leading some authors to consider them nearly interchangeable Arrow (1951) characterized the Pareto principle as a form of unanimity, while Fishburn (1973) referred to the Pareto criterion as a strong form of unanimity Additionally, Buchanan (1967) described the unanimity rule as the “political counterpart” of the Pareto criterion, highlighting their close relationship in decision-making processes.
This chapter aims to achieve two objectives: first, to clearly differentiate between various types of unanimity rules and Pareto concepts, emphasizing the importance of technical precision; second, to explore the Pareto principles applicable for evaluating institutions, while also addressing the limitations of using the Pareto criterion in assessing institutions within large populations.
In a voting population of N individuals, the smallest majority, M, is defined as (N + 1)/2 for odd N and (N + 2)/2 for even N Each individual has preferences among a set of alternatives {w, x, y, z, q}, where q represents the status quo or existing policy For any two alternatives {x, y}, an individual's preferences can be categorized into three types: they prefer x over y (x i y), prefer y over x (y i x), or are indifferent between x and y (x∼ i y).
A voter's individual preferences can be encapsulated in a voter profile, which summarizes their choices For instance, in a three-person voter profile, individual 1 favors option x over option y, individual 2 prefers option y over option x, and individual 3 shows indifference between the two options Additionally, these preferences can be visually represented on a spatial map, as discussed in the following chapter.
Studies in Public Choice 20, DOI 10.1007/978-0-387-98171-0_3, 21
K.L Dougherty and J Edward, The Calculus of Consent and Constitutional Design,
Two variations of k-majority rule have been implemented in practice, as noted by Dougherty and Edward (2004) These variations differ in how they address individuals who choose not to vote compared to those who vote but abstain from expressing a preference.
Definition 3.1.Absolute k-majority rule: alternativexdefeats the status quo,q, by absolutek-majority rule if and only if #yeas≥k, where 1≤k≤N; otherwiseqis chosen.
Three common procedures in decision-making include absolute majority rule, absolute supermajority rule, and absolute unanimity rule, each defined by specific thresholds of affirmative votes For instance, the U.S Supreme Court requires four out of nine justices to grant a writ of certiorari, while the Russian Duma mandates a simple majority for passing proposals Additionally, the Articles of Confederation required unanimous consent from all thirteen states for amendments In these scenarios, the outcomes are determined based on a predetermined threshold, treating non-voters and abstentions similarly to votes against the proposal.
Definition 3.2.Simple k-majority rule: alternative xdefeats the status quo, q, by simplek-majority rule if and only if #yeas+#nays #yeas >k/N; otherwiseqis chosen.
The three primary rules of the simple class of k-majority include the simple majority rule, where a proposal passes if the affirmative votes exceed the negative (k/N = 0.5), the simple supermajority rule (0.5 < k/N < 1), and the simple unanimity rule, which requires all votes to be in favor for a proposal to pass (k/N = 1) Examples of these rules include the simple majority needed to pass legislation in the U.S House of Representatives, the two-thirds majority required in the U.S Senate for presidential impeachment, and the unanimous consent required among permanent members of the U.N Security Council for nonprocedural decisions The distinction between absolute and simple k-majority rule lies in how they handle abstentions; absolute k-majority counts abstentions as votes against, while simple k-majority ignores them, making it more likely for proposals to pass in the simple class when abstentions are present When all participants vote without the option to abstain, both procedures yield equivalent results, although the specific k-majority rule that Buchanan and Tullock analyzed remains ambiguous.
Technical distinctions, such as these, can help us understand subtleties in the relationship between unanimity rule and various Pareto concepts.
1 The terms “absolute” k-majority rule and “simple” k-majority rule are direct extensions of Riker
(1982, pp 44–5) Sen (1979a, pp 71, 181) makes a similar distinction but uses different nomen- clature.
2 See Laruelle and Valenciano (2010) for additional variants of k-majority rule that have been used in practice.
The Pareto criterion, as defined by Sen (1979), states that one alternative, x, is considered Pareto preferred to another alternative, y, if it improves the situation of at least one individual without making anyone else worse off.
Definition 3.4.Weak Pareto criterion: For any two alternativesxandy,xis weakly Pareto preferred toyif and only if everyone strictly prefers xtoy(Arrow, 1951; Sen, 1979b).
Definition 3.5.BT Criterion: Proposalxis BT preferred to status quoqif and only if it is Pareto preferred toq; otherwiseqis BT preferred tox(Buchanan and Tullock, 1962; Head, 1974; Rogowski, 1974; Tsebelis, 1990).
Buchanan and Tullock (1962), Head (1974), and Rogowski (1974) identify Definition 3.5 as the Pareto criterion, while Tsebelis (1990, p 104) refers to it as the "efficiency" criterion It is crucial to recognize that Definition 3.5 is distinct from Definition 3.3 and should be considered independently.
BT criterion in honor of Buchanan and Tullock 4
Definition 3.6.Pareto optimality: Alternativexis Pareto optimal if there does not exist an alternativeythat is Pareto preferred tox(Sen, 1979a).
In decision-making, one alternative \( x \) is considered "Pareto preferred" or "Pareto superior" to another alternative \( y \) if it meets specific criteria Conversely, \( x \) is termed "Pareto dispreferred" or "Pareto inferior" to \( y \) if \( y \) is Pareto preferred to \( x \) When neither alternative is preferred over the other, they are classified as "Pareto indeterminate." Throughout this discussion, we will denote alternatives using specific notations: \( PP(y) \) for any alternative that is Pareto preferred to \( y \), \( PD(y) \) for those that are Pareto dispreferred to \( y \), and \( PO \) for Pareto optimal alternatives It is important to note that Pareto optimality does not depend on any specific alternative, while alternatives that do not achieve this status are referred to as Pareto sub-optimal.
Finally, we define neutrality to help us understand the relationship between sev- eral concepts.
Definition 3.7.Neutrality: If x defeats (ties) yfor one preference profile and all individuals have the same ordinal rankings forzandwas they have forxandy(i.e. x j y→z i w, and so on), thenzdefeats (ties)w(Mueller, 2003, p 134).
Neutrality in voting rules and institutions implies that they should not favor any particular option, regardless of individual preferences The criteria used to determine that option x is socially as good as option y must also apply in reverse, ensuring that if preferences are flipped, option y is equally considered socially as good as option x.
3 Sen (1979a, p 25), Berggren (1996, pp 339–40), and Buchanan (1962) interpret Buchanan and Tullock’s criterion as we do Buchanan and Tullock claim that a desirable change can be made
“only if all persons agree” (Buchanan and Tullock, 1962, pp 92–3) That is, only if everyone is made better off.
4 See Dougherty and Edward (2004, 2010a) for applications of the BT criterion.
Pareto Preference
The differences between the Pareto criterion and the BT criterion, while seemingly subtle, lead to distinct evaluations of individual preferences The Pareto criterion remains neutral regarding the status quo, whereas the BT criterion inherently favors it For instance, in a three-person voter profile, neither alternative is Pareto preferred, resulting in an indeterminate judgment from the Pareto criterion, while the BT criterion endorses the status quo Consequently, in cases of Pareto indeterminacy, these two criteria frequently yield differing outcomes.
Fig 3.1 Pareto Preference vs Pareto Indeterminance
Figure 3.1 illustrates the difference in utility levels for two individuals, normalized to a scale of (0,1) The utility possibility frontier, shown at the top right of the figure, represents the highest achievable utility based on the available resources and technology.
Points located in the top-right area relative to point x, including those on the vertical and horizontal lines through point 5, are considered Pareto preferred to x, as they improve at least one individual's situation without worsening anyone else's This region is represented by the shaded area labeled PP(x) Conversely, points in the bottom-left area, along with those on the dotted line, are deemed Pareto dispreferred to x because x is Pareto preferred to every point within this zone, denoted by the shaded area PD(x) Additionally, points found in the unshaded regions beneath the utility possibility frontier, specifically in the top-left and bottom-right areas relative to x, are classified as Pareto indeterminate concerning x.
Pareto Optimality
In the context of evaluating alternatives, the status quo, denoted as x, is assessed using both the Pareto criterion and the BT criterion Both criteria indicate that alternatives located in the top-right of x are preferred to x However, they differ in their evaluations of alternatives in the unshaded areas Specifically, when comparing the status quo x with the proposal y, neither alternative is deemed Pareto preferred to the other, resulting in Pareto indeterminacy In this scenario, the BT criterion favors the status quo x over y, as it posits that if neither option is Pareto preferred, the status quo should be considered preferable This approach, termed a Pareto extension rule by Sen (1979a), provides a judgment where the Pareto criterion remains silent.
The Pareto criterion remains neutral, while the BT criterion is inherently non-neutral In a scenario where the status quo is maintained, the Pareto criterion yields an indeterminate judgment, whereas the BT criterion favors option y This preference arises because, in the absence of Pareto preferred alternatives, the BT criterion consistently supports the status quo, highlighting its significant role in this evaluation framework.
Figure 3.1 illustrates Pareto optimality, where any point on the utility possibility frontier is deemed optimal, as improving one individual's utility would diminish the other's Points within the frontier are classified as Pareto sub-optimal, indicating that there are alternatives that could enhance one individual's situation without harming the other While the idea of "BT optimality" might arise from the concept of Pareto optimality, it proves challenging to define, as the BT criterion necessitates the status quo to be one of the comparison points.
Refinements to Pareto optimality can enhance our understanding of outcomes that are both Pareto optimal and represent a Pareto improvement over the current situation Scholars should focus on the intersection of alternatives that are Pareto preferred to the status quo and those that are Pareto optimal to ensure desirable results.
A point x is defined as "BT optimal" if no alternative y is BT preferred to it Considering a fixed point x₀ on the utility possibility frontier, its status is influenced by four conditions related to q First, if q equals x₀, then x₀ is the sole BT optimal alternative Second, if q is different from x₀ but still on the utility possibility frontier, x₀ is not BT optimal; instead, q holds that status Third, if q is neither equal to nor on the frontier but is interior to it (Pareto suboptimal), the BT optimality of x₀ remains indeterminate, as no alternatives are BT preferred to it, yet it is not the status quo Lastly, if q does not exist, as in an open seat election, BT comparisons become impossible, leading to a poorly defined concept of BT optimality.
The set of alternatives represented in Figure 3.1, denoted as PP(x) & PO, consists of points on the utility possibility frontier located between the two hash marks These points indicate scenarios where at least one individual experiences an improvement in utility without negatively impacting the other, thereby maximizing the potential for Pareto improvements.
Both Pareto optimality and its combination with Pareto superiority are essential for evaluating policies and institutions in existing literature Research on common pool resources and public goods often emphasizes outcomes that are both Pareto optimal and represent improvements over the initial status quo It is crucial for policymakers to assess whether voluntary contributions enhance the situation compared to a scenario where no contributions are made and to identify if all possible Pareto improvements have been realized.
Lindahl's theory on just taxation, articulated in 1919, emphasizes that when individuals vote on public goods financing through a unanimity rule, they can achieve a Pareto optimal quantity of the public good and a fair distribution of costs This process leads to a Pareto improvement over the current situation of lacking public goods The concepts of Pareto superiority and Pareto optimality are relevant here, as they provide a clear benchmark—the absence of public goods—from which to evaluate improvements.
Scholars have assessed institutions through the lens of Pareto optimality without limiting their analysis to outcomes that improve upon the status quo, as seen in various works within social choice and legislative decision-making literature In these analyses, the status quo is often treated neutrally, either to focus purely on welfare or because it represents just one of many alternatives, thereby avoiding undue influence on outcome evaluations For instance, Aldrich (1995) examines whether different coalition types, such as majority rule or pluralistic coalitions, achieve Pareto optimality without considering if they represent improvements over the status quo, which he deems arbitrary His study employs a "divide the dollar" game to explore party formation in the U.S legislature, highlighting that the initial coalition's formation is fundamentally arbitrary, leading to the conclusion that assessing institutions based on Pareto optimality, rather than their superiority to the status quo, may be a more objective approach.
6 Lindahl’s suggestion had problems, such as not being compatible with an accurate revelation of preferences.
7 See (Grafstein, 1990) for an interesting discussion of unanimity rule and the status quo.
Unanimity Rule and the Pareto Principles
Buchanan and Tullock seem to favor institutional changes that are Pareto im- provements to the status quo However, Buchanan also make statements such as
Unanimous consent serves as the political equivalent of the Pareto criterion for optimality, highlighting the importance of collective agreement in decision-making processes (Buchanan, 1967, p 285) This article explores both criteria in detail, ultimately allowing readers to determine which is more suitable for constitutional and legislative contexts, with further discussion provided in Chapter 4 and the conclusion.
3.4 Unanimity Rule and the Pareto Principles
While some authors equate the unanimity rule with the Pareto criterion, it is important to clarify that neither absolute nor simple unanimity rules are synonymous with the Pareto criterion This distinction is supported by various scholars, highlighting significant differences between these concepts.
Both unanimity rules determine alternatives based on actions, such as voting decisions, while the Pareto criterion assesses options according to the preferences of both voters and nonvoters For instance, in a scenario with three individuals who all prefer an alternative 'x' but choose not to vote—potentially due to the rationality of voting—unanimity rules would uphold the status quo, whereas the Pareto criterion would advocate for the proposed alternative.
Both unanimity rules exhibit a key difference from the Pareto criterion regarding neutrality In a scenario where all individuals vote sincerely based on their true preferences in a three-person voter profile (x1 q, x2 q, q3 x), both unanimity rules favor the status quo In contrast, the Pareto criterion does not provide a definitive choice between the two alternatives.
The absolute unanimity rule contrasts with the Pareto criterion in how it handles voter indifference In a scenario where all three voters participate and vote sincerely, with one voter opting to abstain, the absolute unanimity rule will uphold the status quo, despite the fact that the proposal is preferred by a majority according to the Pareto principle.
Neither class of unanimity rule aligns with the weak Pareto criterion for three key reasons First, unanimity rules focus on the actions of voters, while the weak Pareto criterion bases recommendations on individual preferences Second, even when all individuals vote sincerely, both types of unanimity rules lack neutrality, whereas the weak Pareto criterion maintains neutrality between alternatives Lastly, unlike the previous scenario, both absolute and simple unanimity rules do not fulfill the criteria set by the weak Pareto principle.
Buchanan and Tullock (1962) define a "social state" as Pareto optimal, aligning with our Definition 3.6 However, they also categorize changes as Pareto optimal if they result in Pareto improvements They assert that a transition from point A to point G is Pareto optimal in isolation, despite indicating a move from one nonoptimal state to another This perspective appears to conflate the concepts of Pareto optimality and Pareto improvements.
The Clarifying Concepts rule differs from the weak Pareto criterion in how it handles indifference among individuals In a scenario where all participants sincerely express their preferences in the profile (x1q, x2q, x∼3q), the absolute unanimity rule results in the selection of the status quo, while the simple unanimity rule favors the proposal In contrast, the weak Pareto criterion does not provide a clear decision, leading to an indeterminate outcome.
Neutral definitions of the two unanimity rules suggest that neither voting method will yield a decisive outcome, necessitating a secondary mechanism, like a coin flip, to resolve situations lacking consensus Additionally, the absence of practical examples of such neutral unanimity rules further undermines their feasibility.
The BT criterion, when defined non-neutrally to resemble a simple unanimity rule, demonstrates equivalence with the simple unanimity rule in pair-wise votes when all participants vote sincerely However, it diverges from the absolute unanimity rule due to differing approaches to indifference For instance, in the scenario of (x1q, x2q, x∼3q), the absolute unanimity rule favors the status quo, whereas the BT criterion advocates for the proposed option.
It is important to note that neither version of the unanimity rule aligns with Pareto optimality, as Pareto optimality focuses on the absence of Pareto preferred alternatives, while the unanimity rule is concerned with making decisions between two specific options.
Pareto Indeterminance
Implications
Proposition 1 highlights that if preferences are independent, the Pareto criterion alone is insufficient to establish a comprehensive ranking in constitutional design, legislative decisions, and popular elections It often fails to differentiate between Pareto sub-optimal and optimal points, as well as between any two points within these contexts This limitation also applies to the weak Pareto criterion.
To address the indeterminacy of the Pareto criterion, a Pareto extension rule can be proposed, which favors the status quo in ambiguous cases This approach aligns with the BT criterion but raises concerns, as it is influenced more by the extension rule than by the Pareto criterion itself The notion that a constitution, institution, or public good should only be adopted if it leads to a Pareto improvement is contingent upon the justification of the extension rule rather than the welfare patterns inherent in the Pareto criterion Scholars advocating for the preservation of the status quo during Pareto indeterminate situations often base their arguments on rights, rather than adhering to the judgments suggested by the Pareto criterion.
Proposition 1 suggests that in large populations with two alternatives, both options are likely to be Pareto optimal This is due to the concept of Pareto indeterminacy, which indicates that neither alternative is preferred over the other in terms of Pareto efficiency.
For large populations with a finite number of alternatives, it can be easily demonstrated that all alternatives are likely to be Pareto optimal, extending the conclusions of Proposition 1 for any fixed positive integer.
While the probability of Pareto indeterminacy may not increase as quickly as Table 3.1 indicates when preferences are probabilistically dependent, the assertion that such indeterminacy is likely in large populations remains valid For instance, in the spatial voting models discussed in the following chapter, individual preferences regarding various alternatives exhibit probabilistic dependence Nevertheless, the likelihood of randomly selecting a Pareto optimal outcome from these preferences persists.
The utilities possibility frontier illustrates the concept of Pareto indeterminacy, particularly when the outer edge is flat and vertical, forming a square shape In this scenario, approximately half of the area is Pareto indeterminate regarding the point (0.5,0.5), specifically in the top-left and bottom-right regions Introducing a third individual expands this concept, resulting in about 6/8 of the areas becoming Pareto indeterminate concerning (0.5,0.5,0.5) This principle can be generalized to N individuals in N dimensions, where the likelihood of Pareto indeterminacy aligns with the probabilities outlined in equation (3.1) for p 1 = p −1 = 0.5.
Introduction
Notation
This study examines N individuals, each possessing an ideal point \( I_i \) within a bounded and closed n-dimensional hyper-square Each individual exhibits single-peaked and symmetric utility, indicating a preference for alternatives that are closer to their ideal point over those that are farther away Additionally, the analysis assumes the absence of transaction costs, such as decision-making expenses, and prohibits vote trading, with indifferent voters opting to abstain unless otherwise specified.
In this chapter, we emphasize the complexities of decision-making in two or more dimensions, as opposed to the simpler single-dimensional scenarios While a single dimension allows for a clear median voter and ensures that intransitivity does not occur with single-peaked preferences, multidimensional alternatives lead to qualitatively different outcomes In single-dimensional cases, both majority and unanimity rule cores are Pareto optimal, meaning no alternative can be preferred by a majority or unanimous group over another However, in multidimensional spaces, the absence of a defined median, the prevalence of intransitivities under majority rule, and the typical emptiness of the majority rule core complicate the attainment of Pareto optimal results Thus, we will focus on the more challenging two-dimensional case in our examples.
In our analysis of voting procedures, we utilize a forward agenda approach This means that the initial status quo, referred to as q1, is compared against a proposal, denoted as x1, in the first round The winning alternative from this round is then matched with a new proposal, x2, in the second round, and this process continues for a total of R rounds For rounds beyond the first (r > 1), qr represents the status quo of round r, which is the winning proposal from the previous round (r - 1) Throughout this discussion, we will denote both the proposals and their corresponding positions in the voting process as xr and qr, respectively.
Behavior stemming from nonrational proposals and the absence of transaction costs can often be mistaken for behavior driven by rational proposals with complete information and no transaction costs According to Dougherty and Edward (2010b), rational proposals would consistently yield a Pareto preferred outcome under a unanimity rule, provided that the status quo is not optimal In contrast, nonrational proposals may not guarantee such an outcome.
Constitutional decision-making aligns with Tullock's (1998) and Mueller's (2003) views that unanimity rule can lead to Pareto optimal outcomes By maintaining a consistent number of voting rounds, we ensure that the only variable affecting the results is the voting rule itself.
It would also result from rules that limit the number of amendments, enact time limitations, or create backward agendas 3 All of which are common in deliberative assemblies.
This chapter focuses on the simple k-majority rule rather than the absolute unanimity rule, primarily for three reasons: simple k-majority rules are more prevalent in practice, they are less stringent than absolute k-majority rules, and indifference is nearly nonexistent in spatial voting models When all individuals participate in voting without abstentions, the outcomes under both absolute and simple k-majority rules tend to align Therefore, for the purposes of this chapter, we will refer to simple majority rule and simple unanimity rule simply as majority rule and unanimity rule, respectively.
In our analysis, we denote PP(q1) as any alternative that is Pareto preferred to the initial status quo, q1 Additionally, we use PP(q1) & PO to identify outcomes that are both Pareto preferred to q1 and Pareto optimal These sets of alternatives can be clearly defined within a spatial model.