Elections and Fair Division

Different countries use different electoral systems, all of which aim to help shape a democratic constitutional state. We distinguish cardinal vs ordinal voting on the one hand and proportional vs non-proportional representation on the other hand. In concreto, we discuss the Dutch list system of proportional representation using Plurality Rule (PR) (most votes count) where the whole country is conceived as one district vs the non-proportional British system based on many districts, where in each district Plurality Rule (PR) is used to determine the winner. In both systems voters are supposed to cast only one vote—hence, we speak of cardinal voting—and the party with most votes is the winner. The British district system causes that the distribution of seats over the parties is non-proportional to the number of votes the party received. In countries like Malta and Australia voters are asked to give much more information than just their first preference: voters in these countries may give their individual preference order over the candidates and hence we speak of ordinal voting. The Single Transferable Vote (STV) system, used for instance in Malta, results in proportional representation, while the Alternative Vote (AV) system, used for instance in Australia, gives a non-proportional representation, again because the country is divided into districts. We also discuss the French presidential electoral system. All electoral systems presented, although designed with the best intentions, turn out to be seriously defective.

Finally, we will show that designers of an agenda for decision making can achieve their preferred outcome by a smart agenda design.

In this chapter we will discuss electoral functions and rules which take the individual preference orders over the alternatives by the voters as input. In particular, we shall discuss Plurality Rule (PR), Majority Rule (MR) and the Borda Rule (BR). For each of these we shall establish whether they satisfy a number of (at first sight) desirable properties: unrestricted domain, anonymity, neutrality, monotonicity, completeness, Independence of Irrelevant Alternatives (IIA) and transitivity. We shall see that in the case of only two alternatives the only electoral function satisfying the first five conditions is Majority Rule (May, Econometrica 20:680–684, 1952) and that in the case of three or more alternatives there cannot be an electoral function satisfying all seven conditions (Balinski and Laraki, Majority Judgment vs Majority Rule. Social Choice and Welfare, 2019. https://​doi.​org/​10.​1007/​s00355-019-01200-x). In particular, none of the electoral rules PR, MR and BR satisfies all these properties. We shall also see that the different electoral functions, applied to the same profile, may give very different outcomes. In other words, the result of an election does not depend so much on the preferences of the voters, but rather on the electoral function used (Saari, Decisions and Elections. Cambridge University Press, Cambridge, 2001). We discuss Saari’s geometrical representation of voting profiles and explain how paradoxical results in (minimal) liberalism, judgment aggregation and in the allocation of tenders all boil down to violations of IIA and can be reduced to the Condorcet paradox.

In this chapter we show Arrow’s (Social Choice and Individual Values, 2nd edn. Yale University Press, New Haven, 1951/1963) characterization of the dictatorial ranking function: In case of three or more alternatives, any social ranking function that is transitive, Pareto-optimal and Independent of Irrelevant Alternatives (IIA) is dictatorial. In other words: there is no social ranking function that is transitive, Pareto optimal, IIA and not dictatorial. This result is called Arrow’s Impossibility theorem (1951). We prove this result first for the case of two individuals—we call them Romeo and Julia—because this case shows perfectly the role of the different conditions in making one of the persons in fact a dictator. Next we present a proof for the general case of two or more voters (and n, \(n \geq 3\), alternatives).

Gibbard (Econometrica 41:587–601, 1973) and Satterthwaite (J. Econ. Theory 10:187–217, 1975) proved that in the case of three or more alternatives every social choice function that is Pareto optimal and strategy-proof is dictatorial. In other words, in the case of three or more alternatives there exists no social choice function that is Pareto optimal, strategy-proof and non dictatorial. We shall prove this theorem by reducing it to Arrow’s impossiblity theorem. But, because of its transparency, we also give a proof of this result for the special case of only two individuals/voters and three alternatives.

We also present three theorems due to Balinski and Laraki (Majority Judgment. MIT Press, Cambridge, 2010) (Chapter 4) which show that several desirable properties concerning electing and ranking are incompatible.

Arrow’s and Gibbard-Satterthwaite’s impossibility results assume that the domain of the social ranking function, resp. social choice function, is unrestricted, i.e., that the social ranking function, resp. social choice function, should assign a social order, resp. a social choice, to all logically possible individual preference profiles. But in practice the individual preference profiles frequently satisfy certain restrictions. We shall show Black’s (Theory of Commitees and Elections. Cambridge University Press, Cambridge, 1958) result that under some of these restrictions, in particular in the case of single-peaked preferences, nice (i.e., satisfying a number of desired properties), non dictatorial social choice functions do exist.

In the preceding chapters we have seen that the traditional framework for elections, in which voters are supposed to give preference orders over the alternatives, although adopted by almost everyone in the field, is doomed to failure. It was only around 2010 that Balinski and Laraki (Majority Judgment vs Majority Rule. Social Choice and Welfare, 2019. https://​doi.​org/​10.​1007/​s00355-019-01200-x) pointed out that Plurality and Majority Rule do not respect dominance, i.e., a candidate with the better evaluations may not become the winner. For that reason they developed a new electoral rule, called Majority Judgment, that takes as input the evaluations of the candidates by the voters instead of their preference orders (or first choice in the case of Plurality Rule). By taking the median value of a candidate’s evaluations, Majority Judgment measures the global support this candidate receives from the electorate. Majority Judgment has many nice properties not shared by other voting systems. Contrary to Plurality Rule and Majority Rule, Majority Judgment is transitive, Independent of Irrelevant Alternatives and respects dominance. In addition, Majority Judgment is strategy-proof with respect to the final judgment (majority grade) of a candidate and partially strategy-proof with respect to the social ranking of the candidates. Balinski and Laraki illustrate their findings with recent presidential elections in the USA and in France, in the meantime pointing out that the French electoral system is not monotonic. They also point out that summing or averaging points is easy to manipulate and not consistent with Majority Judgment nor with Majority Rule, while Approval Voting gives arbitrary outcomes and does not respect dominance. This chapter is an extended version of Harrie de Swart, How to choose a president, mayor, chairman: Balinski and Laraki unpacked, published in The Mathematical Intelligencer 2022, https://​doi.​org/​10.​1007/​s00283-021-10124-3; link to the Creative Common license: https://​creativecommons.​org/​licenses/​by/​4.​0/​. As such it can be read independently of the other chapters. I am grateful to Sergei Tabachnikov (editor) and several anonymous referees for suggesting important improvements.

A measure of a priori voting power of a member in a set (assembly, union or group) was introduced by Lionel Penrose in 1946 and rediscovered by Banzhaf in 1965. Henceforth, one may speak of the Penrose-Banzhaf (power) index. In order to satisfy the principle of one man, one vote, Penrose also made clear that the voting power of each nation in a federal assembly of nations, like the Eurpean Union (EU), should be proportional to the square root of the number of citizens entitled to vote. We call this the rule of Penrose. We present the power distribution among the different countries of the European Union for several periods with constantly changing decision rules and shall see that Penrose’s rule is frequently violated.

In practically all western countries one may observe that the government and its agencies do not function properly and even seem to be hostile towards its citizens. What is immediately evident from the outside is that the ministers and politicians are seldomly qualified for the ministry they are supposed to manage, that in most cases they do not have a proper beta education and that frequently they have never worked in real life such as industry, small or big business. Usually, they are only bureaucrats who try to make a career in the administrative circuit, in which they never have to think about the question who pays the bill of their decisions. But there are many more reasons why democracy in western countries is in big trouble. In the first section we shall point out a number of weak aspects of western democracies. In the next section we shall introduce an alternative polity which is due to Rients Hofstra (Een Alternatief Staatsbestel. https://​alternatiefstaat​sbestel.​nl/​), with the intention to make western democracies really democratic again. This chapter is co-authored by Rients Hofstra, in particular Sect. 6.2.

In a fair division problem a scarce good, also called the estate, has to be divided amongst some agents and the problem is to do so in a fair way. The claims-based and preference-based approach to fair division take, respectively claims and preferences of the agents into account for realizing fair divisions. This chapter provides an introduction to the claims-based approach to fair division.

In particular, we will present an overview of various models that are within the scope of the claims-based approach: in a bankruptcy problem, agents have claims with different amounts, whereas in a weighted bankruptcy problem, claims may not only vary in their amount, but also in their strength. First we discuss (weighted) bankruptcy problems for which the estate is continuous, such as an amount of money. Then we discuss versions of these problems for which the estate is discrete, such as a number of parliamentary seats. Finally, we discuss cooperative game theory, which can be used to model problems in which claims belong, first and foremost, to groups of agents.

The presented overview of claims-based models and corresponding allocation rules is, deliberately, biased. For, models and rules are presented with an eye towards the extent to which they can elaborate on and make precise the idea that ‘fairness requires that claims are satisfied in proportion to their strength’. This idea has been advocated for, in the philosophical literature, by a.o. Nicholas Rescher and John Broome. This chapter sketches the philosophical underpinning of ‘proportional claims-based fairness’ and briefly introduces the aforementioned claims-based models. In subsequent chapters, the various claims-based models, their allocation rules, and ‘results on proportionality’ will be discussed in more detail.

In a bankruptcy problem, a continuous estate has to be divided amongst individuals who have claims of varying amounts. There is a vast literature on bankruptcy problems, starting with O’Neill (Math. Soc. Sci. 2:345–371, 1982) and comprehensively reviewed in Thomson (How to Divide When There Isn’t Enough: From Aristotle, the Talmud, and Maimonides to the Axiomatics of Resource Allocation. Cambridge University Press, Cambridge, 2019). In this literature, a multitude of allocation rules have been proposed and studied. In this chapter, we review the most salient bankruptcy rules and their properties. Special attention will be devoted to the proportional rule and to the consistency axiom.

In this chapter we discuss weighted bankruptcy problems, which allow us to represent agents with claims to different amounts and of varying strengths. In Sect. 9.1, we discuss the relation between weighted bankruptcy problems and related resource allocation problems. In Sect. 9.2, we discuss various allocation rules for weighted bankruptcy problems that have been proposed in the literature. These rules are all defined as generalizations of the allocation rules for the bankruptcy problems that we discussed in the previous chapter. After that, we fully concentrate on the absolute priority rule \(P^\dagger \), which has been proposed as a generalization of the proportional bankruptcy rule \(\mathbb {P}\). In Sect. 9.3 we explain how \(P^\dagger \) can be characterized in terms of axioms that can be justified of a conception of fairness that has been articulated by Wintein (A Theory of Fairness from Consistency. Theory and Decision, 2025). Indeed, we argue that fairness requires that, for weighted bankruptcy problems, we apply \(P^\dagger \). We conclude Sect. 9.3 by commenting on alternative characterizations of \(P^\dagger \).

In this chapter we study proportional allocation methods for discrete weighted bankruptcy problems, i.e., weighted bankruptcy problems with a discrete (‘indivisible’) estate, such as horses or seats in a parliament. The major part of this chapter is devoted to an overview of apportionment theory which, in its canonical formulation, is the systematic study of methods to allocate, proportionally, a number of seats to states on the basis of their populations. However, as Balinski and Young (Fair Representation. Brookings Institution Press, Washington D.C., 2001) explain ‘any problem in which objects are to be allocated in non-negative integers proportionally to some numerical criterion belongs to this class, and [apportionment] theory applies to it’. In particular, apportionment theory applies to discrete weighted bankruptcy problems, as we explain in Sect. 10.1. In Sect. 10.2, we discuss various apportionment methods, which output an allocation—or a set of allocations in case there are ties—of the discrete estate. Another way to allocate a discrete estate is via a lottery. In Sect. 10.3, we discuss a lottery method to allocate the estate of a discrete weighted bankruptcy problem in a fair, i.e., ‘proportional’ way.

In order to model a fair division problem as a (weighted) bankruptcy problem, we need to know the exact amount and strength of the claim of each individual agent. For quite a few fair division problems, however, that information is not available. Cooperative game theory can be used to model situations where claims belong, first and foremost, to groups of agents. In this section, we present a concise introduction to the basics of cooperative game theory and we explain the importance of this discipline for claims-based fair division. In particular, we focus on the Shapley value, a prominent and well-known solution value (allocation rule) for cooperative games.

In this chapter we discuss fair division problems based on preferences of the persons involved. How to divide a cake or land fairly, when different persons have different preferences for parts of the cake or land, is the topic of the first section. Important properties are proportionality, envy-freeness, equitability and efficiency. In the second section we present the Adjusted Winner procedure and the Proportional Allocation procedure of Brams and Taylor (Fair Division: From Cake-cutting to Dispute Resolution. Cambridge University Press, Cambridge, 1996), both having attractive properties. In particular the Adjusted Winner procedure is an easy to understand procedure that can solve many disputes in, for instance, divorce settlements, resulting in outcomes that for both partners are better than a fifty-fifty division.