In this post I outline what was included in the first three parts of the Social Choice Theory entry (written by Christian List) in the Stanford Encyclopaedia of Philosophy. Unless otherwise labelled by ‘Thoughts:’, all content is paraphrased or quoted from List’s article.
1 Social Choice Theory is
- The study of collective decision processes and procedures. Concerns the aggregation of individual inputs into collective outputs.
- e.g. How can a group of individuals choose a winning outcome from a given set of options? How can we rank different social alternatives in an order of social welfare?
- A cluster of models vs a single theory
1.1 Condorcet (1743-1794)
Investigated majority rule and found that it has nice properties but is subject to surprising problems.
- Condorcet’s Jury Theorem (Elaborated on in section 2.2)
- Condorcet’s Paradox: Majority preferences can be ‘irrational’ (intransitive) even when individual preferences are ‘rational’ (transitive).
1.2 Arrow (1921-)
Introduced a general approach to the study of preference aggregation. Considered a class of possible aggregation methods (Arrovian social welfare functions) and asked which methods satisfied certain axioms.
- Arrovian social welfare functions: aggregates ordinal preferences
- More a voting rule than a welfare evaluation method.
- Debate on whether one might use interpersonally comparable welfare measurements for welfare evaluation (Sen for, Arrow against)
- Axiomatic method:
- Arrow’s Impossibility Theorem
- Axioms debated.
- Riker interpreted this as a mathematical proof of the impossibility of populist democracy.
- Sen interpreted it as a proof that ordinal preferences are insufficient for making satisfactory social choices.
- Used to examine trade-offs involved in finding satisfactory decision procedures. (‘Possibilist’ interpretation of social choice theory, Sen)
1.3 Borda, Carroll, Black, and others
- Borda count (1733-1799)
- Voting system that avoids Condorcet’s paradox but violates one of Arrow’s axioms, the independence of irrelevant alternatives (IIA).
- Precursor to debates on how to respond to Arrow’s theorem.
- Ramon Llull (c1235-1315) proposed the aggregation method of pairwise majority voting
- Nicolas Cusanus (1401-1464) proposed a variant of the Borda count
2 Three formal arguments for majority rule
Our decision problem: A collective choice between two alternatives
2.1 The concept of an aggregation rule
- Defines Majority Rule, Dictatorship, Weighted Majority Rule.
- Notes that
- An aggregation rule is defined extensionally, not intensionally: it is a mapping (functional relationship) between individual outputs and collective outputs, not a set of explicit instructions that can be extended to inputs outside the function’s domain.
- An aggregation rule is defined for a fixed set of individuals N and a fixed decision problem, so that majority rule in a group of two individuals is a different mathematical object from majority rule in a group of three.
- Thought: then Majority Rule generally is not an aggregation rule, but a group of aggregation rules.
- For a decision problem with alternatives and people, there are possible aggregation rules. That is, the number of possible aggreggation rules grows exponentially with the number of admissible profiles and the number of possible decision outcomes.
2.2 Argument 1: A procedural argument for majority rule
- May (1952) showed that majority rule is the only aggregation rule that satisfies (1) Universal Domain, (2) Anonymity, (3) Neutrality and (4) Positive Responsiveness.
- This theorem helps us characterise other aggregation rules in terms of the axioms they violate. ( possibilities.)
- Thought (To-do): Consider making a table of these 16 possibilities.
2.3 Argument 2: An epistemic argument
- Condorcet’s Jury Theorem
- Follow-up result: If two states of the world have an equal prior probability,majority rule is the most reliable of all aggregation rules, maximising Pr(V=X). (Ben-Yashar and Nitzan 1997).
- This is an epistemic argument insofar as you consider the aggregation rule as a truth-tracking method.
- Competence assumption is not a conceptual claim but an empirical one.
- Theorem can be undermined when each individual’s reliability, though above 1/2, is an exponentially decreasing function approaching 1/2 with increasing jury size (List 2003a, Berend and Paroush 1998).
- Conditional Independence Assumption:
- Theorem is robust to some interdependencies between individual votes, but the structure of these interdependencies matters.
- Bayesian networks have been used to model the effects on voter dependencies on the jury theorem and to distinguish between stronger and weaker variants of conditional independence. (Pearl’s work on causation, 2000)
- Dietrich has argued that Condorcet’s two assumptions are never simultaneously justified, in the sense that, even when they are both true, one cannot obtain evidence to support both at once.
- Game-theoretic work shows that voters may not choose to reveal their judgements truthfully.
- Thoughts: I like how List divides Condorcet’s Jury Theorem into a non-asymptotic conclusion and an asymptotic conclusion.
- TODO: Follow up on structure of interdependencies and using Bayesian networks to model effects of interdependencies.
2.4 Argument 3: A utilitarian argument
- Rae-Taylor Theorem: If each individual has an equal prior probability of preferring each of the two alternatives, majority rule maximises each individual’s expected utility.
- Each voter is given a utility of 1 from a match between his or her vote and the collective outcome and a utility of 0 from a mismatch.
- Assumes equal-stakes. Stake = personal utility difference between a voter’s preferred outcome and dispreferred outcome.
- Majority rule minimises the number of frustrated voters (voters on the losing side).
- Brighouse and Fleurbaey show that when stakes are allowed to vary across voters, total utility is maximised by a weighted majority rule, where each individual’s voting weight is proportional to his or her stake (personal utility difference between a voter’s preferred and dispreferred outcomes).
3 Preference Aggregation
- Formally defines social preference aggregation rules, preference orderings, profiles, pairwise majority voting.
- Discusses the frequency of intransitive majority preferences (see Condorcet’s Paradox). (Frequent.)
- If all possible preference profiles are equally likely to occur (‘impartial culture’), majority cycles (intransitivities) should be probably in large electorates. But the probability of cycles can be significantly lower under certain systematic, small devations from an impartial culture.
- ‘Top-cycles’ vs cycles below a possible Condorcet-winning alternative
3.1 Arrow’s Theorem
- Theorem: If the number of alternatives is greater than two, there exists no preference aggregation rule satisfying universal domain, ordering, the weak Pareto principle, independence of irrelevant alternatives and non-dictatorship.
- How to deal with this: Need to relax at least one of Arrow’s five conditions or give up the restriction of the aggregation rule’s inputs to orderings and defend the use of richer inputs.
- Carries over to
- belief orderings over several hypotheses (ordinal credences),
- multiple criteria that a single decision-maker may use to generate an all-things-considered ordering of several decision opotions, and
- conflicting value ranking to be reconciled.
- Applied to problems such as
- Interpersonal aggregation problems
- Constraint aggregation in optimality theory in linguistics
- Theory choice
- Evidence amalgamation
- Aggregation of multiple similarity orderings into an all-things-considered similarity ordering
3.2 Non-dictatorial preference aggregation (Relaxing different axioms)
Discusses relaxing four of the five axioms.
- Relaxing Universal Domain: Accept as input only preference profiles that satisfy certain ‘cohesion’ conditions, such as single-peakedness. Single-peakedness is plausible for e.g. deciding between tax rates.
- Black proved that, with preferences satisfying single-peakedness, majority cycles cannot occur and (2) the most preferred alternative of the median individual relative to the relevant left-right alignment is the Condorcet winner, assuming n is odd.
- Other domain-restriction conditions with similar implications: single-cavedness (mirror of single-peakedness), separability into two grous, and latin-squarelessness. Sen showed that all these conditions imply a weaker condition, triple-wise value restriction.
- TODO: Examine how rigorous definition of single-peakedness rules out caves. Examine triple-wise value restriction.
- Open Problem: Further empirical work is needed on whether real-world preferences fall into such a restricted domain.
- Group deliberation can induce single-peaked preferences by leading participants to focus on a shared cognitive or ideological dimension. (What?)
- Experimental evidence from deliberative opinion polls is consistent with this hypothesis, though further empirical work is needed.
- Relaxing ordering (rationality)
- Pareto dominance procedure: Produces transitive but incomplete social preferences
- Pareto extension procedure: Produces complete but often intransitive social preferences. (xRy iff x is not Pareto dominated by y for all individuals in N.)
- Each individual has veto power over the (presence or) absence of a weak social preference for x over y.
- Gibbard: Even if we replace the requirement of transitivity with ‘quasi-transitivity’ (strict relation is transitive, indifference relation need not be transitive), the resulting possibilities of aggregation are still limited. If there are more than two alternatives, there exists no preference aggregation rule satisfying universal domain, quasi-transitivity and completeness of preferences, the weak Pareto principle, independence of irrelevant alternatives, and non-oligarchy.
- Is Pareto dominance?
- In an oligarchy, the oligarchs are jointly decisive and have individual veto power.
- Parallel to Arrow’s Impossibility Theorem.
- Relaxing the Weak Pareto Principle
- Spurious unanimity: unanimous preference for x over y is based on mutually inconsistent reasons. (E.g. if there is uncertainty and each person has poor probability estimates.)
- Sen’s ‘moral liberalism’: if we wish to respect individual rights, we may sometimes have to sacrifice Paretian efficiency.
- Thought: I still don’t buy the example. Might discuss this in a post later.
- Relaxing Independence of Irrelevant Alternatives
- Discusses rules that violate IIA: Plurality rule, Borda count.
- Make aggregation potentially vulnerable to strategic voting.
3.3 The Gibbard-Satterthwaite Theorem
- Theorem: The exists no social choice rule satisfying universal domain, non-dictatorship, the range constraint, resoluteness and strategy-proofness.
- Discusses challenges to each condition.
- Social choice rules output one or several winning alternatives (vs preference aggregation rules, which output social preference relations).
- Article discusses social choice rules and social choice rules defined by the Condorcet winner criterion (may produce an empty choice set), plurality rule and the Borda count.
- The latter two always produce non-empty choice sets and satisfying Universal Domain, Non-dictatorship, have a range of at least three distinct alternatives provided at least three alternatives exist (satisfy the range constraint), and – when supplemented with an appropriate tie-breaking criterion – always produces a unique winning alternative (Resoluteness).