Social Choice Theory

Social Preference Aggregation

  • different people may have different preferences over social outcomes
    • in the US, polarized issues include gun control, immigration, civil rights, etc
  • big question: how do we aggregate individual preferences into social preferences?
    • which group's opinion is "correct"?
    • when (and when not to) implement social change?

Social Choice Function

  • input:
    • society \(\{ 1, \ldots, i, \ldots, N \}\) of voters
    • choice set \(X\)
      • example: \(X=\{ \text{Biden}, \text{Harris}, \text{Sanders} \}\)
    • each voter has her own preference relation \(\succsim_i\) over \(X\)
  • output:
    • social preference \(\succsim_S\)

Simple Majority: Condorcet Paradox

voter 1 voter 2 voter 3
most preferred Biden Harris Sanders
middle Harris Sanders Biden
least preferred Sanders Biden Harris
  • how many voters think Biden \(\succ\) Harris? \(\to\) 2 out of 3
  • how many people think Harris \(\succ\) Sanders? \(\to\) 2 out of 3
  • how many people think Sanders \(\succ\) Biden? \(\to\) 2 out of 3

Simple Majority: Agenda Manipulation

voter 1 voter 2 voter 3
most preferred Biden Harris Sanders
middle Harris Sanders Biden
least preferred Sanders Biden Harris
  • problem: agenda manipulation
    • suppose agenda setter wants Biden to win
      • step 1: \(\text{Sanders vs Harris}\) \(\to\) Harris wins
      • step 2: \(\text{winner of step 1 vs Biden}\) \(\to\) Biden wins
    • different winner if different agenda

Simple Majority: Strategic Voting

voter 1 voter 2 voter 3
most preferred Biden Harris Sanders
middle Harris Sanders Biden
least preferred Sanders Biden Harris
  • another problem: strategic voting
    • agenda is \(\text{Sanders vs Harris}\), then \(\text{winner vs Biden}\)
    • focus on voter 2 and her vote at step 1
      • vote sincerely: Harris beats Sanders, Biden beats Harris
      • vote strategically: vote Sanders, Sanders wins at step 1, then Sanders beats Biden at step 2

Desirable Properties of SCF: Completeness

  • full domain / complete ranking (C):
    • returns a social preference for any profile \(\{ \succsim_1, \dots, \succsim_N \}\) of individual preferences

Desirable Properties of SCF: Transitivity

  • transitivity of social ranking / no cycles (T):
    • if \(x \succsim_S y\) and \(y \succsim_S z\), then \(x \succsim_S z\)
  • note: all individual preferences satisfy C and T; we don't want to miss these properties of rationality during aggregation

Desirable Properties of SCF: Pareto

  • Pareto (P):
    • if every person agrees that \(x \succ_i y\), then \(x \succ_S y\)

Dictatorial Preferences

  • so far we want social ranking to be complete, transitive, and Paretian
    • "easy" way to satisfy all three is pick a dictator
      • person \(d\) is a dictator if \(x \succ_d y \Longrightarrow x \succ_S y\)
        • regardless of others' preferences, social ranking is same as dictator's

Desirable Properties of SCF: IIA

  • independence of irrelevant alternatives (IIA):
    • social preferences between \(x\) and \(y\) depend only on the individual preferences between \(x\) and \(y\), and not other alternatives
    • let \(\{\succsim_1, \dots, \succsim_N\}\) and \(\{\succsim_1', \dots, \succsim_N'\}\) be two profiles of individual preferences
      • ranking of \(x\) and \(y\) is the same in both: \(x \succsim_i y \iff x \succsim_i' y\) for every \(i = 1\dots N\)
      • IIA: social preference must be the same as well: \(x \succ_S y \iff x \succ_S' y\)

Why IIA?

  • individual's intensity of preferences should not determine the social ranking
    • intensity of relative political preference is private info and can easily be misrepresented
  • a good SCF should be stable with respect to adding and removing of alternatives

Desirable Properties of SCF: Summary

  • we would like our social choice function to satisfy
    • completeness (C)
    • transitivity (T)
    • Pareto (P)
    • no dictator (ND)
    • independence of irrelevant alternatives (IIA)

Example of SCF: Unanimity Rule Without Status Quo

\[ x \succsim_S y \iff x \succsim_i y \text{ for every person } i = 1\dots N \]
  • if everyone prefers \(x\) to \(y\), then \(x\) is socially preferred to \(y\)
  • if everyone prefers \(y\) to \(x\), then \(y\) is socially referred to \(x\)
  • if people do not unanimously agree that \(x\) is better than \(y\) or vice versa, SCF does not return a social preference
C T P ND IIA
- + + + +

Borda "Scoring" Rule

  • every person provides a ranking of all alternatives, worst \(\to\) best
    • worst choice receives \(1\) point; second worst \(2\) points, and so on
  • social score of an alternative is the sum of individual scores
  • winner is the alternative with highest score

Borda Rule: Example

60 voters 40 voters
Navalny Ivanov
Medvedev Medvedev
Ivanov Zuganov
Zuganov Navalny
  • social score is
    • Navalny: \(4 \cdot 60 + 1\cdot 40 = 280\)
    • Medvedev: \(3 \cdot 60 + 3\cdot 40 = 300^*\)
    • Ivanov: \(2 \cdot 60 + 4\cdot 40 = 280\)
    • Zuganov: \(1 \cdot 60 + 2\cdot 40 = 140\)

Borda Rule: Properties

C T P ND IIA
+ + + + -

Borda Rule: IIA Fails, Example

60 voters 40 voters 60 voters 40 voters
Navalny Ivanov Navalny Ivanov
Medvedev Medvedev Medvedev Medvedev
Ivanov Zuganov Ivanov Navalny
Zuganov Navalny
  • social score is
    • Navalny: \(4 \cdot 60 + 1\cdot 40 = 280 \to 3 \cdot 60 + 1\cdot 40 = 220^*\)
    • Medvedev: \(3 \cdot 60 + 3\cdot 40 = 300^* \to 2\cdot 60 + 2\cdot 40 = 200\)
    • Ivanov: \(2 \cdot 60 + 4\cdot 40 = 280 \to 1\cdot 60 + 3 \cdot 40 = 180\)
    • Zuganov: \(1 \cdot 60 + 2\cdot 40 = 140 \to\) not participating
  • IIA failt: medvedev wins if Zuganov runs, Navalny wins if Zuganov does not run

Borda Rule: Why Failure of IIA is Bad

60 voters 40 voters 60 voters 40 voters
Navalny Ivanov Navalny Medvedev
Medvedev Medvedev Medvedev Ivanov
Ivanov Navalny Ivanov Navalny

  • social score is

    • Navalny: \(3 \cdot 60 + 1\cdot 40 = 220^* \to 3 \cdot 60 + 1\cdot 40 = 220\)
    • Medvedev: \(2\cdot 60 + 2\cdot 40 = 200 \to 2\cdot 60 + 3\cdot 40 = 240^*\)
    • Ivanov: \(1\cdot 60 + 3 \cdot 40 = 180 \to 1\cdot 60 + 2 \cdot 40 = 140\)

  • Borda rule is subject to strategic manipulation: my misrepresenting their preferences, the 40-group elects their favorite candidate

Big Question of Social Choice Theory

  • so far, all our examples of social choice functions failed at least of desirable property
  • is there any social choice function that satisfies C, T, P, ND, IIA?

Arrow's Impossibility Theorem

  • for any social choice problem with at least three alternatives, there does not exist a social choice function that satisfies C, T, P, ND, IIA
    • interpretation: groups of people do not have well-defined preferences, even if each person is rational

Arrow's Theorem: Alternative Statement

  • "for any social choice problem with at least three alternatives, if a SCF satisfies C, T, P, and IIA, then it must be dictatorial
    • resist the urge to interpret as "we need to have a dictator for society to be rational"
    • AIT merely states that non-dictatorial SCFs have some problems

AIT: Interpretation

  • no non-dictatorial SCF can satisfy all 5 axioms of social choice but many can satisfy \(4/5\)
    • unanimity rule without status quo: all but C
    • simple majority: all but T
    • Borda scoring rule: all but IIA
    • "constant rule" (constant social preferences, e.g. alphabetical ranking): all but P

Are Axioms Too Demanding?

  • IIA has weakest moral basis; Borda rule satisfies all but IIA but has problems:
    • how to measure intensity of preferences?
    • how to avoid strategic manipulation of social ranking by addition/removal of candidates?
  • "my rule is intended for honest men only": Borda, probably

Other Ways Around AIT

  • AIT: "or any social choice problem with at least three alternatives, there does not exist a social choice function that satisfies C, T, P, ND, IIA"
    • does not apply if choice is between two alternatives