Single-Peaked Preferences

  • consider one person's preferences over choice set \(X \subset \mathbb{R}\)
  • her preferences are single-peaked if there exists option \(v \in X\) such that
    • \(v \succ x \succ y\) whenever \(y<x<v\) or \(y>x>v\)
    • \(v\) is the bliss point
    • \(!\) this is a restriction on the set of considered preferences

Single-Peaked Preferences: Nice Properties

  • suppose we have \(N\) (odd) individuals with single-peaked preferences
  • then,
    • majority rule satisfies transitivity
    • truthful reporting is a weakly dominant strategy*

Single-Peaked Preferences: (Hidden) Assumptions

  • voters have single-peaked preferences if they have an ideal balance between the two directions of the ideological spectrum and if they dislike policies the farther away they are from their ideal point
    • one dimension
    • further = worse
      • cannot dislike candidates for any reason other than policy

Voters' Ideology as Single-Peaked Preferences

  • if we assume single-peaked preferences, every voter is characterized just by her bliss point
  • we can place all voters' bliss points on the line
  • call the collection of all voters' bliss points the electorate

Black's Median Voter Theorem

  • suppose we have \(N\) (odd) individuals with single-peaked preferences

    • their bliss points are sorted in increasing order: \(v_1 \leq v_2 \leq \dots \leq v_N\)

  • then,

    • bliss point of the median voter is Condorcet winner
      • it beats all other options in pairwise comparisons via majority rule

Identifying the Median Voter

  • median voter splits electorate in half: half of voters are to the left, half are to the right
  • identity of median voter depends on the electorate
    • in conservative states, median voter is more conservative
    • if poorer voters are less likely to vote, then median voter \(\ne\) median citizen
    • many factors affect voter participation: age, education, what's on the ballot, weather, how the sports team is doing, etc
    • franchise restrictions
      • voting rights for women, racial minorities, convicted felons, non-residents, etc

Black's Median Voter Theorem: Interpretations

  • policy closest to the median voter is going to win against any other policy
  • corollary: if status quo is at the median voter's bliss point, then no challenging proposal will be accepted

When MVT Doesn't Apply

  • MVT holds only if there is a single issue
    • if there are two or more issues that parties take stands on, but only one election, there is no guarantee that the median voter's preference will win on any issue
    • even with single-peaked preferences, multiple dimensions make it possible for voting cycles to arise
  • is the MVT useless?
    • possibly so, but IRL platforms empirically boil down to a single dimension -- liberal-conservative spectrum in the US