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

Strategic Political Competition

  • no we know how to represent voters' ideology
  • what if we had multiple political candidates competing in an election?
    • what platforms will they propose?
    • will they moderate or go to extremes?
    • will we see polarization?

A Formal Model of Political Competition

  • key actors (players)
  • what they can do (strategies)
  • their goals (payoffs)

Downsian Model of Electoral Competition

  • players: two candidates (parties) D and R
  • strategies: each candidate chooses a policy platform
    • a number on a real line
  • goals: winning the election under majority rule
    • payoff is \(1\) for the winner, \(0\) for the loser, \(0.5\) if tied
    • these are office-motivated candidates (they don't personally care about policy)

Downsian Model of Electoral Competition: Voting

  • voters are technically also players of this game, but we already studied this part
  • we assume we have an electorate of voters with single-peaked preferences
    • represented by bliss points on the same real line as candidates
  • each voter votes for the candidate whose policy is closest to her bliss point
    • if candidates are equidistant, then the vote is split evenly
  • outcome is determined via majority rule

Downsian Model of Electoral Competition: Result

  • for any electorate, both candidates propose the ideal policy of the median voter
    • AKA full convergence
  • if you are interested in winning an election, you have no reason to polarize
    • to win, you need the majority \(\to\) propose 'moderate' policies to convince more voters