Median Voter Theorem

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

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

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

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

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

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

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

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

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?

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)

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

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