SinglePeaked Preferences
 consider one person's preferences over choice set \(X \subset \mathbb{R}\)
 her preferences are singlepeaked 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
SinglePeaked Preferences: Nice Properties
 suppose we have \(N\) (odd) individuals with singlepeaked preferences
 then,
 majority rule satisfies transitivity
 truthful reporting is a weakly dominant strategy*
SinglePeaked Preferences: (Hidden) Assumptions
 voters have singlepeaked 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 SinglePeaked Preferences
 if we assume singlepeaked 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 singlepeaked 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
 bliss point of the median voter is Condorcet winner
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, nonresidents, 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 singlepeaked 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  liberalconservative 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 officemotivated 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 singlepeaked 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