## 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