1. 1 EC9B6 Voting and Communication Lecture 1 Prof. Francesco Squintani f.squintani@warwick.ac.uk

2. 2 Preview of the lecture We will introduce single-peaked utilities. We will prove Black’s theorem: Majority voting is socially fair when utilities are single-peaked. We will prove Median Voter Convergence in the Downsian model of elections. We will introduce the citizen candidate model.

3. 3 Voting Models Voting models are non-cooperative games that model elections. They can study one-shot elections, or repeated elections. There may be 2 or more candidates. Candidates’ strategic decisions may include whether and when to run in the election, with which promised policy platform, campaign spending, and so on.

4. 4 Downsian Elections There are 2 candidates, i=1,2. Candidates care only about winning the election. Candidates i=1,2 simultaneously commit to policies x i if elected. Policies are real numbers. There is a continuum of voters, with diverse ideologies k, with cumulative distribution F. The utility of a voter with ideology k if policy x is implemented is u(x,k)= L(|x-k|), with L’<0. After candidates choose platforms, each citizen votes, and candidate with the most votes wins. If x 1 = x 2, then the election is tied.

5. 5 For any ideology distribution F, let the median policy m be such that 1/2 of voters' ideologies y > m & 1/2 of ideologies y < m : F (m) = 1/2. Median Voter Theorem: The unique Nash Equilibrium of the Downsian Election model is such that candidates i choose x i = m, and tie the election.

6. 6 Proof. Fix any x 1 = x 2. Because L’<0, each voter with ideology k votes for the candidate i that minimizes |x i -k|. Hence, if x i < x j, candidate i’s vote share is F(½ ( x 1 + x 2 ) ), and candidate j’s is 1- F(½ ( x 1 + x 2 ) ). For either candidate i, if x i = m, then candidate j’s best response is BR j = {x j : |x j - m|< |x i - m|}. Candidate j wins the election. But if j plays any x j such that |x j - m|< |x i - m|, the best response of i cannot be x i.

7. 7 If i plays x i, i loses the election. But i can at least tie by playing m. Hence, there cannot be any Nash equilibrium where either candidate i plays x i = m. Suppose both candidates play x 1 = x 2 = m, then all voters are indifferent between x 1 and x 2, and the election is tied. If either candidate i deviates and plays x i = m, then she loses the election. So, there is unique Nash equilibrium: x 1 = x 2 = m.

8. 8 Assume that F is symmetric around m, i.e., F(x) = 1 – F(2m-x), for all x. Let us compare the voter welfare from profile x 1 = x 2 = m, with any profile x’ 1, x’ 2 = 2m – x’ 1. Each platform x’ 1 and x’ 2 is voted by 1/2 of voters. So, if voters are risk averse, L’’ < 0, then they all prefer the sure outcome x 1 = x 2 = m, to the `bet’ x’ 1, x’ 2 which has expected value equal to m.

9. 9 Downsian Elections with Ideological Candidates Suppose that candidate i’s ideology is k i, with k 1 < m < k 2, and m - k 1 < k 2 – m. The utility of candidate i if policy x is implemented is u(x, k i )= L(|x- k i |), with L’<0. Theorem: The unique Nash Equilibrium is such that candidates i choose x i = m, and tie.

10. 10 Proof. Again, for any x 1 = x 2, if x i < x j, candidate i’s vote share is F(½ ( x 1 + x 2 ) ), and candidate j’s is 1- F(½ ( x 1 + x 2 ) ). Suppose that x 1 < m, then candidate 2 wins and implement x 2 by choosing x 2 in (x 1, 2m - x 1 ). Hence, if x 1 < 2m- k 2, then BR 2 (x 1 )={k 2 }, and if 2m- k 2 < x 1 < m, then BR 2 (x 1 ) is empty. But if x 2 = k 2, then BR 1 (x 2 ) is empty. If m < x 1 < k 2, then BR 2 (x 1 )=[x 1, +∞). If x 1 > k 2, then BR 2 (x 1 )={k 2 }. But if x 2 > x 1 >m or x 2 = k 2, then x 1 not in BR 1 (x 2 ).

11. 11 Hence, we conclude that there cannot be any Nash Equilibrium with x 1 = m or x 2 = m. Suppose that candidate i chooses x i = m. Then, regardless of the choice x j, the implemented policy is m. Hence, BR j (x i ) = (-∞, +∞). We conclude that the unique Nash Equilibrium is such that x 1 = x 2 = m, and the election is tied.

12. 12 Conclusion 12 We have introduced single-peaked utilities. We have proved Black’s theorem: Majority voting is socially fair when utilities are single-peaked. We have proved Median Voter Convergence in the Downsian model of elections. We have introduced citizen candidate models.

13. 13 Preview of the next Lecture We will introduce ideology and uncertainty in the Downsian model, to represent `probabilistic elections with responsible parties.’ We will derive the equilibrium of this model and prove that responsible parties yield higher welfare to the electorate than opportunistic parties.