2. Hotelling Problem: Median Voter & Citizen-Candidate Models

Let Vendor1 = 0.25
Let Vendor2 = 0.75
Therefore, Vendor1 captures: 0 ≤ V1 < 0.5
Therefore, Vendor2 captures: 0.5 < V2 ≤ 1

Movement towards point 0.5 by the first vendor, incrementally.

Strategy becomes to move to point 0.5 for all vendors.

Nash Equilibrium

The payoff for an individual is dependent on the strategy of the other player making this a sequential game.

Nash Equilibrium in this Game: A choice of location by each ice cream vendor such that neither would like to move given the location chosen by the other. 

Nash Equilibrium: set of strategies chosen by agents in a game such that no one agent would want to deviate given the strategies chosen by the others.

Nash Equilibrium: Hotelling

• (1⁄2, 1⁄2) is a Nash equilibrium. Payoff is represented by the vendors market share (1).

• How do we know that?
  There are no other profitable deviations from this point, to change and be better off (profit).

• Start with V1 at 1⁄2 and check for V2 in alternative locations. To the right, less market share, and same for movement to the left. Symmetrical for V2 at 1⁄2 and checking for V1 too.

• Is the equilibrium unique? yes, there are no other points

The Median Voter Theorem

• One-dimensional policy: e.g., tax rate.

• Suppose preferences are single-peaked.

• Suppose there are two candidates, 1 and 2.

• They simultaneously announce their platforms p1 and p2

• Full commitment to platforms.

• Voting is by majority rule.

The Median Voter Theorem: If the two candidates care only about winning, then in equilibrium, p1 = p2 = ideal point of the median voter.

The median-voter model rests on the assumptions:

• That the voters have full information regarding any gains or losses resulting from a particular policy

• That they will actually vote consistent with their preferences, single-peaked preferences

Single-peaked Preferences and Condorcet Cycles

• Single-peaked preferences: The options can be represented as points on a line, and each individual’s utility function has a maximum at some point on the line and slopes away from the maximum on either side. Moving their decision away from this distribution decreases utility.
  The graph below represents three unique (single-peaked) preference points.

• Ideal point and the Median voter

If preferences are single-peaked, then the ideal point of the median voter is a Condorcet Winner.

• Single-peakedness guarantees a Condorcet winner and hence no Condorcet cycles

Condorcet Winner: a policy capable of beating any alternative policy in a pairwise vote.

The existence of a Condorcet winner does not mean this policy will be implemented.

From Hotelling to Downs...

• How does this translate into voting behaviour?

• Downsian competition: policy convergence.

A Downsian model of electoral competition and forward-looking voting indicates that majoritarian, as opposed to proportional, elections increase competition between parties by focusing it into some key marginal districts. This leads to:

• Less public goods

• Less rents for politicians

• More redistribution

• Larger government

Some important issues:

• Politicians are assumed to care only about winning.

• Voters care only about the (winning) policy.

Probabilistic voting

Probabilistic voting: The Standard Setup

• Citizens care about policy and ideology, the latter is the dimension in which they differ. There could be another area in which they differ, for example other attributes such as personal characteristics of the party leadership. The ideological dimension cannot be modified as part of the electoral platform.

• Politicians still only care about winning.

• Say there are two parties: L and R.

• They can (re)distribute resources across groups of voters.

• Each party can commit to such a policy of redistribution.

• Each voter values the resources promised to herself.

• Vote for greater payoff from “resources + ideology”.

Probabilistic voting

• Trade-off between policy and ideology.
  We assume, voters differ in their evaluations of these features. Think about a second policy dimension, orthogonal to fiscal policy, in which electoral candidates cannot make credible commitments but set an optimal strategy after the election according to their ideology. In this setting, some groups of voters may become more attractive prey for office-seeking politicians, who are willing to modify policy in the direction of the favoured group.

• Note, here the preferences of every voter matters (not just the median!). 

  • Swing voters: not too stuck in any particular ideology, making them targeted and 'promised' more. Typical of swing states in the US such as Florida. "cheap to buy"

  • Loyalists: Such as New York as a 'forever' democratic state "nobody worries about loyalists"

What happens in equilibrium?

• Swing voters get more in equilibrium.

• There is (still) policy convergence in equilibrium.

• So eventually people vote on the basis of their ideologies!

Example

• Two parties: L and R.

• Three voters: Tom, Dick and Harry.

• Tom has a strong ideological preference for L.

• Harry has a strong ideological preference for R.

• Dick has no ideological preference.

What promise of resources would L and R make to Tom, Dick and Harry?

• Suppose Tom’s ideological preference is so strong that he would always vote for L.

• Suppose Harry’s ideological preference is so strong that he would always vote for R.

• So L and R promise nothing to Tom and Harry!

• Both parties promise as much as possible to Dick.

• Tom still votes for L, Harry still votes for R and Dick is indifferent between them (and so, perhaps, votes on whim).