## Hotelling (1929) Game

Two ice cream vendors are on a beach that stretches the 0-1 interval. Customers are uniformly distributed along that interval. The vendors simultaneously select a position. Customers go to the closest vendor and split themselves evenly if the vendors choose an identical position. Each vendor wants to maximize the number of customers.

Where do the vendors locate?

## Hotelling (1929) Game Formalized

• players: 2 ice cream trucks
• actions: $$x_i \in [0,1]$$, location of the ice cream truck
• payoffs: $$i$$ receives fraction of consumers who are closer to $$i$$: u_A (x_A,x_B) = \left\{ \begin{aligned} & \frac{x_A + x_B}{2}, & \text{ if } x_A < x_B, \\ &1/2, & \text{ if } x_A = x_B\\ & 1 - \frac{x_A + x_B}{2}, & \text{ if } x_A > x_B,\\ \end{aligned} \right.

## Hotelling Law

• in many markets producers locate as close as possible

## Hotelling (1929) $$\to$$ Downs (1957)

• ice cream sellers $$\to$$ political parties
• maximize sales $$\to$$ maximize share of vote
• OR win election via majority rule
• many other social choice functions lead to same conclusion
• customers $$\to$$ voters
• buy from closest vendor $$\to$$ vote for ideologically closest party

## (Downsian) Electoral Competition: List of Ignored Things

• parties are office motivated and can commit to policies:
• are not policy-motivated
• cannot change policy after announcement
• cannot lie
• voters are sincere and have spatial preferences
• they do not strategize
• no valence
• no retrospective voting
• complete information

## Downsian Competition: Non-Uniform Electorate

• consider standard Downsian model:
• $$[0,1]$$ political spectrum
• 2 parties that compete via selecting policies
• majority rule
• distribution of voters is $$F$$ with density $$f$$

## Median Voter Theorem

• both parties select location of median voter
• $$x^M$$ such that $$F(x^M) = 1/2$$
• can we prove this result without actually solving the game?

## How to Solve a Game Without Solving It

• guess and verify
• (for whatever reason) you predict that a profile of strategies is an equilibrium
• for every player, prove that they do whatever you predict given predicted strategies of others
• check there are no other NE by contradiction
• suppose that there is some other NE
• prove that it cannot be a NE

## Downsian Competition w/ Non-Uniform Electorate: Guess and Verify

• we suspect that both parties select location of median voter
• we guess: $$x_A^* = x_B^* = x^M$$
• we verify:
• if $$x_B^* = x^M$$, then $$x_A^* = x_B^*$$
• if $$x_A^* = x^M$$, then $$x_B^* = x_A^*$$

## Downsian Competition w/ Non-Uniform Electorate: Uniqueness

• only one party at the median?
• no
• both parties to the left of median?
• no
• both parties to the right of median?
• no
• one to the left, one to the right?
• no
• with all other options exhausted, both parties at the median is the unique NE

## Median Voter Theorem: Robustness

• we made many assumptions to get to median voter theorem, let's start relaxing them
• what if parties also care about policies?

## Downsian Model with Policy-Motivated Candidates

• suppose that parties are ideological:
• party $$D$$ is democratic, has ideal policy $$\tilde{x}_D=0$$
• party $$R$$ is republican, has ideal policy $$\tilde{x}_R=1$$
• each party wants the elected policy to be closest to its bliss point
• still majority rule and electorate is $$F$$

## Downsian Model with Policy-Motivated Candidates: Trade-Off

• now there is a trade-off:
• placing at $$x^M$$ is good because it maximizes chances of winning
• placing at $$x^M$$ is not great because it's not the party's ideal policy

## Policy-Motivated Candidates: Solution

• suppose chosen positions are $$x_D < x^M < x_R$$
• then, this cannot be an equilibrium
• suppose $$D$$ is the losing party. Then, $$D$$ can deviate to $$x^M$$ and win the election. Since $$x^M$$ (new winning policy) is closer to $$0$$ (bliss point of $$D$$) than $$x_R$$ (old winning policy), this is a profitable deviation for $$D$$

## Downsian Model: Extentions

• we saw that MVT holds for office- and policy-motivated candidates
• one big assumption we made so far is complete information
• let us introduce some uncertainty

## Current Model: Downsian w/ Uncertainty about Median Voter

• candidates: 2 office-motivated parties, $$D$$ and $$R$$, choosing positions $$x_D$$ and $$x_R$$
• like before, parties' bliss points are $$\tilde{x}_D=0$$ and $$\tilde{x}_D=1$$
• electorate: median voter's position is unknown
• candidates think that $$x^M$$ is uniform on $$\left[\frac{1}{2}-\frac{1}{2b},\frac{1}{2}+\frac{1}{2b}\right]$$
• what is $$b$$? -- measure of uncertainty about the position of the median voter

## Current Model: Expected Utility

• expected utility of party $$D$$ is $$u_D(x_D,x_R) = -\text{prob(*D wins*)} \cdot |x_D - 0| - \text{prob(*R wins*)} \cdot |x_R - 0|$$

## Current Model: Probability of Winning

• given chosen positions $$x_D < x_R$$, probability that $$D$$ wins is p_D(x_D,x_R) = \left\{\begin{aligned} 0, \text{ if } \frac{x_D+x_R}{2} &< \frac{1}{2} - \frac{1}{2b},\\ \frac{b(x_D + x_R - 1)+1}{2}, \text{ if } \frac{x_D+x_R}{2} &\in \left[\frac{1}{2} - \frac{1}{2b},\frac{1}{2} + \frac{1}{2b}\right]\\ 1, \text{ if } \frac{x_D+x_R}{2} &> \frac{1}{2} + \frac{1}{2b},\\ \end{aligned}\right.

## Current Model: Utility Maximization

• recall that expected utility is $$u_D(x_D,x_R) = -p_D(x_D,x_R) \cdot x_D - (1-p_D(x_D,x_R)) \cdot x_R$$
• best response logic: given $$x_R$$, how does $$D$$ choose $$x_D$$ to maximize his expected utility?

## Current Model: First Order Conditions

• we get $$\frac{\partial u_D}{\partial x_D} = -\frac{b x_D}{2} + \frac{b-1}{2} < 0$$
• negative derivative: lower (lefter) $$x_D$$ $$\to$$ higher expected utility
• $$D$$ should pick $$x_D$$ as left as possible
• how left? -- to $$\frac{1}{2} - \frac{1}{2b}$$

## Current Model: Equilibrium

• in (unique) equilibrium, $$x_D^* = \frac{1}{2} - \frac{1}{2b} \text{ and } x_R^* = \frac{1}{2} + \frac{1}{2b}$$

## Current Model: Conclusion

• for the first time, we get polarization:
• polarization = distance between positions announced by candidates, $$x_R^* - x_D^* = \frac{1}{b}$$
• polarization is high when there is a lot of uncertainty about position of median voter
• if median voter could be anywhere ($$b = 1$$), polarization is highest and $$x_D^* = 0$$, $$x_R^* = 1$$
• politicians announce their favorite policies when they have no idea what voters want
• polarization is low when there is little uncertainty about median voter
• if median voter is at $$\frac{1}{2}$$ ($$b \to \infty$$), polarization is lowest and $$x_D^* = \frac{1}{2}$$, $$x_R^* = \frac{1}{2}$$
• MVT holds when there is little uncertainty about position of median voter