Question

Two candidates are competing in a majority rule election with 7 voters. The possible policies are ordered on a number line and creatively labeled {1, 2, 3, 4, 5, 6, 7}. Each policy is the favorite of one voter, and each voter has single peaked preferences. The candidates, L and R, announce policies, and whomever gets the most votes wins and implements the policy she announced. (Each voter votes for whichever candidate they strictly prefer. If a voter is indifferent, she allocates exactly half a vote to each candidate. If the candidates tie, they flip a coin, and the winner of the coin toss wins the election and implements the policy she announced.) Unlike the Downsian model, the candidates also have single peaked policy preferences. Candidate L’s favorite policy is 2. Candidate R’s favorite policy is 6. (Politics is pretty polarized these days.) In addition, the winning candidate obtains 10 jollies from winning the election.

So:

• if L wins with a policy of k in {1,2,3,4,5,6,7}, then L obtains −|2 − k| + 10 jollies, and R obtains −|6 − k| jollies.

• If R wins with a policy of j in {1,2,3,4,5,6,7}, then candidate L obtains −|2 − j| jollies, and R obtains −|6 − j| + 10 jollies.

(a) If L announces a policy of 4, what is R’s best response?

(b) Is it a Nash equilibrium of this game for each candidate to announce 4?

(c) Is it a Nash equilibrium for each candidate to announce her ideal point?

(d) Does your answers change if the candidates each obtain 2 jollies from winning the election?

Answer 1

Because of single-peaked preferences. Voters prefer policies that are closer to their choice. For eg. voter 1 will prefer policy 3 over 4 because 3 is closer to 1 than 4. The jollies awarded are such that winner gets 10 jollies and loses a number of jollies equal to how far their announced policy is from their favourite policy

a) If L announces a policy of 4,R's best response will be to announce a policy of 4 too. If he announces any policy higher than 4 (Let's say 5), then only voter 5,6,7 will vote for him compared to voters 1,2,3 and 4 voting for L thus L winning the election with -2+10=8 jollies and R losing with -1(since 5 is 1 away from 6) jolly (Note R cannot announce policy less than 4 as he will lose even more jollies and also still lose elections because of voter 4,5,6 and 7 voting for L).

This creates a tie and winner will be decided by a coin toss and in the end one will end up with 8 jollies and other with -2 (So expected payoff for announcing 4 is (8-2)/2=3 for R compared to -1 if he announces 5,0 if he announces 6 and -1 if he announces 7,similarly his payoff for announcing 1 will be -5,2 will be -4,3 will be -3)

b) Yes it is a Nash equilibrium of this game for each candidate to announce 4 because neither L nor R have any reason to deviate from this. Whoever changes their policy from 4 will lose the elections and their payoff will be either 0 or negative by losing the election.

c) It is not a nash equilibrium for each candidate to announce their ideal point. Although This creates a tie and a coin toss where the loser loses 0 jollies and the winner gets 10 jollies, so their expected payoff is 5 which is higher than before. Both of them can actually increase their number of jollies(if the other's policy remains unchanged). For eg. R can move from 6 to 5 and ensure that he wins the election by getting votes of 4th,5th 6th and 7th voter(as long as L stays at policy 2) and it ensures he will get -1+10=9 jollies and avoid the coin toss which gives him 50% chance of 10 and 50% chance of 0. Similarly L can also move at a higher policy to either 3 or 4 to ensure winning the election and a higher payoff if the policy of R stays at 6. This creates a non-stable equilibrium and both have a reason to deviate from it thus it is not a Nash equilibrium

d) Yes the answer changes if the candidates each obtain 2 jollies from winning the election

1) If L announces a policy of 4 and the previous answer of R announcing a policy of 4,the loser of the coin toss will get -2 jollies and the winner will get -2+2=0 jollies which means an expected number of jollies= -1. Instead R's best response will be to announce a policy 6. This way The expected payoff of L will be winning the election with 0(-2+2) jollies at policy 4 and R's payoff will be losing the election with 0 jollies at policy 6.

2) No it is not a nash equilibrium of this game for each candidate to announce 4 as both will have a negative expected payoff of -1 (As calculated in 1) ). Both will increase their payoffs to 0 by moving to their ideal point so both have a reason to deviate from this

3) it is a nash equilibrium for each candidate to announce their ideal point. When announcing their ideal point. The winner of the coin toss(since both get equal votes) will get -0+2= 2 jollies and the loser gets 0 jollies so the expected payoff is 1 jolly. Deviating from this does not benefit either so the equilibrium is sustained (i.e. deviating by 1 (either L moving to 3 or R moving to 5) will ensure winning the election but the payoff will still be -1+2=1 jolly so it does not leave them better off)