1)The median voter theorem says that candidates will always locate where the median voter is situated along a linear policy spectrum. An implicit assumption in the model is that all voters will turn out to vote for a candidate closest to their policy position. Let us see what happens if some voters sit out an election if both candidates are too far from their positions.
Suppose there are three types of voters –
Liberal (L) who are located between and in the policy spectrum. These individuals participate in the electoral process if and only if there is at least one candidate within their policy range between and . If they participate, they vote for the candidate closest to them in policy within the to range.
Moderates (M) who are located between and in the policy spectrum. These individuals always participate and vote for the candidate closest to them in policy anywhere on the policy spectrum.
Conservatives (C) who are located between and in the policy spectrum. These individuals participate in the electoral process if and only if there is at least one candidate within their policy range between and . If they participate, they vote for the candidate closest to them in policy within the to range.
Let denote Candidate ’s location choice. There are two candidates standing for election, . For simplicity assume that there are eligible voters located uniformly on the line shown above. Given the location of the three types of individuals, we then have each of liberals, conservatives and moderates.
We will assume that the winner of the election is the one who receives the most votes. If there is a tie, there is an equal probability that either candidate wins. The winner receives a payoff and the losing candidate receives a zero payoff.
The candidate descriptions are depicted in the figure below.
For example, let’s say Candidate 1 chooses and Candidate 2 chooses . Candidate 1 will receive all the votes from conservatives.
Candidate 2 will not receive any votes from liberals or conservatives. Moderates will split their vote between Candidate 1 and Candidate 2 with moderators to the right of the center point between and voting for Candidate 1 and the rest for Candidate 2.
The mid-point between and is
The figure below shows the vote split:
Liberal individuals do not vote in this case.
So, the total votes won by each candidate are:
a.Suppose both candidates locate at the center, i.e. and , how many votes does each candidate get. Who wins?
b.Suppose instead that Candidate 1 locates at the center, i.e. , but Candidate 2 chooses the , how many votes will each candidate receive? Who wins?
c.Is and a Nash Equilibrium? Does the median-voter theorem hold?
d.Can there be a Nash equilibrium where one of the candidates chooses a moderate position (between and )?
e.Show that is a Nash Equilibrium.
[For the next question consider this case: This question illustrates the difference between “truthful voting” and “strategic voting”
There are two competing legislative bills in front of a committee in Congress. The committee has three legislators. Let us call the two bills, and and the three legislators, and . The decision of which bill to pass or whether to pass the bill or not is considered in two stages.
Stage 1: In the first stage, members vote between the two bills. The bill that gets the most votes then moves to the second stage.
Stage 2: In the second stage, the legislators vote between passing the bill under consideration or maintain status quo ().
Each legislator has her own ranking over the three possible outcomes of the voting process, , and . The Table below shows the preferences of the legislators:
Legislator Best Outcome Next Best Outcome Worst Outcome
1
2
3
The payoff to a legislator from each outcome is as follows. She gets if the best outcome is achieved; she gets if her next best outcome is achieved and she gets if her worst outcome occurs.]
2)Consider the truthful vs. strategic voting example considered given above. We know that all legislators will vote truthfully in the second stage. Eliminating the strategies where we do not have truthful voting in stage 2, we have the following strategies possible for each Legislator:
Since there are three players in the game, we depict the payoff matrix in the first stage for Legislator 1 and Legislator 2, given each possible action, A or B, chosen by Legislator 3 in stage 1. The payoffs listed in each cell show player 1’s payoff first, followed by player 2’s payoff and then player 3’s payoff.
a.To find the best response of Player 1, in each payoff matrix above, mark out the highest payoff to L1 along each column. (Mark both payoffs in a column if they are equal.)
b.Similarly, to find the best response of Player 2, in each payoff matrix above, mark out the highest payoff to L2 across each row. (Mark both payoffs in a row if they are equal.)
c.Finally, to get L3’s best response, compare L3’s payoff across matrix in each cell. For instance, you will compare the third number in the top-left cell in the first matrix to the third number in the top-left cell of the second matrix. Again, mark both payoffs if they are equal across the two matrices.
d.Find all the Nash Equilibria (in pure strategies).
e.State the outcome associated with each Nash Equilibrium that you have found in part c).
f.For each equilibrium, state which legislator votes truthfully and which one votes strategically.
g.Comment on the equilibrium where all legislators vote for B in Stage 1.
The post 1)The median voter theorem says that candidates will always locate where the median voter is situated along a linear policy spectrum. An implicit assumption in the model is that all voters will turn out to vote for a candidate closest to their policy position. Let us see what happens if some voters sit out an election if both candidates are too far from their positions. Suppose there are three types of voters – Liberal (L) who are located between and in the policy spectrum. These individuals participate in the electoral process if and only if there is at least one candidate within their first appeared on essaypanel.com.
