Two political candidates, A and B, are competing in an election. Voters have different political views, which can be represented as points along a line from 0 to 1:

0 represents the far left, and 1 represents the far right.

Voters are evenly spread across this political spectrum, meaning that every position along the line has the same number of voters.

Each voter votes for the candidate whose position is closest to their own.

If both candidates are equally close, the voter chooses randomly.

Each candidate chooses a position on the spectrum (a number between 0 and 1) at the same time, aiming to get as many votes as possible.

What position will candidate A and B choose in the Nash equilibrium respectively? For example,
𝑥
𝐴
=
0.2
and
𝑥
𝐵
=
0.6

Question

Two political candidates, A and B, are competing in an election. Voters have different political views, which can be represented as points along a line from 0 to 1:

0 represents the far left, and 1 represents the far right.

Voters are evenly spread across this political spectrum, meaning that every position along the line has the same number of voters.

Each voter votes for the candidate whose position is closest to their own.

If both candidates are equally close, the voter chooses randomly.

Each candidate chooses a position on the spectrum (a number between 0 and 1) at the same time, aiming to get as many votes as possible.

What position will candidate A and B choose in the Nash equilibrium respectively? For example, 
𝑥
𝐴
=
0.2
 and 
𝑥
𝐵
=
0.6

BlackTom AI · Accepted Answer

The question describes a classic spatial voting model with a single-peaked, uniform distribution of voters along the interval [0,1], where each voter supports the candidate closest to them and ties are broken randomly.

Option 1: 0.5 for A and 0.5 for B. 
- Consider a symmetric profile where both candidates choose the same position x. If x is not 0.5, say x < 0.5, then candidate B can move slightly to the right toward 0.5 and capture more voters on the right half, while the left-side voters would still be closer to A or B depending on the exact positions, making the original x unstable. A more formal intuition is that any asymmetric deviation from a non-central position creates a larger share for the deviator by shifting the dividing line toward the opponent, prompting a response. 
- If both are at 0.5, no candidate can increase their vote share by unilateral deviation: moving away from 0.5 pulls closer voters only to one side, reducing overall votes due to the uniform distribution and the existence of a proportional split around the center. In this symmetric, center position, each candidate covers exactly half of the electorate, and any small move would create a larger losing side for the deviator, given the tie-breaking rule. 
- Therefore, 0.5 for A and 0.5 for B constitutes a Nash equilibrium: neither candidate can improve their vote count by a unilateral, cost-free move given the other’s position.

Option 2: 0.5 for A and 0.5 for B. 
- This option presents the same profile as Option 1. Since the scenario is symmetric with two identical choices, the same reasoning applies: at the central position, deviations are unprofitable for either candidate because moving away would reduce the number of voters who are closer to you or would only trade places with the other candidate in a way that does not increase your total votes due to the even voter density and the equal split at the center. 
- There is no inherent asymmetry or external constraint here to favor one candidate over the other at any non-central position, so the central Nash equilibrium persists, and the conclusion remains that both choosing 0.5 is stable under unilateral deviation.

In summary, the central, symmetric positioning at xA = xB = 0.5 aligns with the intuitive and formal predictions of the spatial model with uniform voter distribution: the center candidate balance point yields no profitable unilateral deviation, making it a Nash equilibrium. Note that the two options are effectively the same profile, so both reflect the same equilibrium reasoning and outcome.

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