Alice, Bob and Charlie are candidates in an election. They play the following ga

Question

Alice, Bob and Charlie are candidates in an election. They play the following game:

• Candidates simultaneously announce policy positions, which are simply numbers in the interval [0,1].

• Voters each submit an ordered list of the three candidates. Each voter will rank candidates according to who announced a policy closest to the voter’s opinion, breaking ties by alphabetical order of first name. Voters’ opinions range uniformly from 0 to 1, as in the Hotelling model.

• Election officials tally up who would have won a two-way election between each pair of candidates. For instance, if Alice appears ahead of Bob on more than half the ballots, we say she would have beat him in a two-way election. If someone (called a Condorcet winner ) beats each of the other two candidates, they are the winner. If there is no Condorcet winner, the candidate with the most first-place votes (plurality rule) is the winner.

Find a strategy for Alice which guarantees that she wins, and explain why. Also explain why this strategy would not guarantee that she wins if the winner were determined simply by first-place votes.

Explanation / Answer

Prior to the first round, each player independently selects one of the positions and reveals it at the start of the game. Bob’s position is where the game starts.

Working outside the home was nothing new for poor white, immigrant, and black women. They toiled as housekeepers, factory workers, and in other menial jobs in order to survive. Female factory workers earned only a quarter to a third of what men earned for the same job. There were no sick days or health benefits.

Women were known to have given birth on the floors of factories where they worked. Since they did not have the right to vote, they had little opportunity to pressure lawmakers to pass laws that would have improved their wages and working conditions.

Those opposed to woman suffrage, the so-called “antis,” assembled all their forces to stop ratification. The liquor and brewing industries, factory owners, railroads, banks, and big city political machines all feared women would vote for progressive reforms. Southern whites objected to more black voters. Some argued that the 19th Amendment invaded states’ rights. Others claimed that it would undermine family unity. Besides, the “antis” said, wives were already represented at the ballot box by their husbands.

Both Alice and Bob play optimally: Alice will always choose an option that will lead to her winning the game, if this is possible, and Bob will always try to prevent Alice from winning.

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