2. Whether candidate 1 or candidate 2 is elected depends on the votes of two citizens. The economy may be in one of two states, A and B. The citizens agree that candidate 1 is best if the state is A and candidate 2 is best if the state is B. Each citizen's preferences are represented by the expected value of a Bernoulli payoff function that assigns a payoff of 1 if the best candidate for the state wins (obtains more votes than the other candidate), a payoff of 0 if the other candidate wins, and payoff of 1/2 if the candidates tie. Citizen 1 is informed of the state, whereas citizen 2 believes it is A with probability 0.9 and B with probability 0.1. Each citizen may either vote for candidate 1, vote for candidate 2, or not vote. a. Formulate this situation as a Bayesian game. That is, set out the set of players, the set of states, the set of signals for each player etc. b. Show that the game has exactly two pure Nash equilibria. In one Nash equilibrium, citizen 2 does not vote. In the second Nash equilibrium, citizen 2 votes for 1. c. Show that one of the player's actions in the second of these equilibria is weakly dominated. d. Why is the "swing voter's curse" an appropriate name for the determinant of citizen 2's decision in the second equilibrium?
2. Whether candidate 1 or candidate 2 is elected depends on the votes of two citizens. The economy may be in one of two states, A and B. The citizens agree that candidate 1 is best if the state is A and candidate 2 is best if the state is B. Each citizen's preferences are represented by the expected value of a Bernoulli payoff function that assigns a payoff of 1 if the best candidate for the state wins (obtains more votes than the other candidate), a payoff of 0 if the other candidate wins, and payoff of 1/2 if the candidates tie. Citizen 1 is informed of the state, whereas citizen 2 believes it is A with probability 0.9 and B with probability 0.1. Each citizen may either vote for candidate 1, vote for candidate 2, or not vote. a. Formulate this situation as a Bayesian game. That is, set out the set of players, the set of states, the set of signals for each player etc. b. Show that the game has exactly two pure Nash equilibria. In one Nash equilibrium, citizen 2 does not vote. In the second Nash equilibrium, citizen 2 votes for 1. c. Show that one of the player's actions in the second of these equilibria is weakly dominated. d. Why is the "swing voter's curse" an appropriate name for the determinant of citizen 2's decision in the second equilibrium?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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### Voting Game Theory Exercise
**Problem 2: Election Dynamics and Bayesian Games**
In a simple voting game, the election of candidate 1 or candidate 2 depends on the votes of two citizens. The economic environment can be in one of two possible states, A or B. Both citizens agree:
- Candidate 1 is preferable if the economy is in state A.
- Candidate 2 is preferable if the economy is in state B.
#### Payoff Structure
Each citizen's preference is characterized by the expected value of a Bernoulli payoff function:
- **Payoff of 1**: If the chosen candidate, deemed best for the state, wins.
- **Payoff of 0**: If the other candidate wins.
- **Payoff of 0.5**: If there is a tie.
Citizen 1 has certain knowledge that the state is A. Citizen 2 believes:
- The probability that the state is A is 0.9.
- The probability that the state is B is 0.1.
#### Voting Options
Each citizen can make one of three decisions:
1. Vote for candidate 1.
2. Vote for candidate 2.
3. Abstain from voting.
#### Questions
a. **Formulation as a Bayesian Game**
Define this setup as a Bayesian game, identifying the set of players, the potential states, and the signals available to each player.
b. **Identification of Nash Equilibria**
Demonstrate that the game possesses exactly two pure Nash equilibria:
- In the first equilibrium, citizen 2 chooses not to vote.
- In the second equilibrium, citizen 2 votes for candidate 1.
c. **Dominance of Strategies**
Show how one of the player's strategies in the second equilibrium is weakly dominated.
d. **The "Swing Voter's Curse"**
Analyze why the term "swing voter's curse" aptly describes citizen 2's choice in the second equilibrium.
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This exercise provides a foundational understanding of strategic decision-making in uncertain environments and demonstrates the application of Bayesian and Nash equilibrium concepts in game theory.
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