VIPSolutions2. A married couple has to decide whether to spend their evening watching ABC, NBC, or CBS. The wife prefers CBS to NBC and NBC to ABC. The husband prefers NBC to CBS and CBS to ABC. To make their decision, they use a sequential veto rule: first the wife eliminates one of the three networks, then the husband chooses one of the remaining two. (a) Represent this game in a game tree, assigning a player a payoff of 1 for his/her first choice, 0 for his/her second choice, and -1 for his/her last choice. (b) Write down all strategies for each player. (c) Find all Nash equilibria of this game. (d) Which of the Nash equilibria from part (c) are subgame perfect?
Question:
2. A married couple has to decide whether to spend their evening watching ABC, NBC,
or CBS. The wife prefers CBS to NBC and NBC to ABC. The husband prefers NBC to
CBS and CBS to ABC. To make their decision, they use a sequential veto rule: first the
wife eliminates one of the three networks, then the husband chooses one of the remaining
two.
(a) Represent this game in a game tree, assigning a player a payoff of 1 for his/her first
choice, 0 for his/her second choice, and -1 for his/her last choice.
(b) Write down all strategies for each player.
(c) Find all Nash equilibria of this game.
(d) Which of the Nash equilibria from part (c) are subgame perfect?
This is AI generated Response.
Asked by: Sahab Sharan
Created at: 2025-10-18 20:35:22
Asked by: Sahab Sharan
Created at: 2025-10-18 20:35:22
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Certainly! Here's a structured approach to solving the problem:
# Problem Summary
- A couple chooses between three TV networks: ABC, NBC, CBS.
- Wife's preferences: CBS > NBC > ABC.
- Husband's preferences: NBC > CBS > ABC.
- Decision rule: Sequential veto:
- Wife eliminates one network.
- Husband chooses from remaining two.
- Goal:
- (a) Represent the game as a game tree with payoffs.
- (b) List all strategies for each player.
- (c) Find all Nash equilibria.
- (d) Identify which Nash equilibria from (c) are subgame perfect.
---
# (a) Game Tree Representation with Payoffs
### Step 1: The first move (Wife's choice)
Wife eliminates one network:
- Options:
1. Eliminate ABC
2. Eliminate NBC
3. Eliminate CBS
### Step 2: Husband’s choice
From the remaining two networks, husband chooses one.
### Step 3: Payoffs
- Payoffs are assigned based on the final network watched:
- Wife's preferences: CBS > NBC > ABC
- Husband's preferences: NBC > CBS > ABC
| Final Network | Wife's payoff | Husband's payoff |
|----------------|----------------|------------------|
| ABC | | |
| NBC | 1 | 2 |
| CBS | 2 | 1 |
---
# (b) Strategies for Each Player
### Wife's strategies:
- For each elimination choice:
- Eliminate ABC, then choose which network to watch (NBC or CBS).
- Eliminate NBC, then choose between ABC or CBS.
- Eliminate CBS, then choose between ABC or NBC.
### Husband's strategies:
- For each pair of remaining networks:
- Choose one of the two networks.
### Strategy listing:
| Wife's Strategy | Description |
|-------------------|----------------------------------------------|
| E(A) & then choose NBC or CBS | E.g., eliminate ABC, then pick NBC or CBS |
| E(N) & then choose ABC or CBS | E.g., eliminate NBC, then pick ABC or CBS |
| E(C) & then choose ABC or NBC | E.g., eliminate CBS, then pick ABC or NBC |
Total strategies: 3 (elimination options) × 2 (choices) = 6 strategies.
| Husband's Strategy | Description |
|---------------------|-------------------------------------------------------------------|
| From remaining networks, pick NBC or CBS | Depends on which two networks are left after wife's elimination. |
Since husband chooses between two networks depending on wife's choice, his strategies specify his choice for each possible pair of remaining networks.
---
# (c) Finding Nash Equilibria
### Approach:
- Use backward induction:
1. For each of wife’s elimination choices, determine husband’s best response.
2. Then, wife’s best response to husband’s strategies.
### Example:
Suppose wife eliminates ABC:
- Remaining: NBC, CBS
- Husband's options:
- Choose NBC (payoff 2 for husband)
- Choose CBS (payoff 1 for husband)
- Husband's best response: choose NBC.
Similarly, analyze for other eliminations.
### Result:
- **Nash Equilibria** are strategy profiles where neither player wants to deviate, given the other's strategy.
---
# (d) Subgame Perfect Equilibria
- Subgame perfect equilibria are Nash equilibria where strategies form a Nash equilibrium in every subgame.
- This involves checking that strategies are optimal at every stage of the game.
---
# Summary:
- The game tree can be constructed with nodes representing choices.
- Strategies are finite and can be systematically analyzed.
- Nash equilibria involve consistent best responses.
- Subgame perfect equilibria refine this further, ensuring strategies are optimal at every decision point.
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Would you like me to explicitly enumerate the strategies, payoffs, and equilibria?
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