Consider the wage negotiations between Cricket Australia (CA) and the union that represents the players. Assume that Cricket Australia and the union are bargaining over how much of a $100 surplus will be split. Suppose that CA moves first and makes an offer p. The union (U) may accept or reject the offer. If the offer is rejected neither party gets anything (payoff of 0). If the offer is accepted the Union gets p and the CA gets 100 - p. Assume that the CA can make offers of $1, $50 or $99. Which of the following statements are true?
Group of answer choices

In the credible (subgame perfect) equilibrium, there is a second-mover advantage

There are two credible (subgame perfect) equilibria

There is one credible (subgame perfect) equilibrium

In the credible (subgame perfect) equilibrium, there is a first-mover advantage

CA and the union share the surplus equally in the credible (subgame perfect) equilibrium.

This scenario is a bargaining game where Cricket Australia (CA) makes an offer, and the union (U) can either accept or reject the offer. We are tasked with identifying the credible (subgame perfect) equilibrium, which is the equilibrium found by applying backward induction to ensure rational decisions at every stage of the game.

Let's break down the analysis:

### Available Offers:
CA can make offers of $1, $50, or $99. The Union (U) has two options upon receiving an offer:
- **Accept the offer:** The union receives the proposed amount $ p $, and CA gets $ 100 - p $.
- **Reject the offer:** Both parties get zero.

The **subgame perfect equilibrium** (SPE) is determined by considering the union's reaction to each possible offer and then working backward to determine CA's optimal choice.

### Backward Induction:
1. **If CA offers $99:**
   - The union will compare $99 with the payoff of rejecting the offer (which is $0). Since $99 > $0, the union will accept the offer.
   - CA gets $ 100 - 99 = 1 $, and the union gets $ 99 $.

2. **If CA offers $50:**
   - The union compares $50 with the payoff of rejecting the offer (which is $0). Since $50 > $0, the union will accept this offer.
   - CA gets $ 100 - 50 = 50 $, and the union gets $ 50 $.

3. **If CA offers $1:**
   - The union compares $1 with the payoff of rejecting the offer (which is $0). Since $1 > $0, the union will accept this offer.
   - CA gets $ 100 - 1 = 99 $, and the union gets $ 1 $.

### Analysis of Equilibria:
- Since the union will accept any positive offer (because all offers are better than rejecting and getting zero), CA's decision depends on what maximizes **its** payoff. 
- The best offer for CA is $1, which gives CA a payoff of $99 and the union $1. This is a subgame perfect equilibrium because CA anticipates that the union will accept this offer rather than rejecting it.

### Statements:
- **There is one credible (subgame perfect) equilibrium:** This is true. The unique subgame perfect equilibrium is that CA offers $1, and the union accepts it.
- **In the credible (subgame perfect) equilibrium, there is a first-mover advantage:** This is true. Since CA moves first, it can make the offer that maximizes its payoff ($1), giving it a significant advantage.
- **In the credible (subgame perfect) equilibrium, there is a second-mover advantage:** This is false. The union, as the second mover, is essentially forced to accept the offer, with no real ability to influence the outcome in its favor.
- **There are two credible (subgame perfect) equilibria:** This is false. There is only one subgame perfect equilibrium, where CA offers $1.
- **CA and the union share the surplus equally in the credible (subgame perfect) equilibrium:** This is false. In the equilibrium, CA keeps $99, and the union gets $1, which is far from an equal split.

### Final Conclusion:
The **true statements** are:
- **There is one credible (subgame perfect) equilibrium.**
- **In the credible (subgame perfect) equilibrium, there is a first-mover advantage.**

Would you like further details on any specific part of this reasoning or need clarification?

### 5 Related Questions:
1. What would happen if CA could make more diverse offers, such as any integer between $1 and $100?
2. How would the equilibrium change if the union had the option to counteroffer?
3. What if rejecting the offer resulted in a non-zero payoff for the union (e.g., $20)?
4. How does the concept of Nash equilibrium differ from subgame perfect equilibrium in this scenario?
5. Can this model apply to real-world wage negotiations, or are there limitations?

### Tip:
In any bargaining scenario, understanding the second player's likely reaction helps the first player craft an optimal strategy using backward induction.

Consider the wage negotiations between Cricket Australia (CA) and the union that represents the players. Assume that Cricket Australia and the union are bargaining over how much of a $100 surplus will be split. Suppose that CA moves first and makes an offer p. The union (U) may accept or reject the offer. If the offer is rejected neither party gets anything (payoff of 0). If the offer is accepted the Union gets p and the CA gets 100 - p. Assume that the CA can make offers of $1, $50 or $99. Which of the following statements are true? Group of answer choices In the credible (subgame perfect) equilibrium, there is a second-mover advantage There are two credible (subgame perfect) equilibria There is one credible (subgame perfect) equilibrium In the credible (subgame perfect) equilibrium, there is a first-mover advantage CA and the union share the surplus equally in the credible (subgame perfect) equilibrium.

Explore the wage negotiations between Cricket Australia (CA) and the players' union through the lens of game theory. This problem focuses on finding the subgame perfect equilibrium using backward induction, illustrating key concepts like bargaining strategies and first-mover advantage. Ideal for students studying game theory at an undergraduate level.

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