Question 5

Two siblings, Alice and Bob, found two $10 bills in the street. Their mom decides to play a game with Alice and Bob to determine how to allocate the money between the siblings. In this game, Alice and Bob simultaneously and independently announce how much they want. The amount each can announce is either 0$ or 10$ or 20$.

Their monetary rewards are determined as follows:

• If the sum of their announcement is less than or equal to 20$, then they both receive their individual announcement as a monetary reward.
• If the sum of their announcement is greater than 20$, then their mom keeps the money and siblings do not receive any monetary reward.

Suppose that each sibling cares about their own monetary reward and also partially that of other sibling's monetary reward. In particular, each sibling's payoff equals their own monetary reward + 20% of other sibling's monetary reward.

Example 1: If Alice and Bob both announce $10, then: Alice's payoff is 10+0.2*10=$12 and Bob's payoff is 10+0.2*10=$12.

Example 2: If Alice and Bob both announce $0, then: Alice's payoff is 0+0.2*0=$0 and Bob's payoff is 0+0.2*0=$0.

1. Using the above information, present the normal-form representation of the game via a payoff matrix.
2. Determine whether any of the sibling has a strictly dominated strategy and explain why.
3. Find all Nash equilibria of the above game.

Now, suppose that their mom changes the game and let the siblings make their announcement sequentially.

assume that Alice announces first, and Bob announces thereafter.

1. present the extensive-form representation of the game. What is your prediction regarding the outcome of the game?

[6+3+3+1+8 =21 marks]