Game Theory Questions:

Question

Consider the following bargaining game with three rounds: Players 1 and 2 divide a pie of size 1. Both players have a common discount factor, δ = 0.9. In the first round (player 1 proposes): player 1 proposes x ∈ [0, 1]. If player 2 accepts the offer, then player 1 gets x, and player 2 gets 1 − x. If player 2 rejects the offer, the game proceeds to round 2. In the second round (player 2 proposes): player 2 proposes y ∈ [0, 1]. If player 1 accepts the offer, then player 1 gets δy, and player 2 gets δ − δy. If player 2 rejects the offer, the game proceeds to round 3, the final round. In the third and final round (player 1 proposes): player 1 proposes z ∈ [0, 1]. If player 2 accepts the offer, player 1 gets δ 2 z, and player 2 gets δ 2 − δ 2 z. If player 2 rejects the offer, everyone gets 0. 1. What would be the SPNE of the game?

Two bidders compete in a second price auction (i.e., the winning bidder pays the losing bidder’s bid, and the losing bidder does not pay anything). They submit sealed bids, and the one with the highest bid wins the contract and pays the other bidder’s bid. Each bidder i’s private valuation is vi and is distributed independently and uniformly between 0 and 50. 1. For any given bidder, prove that he has a dominant strategy bid and show what it is. 2. Assuming each bidder bids his dominant strategy noted above, if a bidder with vi = 40 wins, what price does he expect to pay?

Answer 1

I'm only going to focus on the problem in the first paragraph. If you would like more help, please contact me directly and we can setup a session to discuss this problem in more detail and how you could solve the second problem.

The key to this problem in the first paragraph is understanding the setup and the backward induction. I find that in game theory it helps to create a setup (preferably using the notation your instructor uses) that in this multi-step situation converts the problem into a flow chart or tree diagram. This problem involves 6 decisions that flow downhill.

(1) Player 1 chooses x~[0,1]

(2a) Player 2 accepts the offer. Game ends. Rewards [P1 = x; P2 = 1-x]

(2b) Player 2 rejects the offer. Game continues

(3) Player 2 chooses y~(0,1)

(4a) Player 1 accepts the offer. Game ends. Rewards [P1 = δ(1-y); P2= δy]

(4b) Player 1 rejects the offer. Game continues

(5) Player 1 chooses z~[0,1]

(6a) Player 2 accepts the offer. Game ends. Rewards [P1 = δ²z; P2 = δ²(1-z)]

(6b) Player 2 rejects the offer. Game ends. Rewards [P1 = 0; P2=0]

For demonstration purposes, while the problem implies any offer on the continuous interval of [0,1] is appropriate, I'm going to use 4 decimal places to demonstrate the numbers and calculations because it is easier to see the math with rounding. It doesn't change the logic, it does change the notation for the final answer.

Working backwards, what decision will Player 2 make at Step 6?

Given, that Player 2 will get nothing if they reject the offer, Player 2 will accept any offer greater than 0, i.e. δ²(1-z)>0 because something is better than nothing.

Knowing Player 2's situation at Step 6, what decision should Player 1 make at Step 5?

Player 1 can set z such that they get as close to 1 as possible while leaving some tiny amount for player 2. Player 1 should set z = 0.9999, which means after discounting that P1 gets about .8099 and P2 gets .0001.

Working backwards, what decisions should Player 1 make at step 4?

Given that Player 1 knows they will obtain at least .8099 if they reject Player 2's offer, they should only accept offers greater than .8099.

Knowing Player 1's best strategy at Step 4, what decision should Player 2 make at step 3?

Player 2 gets almost nothing if Player 1 rejects the offer at step 4, so Player 2 should choose y, such that

1-y after discounting is just a tiny bit better than .8099, so Player 2 wants δ(1-y) = .8099 + .0001 = .81

using that equation and with δ = .9 we get y = 0.1. With y = .1, the rewards would be P1 = .9(1-.9) = 0.81 and P2 = .9(.1) = 0.09, and because these rewards are greater than if they continued the game, they are the best strategy.

Working backwards, what decisions should Player 2 make at step 2?

Given that Player 2 knows they will obtain at least .09 if they reject Player 1's offer, they should only accept offers greater than .09.

Finally, knowing Player 2's best strategy at Step 2, what decision should Player 1 make at step 1?

Player 1 can set y = .9099, so that 1-y is .0901. Player 2 will accept the offer because .0901 is greater than .09, and Player 1 should make the offer because .9099 is greater than its share (.81) if B were to reject Player 1's offer.

That is the logic. The rest is writing it up using the instructor's preferred game theory notation and style and methodology. I hope this helps.