How to static incomplete game

Question 1

Bayesian game with 2 players cookbook, BNE.

Answer 1

1. Propose equilibrium strategy for player with fewest strategies, Player F. The one player that has no types.
2. Find opponent´s BR to Player F´s equilibrium strategy, by looking at the matrices for the two types.
3. Find players F`s BR to 2), by calculating the expected payoff for his strategies. Remember to conclude wether it is a BNE or not.

Question 2

How should I be aware of which player that have different types.

Answer 2

The assignment would typically say, which player that have different types. That can both be player 1 and 2.

Question 3

Consider a first-price sealed bid auction, with a uniform distribution:
- two identical bidders
- Valuation, v_i 1-3
- private values

Show the symmetric BNE linear strategy b_i(v_i)=cv_i+d. Find c and d.

Answer 3

1. expected payoff, bidder i:
   - Remember specific form when writing it up expectations
   - Use the CDF for the uniformely distributed variabel
   - you want the function to be easy to differentiate wrt. b_i later.
2. FOC wrt. b_i and isolate b_i
   - Again you want to have it in a shape that is close to the linear strategy.
3. Compare BR-function to linear strategy, thus getting c* and d*.
4. conclude that it is a symmetric BNE in linear strategy with b_i(v_i)=cv_i+d*

Question 4

How to calculate the highest valuation, and then get the highest bid, which is equal to the (expected) revenue for the seller?

Answer 4

Step 1) Calculate highest expected valuation, use following equation

a+(b-a)*N/(N+1) for N players.

Step 2) Insert the highest expected valuation into the bidding function (could be the optimal bidding function).

Notice if you have to compare with another valuation with more players. When the amount of players increase then the valuation increase.

Big demand high price.

Question 5

What about games with more than two identical players.

Answer 5

…

Question 6

How do we write the expected payoff of bidder i, when we have two identical bidders (i and j) .

Answer 6

In step 1 we insert bidder j´s bidding function into bidder i´s exp. payoff function, so we get bidder i´s exp. payoff conditional on bidder j´s bidding strategy.

E(u_i(v_i,b_i)=P(b_i>b_j(v_j))

Which says that bidder i wins, but we could flip it around due to the players are identical.