Question

Suppose you own the painting “Fresh Air” that you want to sell. This painting exists only in the artist’s imagination. 1 Suppose there are two potential buyers for this painting. You know that the painting is worth $20,000 to the first buyer and $10,000 to the second buyer. You decide to sell your painting using a closed bid-first price auction. Calculate your revenue for the following scenarios and choose the best scenario.

i) The buyers don’t know each other’s valuation exactly and they each guess that it is a random value with uniform distribution between 0 and $30,000. The buyers will use their Bayes Nash equilibrium bidding functions to bid. Buyer 1 bids: Buyer 2 bids: Revenue from the auction:

ii) You secretly tell the first buyer, the valuation of the second buyer. (second buyer is unaware of this and will stick to his Bayes-Nash equilibrium strategy) Buyer 1 bids: Buyer 2 bids: Revenue from the auction:

iii) You secretly tell the second buyer, the valuation of the first buyer. (first buyer is unaware of this and will stick to his Bayes-Nash equilibrium strategy) Buyer 1 bids: Buyer 2 bids: Revenue from the auction:

Answer 1

i) Bayesian Nash Equilibrium

• Bayesian Nash equilibrium = straightforward extension of NE:
• Each type of player chooses a strategy that maximizes expected utility given the actions of all types of other players and that player’s beliefs about others’ types

• In our BoS variant:
• Action of Player 1 is optimal (maximizes expected utility) given the actions of the two types of Player 2 (and Player 1s beliefs about Player 2s type)
• Action of each type of Player 2 is optimal, given the action of Player 1

You and a friend are playing a 2 × 2 matrix game, but you’re not sure if it’s BoS or PD. Both are equally likely.

A B

A 2,2	0,3
B 3,0	1,1

PD

A B

A 2,1	0,0
B 0,0	1,2