Question

5. Consider a first price auction (with independent private values) where there are two bidders, A and B. There are two possible types of bidders, a bidder with a 60 with probability 0.5 and 100 with probability 0.5. Bids can come only in increments of 10. Consider the following strategy profile: "Each bidder bids 50 if the valuation is 60 and bids 70 if the valuation is 100." Is this strategy pair a Bayes-Nash equilibrium? (*Note: As we assumed in class, if the bids are equal, the auctioneer flips a coin to determine the winner, which means you would become the winner with 50% chance when you are tied with the other player.) [5]

Helpme solving Bayes-Nash equilibrium problem

Answer 1

BAYES-NASH EQUILIBRIUM DEFINITION

It is defined as a strategy profile and beliefs specified for each player about the types of the other players that maximises the expected payoff for each player given their beliefs about each player's type and given the strategies played by the other players.

EXISTENCE OF BAYES-NASH EQUILIBRIA

THEOREM

Consider a finite incomplete information Bayesian game. Then a mixed strategy Bayesian Nash equilibrium exists.

THEOREM

Consider a Bayesian game with continuous strategy spaces and continuous types. If strategy sets and type sets are compact, payoff functions are continuous and concave in own strategies, then a pure strategy Bayesian Nash equilibrium exists.

-A major application to Bayesian games is auctions, which are a common method for allocating scarse goods across individuals with different valuations for these goods. This corresponds to a situation of incomplete information because the violations to different potential buyers are unknown. From the first theorem stated, the auction is obeys Bayes Nash equilibrium.