Bayesian Games
• An Example
• Definitions
• Applications
Example: Auction
• 2 bidders each bidding for a painting
• Each player has a value θi in Θi = {2,3} equally
likely
• First price auction: highest bidder wins, pays
bid
A Symmetric Equilibrium:
• Have to bid whole integers Ai = {0,1,2,3}
• Let us look for a symmetric pure strategy Bayesian
equilibrium
• So, we need to specify s(3)=?, s(2)=? so that no
player wants to deviate if they both bid that way
A Symmetric Equilibrium:
• Have to bid whole integers Ai = {0,1,2,3}
• We can try various combinations and see
• If bid is equal to value, then payoff is zero no
matter whether win or not, so let us try
s(3)<3, s(2)<2
A Symmetric Equilibrium:
• Try: s(3)=2, s(2)=1
• If θi = 3, and other uses this strategy, then
expected utility from
–
–
–
–
bidding 3 = 0
bidding 2 = (3-2) ( 1/2 + 1/4) = 3/4
bidding 1 = (3-1) (1/4) = 1/2
bidding 0 = 0
A Symmetric Equilibrium:
• Try: s(3)=2, s(2)=1
• If θi = 2, and other uses this strategy, then
expected utility from
–
–
–
–
bidding 3 = (2-3) ( 1) = -1
bidding 2 = 0
bidding 1 = (2-1) (1/4) = 1/4
bidding 0 = 0
A Symmetric Equilibrium:
• So, s(3)=2, s(2)=1 is a best response to the
same strategy by the other player
• It is a (symmetric) pure strategy Bayesian
equilibrium
Quiz
• 2 bidders each bidding for a painting
• Each bidder has a value θi in Θi = {2,3} equally likely
and independent of the other bidder’s value
• They bid in a first price auction: the highest bidder
wins, and pays his or her bid
• They have to bid whole integers in Ai = {0,1,2,3}
• If there is a tie, the winner is determined by the flip of a
fair coin.
Quiz
Suppose bidder 2 always bids 1 (whatever his or her
value). Which statement is wrong regarding player 1
when her value is 3:
a) Her expected utility from bidding 3 is (3-3)(1)=0;
b) Her expected utility from bidding 2 is (3-2)(1)=1;
c) Her expected utility from bidding 1 is (3-1)(1/2)=1;
d) Her expected utility from bidding 0 is (30)(1/2)=3/2;
Quiz Explanation
(d) is wrong.
• Please check that (a) (b) and (c) are true by
looking at value minus bid times probability of
winning.
• Given the other player bids 1 regardless of the
type, bidding 0 has a 0 probability of winning thus
the payoff is 0.