Illustrating Bayesian equilibrium
Open
bid
First price
Second price
Decending
bids
Ascending
bids
(Dutch auction)
(English auction)
L
R
L
R
3, 1
2, 0
3, 0
2, 1
D 0, 1
4, 0
0, 0
4, 1
U
Sealed
bid
First - price Second - price
sealed - bid
sealed - bid
(Vickrey auction)
First price sealed-bid auction w/private values
A Bayesian Nash equil. in first price auction
We must show that Bi (vi )  12 (vi  1) maximizes
Bid : bi  Ai  [1, 2], i  1, 2
(vi  bi )  Pr (Other bids less than bi )
Valuations : vi  Ti  [1, 2], i  1, 2, are distributed
independently and uniformly : For each i, Pr(v j  x)  x  1.
ui (bi , b j ; vi ) 
vi  bi if bi is bigger than the other bid.
otherwise.
0
One must bid less than true value in order to earn if one
wins. This must be traded off against the fact that a lower
bid reduces the probability for having a winning bid.
Consider the following strategies: Bi (vi )  12 (vi  1)
Type t2
Type t2
Probability that other bids less than bi: 2(bi  1)
density
2
max(vi  bi )  2(bi  1)
bi
FOC yields: bi  12 (vi  1)
1
1
2
1
3
2
2
This shows that it is a Bayesian
Nash equil. that both bid
bid
1 + (true value – 1)/2.
Second price sealed-bid auction w/private values
Bid : bi  Ai  [1, 2], i  1, 2
Valuations : vi  Ti  [1, 2], i  1, 2, are distributed
independently and uniformly : For each i, Pr(v j  x)  x  1.
ui (bi , b j ; vi ) 
vi  b j if bi is bigger than the other bid.
otherwise.
0
One need not bid less than true value in order to earn if
one wins. Bidding true value weakly dominates all other
strategies. Both bidding the true value is a Baysian Nash
equilibrium: Bi (vi )  vi
1