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First Price Auction

The document discusses a first-price sealed-bid auction involving two bidders, where each bidder submits a bid based on their valuation of the good. It explores the best response strategies and the concept of Bayesian Nash Equilibrium (BNE) in the context of linear bidding strategies. The analysis concludes that the optimal bidding strategies for both players are half of their valuations, reflecting a trade-off between the likelihood of winning and the payoff from winning.

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Tarun Kumar
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25 views2 pages

First Price Auction

The document discusses a first-price sealed-bid auction involving two bidders, where each bidder submits a bid based on their valuation of the good. It explores the best response strategies and the concept of Bayesian Nash Equilibrium (BNE) in the context of linear bidding strategies. The analysis concludes that the optimal bidding strategies for both players are half of their valuations, reflecting a trade-off between the likelihood of winning and the payoff from winning.

Uploaded by

Tarun Kumar
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Consider the following …rst price, sealed bid auction.

There are 2 bidders i = 1; 2:


Each bidder i submits a bid bi 0 simultaneously; and the agent with higher
bid gets the good and pays a price equal to his bid. (If both submit the same
bid, then the good is given to one agent picked randomly.
Bidder i has a valuation vi for the good, i.e., if he gets the good at price p;
then his payo¤ is vi p: For each bidder i; vi is distributed uniformly over [0; 1]:
The bidder valuations are drawn independent of each other (not a great
assumption - we will do the other extreme later)

Therefore, the payo¤ function for player i is


8
< vi bi if bi > bj
vi bi
ui (b1 ; b2 ; v1 ; v2 ) = 2 if bi = bj
:
0 if bi < bj
Strategy of player i is a bid as a function of the valuation, i.e., a function
bi (vi ):
In a BNE, player 1’s bidding strategy b1 (v1 ) is a best response to player 2’s
bidding strategy b2 (v2 ); and vice versa.
Formally, a strategy pro…le (b1 (v1 ); b2 (v2 )) is a BNE if for each vi 2 [0; 1]; bi (vi )
solves
1
max (vi bi ) Pr [bi > bj (vj )] + (vi bi ) Pr [bi = bj (vj )]
bi 2
We now look for a special class of equilibria: linear equilibria where

b1 (v1 ) = a 1 + c 1 v1
b2 (v2 ) = a 2 + c 2 v2

Notice that we are not restricting the strategy spaces, rather we look for equi-
libria where bidding strategies have the linear form. [[Deviations need not be
linear]]
Suppose that player j adopts strategy bj (vj ) = aj + cj vj : Then, for each vi ;
player i0 s best response solves

max (vi bi ) Pr [bi > aj + cj vj ]


bi

We have dropped the second term because, since vi is uniformly distributed, so


is bj (vj ) = aj + cj vj and hence any speci…c bid occurs with zero probability.
Therefore, Pr [bi = aj + cj vj ] = 0:
Solution steps:
First note that cj 0 as the bids must be weakly increasing.
We must have aj bi aj + cj ; since the bid of one player cannot be
below the minimum (pointless!) of the rival and above the maximum (stupid!).
Therefore,
bi aj bi aj
Pr [bi > aj + cj vj ] = Pr vj < = (uniform!)
cj cj

1
Player i’s objective function is therefore,

bi aj
max (vi bi ) ;
bi cj

which gives us the best response function


1
2 (vi + aj ) if vi > aj
bi (vi ) =
aj if vi < aj

Since we are looking for a linear equilibrium, we reject values of aj such that
0 < aj < 1: If aj 1; then we would have bi (vi ) vi ; which cannot be optimal.
Therefore, we must have aj 0; implying that

1
bi (vi ) = (vi + aj ) ;
2
a
implying that ai = 2j and ci = 12 :
al 1
Repeating the same analysis for player j; we have aj = 2 and cj = 2 . This
implies that ai = aj = 0; and therefore, the BNE is
v1 v2
b1 (v1 ) = and b2 (v2 ) = :
2 2
This re‡ects a fundamental tradeo¤ that the bidder faces: for any valuation,
higher the bid, larger the chances of winning, but lower the gain conditional on
winning.

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