ECON 383 Applied Game Theory
Tutorial 4
Equilibrium Bidding in First-Price Auctions
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Finding equilibrium bidding in first­price auctions is simply another form of optimization
problems.
● Any optimization problem follows three steps
● Step 1: Set the objective function in term of (a) choice variable(s).
● Step 2: Find the First­order condition by differentiate the objective function with
respect to the choice variable(s).
● Step 3: Solve for the equilibrium.
● For example,
● In a utility maximization problem, the objective function is a utility function in
term of consumptions on good X and good Y. We derive the F.O.C by
differentiating the utility function with respect to X and Y.
● To find a Cournot equilibrium, the objective function is a firm’s profit function in
term of the firm’s output. We derive the F.O.C., which is also called “the Best
Response function”, by differentiating the profit function with respect to output.
● We can break the process of finding the first­price, or any kind of, auctions into 3 steps.
● Step 0: Assumption
● Step 1: Set the Bidder’s expected profit function in term of value.
● Step 2: Find the F.O.C. by differentiating the expected profit function with respect
to value
● Step 3: solve for the equilibrium by a trick called “guess and verify”
Step 0
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There’re n , where n > 2 , bidders in the auction
We focus only on symmetric equilibrium. (All the bidders use the same strategy)
Notation
○ b∈[0, 1] for “bid”
○ v∈[0, 1] for “any value in general”
○ vj ~ U [0, 1] , j = 1, 2, 3, ..., n for “bidder j’s value”, where v1, v2 , v3, ..., vn are
independent.
Step 1
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Choice Variable: “v” for “value”
● It’s natural to think that the choice variable should be “bid”, which is
b = s(v)
where s is a function mapping value v to bid b .
For example, (take note)
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However, b is not a convenient choice variable since it’s difficult to differentiate
with respect to a function.
● Under the assumption that s in strictly increasing, choosing b is to maximize
profit is equivalent to choosing v to maximize profit at the point v = vi .( vi is bidder
i’s true value).
● This is easier to do calculus later since v is a variable, not a function.
Expected profit function = vn−1(vi − s(v))
● Why?
● By intuition, Bidder i’s expected profit = Prob(bidder i win)x(Profit if win)
1) Profit if win
= vi − s(v)
2) Prob(bidder i win)= vn−1
● Why?
● Since bidder i win if his bid is higher than any other bidders,
Prob(bidder i win) = Prob( s(v) > s(v1) and s(v) > s(v2) and..... and s(v) > s(vn) )
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Since s is a strictly increasing function,
Prob(bidder i win) = Prob( v > v1 and v > v2 and..... and v > vn )
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Since v1, v2 , v3, ..., vn are independent,
Prob(bidder i win) = Prob( v > v1 )Prob( v > v2 ).....Prob( v > vn ),
and there are n­1 term of these.
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Consider Prob( v > vj ), for any j =/ i .
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For any j , vj ~ U [0, 1] .
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Therefore Prob (v > vj) = v . (See the graph)
Since there are n­1 terms of Prob (v > vj) , finally we get
Prob(bidder i win) = vn−1
In conclusion, our problem is to maximize bidder i’s expected profit function =
vn−1(vi − s(v)) by choosing a choice variable v.
Step 2
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Differentiate vn−1(vi − s(v)) with respect to v , evaluated at the point where v = vi
and set it equal to zero.
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F.O.C is
(n − 1)(vi − s(vi)) = vis′(vi)
Step 3
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Guess that s(vi) = cvi k . We have to solve for c and k.
Substitute our guess in the F.O.C. Then the F.O.C is
(n − 1)(vi − cvi k) = vickvi k−1
(n − 1)(vi − cvi k) = ckvi k
Only k=1 satisfies this equation. Therefore
(n − 1)(vi − cvi ) = cvi (n − 1)vi = c(1 + n − 1)cvi c = (n − 1)/n
Since k = 1 and c=(n­1)/n,
s(vi) = [(n − 1)/n]vi Homework (D.I.Y)
1) How the equilibrium change if vj ~ U [0, 3] ?
2) Why in the All­pay auction the object function is vn−1(vi) − s(v) instead of vn−1[vi − s(v)] ?