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Auctions and Mechanism Design

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Auctions and Mechanism Design

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Now · 🎯 Learning Objectives

Auctions and Mechanism Design

🎯 Learning Objectives

Compare first-price, second-price, English, and Dutch auctions
Derive optimal bidding strategies in private-value auctions
State and apply the Revenue Equivalence Theorem
Understand the revelation principle in mechanism design
Design auctions to maximise revenue or efficiency

9.1 Intuition: Selling to the Highest Valuer

Auctions are institutions for allocating resources when the seller doesn't know buyers' valuations. The challenge: design rules that induce buyers to reveal their true values.

🔑 Key Insight: Different auction rules lead to different bidding behaviour. The optimal auction design depends on the information structure and seller's objectives.

9.2 Auction Formats

Classification

Format	Type	Description
English (open ascending)	Open	Auctioneer raises price; bidders drop out; last bidder wins
Dutch (open descending)	Open	Auctioneer starts high, lowers until someone accepts
First-price sealed-bid	Sealed	Highest bidder wins, pays their bid
Second-price (Vickrey)	Sealed	Highest bidder wins, pays second-highest bid

Private vs. Common Values

Value Model	Description	Example
Private values	Each bidder knows their own value	Art for personal enjoyment
Common values	Item has same value to all, but unknown	Oil drilling rights
Affiliated values	Values are correlated but private signals differ	Most real-world auctions

9.3 Second-Price (Vickrey) Auction

Dominant strategy: Bid your true value

v_i

. Why? Your bid determines whether you win, but not what you pay (you pay the second-highest bid). If you bid below your value, you risk losing when you could have profited. If you bid above, you risk winning at a price above your value.

Formal Proof

Let

b_i

be your bid,

v_i

your value. Let

m = \max_{j \neq i} b_j

be the highest competing bid.

Case	Payoff for bidding $v_i$	Payoff for bidding $b_i > v_i$
$m > v_i$	Lose, payoff 0	Lose if $b_i m$
$m < v_i$	Win, payoff $v_i - m > 0$	Same (win)
$m = v_i$	Tie/win, payoff 0	Might win with $v_i - m = 0$

Bidding above

v_i

creates the risk of winning at a loss without any benefit. Bidding below creates the risk of losing a profitable win.

9.4 First-Price Sealed-Bid Auction

No dominant strategy — bid depends on beliefs about others.

Symmetric Equilibrium with Private Values

Assume

n

bidders with values

v_i \sim U[0,1]

. Each bids

b(v) = \frac{n-1}{n} v

. Derivation: Bidder with value

v

bids

b

. Probability of winning =

P(\text{all others bid} < b) = P(v_{(1)} < b^{-1}(b)) = (b^{-1}(b))^{n-1}

. Expected payoff:

(v - b) \cdot (b^{-1}(b))^{n-1}

Maximising and imposing symmetry yields

b(v) = \frac{n-1}{n} v

Key Properties

Bidders shade their bids below true value
Shading decreases as $n$ increases (competition)
With $n \to \infty$ , $b(v) \to v$ (perfect competition)

9.5 Revenue Equivalence Theorem

Theorem: Any auction format that:

Always awards the item to the highest bidder
Gives the lowest-valuing bidder zero expected payoff ...yields the same expected revenue to the seller.

Auction Format	Expected Revenue (2 bidders, U[0,1])
First-price	$\frac{1}{3}$
Second-price	$\frac{1}{3}$
English	$\frac{1}{3}$
Dutch	$\frac{1}{3}$

The expected revenue is

E[v_{(2)}]

, the expected second-highest value.

Revenue Equivalence Formula

For

n

bidders with i.i.d. values from distribution

F

\text{Expected Revenue} = n \int_0^{\bar{v}} [1 - F(v)] F(v)^{n-1} v \ f(v) \ dv

For

U[0,1]

n=2

E[\text{Revenue}] = \int_0^1 (1-v^2) dv = \frac{1}{3}

9.6 Mechanism Design

Goal: Design a game whose equilibrium achieves desired social outcomes.

The Revelation Principle

For any equilibrium of any auction, there exists a truthful (incentive-compatible) direct mechanism that yields the same allocation and payoffs. (Diagram)

Key Mechanism Design Properties

Property	Meaning
Incentive compatibility	Truth-telling is a Nash equilibrium
Individual rationality	Players voluntarily participate
Efficiency	Allocate to highest valuer
Budget balance	Transfers sum to zero (in mechanism)
Revenue maximisation	Seller's expected revenue is maximised

📊 Formula Summary

Concept	Formula
First-price bid (U[0,1], n bidders)	$b(v) = \frac{n-1}{n} v$
Expected revenue (2 bidders, U[0,1])	$\frac{1}{3}$
Second-price dominant strategy	$b_i = v_i$

✅ Practice Questions

Q1: In a second-price auction with values

v_1=80, v_2=60, v_3=40

, what is the winning bid and payment?

Solution

Bidder 1 wins (highest value, bids 80 truthfully). Payment = second-highest bid = 60. Bidder 1's surplus = 80 - 60 = 20. Q2: In a first-price auction with 3 bidders, values uniformly distributed on [0,1], what is the equilibrium bid of a bidder with value 0.7? Solution

$b(0.7) = \frac{3-1}{3} \times 0.7 = \frac{2}{3} \times 0.7 \approx 0.467$ Q3: Explain the Revenue Equivalence Theorem in one paragraph. Solution

The Revenue Equivalence Theorem states that under private values with independent signals and risk-neutral bidders, any auction format that always awards the item to the highest bidder and gives the lowest possible type zero expected surplus yields the same expected revenue. This means first-price, second-price, English, and Dutch auctions all generate identical expected revenue for the seller. The theorem shows that auction format matters less for revenue than the information structure and number of bidders. Join Discord PreviousSignalling GamesNextMechanism Design Deep