Quiz 2

Auctions and Mechanism Design

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# 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 auction...

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

FormatTypeDescription
English (open ascending)OpenAuctioneer raises price; bidders drop out; last bidder wins
Dutch (open descending)OpenAuctioneer starts high, lowers until someone accepts
First-price sealed-bidSealedHighest bidder wins, pays their bid
Second-price (Vickrey)SealedHighest bidder wins, pays second-highest bid

Private vs. Common Values

Value ModelDescriptionExample
Private valuesEach bidder knows their own valueArt for personal enjoyment
Common valuesItem has same value to all, but unknownOil drilling rights
Affiliated valuesValues are correlated but private signals differMost real-world auctions

9.3 Second-Price (Vickrey) Auction

Dominant strategy: Bid your true value viv_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 bib_i be your bid, viv_i your value. Let m=max⁡j≠ibjm = \max_{j \neq i} b_j be the highest competing bid.
CasePayoff for bidding viv_iPayoff for bidding bi>vib_i > v_i
m>vim > v_iLose, payoff 0Lose if bimb_i m
m<vim < v_iWin, payoff vi−m>0v_i - m > 0Same (win)
m=vim = v_iTie/win, payoff 0Might win with vi−m=0v_i - m = 0
Bidding above viv_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 nn bidders with values vi∼U[0,1]v_i \sim U[0,1]. Each bids b(v)=n−1nvb(v) = \frac{n-1}{n} v. Derivation: Bidder with value vv bids bb. Probability of winning = P(all others bid<b)=P(v(1)<b−1(b))=(b−1(b))n−1P(\text{all others bid} < b) = P(v_{(1)} < b^{-1}(b)) = (b^{-1}(b))^{n-1}. Expected payoff: (v−b)⋅(b−1(b))n−1(v - b) \cdot (b^{-1}(b))^{n-1} Maximising and imposing symmetry yields b(v)=n−1nvb(v) = \frac{n-1}{n} v.

Key Properties

  • Bidders shade their bids below true value
  • Shading decreases as nn increases (competition)
  • With n→∞n \to \infty, b(v)→vb(v) \to v (perfect competition)

9.5 Revenue Equivalence Theorem

Theorem: Any auction format that:
  1. Always awards the item to the highest bidder
  2. Gives the lowest-valuing bidder zero expected payoff ...yields the same expected revenue to the seller.
Auction FormatExpected Revenue (2 bidders, U[0,1])
First-price13\frac{1}{3}
Second-price13\frac{1}{3}
English13\frac{1}{3}
Dutch13\frac{1}{3}
The expected revenue is E[v(2)]E[v_{(2)}], the expected second-highest value.

Revenue Equivalence Formula

For nn bidders with i.i.d. values from distribution FF:
Expected Revenue=n∫0vˉ[1−F(v)]F(v)n−1v f(v) dv\text{Expected Revenue} = n \int_0^{\bar{v}} [1 - F(v)] F(v)^{n-1} v \ f(v) \ dv
For U[0,1]U[0,1], n=2n=2: E[Revenue]=∫01(1−v2)dv=13E[\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

PropertyMeaning
Incentive compatibilityTruth-telling is a Nash equilibrium
Individual rationalityPlayers voluntarily participate
EfficiencyAllocate to highest valuer
Budget balanceTransfers sum to zero (in mechanism)
Revenue maximisationSeller's expected revenue is maximised

📊 Formula Summary

ConceptFormula
First-price bid (U[0,1], n bidders)b(v)=n−1nvb(v) = \frac{n-1}{n} v
Expected revenue (2 bidders, U[0,1])13\frac{1}{3}
Second-price dominant strategybi=vib_i = v_i

✅ Practice Questions

Q1: In a second-price auction with values v1=80,v2=60,v3=40v_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)=3−13×0.7=23×0.7≈0.467b(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
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