Auction Theory: The Mathematics of Bidding Wars

In 1994, the U.S. government auctioned radio spectrum licenses using game theory. The result? $7.7 billion in revenue — far more than expected.

In 2000, the UK’s 3G telecom auction raised $34 billion using carefully designed rules.

In 2016, the FCC’s broadcast incentive auction was called “the most complex auction ever conducted” — a reverse auction followed by a forward auction, designed by game theorists.

Auctions are everywhere: eBay, Google Ads, treasury bonds, oil leases, art, real estate. And the mathematics of bidding can mean the difference between winning and overpaying, or losing and leaving money on the table.

Welcome to auction theory — where game theory meets billions of dollars.

The Four Classic Auction Formats

1. English Auction (Ascending-Bid)

How it works:

  • Price starts low
  • Bidders raise the price
  • Last bidder standing wins
  • Pays their final bid

Examples: Art auctions (Sotheby’s, Christie’s), eBay (with proxy bidding)

Strategy: Keep bidding until the price exceeds your valuation, then stop.

2. Dutch Auction (Descending-Bid)

How it works:

  • Price starts high
  • Price drops continuously
  • First bidder to accept wins
  • Pays the current price

Examples: Flower auctions in the Netherlands, some IPOs

Strategy: Accept when the price drops to a level where you expect no one else will accept (game of chicken).

3. First-Price Sealed-Bid Auction

How it works:

  • All bidders submit sealed bids simultaneously
  • Highest bidder wins
  • Pays their bid

Examples: Government contracts, oil leases, procurement auctions

Strategy: Bid below your valuation (to capture surplus), but high enough to win.

4. Second-Price Sealed-Bid Auction (Vickrey Auction)

How it works:

  • All bidders submit sealed bids simultaneously
  • Highest bidder wins
  • Pays the second-highest bid

Examples: eBay proxy bidding, some ad auctions, stamp collecting (named after William Vickrey)

Strategy: Bid your true valuation (dominant strategy!).

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph TD A[Auction Formats] --> B[English: Ascending] A --> C[Dutch: Descending] A --> D[First-Price Sealed] A --> E[Second-Price Sealed] B --> F[Open: See others' bids] C --> F D --> G[Sealed: Submit privately] E --> G B --> H[Pay your final bid] C --> H D --> H E --> I[Pay second-highest bid] style A fill:#2d3748,stroke:#4299e1,stroke-width:3px style E fill:#2d3748,stroke:#48bb78,stroke-width:2px

The Vickrey Auction: A Game-Theoretic Miracle

The second-price sealed-bid auction has a remarkable property: bidding your true valuation is a dominant strategy.

Proof: Why Truthful Bidding is Optimal

Suppose your true valuation is $100.

Case 1: You bid exactly $100

  • If the second-highest bid is $80, you win and pay $80 (surplus: $20)
  • If the second-highest bid is $110, you lose (surplus: $0)

Case 2: You bid more than $100 (say $120)

  • If the second-highest bid is $80, you win and pay $80 (surplus: $20) — same as truthful
  • If the second-highest bid is $110, you win and pay $110 (surplus: -$10) — worse! You overpay
  • If the second-highest bid is $130, you lose (surplus: $0) — same as truthful

Overbidding risks winning and paying more than your valuation.

Case 3: You bid less than $100 (say $80)

  • If the second-highest bid is $70, you win and pay $70 (surplus: $30) — same as truthful
  • If the second-highest bid is $90, you lose (surplus: $0) — worse! You miss a profitable win
  • If the second-highest bid is $110, you lose (surplus: $0) — same as truthful

Underbidding risks losing when you could have won profitably.

Conclusion: Bidding your true value weakly dominates all other strategies.

This is the most elegant result in auction theory.

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph TD A[Your True Value: $100] --> B[Bid Strategy?] B --> C[Bid $100: Truthful] B --> D[Bid $120: Overbid] B --> E[Bid $80: Underbid] C --> F[2nd highest = $80
Win, pay $80
Surplus: $20] C --> G[2nd highest = $110
Lose
Surplus: $0] D --> H[2nd highest = $110
Win, pay $110
Surplus: -$10
DISASTER!] E --> I[2nd highest = $90
Lose
Surplus: $0
MISSED OPPORTUNITY!] style C fill:#2d3748,stroke:#48bb78,stroke-width:3px style H fill:#742a2a,stroke:#f56565,stroke-width:3px style I fill:#742a2a,stroke:#f56565,stroke-width:3px

The Revenue Equivalence Theorem

One of the most surprising results in auction theory:

Under certain conditions, all four auction formats generate the same expected revenue for the seller.

Conditions:

  1. Bidders are risk-neutral
  2. Bidders’ valuations are independent and drawn from the same distribution
  3. Payment depends only on bids
  4. Lowest bidder gets zero surplus

Why it works:

In equilibrium:

  • English and Second-Price: Bidders bid truthfully
  • Dutch and First-Price: Bidders shade their bids (bid below valuation)

The shading in first-price auctions exactly compensates for the second-price mechanism’s tendency to leave more surplus to bidders!

Implication: From a revenue perspective, auction choice doesn’t matter (under these idealized conditions). But other factors do matter: simplicity, resistance to collusion, speed.

First-Price Auctions: How Much to Shade?

In a first-price sealed-bid auction, bidding your true valuation guarantees zero surplus. You need to shade your bid.

But shade too much, and you lose to competitors.

Optimal Shading Formula (Simplified)

Suppose:

  • Your valuation: v
  • You believe others’ valuations are uniformly distributed on [0, v_max]
  • There are n bidders

Optimal bid:

b(v) = v · (n-1)/n

Example with 2 bidders:

  • Your valuation: $100
  • Optimal bid: $100 × (2-1)/2 = $50

Example with 10 bidders:

  • Your valuation: $100
  • Optimal bid: $100 × (10-1)/10 = $90

Insight: More competition → less shading (must bid closer to true value to win).

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph LR A[Number of Bidders] --> B[2 bidders: Bid 50% of value] A --> C[3 bidders: Bid 67% of value] A --> D[5 bidders: Bid 80% of value] A --> E[10 bidders: Bid 90% of value] A --> F[100 bidders: Bid 99% of value] style B fill:#2d3748,stroke:#ed8936,stroke-width:2px style F fill:#2d3748,stroke:#48bb78,stroke-width:2px

Common Value vs Private Value Auctions

Private Value Auctions

Each bidder has their own valuation that doesn’t depend on others’ information.

Example: You’re bidding on a painting you’ll hang in your home. Its value to you is independent of what others think.

Strategy: Bid based on your own valuation.

Common Value Auctions

The item has the same value to everyone, but bidders have different estimates of that value.

Example: Offshore oil drilling rights. The oil is worth the same to all companies, but they have different geological surveys.

Danger: The winner’s curse

The Winner’s Curse

In common value auctions, winning often means you overpaid.

Why?

Suppose the oil tract is truly worth $50M, but estimates range from $30M to $70M due to uncertainty.

If you bid $60M and win, it means everyone else bid less. This suggests their estimates were lower. Your estimate was probably too optimistic.

On average, the winner is the most overoptimistic bidder, and thus likely overpaid.

%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph TD A[True Value: $50M] --> B[Bidder Estimates] B --> C[Bidder 1: $35M] B --> D[Bidder 2: $45M] B --> E[Bidder 3: $50M] B --> F[Bidder 4: $55M] B --> G[Bidder 5: $65M - WINS] G --> H[Winner's Curse!] H --> I[Most optimistic bidder wins] I --> J[Likely overestimated value] J --> K[May have overpaid] style G fill:#742a2a,stroke:#f56565,stroke-width:3px style K fill:#742a2a,stroke:#f56565,stroke-width:3px

Solution: Sophisticated bidders adjust their bids downward to account for the winner’s curse.

Real-world impact:

  • Oil companies lost billions in 1960s-70s offshore auctions before learning to adjust
  • Book publishing auctions (advance payments to authors)
  • Corporate takeovers (acquiring companies often overpay)

Strategic Bidding: Advanced Tactics

1. Jump Bidding (English Auctions)

Instead of incrementing by $1, jump your bid dramatically.

Why?

  • Signal strength: Intimidate weaker bidders into dropping out
  • Reduce information leakage: Fewer rounds means competitors learn less about your valuation

Risk: Paying more than necessary.

2. Sniping (Online Auctions)

Wait until the last second to submit your bid.

Why it works on eBay:

  • Prevents competitors from responding
  • Exploits fixed end time

Why it doesn’t work theoretically:

  • Vickrey logic says you should bid your max and walk away
  • But in practice, people use incremental bidding, making sniping effective

3. Collusion and Bidding Rings

Bidders coordinate to keep prices low, then split the gains.

Example:

  • Ring members agree one will bid, others stay silent
  • Winner pays low price
  • Ring holds private “knockout auction” to determine who really wanted it and how to split the gains

Illegal in most jurisdictions, but hard to detect and prosecute.

Auction design countermeasures:

  • Random auction timing
  • Anonymous bidding
  • Reserve prices

Google’s ad auction runs billions of times per day. Here’s how it works:

Generalized Second-Price (GSP) Auction

When you search “running shoes,” advertisers bid for placement.

Not quite a Vickrey auction:

  • Winner pays slightly more than the next bid (per click)
  • But position matters (top ad gets more clicks)

Equilibrium bidding:

  • Complex, because value-per-click varies by position
  • Google uses Ad Rank = Bid × Quality Score
  • Higher quality ads pay less per click (incentivizes better ads)

Why it’s brilliant:

  • Simple for advertisers
  • Maximizes revenue while ensuring ad quality
  • Truthful bidding is approximately optimal
%%{init: {'theme':'dark', 'themeVariables': {'primaryTextColor':'#fff','secondaryTextColor':'#fff','tertiaryTextColor':'#fff','textColor':'#fff','nodeTextColor':'#fff'}}}%% graph TD A[User Search] --> B[Ad Auction Runs] B --> C[Advertisers submit bids] C --> D[Ad Rank = Bid × Quality Score] D --> E[Rank ads] E --> F[Top ad: Highest Ad Rank] F --> G[User clicks?] G -->|Yes| H[Advertiser pays second-price] G -->|No| I[No payment] J[Billions of auctions per day] K[Revenue: ~$200B/year for Google] style B fill:#2d3748,stroke:#4299e1,stroke-width:3px style K fill:#2d3748,stroke:#48bb78,stroke-width:2px

Spectrum Auctions: Game Theory in Government

Governments use auctions to allocate radio spectrum for telecom.

Challenges:

  • Multiple licenses sold simultaneously
  • Complementarity: Some licenses are more valuable together (e.g., adjacent regions)
  • Substitutability: Some licenses are alternatives

Solution: Simultaneous Multiple Round Auction (SMRA)

  • All licenses auctioned at once
  • Bidders can see others’ bids and adjust
  • Continues until no new bids

Game theory considerations:

  • Signaling: Bidders use bids to communicate (e.g., “I want Region A, you take Region B”)
  • Tacit collusion: Coordinating to keep prices low
  • Predatory bidding: Driving up prices on licenses you don’t want to drain competitors’ budgets

Result: Raised hundreds of billions globally, vastly more than previous methods.

Auction Design: Mechanism Design

Mechanism design is “reverse game theory” — design rules that lead to desired outcomes.

Goals:

  1. Efficiency: Winner is the bidder who values it most
  2. Revenue maximization: Seller gets high price
  3. Truthfulness: Bidders want to bid honestly
  4. Simplicity: Easy for bidders to understand
  5. Robustness: Resistant to collusion and manipulation

Trade-offs:

  • Vickrey auctions are truthful and efficient, but not revenue-maximizing
  • First-price auctions can generate more revenue, but are more complex
  • English auctions are simple, but vulnerable to collusion

Optimal auction design (Myerson, 1981): Set a reserve price below which you won’t sell, and use a second-price format.

Real-World Applications

1. eBay

Uses proxy bidding (essentially Vickrey auction):

  • You submit your max bid
  • eBay automatically outbids others up to your max
  • You pay just above the second-highest bid

But: Fixed end time enables sniping, reducing efficiency.

2. Treasury Bonds

U.S. Treasury uses discriminatory (first-price) and uniform (second-price) auctions.

  • Uniform: All winners pay the same price (lowest accepted bid)
  • Encourages more aggressive bidding
  • More efficient price discovery

3. Electricity Markets

Power plants bid to supply electricity.

  • Day-ahead markets: Generators bid for next day
  • Real-time markets: Adjust for demand fluctuations
  • Auctions run continuously, clearing the market

4. Keyword Advertising

Google, Facebook, Amazon all use auction-based ad systems.

  • Billions in revenue
  • Complex equilibria
  • Game theory Ph.D.s optimize bids

Common Mistakes in Auctions

Mistake 1: Bidding Emotionally

Problem: “Auction fever” — you keep bidding to win, exceeding your valuation

Solution: Set a maximum bid beforehand and stick to it

Mistake 2: Ignoring the Winner’s Curse

Problem: In common value auctions, winning suggests you overestimated

Solution: Shade your bid to account for adverse selection

Mistake 3: In Vickrey Auctions, Not Bidding Your Max

Problem: Trying to “game” the system by bidding low

Solution: Bid your true value — it’s a dominant strategy!

Mistake 4: In First-Price Auctions, Bidding Your True Value

Problem: Guarantees zero surplus

Solution: Shade your bid appropriately based on competition

Key Takeaways

  1. Four classic auction formats: English, Dutch, First-Price Sealed, Second-Price Sealed (Vickrey)
  2. Vickrey auctions have a dominant strategy: Bid your true valuation
  3. Revenue Equivalence Theorem: Under ideal conditions, all formats yield the same revenue
  4. First-price auctions require bid shading: Optimal bid = v × (n-1)/n
  5. Winner’s curse: In common value auctions, winning often means overpaying
  6. Google Ads uses auctions to allocate billions in ad spending
  7. Mechanism design uses game theory to design auctions with desired properties
  8. Real-world auctions involve billions in spectrum, art, ads, and government contracts

Practice Problem

You’re bidding in a first-price sealed-bid auction for a contract worth $100,000 to you. There are 4 other bidders with similar valuations.

Using the simplified shading formula, what should you bid?

Solution

Formula: b(v) = v × (n-1)/n

Where:

  • v = $100,000 (your valuation)
  • n = 5 (total number of bidders, including you)

Calculation:

b = $100,000 × (5-1)/5 b = $100,000 × 4/5 b = $100,000 × 0.8 b = $80,000

Answer: Bid $80,000.

Reasoning: With 5 bidders, you need to shade your bid by 20%. This balances the risk of losing (by bidding too low) with the benefit of winning with surplus (by not bidding your full value).

If you win, your expected surplus is approximately $20,000. But you’ll only win when competitors value it less than you, which is why shading is optimal.

What’s Next?

You’ve now completed the intermediate level of the Game Theory Series! You’ve learned:

  • Why unpredictability matters (mixed strategies)
  • How cooperation emerges (repeated games)
  • How tragedy strikes shared resources (tragedy of the commons)
  • How to reason backwards (backward induction)
  • Why humans aren’t perfectly rational (ultimatum game)
  • How auctions allocate billions (auction theory)

These concepts power decision-making in business, politics, economics, and everyday life.

Continue exploring game theory, and you’ll see strategic interactions everywhere — from negotiations to sports, from markets to evolution.

This post is part of the Game Theory Series, where we explore the mathematics of strategic decision-making.