Derive the Optimal Bid in a Two-Bidder Second-Price Auction
Point72Quantitative Researcher · Technical Screen
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Two risk-neutral bidders compete for one item in a sealed-bid second-price auction. Explain the assumptions and derivation clearly, check edge cases, and show how the result changes when those assumptions no longer hold.
Derive the Optimal Bid in a Two-Bidder Second-Price Auction
Two risk-neutral bidders compete for one item in a sealed-bid second-price auction. Your private value is v with 0 < v < 1. The other bidder's private value is uniformly distributed on [0, 1], and assume that bidder bids their value. Determine your optimal bid, prove the result by considering possible opponent bids, and contrast it with a first-price auction under the same value distribution.
Constraints & Assumptions
- Highest bid wins and pays the other bidder's bid.
- Ties may be broken arbitrarily and have probability zero under the continuous distribution.
- Utility is value minus payment when winning and zero when losing.
- The first-price comparison should state any symmetry or equilibrium assumption it uses.
Clarifying Questions to Ask
- Are bidders risk-neutral and values private and independent?
- Is the goal a dominant strategy or a Bayesian best response?
- How are ties handled if bids can have atoms?
Hints
Hint 1
Condition on the opponent bid: Compare outcomes when the opponent bid is below, above, or between your value and an alternative bid.
Hint 2
Separate payment rules: In a first-price auction your own bid sets the payment, creating a trade-off between win probability and surplus.
Follow-up questions
- How does risk aversion change first-price bid shading?
- Does truthful bidding remain dominant with more than two bidders in a standard second-price auction?
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