Derive the Optimal Bid in a Two-Bidder Second-Price Auction

Two risk-neutral bidders are competing for a single item in a sealed-bid second-price auction. Your private valuation is $v$, where $0 < v < 1$. The rival's private value is drawn uniformly from $[0,1]$, and suppose that bidder bids exactly that value. Find your optimal bid, justify it by examining the possible bids of the opponent, and compare it with a first-price auction under the same value distribution.

Constraints & Assumptions

• The highest bid wins, and the winner pays the other bidder's bid.
• Ties can be broken in any way and occur with probability zero because the distribution is continuous.
• If you win, utility equals value minus payment; if you lose, utility is zero.
• The first-price comparison should specify any symmetry or equilibrium assumption it relies on.

Clarifying Questions to Ask

• Are the bidders risk-neutral, and are values private and independent?
• Is the objective a dominant strategy or a Bayesian best response?
• If bids can have atoms, how are ties handled?

Hint — Condition on the opponent's bid: Compare what happens when the opponent's bid is below, above, or between your value and an alternative bid.

Hint — Separate the payment rules: In a first-price auction, your own bid determines the payment, so there is a trade-off between the probability of winning and the surplus.

What a Strong Answer Covers

• A pointwise argument that bidding $v$ weakly dominates every overbid and every underbid.
• A clear distinction between auction strategy and the financial notion of arbitrage.
• A derivation of bid shading for the symmetric first-price case.
• The assumptions under which these conclusions hold.

Follow-up Questions

• How does risk aversion alter bid shading in a first-price auction?
• With more than two bidders in a standard second-price auction, does truthful bidding remain dominant?

Overview: Two risk-neutral bidders compete for one item in a sealed-bid second-price auction. State the assumptions and derivation clearly, examine edge cases, and show how the result changes when those assumptions are relaxed.