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

# 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? ```hint Condition on the opponent bid Compare outcomes when the opponent bid is below, above, or between your value and an alternative bid. ``` ```hint Separate payment rules In a first-price auction your own bid sets the payment, creating a trade-off between win probability and surplus. ``` ### What a Strong Answer Covers - A pointwise argument showing that bidding v weakly dominates every overbid or underbid. - Correct separation between auction strategy and the financial concept of arbitrage. - A derivation of bid shading in the symmetric first-price case. - Assumptions under which the conclusions hold. ### 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?