21. Mechanism Design

Focus on direct mechanisms via the revelation principle, design the optimal auction with virtual values, then verify numerically why truth-telling is dominant in a second-price auction.

• The Revelation Principle and Optimal Auctions — This scene finds the best rule for selling one item. We skip complicated games and only ask directly. The revelation principle says honest rules are enough. Ask the buyer what the item is truly worth. Reports below the reserve get nothing. Reports at or above it pay exactly the reserve. A well-chosen reserve maximizes expected revenue. The virtual value in the formula below marks the spot. For example, you sell one ⚽ soccer ball. Your friend values it at 8,000 won. You set the reserve at 5,000 won. If your friend reports at least 5,000, the ball is theirs. They pay exactly 5,000 💰 won. Lying gains nothing, so honesty wins.
• Second-Price Auction: Truth-Telling Is Dominant — This scene shows why honest bidding wins in a second-price auction. There is one opponent, and they bid honestly. Your bid does not set the price you pay. Winning means paying the opponent's bid. So bidding above your value gains nothing. It only creates wins that cost too much. Bidding lower throws away wins you could afford. As the formula below shows, the loss grows with the square of the gap. For example, your class auctions off one ⚽ ball. You value it at 6,000 won. A friend bids 4,000 won. Bid your honest 6,000 and you win. You pay only the friend's 4,000 💰 won. Bid 8,000 and the price stays the same. But if the friend had bid 7,000, you would overpay.