Question

Suppose that a person (the "seller") wishes to sell a single desk. Ten people are interested in buying the desk: Ann, Bill, Colin, Dave, Ellen, Frank, Gale, Hal, Irwin, and Jim. Each of the potential buyers would derive some utility from owning the desk, and this utility is measured in dollar terms by the buyer's "value." The valuations of the 10 potential buyers are shown in the following table. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text { Ann } & \text { Bill } & \text { Colin } & \text { Dave } & \text { Ellen } & \text { Frank } & \text { Gale } & \text { Hal } & \text { Irwin } & \text { Jim } \\ 45 & 53 & 92 & 61 & 26 & 78 & 82 & 70 & 65 & 56 \\ \hline \end{array} $$ Each bidder knows his or her own valuation of owning the desk. Using the appropriate concepts of rationality, answer these questions: (a) If the seller holds a second-price, sealed-bid auction, who will win the auction and how much will this person pay? (b) Suppose that the bidders' valuations are common knowledge among them. That is, it is common knowledge that each bidder knows the valuations of all of the other bidders. Suppose that the seller does not observe the bidders' valuations directly and knows only that they are all between 0 and 100 . If the seller holds a first-price, sealed-bid auction, who will win the desk and how much will he or she have to pay? (Think about the Nash equilibrium in the bidding game. The analysis of this game is a bit different from the [more complicated] analysis of the first-price auction in this chapter because here the bidders know one another's valuations.) (c) Now suppose that the seller knows that the buyers' valuations are 45,53 , \(92,61,26,78,82,70,65\), and 56 , but the seller does not know exactly which buyer has which valuation. The buyers know their own valuations but not one another's valuations. Suppose that the seller runs the following auction: She first announces a reserve price \(p\). Then simultaneously and independently the players select their bids; if a player bids below \(\underline{p}\), then this player is disqualified from the auction and therefore cannot win. The highest bidder wins the desk and has to pay the amount of his or her bid. This is called a "sealed-bid, first price auction with a reserve price." What is the optimal reserve price \(p\) for the seller to announce? Who will win the auction? And what will the winning bid be?

Short answer

(a) Colin wins, paying $82. (b) Colin wins, paying $83. (c) Optimal reserve price is $80; Colin wins with $83.

Step-by-step solution

Understanding Second-Price Sealed-Bid Auction

In a second-price sealed-bid auction, each bidder submits a bid without knowing the others' bids. The highest bidder wins, but the price paid is the second-highest bid. Hence, the bidder with the highest valuation should bid their true value.

Determine Winner and Payment for Second-Price Auction

From the given valuations, Colin values the desk the most at $92. In a second-price auction, Colin wins because he has the highest valuation and would probably bid $92. However, he pays the second highest bid, which is Gale's valuation of $82.

Understanding First-Price Sealed-Bid Auction with Common Knowledge

In a first-price auction with common knowledge, each bidder knows everyone's valuations. Bidders strategically choose lower bids than their valuations to maximize profit, knowing others' values. This involves predicting other bidders' strategies.

Determine Winner and Payment for First-Price Auction

Given Colin's highest valuation of $92 and strategic thinking, Colin would likely place a slightly higher bid than the second-highest valuation, Gale's $82, to outbid him with minimal cost. Colin might bid around $83 to secure the desk at minimal cost.

Understanding Sealed-Bid, First-Price Auction with Reserve Price

Here, bidders must bid above a reserve price to participate. The seller maximizes revenue by setting a reserve price low enough to ensure competition but high enough to ensure a worthwhile sale.

Determine Optimal Reserve Price and Winner with Highest Bid

Given the highest valuation is $92 and the competitive range is between alternative high valuations, a reserve price slightly below the second-highest valuation, say $80, ensures competition. Colin would still likely win with a bid around $83, knowing Gale might stop at $82 to avoid overshooting his optimal strategy.

Key concepts

Second-Price Auction

In a second-price auction, each bidder submits a bid without knowing the bids of others. The unique aspect of this auction is that the highest bidder wins, but the payment required is the second-highest bid. This setup means that bidders can safely bid their true valuation without fear of overpaying.

For example, if Colin values the desk at $92, he can bid exactly that. If his bid is the highest, he will win, but he will only pay the amount of the second-highest bid. In the case from the exercise, Gale has the second-highest valuation at $82, meaning Colin wins the auction but pays $82. This encourages bidders to bid honestly, aligning their bids with their true valuations since they know they won't overpay.

First-Price Auction

In a first-price auction, also known as a sealed-bid auction, each participant submits their bid without knowing the others' bids. Unlike the second-price auction, the winner must pay the amount they bid. This adds a layer of strategy, as each bidder must balance bidding their true value against the risk of overpaying.

When bids are common knowledge, players can anticipate each other's strategies. In these conditions, Colin, with the highest valuation, must decide how to ensure he bids just enough to win. Understanding that the highest bid against him would be Gale's $82 valuation, Colin could choose to bid slightly higher, perhaps $83, to win at the lowest possible price while still outbidding Gale.

Sealed-bid Auction

Sealed-bid auctions require bidding without any advance knowledge of competitors' bids. Under both first-price and second-price formats, each bid is kept secret until all are submitted.

These auctions feature distinctive bidding tactics due to the sealed nature. Participants emphasize strategy, particularly in predicting competitors' bids and strategically setting their own to win without exceeding personal valuation limits. This secrecy can encourage more honest bidding since guessing can be difficult.

Reserve Price

The reserve price acts as a minimum threshold below which an item won't be sold in an auction. It protects sellers by ensuring they don't sell at a price lower than deemed acceptable.

Setting an optimal reserve price is crucial in auctions, especially when bidders' valuations are known. A reserve price should be low enough to attract bidders but high enough to ensure a satisfactory sale for the seller. In the exercise, a good strategy is to set the reserve slightly below the known second-best value, maybe at $80, to stimulate bidding while ensuring a reasonable sale.

Nash Equilibrium

Nash equilibrium is a foundational concept in game theory where each player's strategy is optimal given the strategies of all other players. No participant can gain by changing only their own strategy unilaterally.

In the context of auctions, Nash equilibrium can be seen in the strategic decision-making process. For instance, when all players know each other's valuations, they aim to anticipate actions, such as Colin predicting Gale's bidding strategy. Each bidder, by finding the best response strategy, contributes to reaching a Nash equilibrium, ensuring that no one benefits from deviating from their chosen strategy given others' choices.