Question

Consider the two-bidder auction environment discussed in this chapter, where the bidders' values are independently drawn and distributed uniformly on the interval \([0,900]\). Demonstrate that, by setting a reserve price, the auctioneer can obtain an expected revenue that exceeds what he would expect from the standard first- and second-price auctions. You do not have to compute the optimal reserve price or the actual equilibrium strategies to answer this question.

Short answer

Setting a reserve price can increase expected revenue compared to standard auctions by excluding low bids.

Step-by-step solution

Understanding the Auction Format

To solve this problem, we must understand that in both first-price and second-price auctions, bidders submit bids in a sealed envelope. In a first-price auction, the highest bidder pays their bid, while in a second-price auction, the highest bidder pays the second-highest bid. A reserve price is a minimum acceptable bid set by the auctioneer.

Analyzing Bidders’ Valuation Distribution

The bidders' values are drawn from a uniform distribution on the interval \([0,900]\). This means that any value between 0 and 900 is equally likely to be the bidder's valuation of the auctioned item.

Concept of Reserve Price

A reserve price is set by the auctioneer as a minimum price that must be met for the auction item to be sold. If the highest bid is below the reserve price, the item remains unsold. This approach can potentially increase revenue because any winning bid must at least match the reserve price.

Impact of Reserve Price on First and Second-Price Auctions

In both first and second-price auctions without a reserve price, the expected revenue reflects the competition between bidders, but may not reach a bidder's full valuation due to strategic underbidding (especially in first-price auctions). By introducing a reserve price, bids near or below this amount are excluded, potentially pushing winning bids higher in expectation and ensuring the auctioneer receives at least the reserve price.

Expected Revenue with Reserve Price

When a reserve price is applied, the scenario shifts from purely competitive bidding to one where the reserve price acts as a "safety net" for the auctioneer's revenue. If set appropriately, the reserve price can increase the auctioneer's expected revenue compared to when no reserve price is set, because low bids that might have won without reserve are now insufficient, compelling the bidders to bid higher to win the item.

Key concepts

Reserve Price

In auctions, the reserve price is the minimum acceptable price set by the seller or auctioneer. It's crucial because it acts as a threshold that bids must meet or exceed for the auction to result in a sale. Think of it as a protective measure ensuring that the item doesn't sell for less than what the auctioneer values it.

Implementing a reserve price can significantly influence the outcome of an auction.

• If the highest bid falls below the reserve price, the auction concludes with no winner, and the item remains unsold.
• When a reserve price is in place, it guarantees that the auctioneer will not accept a low bid, potentially increasing overall revenue.

By setting a sensible reserve price, the auction owner can enhance the likelihood of obtaining bids closer to the actual value of the item, rather than the bidding starting from zero. It's a strategic tool that not only safeguards the seller's interest but also encourages more competitive bidding if buyers perceive value at or above the reserve price.

First-Price Auction

In a first-price auction, bidders submit their bids in secrecy, often in sealed envelopes. After all bids have been submitted, the highest bidder wins the auction, and they pay their actual bid amount. This type of auction requires bidders to make strategic decisions about their bidding strategy.

One strategy bidders might use is to bid lower than their actual valuation to avoid paying more than necessary – much like dealing in other competitive scenarios. Here, understanding the other bidders' likely valuations becomes key. The goal is to win the auction while spending the least amount possible.

• Bidders in a first-price auction must estimate others' bids and decide whether to bid their true valuation or something lower.
• The reserve price, when applied to a first-price auction, acts as a floor for bids, making sure the winning bid reaches the minimum desired by the auctioneer.

Setting a reserve price helps avoid the situation where the winning bid is significantly lower than one or more bidders’ valuations, potentially increasing the auctioneer's revenue.

Second-Price Auction

Second-price auctions have a slightly different mechanism. In this scenario, after bids are submitted, the highest bidder wins but only pays the amount of the second-highest bid. This type of auction is often more straightforward for participants, as they are incentivized to bid their true valuation.

Why? Because the amount they pay doesn't depend on their actual bid but on the second-highest bid. Therefore, they have no advantage in underbidding, unlike in first-price auctions.

• The second-price auction removes the incentive to "shade" or lower their bids to pay less.
• A reserve price still plays a vital role, ensuring the auctioneer doesn't accept any final bid less than they anticipated receiving.

With a reserve price set, the auctioneer increases the likelihood that the final sale price aligns more closely with the true market and bid valuations, enhancing revenue beyond what might be achieved without it.

Bidders' Valuation

Bidders' valuation refers to the perceived worth of the auctioned item by each bidder. This is typically based on personal preference, budget, and potential resale value. In the case of a uniformly distributed valuation, such as in our example from [0,900] , any value within this range is equally probable for any bidder.

Understanding bidders’ valuations is fundamental to auction strategy because it dictates how much each bidder is willing to offer. For example:

• A bidder's valuation determines their maximum willingness to pay.
• It acts as a critical element in both first and second-price auctions, affecting bidding behavior and strategies.

Sellers, like auctioneers, must consider bidders' valuations when setting a reserve price. They aim to find a balance where the reserve is not so high that it deters all bids but not so low that it doesn't protect the item's value. Analyzing and estimating these valuations can give the auctioneer an advantage in setting initial auction terms that drive better revenue outcomes.