Question

You are one of five risk-neutral bidders participating in an independent private values auction. Each bidder perceives that all other bidders’ valuations for the item are evenly distributed between $\mathbf{\text{(10,000}}$and$\mathbf{\text{)30,000}}$. For each of the following auction types, determine your optimal bidding strategy if you value the item at$\mathbf{\text{\$22,000}}$.

a. First-price, sealed-bid auction.

b. Dutch auction.

c. Second-price, sealed-bid auction.

d. English auction.

Short answer

a. The optimal bid for First-price, sealed-bid auction is $\text{\$19,600}$.

b. The optimal bid for Dutch auction is $\text{\$19,600}$.

c. The optimal bid for Second-price, sealed-bid auction is $\text{\$22,000}$.

d. The optimal bid for English auction is $\text{\$22,000}$.

Step-by-step solution

Concept Introduction

There are four types of auctions: First-price, sealed-bid auction, Dutchauction, second-price and English auction. These will differ with respect to the timing of bidder decisions (if they are taken simultaneously or sequentially) or in the amount of money that is necessary to pay.

Bid for First-price, sealed-bid auction

First-price, sealed-bid auction: In this type of auction, each bidder writeshis bid privately and the auctioneer will be in charge of reviewing them andgiving the winner who has bid the highest amount. The Optimal BiddingStrategy for a First-Price is the following –

$\text{b=v-}\frac{\text{v-L}}{\text{n}}$

Where,$\text{n=5}$ since it is the number of participants in the auction.$\text{L}$ which is the lowest possible valuation of other bidders will be$\text{\$10,000}$ and$\text{v}$ is our valuation which is $\text{\$22,000}$. Therefore, with this information substitute in the previous equation –

$\begin{array}{c}\mathrm{b}=\mathrm{\$}22,000-\frac{\mathrm{\$}22,000-\mathrm{\$}10,000}{5}\\ \mathrm{b}=\mathrm{\$}22,000-\mathrm{\$}2400\\ \mathrm{b}=\mathrm{\$}19,600\end{array}$

Therefore, the value is obtained as $\text{\$19,600}$.

Bid for Dutch Auction

Dutch auction: This type of auction consists of the auctioneer starting by asking for a very high price and graduallydecreasing it to the point where a buyer is willing to accept that price and then, the auction ends.

With respect to the information available by the bidders, it will be the same as in the first-place auction type, the biddersdo not know what the maximum price of each bidder is until the auction is over and the winner announces it with his bid.

Thus, the optimal strategy of the Dutch auction will be very similar to the First - Price Auctions and the optimal biddingstrategy will also be the result $\text{\$19,600}$.

Therefore, the value is obtained as $\text{\$19,600}$.

Bid for Second-price, sealed-bid auction

Second-price, sealed-bid auction: This type of auction is very similar to First-Price auctions in that the bidders do notknow the bids of the other participants only until the auction concludes. However, it differs in a way that the bidder is the winner and will not pay the amount that they offered but will pay the amount that the second-best bidder bid.

Therefore, the optimal bidding strategy for a second-price will be the value that a person has given to the auctioned product, in case of being a winner, the offer of the second-best bidder will be paid, which will be less than our own. In this case, the best strategy will be $\text{\$22,000}$.

Therefore, the value is obtained as $\text{\$22,000}$.

Bid for English auction

English auction: This auction is the most common. It starts with a starting price by the auctioneer and the price willincrease as the bidders increase their bids until no more bids are made and whoever made the highest bid is the winner.

This type of auction differs from the other three in that all bidders can gradually learn the information about the offers of the other bidders.

Therefore, the optimal strategy is to increase the bids until the auction value reaches our estimated value, in this case $\text{\$22,000}$. If the auction exceeds this amount, a person will not be able to continue and another bidder will be the winner, while if the maximum bid is equal to or less than $\text{\$22,000}$, the person bidding this much will be the winner.

Therefore, the value is obtained as $\text{\$22,000}$.