Question

You are a bidder in an independent private values auction, and you value the object at $\cancel{\mathbf{S}}\mathbf{4}\mathbf{,}\mathbf{000}$. Each bidder perceives those valuations are uniformly distributed between $\cancel{\mathbf{S}}\mathbf{1}\mathbf{,}\mathbf{500}$and $\cancel{\mathbf{S}}\mathbf{9}\mathbf{,}\mathbf{000}$. Determine your optimal bidding strategy in a first price, sealed-bid auction when the total number of bidders (including you) is:

a. $\mathbf{2}$.

b. $\mathbf{10}$.

c. $\mathbf{100}$.

Short answer

a.The optimal sealed bid for just bidders will be $\cancel{S}2,750$.

b. The optimal sealed bid for just bidders will be $\cancel{S}3,750$.

c. The optimal sealed bid for just bidders will be $\cancel{S}3,975$.

Step-by-step solution

Concept Introduction

An auction is a sale procedure where the interested parties compete with each other to be awarded the good or service to be auctioned.

In an auction the optimal strategy of the bidder is to obtain the product for the lowest possible value, which meansbidding less than the real value of the product. Therefore, the Optimal Bidding Strategy for a First-Price, Sealed-Bid

Auction equation can be represented as follows –

$b=v‐\frac{v‐L}{n}$

Where,

$b$: is the player's optimal bid.

$v$: Actual valuation of the item.

$L$: lowest possible valuation.

$n$: number of bidders.

Calculation for bidders

a.

The given values can be substituted the in the above formula.

Given the number of bidders is $2$, according to the uniformly distributed valuation the minimum value is $\cancel{S}1,500$. Concerned valuation establishes a price of $\cancel{S}4,000$. Thus, the optimal sealed bid will be –

$$\begin{gathered} b=4000-\left(\frac{4000-1500}{2}\right) \\ b=4000-1250 \\ b=2750 \end{gathered}$$

Therefore, the value is obtained as $\cancel{S}2,750$.

Calculation for  bidders

b.

Keeping the previous data constant but assuming that the number of bidders increases to $10$., the optimal sealed can be calculated as follows-

$$\begin{gathered} b=4000-\left(\frac{4000-1500}{10}\right) \\ b=4000-250 \\ b=3750 \end{gathered}$$

Therefore, the value is obtained as $\cancel{S}3,750$.

Calculation for  bidders

c.

Keeping the previous data constant but assuming that the number of bidders increases to 100, the optimal sealed can be calculated as follows-

$$\begin{gathered} b=4000-\left(\frac{4000-1500}{100}\right) \\ b=4000-25 \\ b=3975 \end{gathered}$$

Therefore, the value is obtained as $\cancel{S}3,975$.