Any Questions from Last Class?

Chapter 15Making Decisions with Uncertainty

COPYRIGHT © 2008Thomson South-Western, a part of The Thomson Corporation. Thomson, the Star logo, and South-Western are trademarks used herein under license.

Chapter 15 – Take Aways

When you’re uncertain about the costs or benefits of a decision, assign a simple probability distribution to the variable and compute expected costs and benefits.

When customers have unknown values, you face a familiar trade-off: Price high and sell only to high-value customers, or price low and sell to all customers.

If you can identify high-value and low-value customers, you can price discriminate and avoid the trade-off. To avoid being discriminated against, high-value customers will try to mimic the behavior and appearance of low-value customers.

In oral or English auctions, the highest-bidder wins but only has to outbid the second-highest bidder. Losing bidders determine the price.

A Vickrey or second-price auction is a sealed-bid auction in which the high bidder wins but pays only the second-highest bid. These auctions are well suited for use on the Internet.

Chapter 15 – Take Aways

In a sealed-bid first-price auction, the high bidder wins and pays her value. Bidders must balance the probability of winning against the profit they will make if they do win. Optimal bids are less than bidders’ private values.

Bidders can raise profit by agreeing not to bid against one another. Such collusion or bid rigging is more likely to occur in open auctions and in small, frequent auctions. If collusion is suspected, do not hold open auctions; do not hold small and frequent auctions; do not disclose information to bidders—do not announce who the winners are,

who else may be bidding, or what the winning bids were.

In a common-value auction, bidders bid below their estimates to avoid the winner’s curse. Oral auctions return higher prices in common-value auctions because they release more information.

Review of Chapter 14

Bargaining can be modeled as either a simultaneous game with two equilibria or as a sequential game where the ability to commit to a position gives one player bargaining power over its rival

The best threat is one you never have to use Nonstrategic view of bargaining: the alternatives to an

agreement determine the terms of any agreement To improve your own bargaining position

Increase your opponent’s gain from reaching agreement, or Reduce your own gain from reaching agreement

Telecom Anecdote

Large telecom supplier uncertain about whether to deal directly with large customers May lose large customers if doesn’t May lose distributors (and their small customers) if does

Lower probability of losing dealers (because they would have to incur costs to change suppliers)

But much larger impact on profit How should we analyze decisions like this?? Benefits of modeling uncertainty

Better decisions Identify source of risk Value of gathering information

Modeling Uncertainty

Definition: a random variable is simply a way of representing numerical outcomes that occur with different probabilities

Definition: a binomial random variable, X; E[X]=p*x1+(1-p)x2

Definition: a trinomial random variable, X; E[X]= p1*x1+ p2*x2+(1- p1-p2) x3

Modeling Uncertainty

“Wheel of Cash” example Wheel divided like a pie into thirds, with values of

$100, $75, and $5 painted on each of the slices Cost to play is $50.00 Should you play the game?

Three possible outcomes: $100, $75, and $5 with equal probability of occurring (assuming wheel is “fair”)

Expected value of playing the game is 1/3($100)+1/3($75)+1/3($5)=$60

If wheel biased toward $5 outcome, expected value is 1/6($100)+1/6($75)+2/3($5)=$32.50

Telecom Decision

Probability of losing customers is 0.6 Probability of losing distributors is 0.2

Telecom Firm

Sell directly to large customers

(.20) × $30 + (.80) × $130 = $110

Sell only through dealers

(.60) × $100 + (.40) × $130 = $112

Distributors leave

(probability = .20)Firm profit = $30

Distributors stay

(probability = .80)Firm profit = $130

Large customers leave

(probability = .60)Firm profit = $100

Large customers stay

(probability = .40)Firm profit = $130

Entry Decision with Uncertainty The probability of retaliation (no accommodation) is 0.5

Discussion: Invitation to invest in real estate venture that depends on uncertain future demand and interest rates

Entrant

Enter

(.50) × $60 + (.50) × $-40 = $10

Stay Out

(.50) × $0 + (.50) × $0 = $0

Incumbent prices high

(probability = .50)Entrant profit = $60

Incumbent prices low

(probability = .50)Entrant profit = $-40

Incumbent prices high

(probability = .50)Entrant profit = $0

Incumbent prices low

(probability = .50)Entrant profit = $0

Uncertainty in Pricing

Two types of customers High-value willing to pay $8 Low-value willing to pay $5

Probability of each is 0.5 Discussion: If MC= $3, what is optimal price?

Uncertainty in Pricing Answer: Price High

Pricing Decision

Price High

(.50) × $5 + (.50) × $0 = $2.50

Price Low

(.50) × $2 + (.50) × $2 = $2

Get high-value customer

(probability = .50)Profit = $5

Get low-value customer

(probability = .50)Profit = $0

Get high-value customer

(probability = .50)Profit = $2

Get low-value customer

(probability = .50)Profit = $2

Price Discrimination Opportunity If you can identify the two types of customers, set

different prices to the each group, and prevent arbitrage between them, then you can price discriminate. Price of $8 to the high-value customers

Price of $5 to the low-value customers.

Discussion: When buying a new car, sales people discriminate against customers. How do they do this?

Discussion: What can you do to defeat this?

Oral Auctions

Proposition: In an oral auction, the item is awarded to the high-value bidder at a price determined by the highest value among the losing bidders

Example: Suppose there are five bidders with values equal to {$5, $4, $3, $2, $1}.

Oral Auctions Proposition: More bidders raise price.

Bidder 1 Bidder 2 Probability Winning bid $5 $5 .25 $5 $5 $8 .25 $5 $8 $5 .25 $5 $8 $8 .25 $8

Bidder 1 Bidder 2 Bidder 3 Probability Winning bid $5 $5 $5 .125 $5 $5 $5 $8 .125 $5 $5 $8 $5 .125 $5 $8 $5 $5 .125 $5 $5 $8 $8 .125 $8 $8 $5 $8 .125 $8 $8 $8 $5 .125 $8 $8 $8 $8 .125 $8

Second Price Auctions

Definition: A Vickrey or second-price auction is a sealed-bid auction where the winning bidder is charged only the second-highest bid.

Proposition: A second-price auction is equivalent to an oral auction

Discussion: Why are eBay auctions equivalent to second-price auctions

Discussion: Why does eBay use second-price auctions?

Sealed-Bid Auctions

Definition: In a sealed-bid first price auction, the item is awarded to the highest bidder at a price equal to the highest bid

Bidding trade-off: higher bid lowers profits but raises probability of winning

Bidders balance these two effects by bidding below value (“shading”)

Bid Rigging

Discussion: oral auction with bidder have values of {$5, $4, $3, $2, $1}.

Bid-rotation schemes

Proposition: Collusion is more likely in oral auctions

Proposition: Collusion is more likely in small, frequent auctions

Frozen Fish Conspiracy

Discussion: Why is the Government such a frequent victim of bid-rigging conspiracies?

Discussion: What can we learn from this?

Common-Value Auctions Definition: In a common-value auction, the true

value is the same for each bidder Winner’s curse problem in auctions

If you win, you learn that you were the one who had the highest and most optimistic estimate of the unknown value of the item Bidders should reduce their value estimates to protect against

this If you are auctioneer, release info to mitigate winners’

curse Winner’s curse worse when

More bidders Other bidders have better information

Common-Value Auctions (cont.) Oral auctions return higher prices in common-value

settings (bidders reveal information with their bids) But more vulnerable to collusion

Discussion: Why do bidders wait until the last minute of the auction to submit bids on eBay?

Alternate Intro Anecdote Since 1994, the Federal Communications Commission (FCC)

has conducted auctions of licenses for electromagnetic spectrum Auctions are conducted electronically and are accessible over

the Internet Anyone with access to a computer with a web browser can follow

the progress of an auction and view the results of each round Commission has found that spectrum auctions more effectively

assign licenses than either comparative hearings or lotteries. The auction approach is intended to award the licenses to those who will use them most effectively.

Commission has reduced the average time from initial application to license grant to less than one year, and the public is now receiving the direct financial benefit from the award of licenses