Any Questions from Last
Class?
Chapter 15
Making Decisions with
Uncertainty
COPYRIGHT © 2008
Thomson South-Western, a part of The Thomson
Corporation. Thomson, the Star logo, and SouthWestern are trademarks used herein under license.
Chapter 15 – Take Aways
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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, highvalue 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,
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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
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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

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May lose large customers if doesn’t
May lose distributors (and their small customers) if does
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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

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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


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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?
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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

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Probability of losing customers is 0.6
Probability of losing distributors is 0.2
Telecom Firm
Sell directly to large customers
Sell only through dealers
(.20) × $30 + (.80) × $130 = $110
(.60) × $100 + (.40) × $130 = $112
Distributors leave
Distributors stay
Large customers leave
Large customers stay
(probability = .20)
Firm profit = $30
(probability = .80)
Firm profit = $130
(probability = .60)
Firm profit = $100
(probability = .40)
Firm profit = $130
Entry Decision with Uncertainty

The probability of retaliation (no
accommodation) is 0.5
Entrant

Enter
Stay Out
(.50) × $60 + (.50) × $-40 = $10
(.50) × $0 + (.50) × $0 = $0
Incumbent prices high
Incumbent prices low
Incumbent prices high
Incumbent prices low
(probability = .50)
Entrant profit = $60
(probability = .50)
Entrant profit = $-40
(probability = .50)
Entrant profit = $0
(probability = .50)
Entrant profit = $0
Discussion: Invitation to invest in real estate venture
that depends on uncertain future demand and interest
rates
Uncertainty in Pricing
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Two types of customers
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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
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Answer: Price High
Pricing Decision
Price High
Price Low
(.50) × $5 + (.50) × $0 = $2.50
(.50) × $2 + (.50) × $2 = $2
Get high-value customer
Get low-value customer
Get high-value customer
Get low-value customer
(probability = .50)
Profit = $5
(probability = .50)
Profit = $0
(probability = .50)
Profit = $2
(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.
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Discussion: When buying a new car, sales people
discriminate against customers. How do they do
this?
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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
$5
$5
$8
$8
Bidder 1
$5
$5
$5
$8
$5
$8
$8
$8
Bidder 2
$5
$8
$5
$8
Bidder 2
$5
$5
$8
$5
$8
$5
$8
$8
Probability
.25
.25
.25
.25
Bidder 3
$5
$8
$5
$5
$8
$8
$5
$8
Winning bid
$5
$5
$5
$8
Probability
.125
.125
.125
.125
.125
.125
.125
.125
Winning bid
$5
$5
$5
$5
$8
$8
$8
$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
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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
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Proposition: Collusion is more likely in oral
auctions
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Proposition: Collusion is more likely in small,
frequent auctions
Frozen Fish Conspiracy
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Discussion: Why is the Government such a frequent
victim of bid-rigging conspiracies?
Discussion: What can we learn from this?
Common-Value Auctions
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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

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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

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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
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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