A Nash equilibrium is a pair of strategies, one for each player, in which each strategy is a best response against the other. When players act rationally, optimally, and in their own self-interest, it’s possible to compute the likely outcomes of games. By studying games, we learn where the pitfalls are and how to avoid them. Sequential games include a potential first-mover advantage, or disadvantage, and players can change the outcome by committing to a future course of action.. Credible commitments are difficult to make because they require that players threaten to act in an unprofitable way—against their self-interest. In simultaneous-move games, players move at the same time. In the prisoners’ dilemma, conflict and cooperation are in tension—self-interest leads to outcomes that no one likes. Studying the games can help you figure a way to avoid these bad outcomes. In repeated games, it is much easier to get out of bad situations. Here are some general rules of thumb: Be nice: No first strikes. Be easily provoked: Respond immediately to rivals. Be forgiving: Don’t try to punish competitors too much. Don’t be envious: Focus on your own slice of the profit pie, not on your competitor’s. Be clear: Make sure your competitors can easily interpret your actions. A significant portion of our studies so far have revolved around situations in which the firm making the decision does not need to consider the effects of the decisions of other competitors in the market. In some markets, the firm we were examining was the only firm. In other markets, the firm was one of many firms and any individual firm’s decision would not affect the market significantly. In this chapter, we begin our study of situations in which there are a relatively small (but more than one!) number of actors in the market, and one actor’s decision will affect the outcomes of the other actors. A new tool is developed, the Nash Equilibrium, which will be used to study such situations. Definition of a Game A game (in the Game Theory sense) is any situation that has participants, a set of rules, and payoffs that depend on the actions chosen by all participants. The rules include such things as what actions are available to the participants, what information is available to each participant when they make their decisions, and the order in which participants make their decisions. The definition of a game is quite broad, and intentionally so. Chess, poker, checkers, baseball, football, Jeopardy!, Go, auctions, negotiations, competition in business, behavior in cartels, how to monitor employees, what to do when serving or receiving in tennis, which way to shoot when taking a penalty shot in soccer, and what to do when approaching a stoplight either are games or can be modeled using Game Theory. Nash Equilibrium The idea behind Nash Equilibrium is that if and when all the participant’s actions are revealed, no participant wishes to change their action. In other words, every participant’s .

1. A Nash equilibrium is a pair of strategies, one for each player,
in which each strategy is a best response against the other.
When players act rationally, optimally, and in their own self-
interest, it’s possible to compute the likely outcomes of games.
By studying games, we learn where the pitfalls are and how to
avoid them.
Sequential games include a potential first-mover advantage, or
disadvantage, and players can change the outcome by
committing to a future course of action.. Credible commitments
are difficult to make because they require that players threaten
to act in an unprofitable way—against their self-interest.
In simultaneous-move games, players move at the same time.
In the prisoners’ dilemma, conflict and cooperation are in
tension—self-interest leads to outcomes that no one likes.
Studying the games can help you figure a way to avoid these
bad outcomes.
In repeated games, it is much easier to get out of bad situations.
Here are some general rules of thumb:
Be nice: No first strikes.
Be easily provoked: Respond immediately to rivals.
Be forgiving: Don’t try to punish competitors too much.
Don’t be envious: Focus on your own slice of the profit pie, not
on your competitor’s.
Be clear: Make sure your competitors can easily interpret your
actions.
A significant portion of our studies so far have revolved around
situations in which the firm making the decision does not need
to consider the effects of the decisions of other competitors in
the market. In some markets, the firm we were examining was
the only firm. In other markets, the firm was one of many firms
and any individual firm’s decision would not affect the market
significantly. In this chapter, we begin our study of situations
in which there are a relatively small (but more than one!)
number of actors in the market, and one actor’s decision will

2. affect the outcomes of the other actors. A new tool is
developed, the Nash Equilibrium, which will be used to study
such situations.
Definition of a Game
A game (in the Game Theory sense) is any situation that has
participants, a set of rules, and payoffs that depend on the
actions chosen by all participants. The rules include such things
as what actions are available to the participants, what
information is available to each participant when they make
their decisions, and the order in which participants make their
decisions. The definition of a game is quite broad, and
intentionally so. Chess, poker, checkers, baseball, football,
Jeopardy!, Go, auctions, negotiations, competition in business,
behavior in cartels, how to monitor employees, what to do when
serving or receiving in tennis, which way to shoot when taking
a penalty shot in soccer, and what to do when approaching a
stoplight either are games or can be modeled using Game
Theory.
Nash Equilibrium
The idea behind Nash Equilibrium is that if and when all the
participant’s actions are revealed, no participant wishes to
change their action. In other words, every participant’s choices
are a best response to the strategies of all other participants;
given the other participants’ strategies no participant can
change their action and receive a better outcome. Note the use
of the word strategy; sometimes mixed strategies are used in
which the participant chooses randomly between several
actions. In the tennis example, if the server always serves down
the middle, then the receiver’s best response is to defend that
area. That is not an equilibrium, since the server’s choice must
be a best response to what the receiver does. In this case, if the
receiver defends the middle then the server can do better by
serving to the outside. We will see later in the lecture that there
is a Nash equilibrium in which the server chooses to serve down

3. the middle with some probability and serves to the outside the
remaining portion of the time. In that equilibrium the receiver
will defend the middle with some (usually different) probability
and defend the outside the remainder of the time. Since neither
the server or receiver can predict the action taken by the other,
the Nash equilibrium in this case is based on the probabilities
rather than the particular choice of either participant.
Example
Consider a hypothetical example of political negotiation. In this
example, there are Democrats and Republicans negotiating over
how to balance the federal budget. Democrats have their
preferred way of balancing the budget (say, raising taxes) and
Republicans have a different preference (reducing spending).
Both sides decide whether to take a strong stance at the
negotiating table or to seek compromise. If both sides take a
strong stance, no agreement is reached and the government
defaults on its debt. This is the worst result for everybody.
Assign a payoff of −100 to both sides for this outcome. If one
side takes a strong stance and the other seeks compromise, the
side taking the strong stance dictates how the budget is
balanced. That is the best outcome for the side taking the
strong stance (assign a payoff of 1) and the worst outcome
except for a government shutdown for the side seeking
compromise (assign a payoff of −1). If both sides seek
compromise, then the budget is balanced through both tax
increases and spending cuts. This is worse for each participant
than getting their way, but it is better outcome than letting the
other side get their way. We will assign a payoff of 0 – or
something between 1 and −1 – to this outcome.
To determine the Nash equilibria, we can use the underlining
method discussed in the book. We draw a grid with the actions
of each player labeling the columns or rows (one player is the
row player, one player is the column player) and fill in each box
in the grid with the payoffs to each participant based on the

4. outcomes determined by the strategies of the row and column
associated with the box. The game can be represented as
follows, where S stands for “Strong negotiating stance” and C
stands for “Compromise”:
Republicans
S
C
Democrats
S
-100, -100
1, -1
C
-1, 1
0, 0
The first payoff is for the Democrats, the second in the pair is
for the Republicans. To find the Nash equilibria, examine the
Democrats’ best action given what the Republicans do and
underline those payoffs. Following this procedure would
underline the −1 in the lower left cell, as Democrats would
rather compromise (-1) than negotiate strongly (-100) if the
Republicans negotiate strongly (this is the first column).

5. Similarly, if the Republicans choose compromise, then the
Democrats would prefer to negotiate strongly (1) instead of
compromising (0) so the 1 in the upper right box would be
underlined. These underlines represent the Democrats’ best
response to what the Republicans choose. The Republicans’
best responses can be found in a similar fashion. The 1 in the
lower left box and the −1 in the upper right box would receive
underlines. If both payoffs are underlined, that means the
action of each party is a best response to the actions of the other
party and therefore is a Nash Equilibrium. In this case, the two
equilibria are where one party negotiates strongly and the other
side gives in. Obviously the Democrats prefer one Nash
equilibrium over the other, and the Republicans have the
opposite preference. How is the particular equilibrium chosen?
Game theory does not give an answer to that question. It only
predicts that one of the equilibria is chosen. However, we can
use game theory to determine ways of influencing which
equilibrium is chosen.
Changing the Rules
In the budget negotiation example above, both sides would like
the other to believe that they will negotiate strongly and will
not deviate from that strategy. If one side believes the other will
negotiate strongly, then their best response will be to
compromise. However, absent any other considerations, the
threat to negotiate strongly and maintain the strong stance (if
the game is repeated over several days or weeks in the case
where both sides chose “strong”) is not credible because it is
not in the best interest of either party to maintain a strong
negotiating position if the other side remains strong. One side
simply saying they will take a strong negotiating stance is
meaningless.
However, if one could somehow commit to negotiating strongly
then the threat may become credible. For instance, Republicans
could sign Grover Norquist’s pledge to not raise taxes. One

6. could view this pledge as removing the compromise option from
the Republicans’ possible choices if the politicians are
committed to keeping their word. Alternatively, this could
adjust the payoffs that the Republicans receive if they
compromise and the Democrats negotiate strongly. If those
strategies were chosen, with or without the pledge the
Republicans would not get their way. However, the pledge
results in additional negative repercussions for Republicans,
including the possibility of much stronger competition in their
next primary which significantly reduces the payoffs associated
with compromising. If the payoffs are reduced enough, it may
be that the Republicans’ best response to Democrats negotiating
strongly is to adopt a strong negotiation stance as well. In that
case, a Republican threat to negotiate strongly is credible; the
strong stance is in the Republicans’ best interests.
The game of “Chicken” follows the payoffs in the matrix above.
If the strategy “S” stands for “go Straight” and “C” stands for
“Chicken out”, then the payoffs from Chicken follow the same
payoff structure. One way to demonstrate that you have
restricted your actions is to detach your steering wheel and
throw it out your window to show your opponent that your
options are now limited. The clip of “Dr. Strangelove”
regarding the Doomsday device
(https://www.youtube.com/watch?v=2yfXgu37iyI) represents
the Russians attempting such a strategy, but as Dr. Strangelove
points out, not letting the Americans know about the Doomsday
decide defeats the purpose.
An example of payoffs changing is demonstrated in the
negotiations between the NFL referee association and the
league. The league and referees were engaged in a labor dispute
and the NFL was using replacement referees to start the season.
Entering the Monday night game of week three, there had been a
few questionable calls by the replacement referees but there
were no major missed calls. The league and the referees

7. remained in a stalemate. Then on the last play of the Green Bay-
Seattle game on Monday night – a nationally televised game –
Seattle was behind and threw a desperation pass into the end
zone as time was running out. The ball came down in someone’s
hands. Replays would clearly show that the Green Bay defender
caught the ball, but after looking at each other for a couple
seconds, one referee signaled a touchdown while another referee
signaled interception. After some discussion the referees
awarded Seattle a touchdown and the win. The play and call
were ubiquitous – it was even discussed on the morning news
programs. The payoff for the NFL choosing a “strong
negotiating stance” was greatly reduced because of the disgust
that fans of the NFL felt towards the skill level of the
replacement referees. Two days later the league and referees
reached an agreement, most likely with the referees receiving
most of their requests. Click on the following link for some
commentary after the call but before the agreement:
Video: Green Bay Packers, Seattle Seahawks Blown Call
While the referee’s union did not change the payoff themselves,
perhaps they had the foresight to predict an event similar to the
one that occurred.
Randomization
Sometimes no pure strategy equilibria exist. Pure strategy
equilibria are equilibria in which one person chooses some
action every time and the other person takes some action all the
time. In the negotiation example, there were two pure strategy
Nash equilibria – one side cooperates and the other side
negotiates strongly. There are be cases where a mixed strategy
in which each participant randomly choses from several actions
is one equilibrium of a game, and sometimes the only
equilibrium of a game. To solve such equilibria, we will
consider the initial concept of an equilibrium that was discussed
earlier: both sides are content with their strategy given what the
other person is doing.

8. If we assume both participants are using mixed strategies, then
for one person to be happy choosing one strategy some of the
time and the other strategy the rest of the time, the payoffs from
the two strategies must be equal (in expectation – this is
addressed a little more in the uncertainty chapter this week. You
may want to skip this section for now and come back to it after
you read the chapter on uncertainty.). If the payoffs are
unequal, then the participant would not be willing to play both
strategies. The expected payoff of one player’s strategy will
depend on how often the other player chooses each available
option. So we can use the indifference condition for one player
to determine the other player’s strategy in order to be an
equilibrium. As an example, consider a soccer penalty shot. The
shooter can choose to kick left or right. The goalie can also
choose to dive left or right. Suppose the payoffs are as given in
the payoff matrix below (the payoff to the shooter is the
probability of a goal, the payoff to the goalie is the probability
that the goalie saves the shot).
Goalie
L
R
Shooter
p
(1-p)

9. q L
10, 90
95, 5
(1-q) R
80, 20
15, 85
The goalie will dive left p percent of the time, and right (1-p)
percent of the time. The shooter will shoot left q percent of the
time and right (1-q) percent of the time. For the shooter to be
willing to randomly choose between shooting left or right, the
payoffs of each action must be equal. The payoff for shooting
left is the payoff associated with the goalie diving left times the
probability the goalie dove left plus the payoff associated with
the goalie diving right times the probability the goalie goes
right. That is, 10p + 95 (1 - p) = 95 − 85 p. The payoff for
shooting right is 80p + 15 (1 - p) = 65 p + 15. Setting these
equal to each other and solving for p results in p = 80/150. This
means if the goalie dives left with a probability of 53.33% the
shooter is indifferent between shooting left or right. Similarly,
the payoff for the goalie diving left is 90q + 20 (1 - q) and the
payoff for the goalie diving right is 5q + 85 (1 - q). Setting
those payoffs equal and solving for q results in q = 65/150.
Thus if the shooter kicks left 43.33% of the time, the goalie is
just as well off diving left as diving right. Thus if the goalie
dives left 53.33% of the time and the shooter kicks left 43.33%
of the time neither the kicker or goalie would want to change
their actions.
Lecture 2
Main Points

10. · Bargaining can be modeled as either a simultaneous-move or
sequential-move game.
· A player can gain an advantage by 1) turning a simultaneous-
move game into a sequential move game with a first-mover
advantage; or by 2) committing to a position.
· Credible commitments are difficult to make because they
require players to commit to a course of action against their
self-interest. Thus, the best threat is one you never have to use.
· The strategic view of bargaining focuses on how the outcome
of bargaining games depends on who moves first and who can
commit to a bargaining position, as well as whether the other
player can make a counteroffer.
· The non-strategic view of bargaining does not focus on the
explicit rules of the game to understand the likely outcome of
the bargaining. This view focuses on the gains from bargaining
relative to alternatives.
· The gains from agreement relative to the alternatives to
agreement determine the terms of any agreement.
· Anything you can do to increase your opponent’s gains from
reaching agreement or to decrease your own will improve your
bargaining position.
Game Theoretical View of Negotiations
In the previous chapter, we discussed some examples of
negotiations to illustrate how and where games can be used to
analyze economic situations. We can add a little more realism to
the game by considering a sequential move game – a game in
which one side moves first and the other side observes and may
base its actions on what the first side chose. To these games we
apply a concept called Subgame Perfect Equilibrium (SPE), a
modification of Nash equilibrium. This modification of a Nash
equilibrium accounts for the order of decisions or actions. To
analyze a game using SPE, find the last decisions that are made
and identify the choice that maximizes the decision maker’s
payoff. Replace the decision with the payoffs – since we know
what the decision-maker will choose, we may make the decision
for him. With the last decisions replaced with the payoffs, find

11. the decisions that are now the last ones to be made and repeat
the process until every decision has been replaced by the
payoffs.
In the toy game illustrated in the text, the Union gets to present
Management with an offer, and Management may accept or
reject the offer. Management has the last decision, so to apply
SPE to this game we analyze Management’s two decisions – one
when the Union decided to extend a low offer and the other one
when the Union decided to extend a generous offer. In either
case, accepting the offer results in a positive payoff for
management and rejecting the offer results in a payoff of zero,
so for both decisions Management will accept the offer. Now
the Union’s decision can be analyzed; if they make a low offer,
they will receive a payoff of 75 while a generous offer will
result in a payoff of 25. Seventy-five is better than 25, so they
should make the low offer.
In this game, the Union acted first and the result was the
equilibrium most beneficial for the Union. In a sense, the Union
was able to commit to a hard bargaining stance. This is an
example of the first-mover advantage. Note that the
Management was not able to commit to bargain hard and reject
a low offer. If it could somehow do so before the Union made
their offer, then Management would then have the upper hand.
Perhaps that may be accomplished through a rainy-day fund so
that the firm’s unavoidable expenses can be covered. It may be
through statements accepting the inevitability of a strike or lock
out and vowing to remain firm in their stance so that eventually
the Union will capitulate. It may be through statements to the
public to engender support. Of course, the Union may take
similar actions. As I am writing this, I am reminded of the
dispute between Time Warner Cable and CBS. The agreement
between Time Warner Cable and CBS has expired and they have
been negotiating a new agreement for a while. Currently, no
agreement has been reached and Time Warner customers in New
York, Los Angeles and Dallas can not watch CBS on their cable
system (Reuters). Both sides have been issuing press releases

12. trying to position themselves as the “good guys” who are
negotiating in good faith and trying to be fair. The particular
article I cite below discusses how the standoff may end – when
the football season begins. Note the payoffs from bargaining
change at that time as well – Time Warner customers may
endure the inconvenience during the last part of summer re-
runs, but once the football and the fall TV season begin Time
Warner may face many customers leaving if no deal has been
reached.
Click the following link:
· Article: Time Warner-CBS Blackout May Last Until Sept:
Analysts
Practical Uses
The textbook presents a stylized look at bargaining using
something called the BATNA, or Best Alternative to a
Negotiated Agreement. The textbook postulates that the result
from bargaining equally splits the gains from bargaining. In
other words, if side A’s BATNA is worth $50 and side B’s
BATNA is worth $30, and an agreement will be worth a
combined total of $100, then the gains from a successful
negotiation are $100 - ($50 + $30) = $20, so the agreement
should increase both side’s payoffs by $10. Other economic
models modify this result slightly – the split may depend on
how patient the sides are relative to each other or other similar
factors. During a strike or lockout, if one side is more patient
than the other, the final agreement will be more favorable to
that side.
Thus, there are several options that immediately present
themselves in order to obtain a more favorable outcome. Being
more patient, or at least appearing more patient and out-waiting
the other side will increase the benefits of a successful
negotiation. Increasing one’s BATNA will also increase the
payoff of a successful negotiation. Suppose side B increases
their BATNA to $40. In that case, only $10 of value is added by
a successful negotiation and thus according to the book’s
model, $5 of value would be earned by either side. This results

13. in side B obtaining a payoff of $45, which is $5 more than when
their BATNA was $30 (they would receive $40—their BATNA
of $30 plus $10).
In a real-world example of this, Gneezy, List, and Price discuss
an experiment in which handicapped and non-handicapped
people go to an automobile repair shop and ask for estimates to
repair a car. Their results show that “disabled agents receive
roughly a $213 (or 30 percent) higher quote price than the non-
disabled” which was statistically significant (not likely to have
occurred from random factors). This could be because the
customers in wheelchairs face a higher cost to obtain other
estimates and would therefore be expected to have fewer
competing estimated. In fact, a survey reported in Gneezy et. al.
show that disabled consumers visit on average 1.67 mechanics
while non-disabled consumers visit on average more than twice
that number. Mechanics are cognizant of these patterns, and so
can infer that the BATNA for a disabled person is worse than
that of a non-disabled person and make their offers based on
this inference. In a follow-up experiment, Gneezy et. al.
repeated the experiment except had “both the disabled and abled
agents explicitly stated to the mechanic that they are ‘getting a
few price quotes.’” This should indicate to the mechanic that
the disabled customer’s BATNA has now increased to be
commensurate with the non-disabled consumer’s BATNA. If
this is the case, the estimates should be the same between the
two groups and that is exactly what was found.
Gneezy, List, and Price. “Toward an Understanding of Why
People Discriminate: Evidence from a Series of Natural Field
Experiments.” NBER working paper 17855.
Main Points
· 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

14. 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.
· Difference-in-difference estimators are a good way to gather
information about the benefits and costs of a decision. The first
difference is before versus after the decision or event. The
second difference is the difference between a control and an
experimental group.
· If you are facing a decision in which one of your alternatives
would work well in one state of the world, and you are
uncertain about which state of the world you are in, think about
how to minimize expected error costs.
Another wrinkle that has not been discussed much in this course
is uncertainty and how it can affect the decision-making
process. Typically, many quantities in a forward looking
projection will not be exact – there will be a range over which
the parameter may vary. For an example, let us slightly modify
the hotel booking example from last week. If the price for a
room is set at $30, then assume the hotel will receive 98 people
arriving at the hotel with reservations 15% of the time, 99
people 25% of the time, 100 people 30% of the time, 101 people
20% of the time and 102 people the remaining 10% of the time.
In this case, there is uncertainty regarding how many visitors
will arrive at the hotel.
Expected Value
The expected value of guests is the average number of guests
that would arrive at the hotel if we repeat the situation many
times. If we observed the hotel for 100 days, then 15% of the
time we would see 98 people arrive. Since 15% of 100 days is
15 days, on 15 of the 100 days we would observe 98 guests
arriving. On 25 days we would see 99 people, and so on. We
could then add the total number of guests that arrive during
those 100 days and then divide by the number of days to
calculate the average number of guests per day which I will call

15. X. This calculation would look like:
X = (98 * 15 + 99 * 25 + 100 * 30 + 101 * 20 + 102
* 10) / 100
We could calculate this to determine the numerical value.
However, we can also distribute the 100 in the denominator and
obtain
X = (98 * 15/100) + (99 * 25/100) + (100 * 30/100) + (101 *
20/100) + (102 * 10/100)
= (98 * 0.15) + (99 * 0.25) + (100 * 0.30) + (101 *
0.20) + (102 * 0.10)
Or
X = (98 * 15%) + (99 * 25%) + (100 * 30%) + (101 * 20%) +
(102 * 10%).
This is a simpler formula to apply generally; instead of needing
to find a particular number of days such that each case occurs a
whole number of days we can apply a formula similar to above.
For each potential outcome, multiply the outcome by the
probability of the outcome. Then add together those
intermediate calculations to determine the average, or expected,
value of the outcome.
Using Uncertainty in Calculations
In our hotel example the quantity of interest to the hotel is not
the number of guests who arrive, but rather the profit. Profit can
be calculated by examining the revenues and costs. Let us
assume that the additional costs incurred from a single
overbooking is $50. This number includes costs associated with
finding the guest other accommodations, transportation costs to
sent the guest to the other accommodations, an estimate of the
loss of revenue from bad word-of-mouth advertising, and
whatever other costs may be created by an overbooking. With
this information we can determine expected revenues and
expected costs. The expected revenue will be the sum of the
room rate times the number of rooms rented times the
probability of that many rooms being rented, or:
ER = $30 * 98 * 15% + $30 * 99 * 25% + $30 * 100 * 60% =
$2983.50

16. Note that I combined the cases where 100, 101 and 102 guests
arrive, since in all three of those cases exactly 100 rooms will
be rented and the revenues will be identical. We could
calculate the marginal costs associated with cleaning, guest
services, and other such costs in a similar way. If the variable
cost per occupied room is $5, then the expected variable costs
will be $497.25. The expected losses from overbooking can be
estimated as
EC = $0 * 70% + $50 * 20% + $100 * 10% = $20.
Here there is a 70% chance the hotel is not overbooked, or in
other words only 98, 99 or 100 guests arrive and thus there is
room for all. A 20% chance exists that the hotel will be
overbooked by one room and a 10% chance two people will not
find lodgings available.
The profit maximizing price is calculated by determining the
marginal revenue and marginal cost associated with a small
change in the price and finding the price at which the marginal
cost is equal to the marginal revenue. When uncertainty is
present, this rule is changed to finding the expected values of
the marginal cost and marginal price. If the price is adjusted
slightly, the probability that a certain number of guests arrive is
likely to change. Therefore, if the price of the room is changed,
the calculations outlined above can be recomputed with the new
outcomes and probabilities and the change in expected revenue
and expected costs can be determined.
For example, suppose lowering the price to $29.90 results in 99
guests arriving 20% of the time, 100 guests 35% of the time,
101 guests 20% of the time, 102 guests 15% of the time, and
103 guests 10% of the time. In that case, the expected revenue
would be
ER = $29.90 * 99 * 20% + $29.90 * 100 * 80% = $2984.02.
So the expected marginal revenue is positive since the expected
revenue in this case is larger. The variable costs per occupied
room would be $499. Immediately we can see that the expected
marginal cost ($499 - $497.25 = $1.75) is larger than the
expected marginal revenue ($0.52). The change in expected

17. costs from overbooking simply adds more to the marginal cost
of decreasing the room rate. To be complete, the expected costs
from overbooking are now:
EC = $0 * 55% + $50 * 20% + $100 * 15% + $150 * 10% =
$40.
Thus the motel should not lower its price to increase revenue.
This analysis suggests that the room rate should be decreased in
order to increase profit but the probability of particular numbers
of guests arriving would need to be known before definitively
making that statement.
Managing Uncertainty
The text discusses several ways to manage uncertainty or risk.
I’ll comment or expand on a couple. The “Uncle Joe” example
on page 203 of your text is reminiscent of the
“Best/Worst/Average Case” analysis. As the name implies, in a
Best/Worst/Average case analysis, the best case, worst case, and
expected case outcomes are examined in order to determine the
potential range of profits. For instance, if a firm was
considering a project to build jet engines, several factors would
determine the profitability of the project. Among them would be
the costs associated with materials, the number of competitors
and the size of the market. The best case scenario would use the
most favorable projections for the costs, number of competitors,
demand for the product, and any other factor affecting the
project’s profitability that are likely. The worst case would use
the least favorable projections – high costs, many competitors,
few demanders – and the expected case would use the expected
or average projections. Note that this analysis is only used to
give a range of potential outcomes; once the best, worst and
average case are determined the analysis ends; they are not
combined into a single projection.
A second way to manage uncertainty is to perform regional
experiments to obtain better estimates of the unknown variables.
For instance, suppose Taco Bell is considering a new product,
the “Doritos Locos taco.” Based on existing products, Taco Bell
should have a very good idea of the marginal costs associated

18. with the new product. However, while Taco Bell likely has
some estimates of the demand for the product, there is
significant uncertainty surrounding the demand for the new
taco. Taco Bell could decide to begin a national distribution of
the taco, but a more safe route may be to introduce the taco to a
small number of regional markets like Toledo. It can then
observe the demand for the product in these representative
markets and make a more informed decision whether to roll out
the product nationally, potentially avoiding a costly mistake if
it has overestimated the product’s demand. Note that Taco Bell
should select several regional markets that represent a cross-
section of potential consumers; testing the product in Toledo,
Detroit, Cleveland, Cincinnati and Pittsburgh would not be as
good as testing in Toledo, Dallas, Sacramento, Atlanta and
Boston since the latter combination better represents the
national market. That is, if there is something about the product
that appeals to consumers in Ohio and the immediately
surrounding areas, the first combination would give an overly
optimistic prediction of the demand.