Managerial Economics

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Deep Thought
Why cant the ant and the caterpillar just get
along? One eats grass, the other eats
Caterpillars Oh, I see now. --- by Jack Handey.
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Overview
Overview
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Lesson Overview
References Baye Chapters 9 and 10 and Dixit
Chapter 4 Lesson II.4. Simultaneous Price
Competition Example 1 Forming Beliefs Example 2
A Normal Form Example 3 Dominate
Strategies Example 4 Weakly Dominate
Strategies Example 5 Dominated
Strategies Example 6 Weakly Dominated
Strategies Example 7 Rationalizable
Strategies Summary Review Questions
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Lesson Overview
Lesson 1 formulates and solves the following
games Example 1 2/3 of the Average Game. Has a
simple solution. Examples 2 and 3 and 4 and 6
Price Competition Game. Has a complex solution
requiring a game table to predict current actions
by the other players. Example 5 Budget Balance
Game. Example 7 Location Game.
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Example 1 Forming Beliefs
Example 1 Forming Beliefs
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Example 1 Forming Beliefs
Comment Sequential move games with perfect
information typically have a unique rollback
solution to choose your optimal strategy and to
predict how others would have responded to every
possible move you might have made. When your
opponents strategies are chosen simultaneously
with yours, rather than sequentially, choosing
your optimal strategies and forming beliefs about
your opponents strategies can be harder since
your opponents are simultaneously forming beliefs
about you. There are various solutions offered
by game theory depending on then extent of
players rationality and of assumptions about the
rationality of other players.
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Example 1 Forming Beliefs about Current
Strategies

• Question How should you play the Guess 2/3 of
  the Average Game?
• Rules
• No talking or other communication between
  players.
• Players secretly write a real number between 0
  and 100.
• The winner is the one closest to 2/3 of the
  average.
• What number should you guess?

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Example 1 Forming Beliefs about Current
Strategies

• Answer Your optimal guess is 2/3 of the average
  of your beliefs about the guesses of the other
  players.
• Game theory provides steps to form those beliefs
  based on a logical process of thinking through
  the thinking of the other players. You will put
  yourself in the position of other players and
  think through the others thinking, which of
  course includes their putting themselves in your
  position and thinking what you are thinking.

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Example 1 Forming Beliefs about Current
Strategies

• Step 1 Any guess higher than 66.67 (2/3rds of
  100) is worse for any player than guessing 66.67
  since higher numbers cannot possibly be 2/3rds of
  the average of any guesses. Those higher numbers
  should be eliminated by any rational player.
• Step 2 Once those guesses are eliminated for
  every player, any guess higher than 44.45 (2/3rds
  of 66.67) is worse for any player than guessing
  44.45 since higher numbers cannot possibly be
  2/3rds of the average of any remaining guesses (0
  to 66.67) . Those higher numbers should be
  eliminated.
• Step 3 Once those guesses are eliminated for
  every player, any guess higher than 29.64 (2/3rds
  of 44.45) is worse for any player than guessing
  29.64 since higher numbers cannot possibly be
  2/3rds of the average of any remaining guesses (0
  to 44.45) . Those higher numbers should be
  eliminated.

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Example 1 Forming Beliefs about Current
Strategies

• That process can continue until any particular
  number above 0 is eliminated. So, guess 0.

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Example 1 Forming Beliefs about Current
Strategies

• Comment Suppose you doubt the assumption of the
  common knowledge of the rationality of all
  players. For example, suppose you assume other
  players are rational, and you assume other
  players assume other players are rational, but
  you make no assumptions about the assumptions
  other players make about the assumptions of other
  players. What number should you guess?
• Because you are rational, eliminate any guess
  higher than 66.67.
• Because you assume other players are rational,
  eliminate any guess higher than 44.45.
• Because you assume other players assume other
  players are rational, eliminate any guess higher
  than 29.64.
• Without further assumptions, all guesses from 0
  to 29.64 are viable.

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Example 2 A Normal Form
Example 2 A Normal Form
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Example 2 A Normal Form
Comment Game tables or normal forms condense the
information in a game tree or extensive form.
Like the extensive form, the normal form
specifies strategies for every player and the
outcomes of the actions taken by all players.
But unlike the extensive form, the normal form
does not specify the order of the actions.
Normal forms are the simplest way to model games
where actions are simultaneous.
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Example 2 A Normal Form
Question Sams Club and Costco both sell
emergency food supplies in a weather-proof bucket
that provides 275 delicious easy-to-prepare
meals, including potato soup and corn chowder.
The unit cost to both retailers is 75. The
retailers compete on price the low-price
retailer gets all the market and they split the
market if they have equal prices. Suppose they
consider prices 75, 85, and 95, and suppose
market demands at those prices are 140, 100, and
80. Define the normal form for this Price
Competition Game.
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Example 2 A Normal Form
Answer To begin, at Sam's Club price 95 and
Costco price 85, Costco gets the entire market
demand of 100. Hence, Sam's makes 0 and Costco
makes (85-75)x100 1,000.
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Example 3 Dominate Strategies
Example 3 Dominate Strategies
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Example 3 Dominate Strategies
Comment The simplest simultaneous move games to
solve do not require you to predict what your
opponent will do now since your best response is
the same no matter what you believe other players
choose for their strategies. A dominate
strategy for a player gives better payoffs for
that player compared with any other strategy, no
matter what other players choose for their
strategies. Any rational player should choose a
dominate strategy.
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Example 3 Dominate Strategies
Question Restrict Sams Club and Costco in the
Price Competition Game to choose between prices
85 and 95, but keep the unit cost to both
retailers at 75, keep the assumption that the
low-price retailer gets all the market and they
split the market if they have equal prices, and
keep market demands at 100 and 80 for prices 85
and 95. Define the normal form for this reduced
Price Competition Game, and find optimal
strategies.
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Example 3 Dominate Strategies
Answer 85 is a dominate strategy for each
player since it gives better payoffs for that
player compared with 95, no matter whether the
other player chooses 85 or 95.
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Example 3 Dominate Strategies
Comment The Reduced Price Competition Game is
like the famous prisoners dilemma. The
prisoner's dilemma is a fundamental problem in
game theory that demonstrates why two people
might not cooperate even if it is in both their
best interests to do so. Two suspects are
arrested. Each is told by the police they are
best off if they confess, making confession a
dominate strategy. But both prisoners
confessing is worse for each than both not
confessing.
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Example 4 Weakly Dominate Strategies
Example 4 Weakly Dominate Strategies
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Example 4 Weakly Dominate Strategies
Comment 1 A weakly dominate strategy for a
player gives at least as good payoffs for that
player compared with any other strategy, no
matter what other players choose for their
strategies, and better payoffs for at least one
choice of strategies for the other players. Any
rational player has no reason not to choose a
weakly dominate strategy. And a rational player
should definitely choose it if there is any
positive probability belief attached to those
strategies for the other players that make the
weakly dominate strategy give better payoffs.
Thus, a rational player should definitely choose
a weakly dominate strategy if there is any
positive probability belief that the other
players, through a slip of the hand or tremble,
may choose unintended strategies.
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Example 4 Weakly Dominate Strategies
Comment 2 Sams and Costco in the price
competition game both gain by monopolizing or
cartelizing the membership warehouse club
industry and keeping prices high, but to do so
requires playing a dominated strategy. The
problem is that the groups success in resolving
their dilemma and fixing high prices harms the
general publics interest (as measured by total
surplus). In the United States, the Sherman
Antitrust Act prohibits such price or quantity
fixing in restraint of trade or commerce.
Violations can lead to jail terms for the firms
executives, not just fines for the corporations.
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Example 4 Weakly Dominate Strategies
In the industry for large turbines that generate
electricity, GE was the largest producer in the
1950s, with 60 percent of the market.
Westinghouse has 30 percent, and Allied-Chambers
had 10 percent. They kept those shares and
obtained high prices though a cleaver
coordination device. Electric utilities
invited bids for the turbines they intended to
buy. If the bid was issued during days 1-17 of a
lunar month, Westinghouse and Allied-Chambers had
to put in very high bids that would be sure
losers, and GE was the chosen winner. Similarly,
Westinghouse was the chosen winner for days
18-25, and Allied-Chambers for days 26-28.
Eventually the Department of Justice figured it
out, and some executives went to jail.
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Example 4 Weakly Dominate Strategies
In the retail industry detection of price cuts
that violate price-setting agreements And the
punishment of such violations can be simplified
and retaliation made quick and automatic by low
price guarantees. At first sight, low price
guarantees seem to guarantee low prices. But
game-theoretic thinking shows that in reality
they can have exactly the opposite effect.
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Example 4 Weakly Dominate Strategies
Question Sams Club and Costco consider
modifying the price competition as described in
Example 2 with the following low-price guarantee
We guarantee lower prices than any other store,
and we do everything in our power to ensure that
youre not paying too much for your purchase.
Thats why we offer our Low Price Guarantee. If
you find a lower advertised price, simply let us
know and well gladly meet that price! To
decide the effect of that guarantee, define the
normal form for the Price Competition Game
modified by the Low Price Guarantee, and check
for dominate strategies in that game and in the
original Price Competition Game.
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Example 4 Weakly Dominate Strategies
Answer The Price Competition Game has a weakly
dominate strategy for each player Sams price
85 gives at least as good payoffs for Sams
compared with 75 or 95, no matter Costcos
price, and better payoffs if Costco picks 85.
Costcos price 85 gives at least as good
payoffs for Costco compared with 75 or 95, no
matter Sams price, and better payoffs if Sams
picks 85. Conclusion Both choose 85.
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Example 4 Weakly Dominate Strategies
To define the normal form for the Modified Price
Competition Game, at Sams price 95 and Costco
price 85, Sams reduces price to 85 and splits
the market demand of 100 hence, both make
(85-75)x50 500. At Sams price 95 and
Costco price 75, Sams reduces price to 75 and
splits the market demand of 140 hence, both make
(75-75)x70 0. At Sams price 85 and Costco
price 75, Sams reduces price to 75 and splits
the market demand of 140 hence, both make
(75-75)x70 0.
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Example 4 Weakly Dominate Strategies
The Price Competition Game modified by The Low
Price Guarantee has a weakly dominate
strategy for each player Sams price 95
gives at least as good payoffs for Sams compared
with 75 or 85, no matter Costcos price, and
better payoffs if Costco picks 95. Costcos
price 95 gives at least as good payoffs for
Costco compared with 75 or 85, no matter Sams
price, and better payoffs if Sams picks 95.
Conclusion The Low Price Guarantee
guarantees high prices.
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Example 5 Dominated Strategies
Example 5 Dominated Strategies
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Example 5 Dominated Strategies
Comment A dominated strategy for a player gives
worse payoffs for that player compared with some
other strategy, no matter what other players
choose for their strategies. While dominate
strategies are the recommended choice to play
games, dominated strategies should never be
chosen. Eliminating dominated strategies reduces
the game, and the new game may have further
dominated strategies, which can be eliminated,
and so on.
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Example 5 Dominated Strategies
Question Congress is under pressure to lower
taxes and raise spending and, thereby, run a
budget deficit. The Federal Reserves primary
task is to prevent inflation, but it is also
under pressure to lower interest rates. The Fed
prefers lower rates but only if inflation is not
a treat, such as when Congress balances its
budget.
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Example 5 Dominated Strategies
Define the normal form for a simultaneous move
game between Congress and the Fed. Congress
likes best (payoff 4) a budget deficit and low
rates, next (payoff 3) budget balance and low
rates, next (2) a budget deficit and high rates,
and worst (1) budget balance and high rates. The
Fed likes best (payoff 4) budget balance and low
rates, next (payoff 3) budget balance and high
rates, next (2) a budget deficit and high rates,
and worst (1) a budget deficit and low
rates. Find optimal strategies for Congress and
the Fed.
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Example 5 Dominated Strategies
Answer The Federal Reserve has no dominate nor
weakly dominate nor dominated nor weakly
dominated strategies. But Congress has Budget
Deficit as dominate, and so Budget Balance as
dominated. After eliminating the latter, the
Federal Reserve now has High Rates as
dominate. Thus, the optimum for Congress is
Budget Deficit and the optimum for Federal
Reserve is High Rates. Comment Those optima
are for each individual player. If the players
colluded, then Budget Balance and Low Interest
Rates are better for both players. But that is
hard to enforce since Congress would be playing a
dominated strategy.
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Example 6 Weakly Dominated Strategies
Example 6 Weakly Dominated Strategies
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Example 6 Weakly Dominated Strategies
Comment A weakly dominated strategy for a player
gives at least as bad payoffs for that player
compared with some other strategy, no matter what
other players choose for their strategies, and
worse payoffs for at least one choice of
strategies for the other players. Eliminating
weakly-dominated strategies reduces the game, and
the new game may have further weakly-dominated
strategies, which can be eliminated, and so on.
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Example 6 Weakly Dominated Strategies
Question Modify the Price Competition Game
between Sams Club and Costco by supposing they
consider prices 75, 76, 77, 78, 79, 80,
85, and 95, and suppose market demands at
those prices are quantities 140, 136, 132, 128,
124, 120, 100, and 80. Keep unit cost 75.
Fill in the empty cells below, and find optimal
prices.
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Example 6 Weakly Dominated Strategies
Step 1 For each firm, 75 is weakly dominated by
any other strategy, and 95 is weakly dominated
by 85. Hence, eliminate 75 and 95 and reduce
the game.
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Example 6 Weakly Dominated Strategies
Step 2 For each firm, 85 is weakly dominated by
80. Hence, eliminate 85 and further reduce the
game.
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Example 6 Weakly Dominated Strategies
Step 3 For each firm, 80 is weakly dominated by
79. Hence, eliminate 80 and further reduce the
game.
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Example 6 Weakly Dominated Strategies
Step 4 For each firm, 79 is weakly dominated by
78. Hence, eliminate 79 and further reduce the
game.
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Example 6 Weakly Dominated Strategies
Step 5 For each firm, 78 is weakly dominated by
77. Hence, eliminate 78 and further reduce the
game.
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Example 6 Weakly Dominated Strategies
Step 6 For each firm, 77 is weakly dominated by
76. Hence, eliminate 77 and solve the game with
prices 76 for each firm.
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Example 7 Rationalizable Strategies
Example 7 Rationalizable Strategies
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Example 7 Rationalizable Strategies
Comment Rationalizable strategy choices in a
game can be justified purely on the basis of
rationality and the common knowledge of
rationality.
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Example 7 Rationalizable Strategies
Question Sams Club and Costco are each planning
to open a new store somewhere in Los Angeles
(Northridge, North Hollywood, Brentwood, or San
Pedro) in January of next year. They face a
tension between locating far apart, giving each
some local market power, and locating where more
customers live. That tension between monopoly
power and competition results in the profit
payoffs in the normal form below. Where should
the stores locate?
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Example 7 Rationalizable Strategies
Answer There are no dominate strategies nor
dominated strategies in the normal form.
However, no matter what Costco believes about
Sams, Costco would not choose location C4 as a
best response. Likewise, no matter what Sams
believes about Costco, Sams would not choose
locations S1 or S2 as a best response. For that
reason, S1 and S2 and C4 are not rationalizable,
and can thus be eliminated.
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Example 7 Rationalizable Strategies
Answer After those eliminations, S3 is now
dominate for Sams, and C3 is Costcos best
response to S3. Thus, the combination (S3,C3) is
the dominance solution to the location game.
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Summary
Summary
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Summary
When your opponents strategies are chosen
simultaneously with yours, choosing your optimal
strategies and forming beliefs about your
opponents strategies can be hard since your
opponents are simultaneously forming beliefs
about you. There are 5 ways to choosing your
optimal strategies and forming beliefs about your
opponents strategies. And these can be used in
any combination.
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Summary

• Choose
• Dominate Strategies. Those are strategies for a
  player that give better payoffs for that player
  compared with any other strategy, no matter what
  other players choose for their strategies.
• Weakly Dominate Strategies. Those are strategies
  for a player that give at least as good payoffs
  for that player compared with any other strategy,
  no matter what other players choose for their
  strategies, and better payoffs for at least one
  choice of strategies for the other players.

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Summary

• Eliminate
• Dominated Strategies. Those are strategies for a
  player that give worse payoffs for that player
  compared with some other strategy, no matter what
  other players choose for their strategies.
• Weakly Dominated Strategies. Those are strategies
  for a player that give at least as bad payoffs
  for that player compared with some other
  strategy, no matter what other players choose for
  their strategies, and worse payoffs for at least
  one choice of strategies for the other players.
• Non-Rationalizable Strategies. Those are
  strategies for a player that are never a best
  response for that player no matter what that
  player believes the other players choose for
  their strategies.

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Summary
The dominance solution to a game is the unique
result from a sequence of selecting Dominate
Strategies or Weakly Dominate Strategies and of
eliminating Dominated Strategies and Weakly
Dominated Strategies and Non-Rationalizable
Strategies.
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Review Questions

• Review Questions
• You should try to answer some of the following
  questions before the next class.
• You will not turn in your answers, but students
  may request to discuss their answers to begin the
  next class.
• Your upcoming Exam 1 and cumulative Final Exam
  will contain some similar questions, so you
  should eventually consider every review question
  before taking your exams.

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Questions about Dominate Strategies
Questions about Dominate Strategies
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Review Questions
Question 1 (A Prisoners Dilemma) Suppose there
is a street on which 25 small businesses are run,
and which suffers from a serious litter problem
that detracts customers. It costs 100 annually
for each business to keep the front of their
store clean. If a store owner decides to keep the
front of their store clean, all businesses on the
street will have improved sales. Suppose every
business on the street will have a 10 increase
in annual sales for each business that decides to
keep the front of their store clean. If more than
ten businesses clean their storefronts, then all
of the businesses will make more money, including
the businesses that clean. If some businesses
clean but fewer than ten do so, then the
businesses that clean will lose money, while the
businesses that do not clean will gain money.
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Review Games
Game 1 (continued) If everyone were to keep the
front of their store clean, every business would
benefit a 250 increase in sales with a cost of
100 yields a 150 gain. However, an individual
business could save 100 by not doing the
cleaning, yet suffer only 10 for their
defection, yielding a 240 gain, which is greater
than they would have if they cooperated. Despite
the fact they may be prepared to contribute 100,
they can avoid doing so in hope that others in
the street will clean anyway, and they receive
the benefit for no personal expense. Will anyone
keep the front of their store clean?
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Review Games
Answer 1 Not Cleaning is a dominate strategy.
For any strategies by each of the other 24
stores, the Payoff to Store X for Not Cleaning is
90 more than the payoff for Cleaning. When each
of the 25 stores follows its dominate strategy,
no one cleans, and the payoff to each store is 0.
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End of Lesson I.4
BA 445

Managerial Economics