Deep Thought
As the light changed from red to
green to yellow and back to red
again, I sat there thinking about life.
Was it nothing more than a bunch of
honking and yelling? Sometimes it
seemed that way. ~ Jack Handey.
(Translation: Today’s lesson teaches how to anticipate the actions
of other players, taking account that they are trying to anticipate
your actions.)
BA 210 Lesson II.4 Simultaneous Price Competition
1
Overview
Overview
BA 210 Lesson II.4 Simultaneous Price Competition
2
Lesson Overview
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
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 1: Forming Beliefs
Example 1: Forming Beliefs
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 1: Forming Beliefs
Comment: Sequential move games 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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:
1. No talking or other communication between players.
2. Players secretly write a real number between 0 and 100.
3. The winner is the one closest to 2/3 of the average.
What number should you guess?
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 2: A Normal Form
Example 2: A Normal Form
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 2: A Normal Form
Question: Sam’s 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
Costco
Sam's
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
0,1000
BA 210 Lesson II.4 Simultaneous Price Competition
$95
0,0
1000,0
800,800
15
Example 3: Dominate Strategies
Example 3: Dominate Strategies
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 3: Dominate Strategies
Question: Restrict Sam’s 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
Costco
Sam's
$85
$95
$85
500,500
0,1000
$95
1000,0
800,800
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 3: Dominate Strategies
Comment: The Reduced Price
Competition Game is like the
famous prisoners’ dilemma.
Costco
Sam's
$85
$95
The prisoner's dilemma is a
fundamental problem in game
Confess
theory that demonstrates why
Prisoner 1
Don't C.
two people might not cooperate
even if it is in both their best interests to do so.
$85
500,500
0,1000
$95
1000,0
800,800
Prisoner 2
Confess
500,500
0,1000
Don't C.
1000,0
800,800
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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Example 4: Weakly Dominate
Strategies
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Costco
Comment 2: Sam’s and Costco
$85
$95
in the price competition game
$85
500,500 1000,0
both gain by monopolizing
Sam's
$95
0,1000 800,800
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
group’s success in resolving their dilemma and fixing high prices
harms the general public’s 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Costco
In the industry for large turbines
$85
$95
that generate electricity,
$85
500,500 1000,0
GE was the largest producer
Sam's
$95
0,1000 800,800
in the 1950s, with 60 percent
of the market. Westinghouse has 30 percent, and AlliedChambers 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Costco
In the retail industry detection
$85
$95
of price cuts that violate
$85
500,500 1000,0
price-setting agreements
Sam's
$95
0,1000 800,800
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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Question: Sam’s Club and Costco consider modifying the price
competition as described in Example 2 with the following lowprice guarantee: “We guarantee lower prices than any other store,
and we do everything in our power to ensure that you’re not
paying too much for your purchase. That’s why we offer our Low
Price Guarantee. If you find a lower advertised price, simply let
us know and we’ll 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Answer: The Price Competition Game
has a weakly dominate strategy
Sam's
for each player:
Costco
$75
$85
$95
$75
0,0
0,0
0,0
$85
0,0
500,500
0,1000
$95
0,0
1000,0
800,800
Sam’s price = $85 gives at least as good payoffs for Sam’s
compared with $75 or $95, no matter Costco’s price, and better
payoffs if Costco picks $85.
Costco’s price = $85 gives at least as good payoffs for Costco
compared with $75 or $95, no matter Sam’s price, and better
payoffs if Sam’s picks $85.
Conclusion: Both choose $85.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
Costco
To define the normal form for the
$75
$85
$95
Modified Price Competition
$75
0,0
0,0
0,0
Sam's
$85
0,0
500,500 500,500
Game, at Sam’s price $95 and
$95
0,0
500,500 800,800
Costco price $85, Sam’s reduces
price to $85 and splits the market demand of 100; hence, both
make $(85-75)x50 = $500.
At Sam’s price $95 and Costco price $75, Sam’s reduces price to
$75 and splits the market demand of 140; hence, both make
$(75-75)x70 = $0.
At Sam’s price $85 and Costco price $75, Sam’s reduces price to
$75 and splits the market demand of 140; hence, both make
$(75-75)x70 = $0.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 4: Weakly Dominate Strategies
The Price Competition Game modified by
The Low Price Guarantee
$75
has a weakly dominate strategy Sam's $85
$95
for each player:
Costco
$75
0,0
0,0
0,0
$85
0,0
500,500
500,500
$95
0,0
500,500
800,800
Sam’s price = $95 gives at least as good payoffs for Sam’s
compared with $75 or $85, no matter Costco’s price, and better
payoffs if Costco picks $95.
Costco’s price = $95 gives at least as good payoffs for Costco
compared with $75 or $85, no matter Sam’s price, and better
payoffs if Sam’s picks $95.
Conclusion: The “Low Price Guarantee” guarantees high prices.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 5: Dominated Strategies
Example 5: Dominated Strategies
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
31
Example 5: Dominated Strategies
Question: Congress is under pressure to lower taxes and raise
spending and, thereby, run a budget deficit. The Federal
Reserve’s 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.
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 5: Dominated Strategies
Federal Reserve
Answer: The Federal Reserve
Low RatesHigh Rates
has no dominate nor weakly
B. Balance
3,4
1,3
dominate nor dominated nor
Congress
B. Deficit
4,1
2,2
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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 6: Weakly Dominated Strategies
Example 6: Weakly Dominated
Strategies
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
36
Example 6: Weakly Dominated Strategies
Question: Modify the Price Competition Game between Sam’s
Club and Costco by supposing they consider
prices $75, $79, $80, $85, and $95,
and suppose market demands at those prices are
quantities 140, 124, 120, 100, and 80. Keep unit cost = $75. Fill
in the empty cells below, and find optimal prices.
11
Sam's
2
2
75
79
80
85
95
Costco
75
0,0
0,0
0,0
0,0
0,0
79
0,0
?,?
?,?
0,496
0,496
80
0,0
496,0
300,300
0,600
0,600
85
0,0
496,0
600,0
500,500
0,1000
BA 210 Lesson II.4 Simultaneous Price Competition
95
0,0
496,0
600,0
1000,0
800,800
37
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.
11
Sam's
2
2
75
79
80
85
95
Costco
75
0,0
0,0
0,0
0,0
0,0
79
0,0
248,248
0,496
0,496
0,496
80
0,0
496,0
300,300
0,600
0,600
85
0,0
496,0
600,0
500,500
0,1000
BA 210 Lesson II.4 Simultaneous Price Competition
95
0,0
496,0
600,0
1000,0
800,800
38
Example 6: Weakly Dominated Strategies
Step 2: For each firm, $85 is now weakly dominated by $80.
Hence, eliminate $85 and further reduce the game.
11
Sam's
2
2
75
79
80
85
95
Costco
75
0,0
0,0
0,0
0,0
0,0
79
0,0
248,248
0,496
0,496
0,496
80
0,0
496,0
300,300
0,600
0,600
85
0,0
496,0
600,0
500,500
0,1000
BA 210 Lesson II.4 Simultaneous Price Competition
95
0,0
496,0
600,0
1000,0
800,800
39
Example 6: Weakly Dominated Strategies
Step 3: For each firm, $80 is now weakly dominated by $79.
Hence, eliminate $80 and further reduce the game to its single
solution of prices $79 for each firm.
11
Sam's
2
2
75
79
80
85
95
Costco
75
0,0
0,0
0,0
0,0
0,0
79
0,0
248,248
0,496
0,496
0,496
80
0,0
496,0
300,300
0,600
0,600
85
0,0
496,0
600,0
500,500
0,1000
BA 210 Lesson II.4 Simultaneous Price Competition
95
0,0
496,0
600,0
1000,0
800,800
40
Example 7: Rationalizable Strategies
Example 7: Rationalizable Strategies
BA 210 Lesson II.4 Simultaneous Price Competition
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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.
BA 210 Lesson II.4 Simultaneous Price Competition
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Example 7: Rationalizable Strategies
Question: Sam’s 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?
11
Sam's
2
2
S1
S2
S3
S4
Costco
C1
1,7
1,2
2,0
0,0
C2
2,5
0,3
3,5
0,-2
C3
0,0
1,2
2,7
0,0
BA 210 Lesson II.4 Simultaneous Price Competition
C4
1,1
2,1
0,1
4,-1
43
Example 7: Rationalizable Strategies
11
2
Costco
Answer: There are no dominate
2
C1
C2
C3
C4
S1
1,7
2,5
0,0
1,1
strategies nor dominated
S2
1,2
0,3
1,2
2,1
Sam's
strategies in the normal form.
S3
2,0
3,5
2,7
0,1
S4
0,0
0,-2
0,0
4,-1
However, no matter what Costco
believes about Sam’s, Costco would not choose location C4 as a
best response. Likewise, no matter what Sam’s believes about
Costco, Sam’s 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.
BA 210 Lesson II.4 Simultaneous Price Competition
44
Example 7: Rationalizable Strategies
11
Answer: After those eliminations,
S3 is now dominate for Sam’s,
and C3 is Costco’s best response Sam's
to S3.
2
2
S1
S2
S3
S4
Costco
C1
1,7
1,2
2,0
0,0
C2
2,5
0,3
3,5
0,-2
C3
0,0
1,2
2,7
0,0
C4
1,1
2,1
0,1
4,-1
Thus, the combination (S3,C3) is the dominance solution to the
location game.
BA 210 Lesson II.4 Simultaneous Price Competition
45
Summary
Summary
BA 210 Lesson II.4 Simultaneous Price Competition
46
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.
BA 210 Lesson II.4 Simultaneous Price Competition
47
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.
BA 210 Lesson II.4 Simultaneous Price Competition
48
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.
BA 210 Lesson II.4 Simultaneous Price Competition
49
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.
BA 210 Lesson II.4 Simultaneous Price Competition
50
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 2 and cumulative Final Exam will
contain some similar questions, so you should eventually
consider every review question before taking your exams.
BA 210 Lesson II.4 Simultaneous Price Competition
51
BA 210
Introduction to Microeconomics
End of Lesson II.4
BA 210 Lesson II.4 Simultaneous Price Competition
52