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 445 Lesson B.4 Simultaneous Price Competition
1
Readings
Readings
Baye “Bertrand Oligopoly” (see the index)
Dixit Chapter 4
BA 445 Lesson B.4 Simultaneous Price Competition
2
Overview
Overview
BA 445 Lesson B.4 Simultaneous Price Competition
3
Overview
Forming Beliefs about others’ strategies that are chosen simultaneously with
yours depends on assumptions of each player’s rationality, and each player’s
beliefs of the rationality of other players.
Dominate Strategies for a player give better payoffs compared with any other
strategy, no matter what other players choose for their strategies. Any rational
player should choose a dominate strategy.
Weakly Dominate Strategies give at least as good payoffs compared with any
other strategy, no matter what other players choose, and better payoffs for at
least one choice of the other players.
Dominated Strategies for a player give worse payoffs compared with any other
strategy, no matter what other players choose for their strategies. No rational
player should choose a dominated strategy.
Weakly Dominated Strategies give at least as bad payoffs compared with any
other strategy, no matter what other players choose, and worse payoffs for at
least one choice of the other players.
Rationalizable Strategies are a best response to some belief about what other
players choose, where the other players are, themselves, believed to choose
rationalizable strategies.
BA 445 Lesson B.4 Simultaneous Price Competition
4
Example 1: Forming Beliefs
Example 1: Forming Beliefs
BA 445 Lesson B.4 Simultaneous Price Competition
5
Example 1: Forming Beliefs
Overview
Forming Beliefs about others’ strategies that are chosen
simultaneously with yours depends on assumptions of each
player’s rationality, and each player’s beliefs of the
rationality of other players.
BA 445 Lesson B.4 Simultaneous Price Competition
6
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.
BA 445 Lesson B.4 Simultaneous Price Competition
7
Example 1: Forming Beliefs about Current Strategies
Question: How should you play the following 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 445 Lesson B.4 Simultaneous Price Competition
8
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 445 Lesson B.4 Simultaneous Price Competition
9
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 445 Lesson B.4 Simultaneous Price Competition
10
Example 1: Forming Beliefs about Current Strategies
That process can continue until any particular number
above 0 is eliminated. So, guess 0.
BA 445 Lesson B.4 Simultaneous Price Competition
11
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 445 Lesson B.4 Simultaneous Price Competition
12
Example 2: Dominate Strategies
Example 2: Dominate Strategies
BA 445 Lesson B.4 Simultaneous Price Competition
13
Example 2: Dominate Strategies
Overview
Dominate Strategies for a player give better payoffs
compared with any other strategy, no matter what other
players choose for their strategies. Any rational player
should choose a dominate strategy.
BA 445 Lesson B.4 Simultaneous Price Competition
14
Example 2: Dominate Strategies
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 445 Lesson B.4 Simultaneous Price Competition
15
Example 2: Dominate Strategies
Question: Sam’s Club and Costco control a large share of
the US retail wholesale market. 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 simultaneously consider prices $85 and $95,
and suppose market demands at those prices are 100 and
80.
Determine optimal prices in this Price Competition Game.
BA 445 Lesson B.4 Simultaneous Price Competition
16
Example 2: 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 445 Lesson B.4 Simultaneous Price Competition
17
Example 2: Dominate Strategies
Comment: The Reduced Price
Competition Game is like the
famous prisoners’ dilemma.
Costco
Sam's
$85
$95
$85
500,500
0,1000
$95
1000,0
800,800
Prisoner 2
The prisoner's dilemma is a
Confess Don't C.
fundamental problem in game
theory that demonstrates why Prisoner 1 Confess 500,500 1000,0
Don't C. 0,1000 800,800
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.
BA 445 Lesson B.4 Simultaneous Price Competition
18
Example 3: Weakly Dominate Strategies
Example 3: Weakly Dominate
Strategies
BA 445 Lesson B.4 Simultaneous Price Competition
19
Example 3: Weakly Dominate Strategies
Overview
Weakly Dominate Strategies give at least as good payoffs
compared with any other strategy, no matter what other
players choose, and better payoffs for at least one choice
of the other players.
BA 445 Lesson B.4 Simultaneous Price Competition
20
Example 3: 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 445 Lesson B.4 Simultaneous Price Competition
21
Example 3: 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 445 Lesson B.4 Simultaneous Price Competition
22
Example 3: Weakly Dominate Strategies
Costco
In the industry for large turbines
that generate electricity,
$85
$95
GE was the largest producer
$85
500,500 1000,0
Sam's
in the 1950s, with 60 percent
$95
0,1000 800,800
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.
BA 445 Lesson B.4 Simultaneous Price Competition
23
Example 3: 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 445 Lesson B.4 Simultaneous Price Competition
24
Example 3: Weakly Dominate Strategies
Question: Sam’s Club and Costco reconsider
their price competition. As before, 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 simultaneously consider prices $75, $85,
and $95, and suppose market demands at those prices are
120, 100, and 80.
BA 445 Lesson B.4 Simultaneous Price Competition
25
Example 3: Weakly Dominate Strategies
But now suppose each store offers the following low-price
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, redefine the normal
form for the Price Competition Game modified by the Low
Price Guarantee, and check for dominate strategies in that
game, and compare with the original Price Competition
Game.
BA 445 Lesson B.4 Simultaneous Price Competition
26
Example 3: Weakly Dominate Strategies
Answer: The Price Competition Game
has a weakly dominate strategy
$75
Sam's
$85
for each player:
$95
Costco
$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 445 Lesson B.4 Simultaneous Price Competition
27
Example 3: Weakly Dominate Strategies
Costco
To define the normal form for the
$75
$85
$95
Modified Price Competition
$75
0,0
0,0
0,0
0,0
500,500 500,500
Game, at Sam’s price $95 and Sam's $85
$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 445 Lesson B.4 Simultaneous Price Competition
28
Example 3: Weakly Dominate Strategies
The Price Competition Game
modified by The Low Price
Sam's
Guarantee has a weakly
dominate strategy for each player:
Costco
$75
$85
$95
$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 445 Lesson B.4 Simultaneous Price Competition
29
Example 4: Dominated Strategies
Example 4: Dominated Strategies
BA 445 Lesson B.4 Simultaneous Price Competition
30
Example 4: Dominated Strategies
Overview
Dominated Strategies for a player give worse payoffs
compared with any other strategy, no matter what other
players choose for their strategies. No rational player
should choose a dominated strategy.
BA 445 Lesson B.4 Simultaneous Price Competition
31
Example 4: 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 445 Lesson B.4 Simultaneous Price Competition
32
Example 4: Dominated Strategies
Question: Woolworths and Coles control a large share of the Australian
supermarket market. In January of next year, they are each planning to
open a new store somewhere in Sydney, either The Central Business
District (CBD) or King’s Cross. They face a tension between locating
far apart, giving each some local market power, and locating near
customers. That tension between monopoly power and competition
results in the profit payoffs in the normal form below. Where should the
stores locate?
Coles
Wool
CBD
King's
CBD
3,4
4,1
King's
1,3
2,2
BA 445 Lesson B.4 Simultaneous Price Competition
33
Example 5: Dominated Strategies
Coles
Answer: Coles has no dominate
nor weakly dominate nor dominated
CBD
King's
nor weakly dominated strategies.
CBD
3,4
1,3
Wool
But Woolworths has King’s Cross
King's
4,1
2,2
as dominate, and so CBD as
dominated. After eliminating the latter, Coles now has King’s Cross as
dominate.
Thus, the optimum for Woolworths is King’s Cross and the optimum for
Coles is King’s Cross.
Comment: Those optima are for each individual player. If the players
colluded, then CBD is better for both players. But that is hard to
enforce since Woolworths would be playing a dominated strategy.
BA 445 Lesson B.4 Simultaneous Price Competition
34
Example 5: Weakly Dominated Strategies
Example 5: Weakly Dominated
Strategies
BA 445 Lesson B.4 Simultaneous Price Competition
35
Example 5: Weakly Dominated Strategies
Overview
Weakly Dominated Strategies give at least as bad payoffs
compared with any other strategy, no matter what other
players choose, and worse payoffs for at least one choice
of the other players.
BA 445 Lesson B.4 Simultaneous Price Competition
36
Example 5: 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 weaklydominated strategies reduces the game, and the new game
may have further weakly-dominated strategies, which can
be eliminated, and so on.
BA 445 Lesson B.4 Simultaneous Price Competition
37
Example 5: Weakly Dominated Strategies
Question: FedEx and UPS control a large share of the
express small package delivery market. The unit cost to
both firms is $75. The retailers compete on price: the lowprice retailer gets all the market and they split the market if
they have equal prices.
BA 445 Lesson B.4 Simultaneous Price Competition
38
Example 5: Weakly Dominated Strategies
Suppose FedEx and UPS simultaneously 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.
Fill in the empty cells of the profit table below, and find
optimal prices.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
79
0,0
136,0
264,0
0,384
0,384
0,384
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
39
Example 5: Weakly Dominated Strategies
Step 1: For each firm, $75 is weakly dominated by any
other strategy. Hence, eliminate $75 and reduce the game.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
40
Example 5: Weakly Dominated Strategies
Step 2: For each firm, $95 is weakly dominated by $85.
Hence, eliminate $95 and reduce the game.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
41
Example 5: Weakly Dominated Strategies
Step 3: For each firm, $85 is weakly dominated by $80.
Hence, eliminate $85 and further reduce the game.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
42
Example 5: Weakly Dominated Strategies
Step 4: For each firm, $80 is weakly dominated by $79.
Hence, eliminate $80 and further reduce the game.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
43
Example 5: Weakly Dominated Strategies
Step 5: For each firm, $79 is weakly dominated by $78.
Hence, eliminate $79 and further reduce the game.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
44
Example 5: Weakly Dominated Strategies
Step 6: For each firm, $78 is weakly dominated by $77.
Hence, eliminate $78 and further reduce the game.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
45
Example 5: Weakly Dominated Strategies
Step 7: For each firm, $77 is weakly dominated by $76.
Hence, eliminate $77 and solve the game with prices $76
for each firm.
11
FedEx
2
2
75
76
77
78
79
80
85
95
UPS
75
0,0
0,0
0,0
0,0
0,0
0,0
0,0
0,0
76
0,0
68,68
0,136
0,136
0,136
0,136
0,136
0,136
77
0,0
136,0
132,132
0,264
0,264
0,264
0,264
0,264
78
0,0
136,0
264,0
192,192
0,384
0,384
0,384
0,384
79
0,0
136,0
264,0
384,0
248,248
0,496
0,496
0,496
80
0,0
136,0
264,0
384,0
496,0
300,300
0,600
0,600
85
0,0
136,0
264,0
384,0
496,0
600,0
500,500
0,1000
BA 445 Lesson B.4 Simultaneous Price Competition
95
0,0
136,0
264,0
384,0
496,0
600,0
1000,0
800,800
46
Example 5: Weakly Dominated Strategies
Comment: The dominance solution where managers
from each firm simultaneously charge price $76 barely
above marginal cost is the result of each firm successively
eliminating prices $95, $85, $80, $79, $78, and $77. It is
like the affect of a sequential price war between two firms,
but with the dominance solution, each firm simultaneously
jumps right to the end and charges $76. An outside
observer would thus never see the sequence of prices in
the “price war” in the dominance solution because the
“price war” and the sequence of eliminations of dominated
strategies is in the minds of the managers.
BA 445 Lesson B.4 Simultaneous Price Competition
47
Example 6: Rationalizable Strategies
Example 6: Rationalizable
Strategies
BA 445 Lesson B.4 Simultaneous Price Competition
48
Example 6: Rationalizable Strategies
Overview
Rationalizable Strategies are a best response to some
belief about what other players choose, where the other
players are, themselves, believed to choose rationalizable
strategies.
BA 445 Lesson B.4 Simultaneous Price Competition
49
Example 6: 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 445 Lesson B.4 Simultaneous Price Competition
50
Example 6: Rationalizable Strategies
Question: Wal-Mart and Target control a large share of the SoCal
discount department store market. In June of next year, they are each
planning to open a new store somewhere in Los Angeles (Northridge,
North Hollywood, Brentwood, or San Pedro). They face a tension
between locating far apart, giving each some local market power, and
locating near customers. That tension between monopoly power and
competition results in the profit payoffs in the normal form below.
Where should the stores locate?
11
Wal-Mart
2
2
S1
S2
S3
S4
Target
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 445 Lesson B.4 Simultaneous Price Competition
C4
1,1
2,1
0,1
4,-1
51
Example 6: Rationalizable Strategies
11
2
Target
Answer: There are no
2
C1
C2
C3
C4
S1
1,7
2,5
0,0
1,1
dominate strategies nor
S2
1,2
0,3
1,2
2,1
Wal-Mart
dominated strategies in the
S3
2,0
3,5
2,7
0,1
S4
0,0
0,-2
0,0
4,-1
normal form. However, no
matter what Target believes about Wal-Mart, Target would
not choose location C4 as a best response. Likewise, no
matter what Wal-Mart believes about Target, Wal-Mart
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 445 Lesson B.4 Simultaneous Price Competition
52
Example 6: Rationalizable Strategies
Answer: After those
eliminations, S3 is now
dominate for Wal-Mart, and
C3 is Target’s best response
to S3.
11
Wal-Mart
2
2
S1
S2
S3
S4
Target
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 445 Lesson B.4 Simultaneous Price Competition
53
Summary
Summary
BA 445 Lesson B.4 Simultaneous Price Competition
54
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 445 Lesson B.4 Simultaneous Price Competition
55
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 445 Lesson B.4 Simultaneous Price Competition
56
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 445 Lesson B.4 Simultaneous Price Competition
57
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 NonRationalizable Strategies.
BA 445 Lesson B.4 Simultaneous Price Competition
58
Review Questions
Review Questions
 You should try to answer some of the review
questions (see the online syllabus) 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 445 Lesson B.4 Simultaneous Price Competition
59
BA 445
Managerial Economics
End of Lesson B.4
BA 445 Lesson B.4 Simultaneous Price Competition
60