BUS 525: Managerial
Economics
Lecture 10
Game Theory: Inside Oligopoly
10-2
Overview
I. Introduction to Game Theory
II. Simultaneous-Move, One-Shot
Games
III. Infinitely Repeated Games
IV. Finitely Repeated Games
V.
Multistage Games
10-3
Game Environments
• Players are individuals who make decisions
• Players’ planned decisions are called strategies.
• Payoffs to players are the profits or losses
resulting from strategies.
• Order of play is important:
– Simultaneous-move game: each player makes decisions
without knowledge of other players’ decisions.
– Sequential-move game: one player observes its rival’s
move prior to selecting a strategy.
• Frequency of rival interaction
– One-shot game: game is played once.
– Repeated game: game is played more than once; either
a finite or infinite number of interactions.
Simultaneous-Move, One-Shot
Games: Normal Form Game
• A Normal Form Game consists of:
– Set of players i = {1, 2, … n} where n is a
finite number.
– Each players strategy set or feasible actions
consist of a finite number of strategies.
• Player 1’s strategies are S1 = {a, b, c, …}.
• Player 2’s strategies are S2 = {A, B, C, …}.
– Payoffs, to the players are the profits or
losses that results from the strategies.
– Due to the interdependence, payoff to a
player depends not only the player’s
strategy but also on the strategies employed
by other players
• Player 1’s payoff: π1(a,B) = ?
• Player 2’s payoff: π2(b,C) = ?
10-4
10-5
Normal Form Game
Normal form game: A representation of a game indicating
the players, their possible strategies, and the payoffs
resulting from alternative strategies
In game theory, Strategy, is a decision rule that describes
the actions a player will take at each decision point
Player 1
Player 2
Strategy
a
b
c
A
B
C
12,11
11,10
10,15
11,12
10,11
10,13
14,13
12,12
13,14
10-6
A Normal Form Game
Player A
Player B
Strategy
Up
Down
Left
10,20
-10,7
Right
15,8
10,10
10-7
Dominant Strategy
A strategy that results in highest
payoff to a player regardless of
opponents action
Player A
Player B
Strategy
Up
Down
Left
10,20
-10,7
Right
15,8
10,10
Principle: Check to see if you have a dominant
strategy. If you have one, play it.
10-8
Dominant Strategy in a SimultaneousMove, One-Shot Game
• A dominant strategy is a strategy
resulting in the highest payoff
regardless of the opponent’s action.
• If “Up” is a dominant strategy for
Player A in the previous game, then:
– πA(Up,Left) ≥ πA(Down,Left);
– πA(Up,Right) ≥ πA(Down,Right).
Secure Strategy
Does B have a dominant strategy?
Play secure strategy, i.e. A strategy that guarantees
the highest payoff under the worst possible scenario.
i.e choose Right; 8>7
Could you do better? Yes, choose left. Because A will
choose Up, A’s dominant strategy; 20>8
Player A
Player B
Strategy
Up
Down
Left
10,20
-10,7
Right
15,8
10,10
Principle: Put yourself in rivals shoes. If your rival has a
dominant strategy, anticipate that he or she will play it.
10-9
Nash Equilibrium
1010
The Nash equilibrium is a condition in which no
player can improve her payoff by unilaterally
changing her own strategy, given the other
players’ strategy
Suppose A chooses Up, B chooses Left. Either player
will have no incentive to change his or her strategy.
Given that A’s strategy is Up, the best player B can do
is to choose Left. Given that B’s strategy is Left, the
best player A can do is to choose Up.
The Nash equilibrium strategy for player A is Up, and
for player B it is Left.
Player A
Player B
Strategy
Up
Down
Left
10,20
-10,7
Nash Equilibrium
Right
15,8
10,10
Key Insights
• Look for dominant strategies.
• Put yourself in your rival’s shoes.
10-11
10-12
A Pricing Game
• Two managers want to maximize
market share: i = {1,2}.
• Strategies are pricing decisions
– SA = {Low Price, High Price}.
– SB = {Low Price, High Price}.
• Simultaneous moves.
• One-shot game.
1013
A Pricing Game
Firm A
Firm B
Strategy Low Price High Price
Low Price
0,0
50,-10
High Price -10,50
10,10
•Is there Nash equilibrium?
•Is Nash equilibrium the best?
•Cheating
1014
An Advertising game
Firm A
Firm B
Strategy
Advertise
Don't Advertise
Advertise
$4,$4
$1, $20
Don't Advertise
$20,$1
$10, $10
Dominant strategy for each firm is to advertise. Nash
equilibrium for the game is to advertise
Key Insight:
• Game theory can also be used to
analyze situations where “payoffs”
are non monetary!
• We will, without loss of generality,
focus on environments where
businesses want to maximize
profits.
– Hence, payoffs are measured in
monetary units.
10-15
10-16
Coordination Games
• In many games, players have
competing objectives: One firm gains
at the expense of its rivals.
• However, some games result in
higher profits by each firm when
they “coordinate” decisions.
1017
A Coordination Game
Firm A
Firm B
Strategy
220-Volt Outlets 110-Volt Outlets
220-Volt Outlets
$100,$100
$0,$0
110-Volt Outlets
$0,$0
$100,$100
The game has two Nash equilibria. Better outcome by
coordination. If firms produce different outlets, lower
demand as consumers needs to spend more money in wiring
their houses
10-18
Key Insights:
• Not all games are games of conflict.
• Communication can help solve
coordination problems.
• Sequential moves can help solve
coordination problems.
Games With No Pure Strategy
Nash Equilibrium
Manager
10-19
Strategy
Worker
Work
Shirk
Monitor
-1,1
1,-1
Don’t,Monitor
1,-1
-1,1
Suppose Managers strategy is to monitor the
worker. Then the best choice for the worker is to
work. But if the worker works, the best strategy
for the manager not to monitor. Therefore,
monitoring is not a Nash equilibrium strategy. Not
monitoring is also not a Nash equilibrium strategy.
Because, if manager’s strategy is “don’t monitor;”
The worker would choose shirking. Manager should
monitor.
Strategies for Games With No
Pure Strategy Nash Equilibrium
10-20
• In games where no pure strategy
Nash equilibrium exists, players find
it in there interest to engage in
mixed (randomized) strategies.
– This means players will “randomly”
select strategies from all available
strategies.
10-21
A Nash Bargaining Game
Management
Union
Strategy
0
50
100
0
0,0
50,0
100,0
50
0,50
50,50
-1,-1
100
0,100
-1,-1
-1,-1
How much of a $100 surplus is to be given to the union?. The
parties write the amount (either 0,50 or100). If the sum exceeds
100, the bargaining ends in a stalemate. Cost of delay -$1 both
for the union and the management. Three Nash equilibrium to this
bargaining game(100,0), (50,50), and (0,100).
Six of the nine potential payoffs are inefficient as they result in
total payoff lesser than amount to be divided. Three negative
payoffs. Settle for 50-50 split which is fair, look at history
10-22
Lessons in
Simultaneous Bargaining
• Simultaneous-move bargaining
results in a coordination problem.
• Experiments suggests that, in the
absence of any “history,” real
players typically coordinate on the
“fair outcome.”
• When there is a “bargaining
history,” other outcomes may
prevail.
10-23
Infinitely Repeated Games
• Infinitely Repeated Games
• A game that is played over and over again forever and in
which players receive payoffs during each play of the
game
• Recall key aspects of present value analysis
– Value of the firm for T period with different payoffs at
different periods
– Value of the firm for ∞ period with same payoffs at
different periods
• Trigger strategy
– A strategy that is contingent on the past play of a game and
in which some particular past action “triggers” a different
action by a player
A Pricing Game that is Repeated
Firm A
Firm B
Strategy Low Price High Price
Low Price
0,0
50,-40
High Price -40,50
10,10
Nash equilibrium in one shot play of this game is for each
firm to charge low prices and earn zero profits
Impact of repeated play on the equilibrium outcome of the
game
Trigger strategy: ”Cooperate provided no player has ever
cheated in the past. If any players cheats, punish the player
by choosing one shot Nash equilibrium strategy forever
after.”
1024
A Pricing Game that is Repeated
•
10-25
The benefit of cheating today on the collusive agreement is earning $50
instead of $10 today. The cost of cheating today is earning $0 instead of
$10 in each future period. If the present value of cost of cheating
exceeds one time benefit of cheating, it does not pay for a firm to cheat,
and high prices can be sustained.
• In the example, if the interest rate is less than 25 percent both firms A
and B will loose more (in present value) from cheating. That’s why we
see collusion in oligopoly.
• Condition for sustaining cooperative outcome with trigger strategies
(ΠCheat- πCoop)/(πCoop- πN)≤ 1/i or
ΠCheat- πCoop ≤ 1/i(πCoop- πN)
• The LHS represents the one time gain from cheating. The RHS
represents the present value of what is given up in the future by
cheating today.
In one-shot games there is no tomorrow, and threats have no bite.
When interest rate is low firms may find it in their interest to collude and
charge high prices
Factor Affecting Collusion in Pricing
Games
To sustain collusive arrangements via
punishment strategies firms must know: (i)
who the rivals are; (ii) who the rivals
customers are; (iii) When the rivals deviate
from collusive arrangement; and (iv) must
be able to successfully punish rivals for
cheating.
These factors are related to:
(1) Number of firms; (2) Firm size; (3)
History of the market; (4) Punishment
mechanism
10-26
10-27
Finitely Repeated Games
• Finitely Repeated Games
– Games in which players do not know when the game will
end
– Games in which players know when the game will end
• Games with an uncertain final period
– When the game is repeated a finite but uncertain times, the
benefits of cooperating look exactly the same as benefits
from cooperating in an infinitely repeated game except that
1-θ plays the role of 1/(1+i); PLAYERS DISCOUNT THEIR FUTURE
NOT BECAUSE OF THE INTEREST RATE BUT BECAUSE THEY ARE NOT
CERTAIN FUTURE PLAYS WILL OCCUR
• In our example Firm A has no incentive to cheat if
ΠFirm ACheat=50 ≤10/θ
=
θ ≤1/5, probability
the game will end is less than 20 percent, If θ=1, one shot game,
dominant strategy for both firm is to charge the low price
ΠFirm ACoperate, which is true if
Repeated Games with a Known
Final period
• Provided the players know when the game will end, the
game has only one Nash equilibrium.
– Same outcome as one shot game
• Trigger strategies would not work
– Rival cannot be punished in the last period
– Trigger strategies wont work
• Backward unraveling.
• End of period problem
– Workers work hard due to threat of firing
– Announces plans to quit, the cost of shirking is reduced
– Manager’s response, fire immediately after announcements
– Worker responds by telling you at the very end
– What to do? Incentives, gratuity
Same outcome as one shot game
10-28
10-29
Multistage Games
• Multistage framework permits players to make sequential
rather than simultaneous decisions
• Differs from one-shot and infinitely repeated games
• Extensive-form game
A representation of a game that summarizes the players,
the information available to them at each stage, the
strategies available to them, the sequence of moves, and
the payoff resulting from alternative strategies
Pricing to Prevent Entry: An
Application of Game Theory
• Two firms: an incumbent and potential
entrant.
• Potential entrant’s strategies:
– Enter.
– Stay Out.
• Incumbent’s strategies:
–
–
–
–
{if
{if
{if
{if
enter, play hard}.
enter, play soft}.
stay out, play hard}.
stay out, play soft}.
• Move Sequence:
– Entrant moves first. Incumbent observes entrant’s
action and selects an action.
10-30
The Pricing to Prevent Entry
Game in Extensive Form
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A(Entrant)
Out
0, 10
10-31
Identify Nash and
Subgame Perfect Equilibria
10-32
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A (Entrant)
Out
0, 10
10-33
Two Nash Equilibria
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A (Entrant)
Out
0, 10
One Subgame Perfect
Equilibrium
-1, 1
Price war
B (Incumbent)
Enter
Share
5, 5
A (Entrant)
Out
0, 10
Subgame Perfect Equilibrium Strategy:
{enter; If enter, Share market}
10-34
10-35
Insights
• Establishing a reputation for being
unkind to entrants can enhance longterm profits.
• It is costly to do so in the shortterm, so much so that it isn’t optimal
to do so in a one-shot game.
An Innovation Game: An
Application of Game Theory
• Class Exercise
• Two firms: Innovator and Rival
• Innovator’s strategies:
– Introduce.
– Don’t introduce.
• Rival’s strategies:
– {if introduced, clone}.
– {if introduced, don’t clone}.
• Move Sequence:
– Innovator moves first. Rival observes innovator’s
action and selects an action.
10-36
Class Exercise
• If you do not introduce the new product, you and your rival
will earn $1 million each
• If you introduce the new product
– Rival clones : you will loose $ 5 million and your rival will earn
$20million
– Rival does not clone: you will make $100 million and your rival
will make $0 million
• Set up the extensive form of the game
• Should you introduce the new product?
• How would your answer change if your rival has promised
not to clone the product?
• What would you do if patent law prevented your rival from
cloning the product?
An Innovation Game: An
Application of Game Theory
10-38
-5,20
Clone
B (Rival)
Introduce
Don’t clone
100, 0
A (Innovator)
Don’t
Introduce
1, 1
Simultaneous-Move
Bargaining
10-39
• Management and a union are negotiating a
wage increase.
• Strategies are wage offers & wage
demands.
• Successful negotiations lead to $600 million
in surplus, which must be split among the
parties.
• Failure to reach an agreement results in a
loss to the firm of $100 million and a union
loss of $3 million.
• Simultaneous moves, and time permits only
one-shot at making a deal.
Fairness: The “Natural”
Focal Point
Management
Union
Strategy
W = $10
W=$5
W=$1
W = $10
100, 500
-100, -3
-100, -3
W = $5
-100, -3
300, 300
-100, -3
W = $1
-100, -3
-100, -3
500, 100
10-40
10-41
Single Offer Bargaining
• Now suppose the game is sequential
in nature, and management gets to
make the union a “take-it-or-leaveit” offer.
• Analysis Tool: Write the game in
extensive form
– Summarize the players.
– Their potential actions.
– Their information at each decision point.
– Sequence of moves.
– Each player’s payoff.
Step 1: Management’s
Move
10
Firm
5
1
10-42
10-43
Step 2: Add the Union’s Move
Accept
Union
Reject
10
Firm
5
Accept
Union
Reject
1
Accept
Union
Reject
Step 3: Add the Payoffs
10-44
Accept
100, 500
Reject
-100, -3
Accept
300, 300
Union
10
Firm
5
Union
Reject
-100, -3
Accept
500, 100
Reject
-100, -3
1
Union
10-45
The Game in Extensive Form
Accept
100, 500
Reject
-100, -3
Accept
300, 300
Union
10
Firm
5
Union
Reject
-100, -3
Accept
500, 100
Reject
-100, -3
1
Union
Step 4: Identify the Firm’s
Feasible Strategies
10-46
• Management has one information set
and thus three feasible strategies:
–Offer $10.
–Offer $5.
–Offer $1.
10-47
Step 5: Identify the
Union’s Feasible Strategies
• The Union has three information set and
thus eight feasible strategies:
–
–
–
–
–
–
–
–
Accept $10, Accept $5, Accept $1
Accept $10, Accept $5, Reject $1
Accept $10, Reject $5, Accept $1
Accept $10, Reject $5, Reject $1
Reject $10, Accept $5, Accept $1
Reject $10, Accept $5, Reject $1
Reject $10, Reject $5, Accept $1
Reject $10, Reject $5, Reject $1
10-48
Step 6: Identify Nash
Equilibrium Outcomes
• Outcomes such that neither the
firm nor the union has an incentive
to change its strategy, given the
strategy of the other.
Finding Nash
Equilibrium Outcomes
Union's Strategy
Accept $10, Accept $5, Accept $1
Accept $10, Accept $5, Reject $1
Accept $10, Reject $5, Accept $1
Reject $10, Accept $5, Accept $1
Accept $10, Reject $5, Reject $1
Reject $10, Accept $5, Reject $1
Reject $10, Reject $5, Accept $1
Reject $10, Reject $5, Reject $1
10-49
Firm's Best Mutual Best
Response
Response?
$1
$5
$1
$1
$10
$5
$1
$10, $5, $1
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Step 7: Find the Subgame
Perfect Nash Equilibrium
Outcomes
10-50
• Outcomes where no player has an
incentive to change its strategy,
given the strategy of the rival, and
• The outcomes are based on “credible
actions;” that is, they are not the
result of “empty threats” by the
rival.
Checking for Credible
Actions
Union's Strategy
Accept $10, Accept $5, Accept $1
Accept $10, Accept $5, Reject $1
Accept $10, Reject $5, Accept $1
Reject $10, Accept $5, Accept $1
Accept $10, Reject $5, Reject $1
Reject $10, Accept $5, Reject $1
Reject $10, Reject $5, Accept $1
Reject $10, Reject $5, Reject $1
Are all
Actions
Credible?
Yes
No
No
No
No
No
No
No
10-51
The “Credible” Union
Strategy
Union's Strategy
Accept $10, Accept $5, Accept $1
Accept $10, Accept $5, Reject $1
Accept $10, Reject $5, Accept $1
Reject $10, Accept $5, Accept $1
Accept $10, Reject $5, Reject $1
Reject $10, Accept $5, Reject $1
Reject $10, Reject $5, Accept $1
Reject $10, Reject $5, Reject $1
Are all
Actions
Credible?
Yes
No
No
No
No
No
No
No
10-52
Finding Subgame Perfect
Nash Equilibrium Strategies
Union's Strategy
Accept $10, Accept $5, Accept $1
Accept $10, Accept $5, Reject $1
Accept $10, Reject $5, Accept $1
Reject $10, Accept $5, Accept $1
Accept $10, Reject $5, Reject $1
Reject $10, Accept $5, Reject $1
Reject $10, Reject $5, Accept $1
Reject $10, Reject $5, Reject $1
Nash and Credible
Nash Only
Firm's Best
Response
$1
$5
$1
$1
$10
$5
$1
$10, $5, $1
10-53
Mutual Best
Response?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Neither Nash Nor Credible
10-54
To Summarize:
• We have identified many
combinations of Nash equilibrium
strategies.
• In all but one the union does
something that isn’t in its self
interest (and thus entail threats that
are not credible).
• Graphically:
There are 3 Nash
Equilibrium Outcomes!
10-55
Accept
100, 500
Reject
-100, -3
Accept
300, 300
Union
10
Firm
5
Union
Reject
-100, -3
Accept
500, 100
Reject
-100, -3
1
Union
Only 1 Subgame-Perfect Nash
Equilibrium Outcome!
10-56
Accept
100, 500
Reject
-100, -3
Accept
300, 300
Union
10
Firm
5
Union
Reject
-100, -3
Accept
500, 100
Reject
-100, -3
1
Union
10-57
Bargaining Re-Cap
• In take-it-or-leave-it bargaining,
there is a first-mover advantage.
• Management can gain by making a
take-it-or-leave-it offer to the
union. But...
• Management should be careful;
real world evidence suggests that
people sometimes reject offers on
the the basis of “principle” instead
of cash considerations.
10-58
An Advertising Game
• Two firms (Kellogg’s & General Mills)
managers want to maximize profits.
• Strategies consist of advertising
campaigns.
• Simultaneous moves.
– One-shot interaction.
– Repeated interaction.
10-59
A One-Shot Advertising
Game
Kellogg’s
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
10-60
Equilibrium to the OneShot Advertising Game
Kellogg’s
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
Nash Equilibrium
High
-1, 15
0, 9
2, 2
10-61
Can collusion work if the
game is repeated 2 times?
Kellogg’s
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
No (by backwards
induction).
• In period 2, the game is a one-shot
game, so equilibrium entails High
Advertising in the last period.
• This means period 1 is “really” the last
period, since everyone knows what will
happen in period 2.
• Equilibrium entails High Advertising by
each firm in both periods.
• The same holds true if we repeat the
game any known, finite number of
times.
10-62
10-63
Can collusion work if firms play
the game each year, forever?
• Consider the following “trigger
strategy” by each firm:
– “Don’t advertise, provided the rival has not
advertised in the past. If the rival ever
advertises, “punish” it by engaging in a
high level of advertising forever after.”
• In effect, each firm agrees to
“cooperate” so long as the rival hasn’t
“cheated” in the past. “Cheating”
triggers punishment in all future
periods.
10-64
Suppose General Mills adopts this
trigger strategy. Kellogg’s profits?
Cooperate = 12 +12/(1+i) + 12/(1+i)2 + 12/(1+i)3
Value of a perpetuity of $12 paid
+…
at the end of every year
= 12 + 12/i
Kellogg’s
Cheat = 20 +2/(1+i) + 2/(1+i)2 + 2/(1+i)3 + …
= 20 + 2/i
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
Kellogg’s Gain to Cheating:
10-65
• Cheat - Cooperate = 20 + 2/i - (12 + 12/i) = 8 - 10/i
– Suppose i = .05
• Cheat - Cooperate = 8 - 10/.05 = 8 - 200 = -192
• It doesn’t pay to deviate.
– Collusion is a Nash equilibrium in the infinitely repeated
game!
Kellogg’s
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
Benefits & Costs of
Cheating
• Cheat - Cooperate = 8 - 10/i
– 8 = Immediate Benefit (20 - 12 today)
– 10/i = PV of Future Cost (12 - 2 forever after)
• If Immediate Benefit - PV of Future Cost > 0
– Pays to “cheat”.
• If Immediate Benefit - PV of Future Cost  0
– Doesn’t pay to “cheat”.
Kellogg’s
General Mills
Strategy
None
Moderate
High
None
12,12
20, 1
15, -1
Moderate
1, 20
6, 6
9, 0
High
-1, 15
0, 9
2, 2
10-66
Key Insight
10-67
• Collusion can be sustained as a Nash
equilibrium when there is no certain
“end” to a game.
• Doing so requires:
– Ability to monitor actions of rivals.
– Ability (and reputation for) punishing
defectors.
– Low interest rate.
– High probability of future interaction.