Introduction to Game Theory

Introduction to Game Theory
Prepared by César R. Sobrino
Universidad del Turabo
August 16, 2018
Prepared by César R. Sobrino Introduction to Game Theory

Outline
1 Description of a Game
Complete & Incomplete Information
Perfect & Imperfect Information
2 Simultaneous Decisions (Static Games)
Pure Strategies (One-shot game)
Dominant Strategy
Dominated Strategy
Mixed Strategies (One-shot game)
Repeated Games
3 Sequential Decisions (Dynamic Games)

Introduction
Game theory models strategic behavior by agents who
understand that their actions affect the actions of
other agents.
This scheme is useful to model interactions among
firms, countries, individuals, etc.
Useful to analyze the interactions in oligopolistic
markets.
Distinguishing feature of oligopoly is interdependence
of firms’ profits.
Arises when number of firms in market is small
enough that every firms’ price and output decisions
affect demand and marginal revenue conditions of
every other firm in market.

The Elements of a Game
A game consists of
A set of players.
A set of strategies for each player.
Set of rules such as: simultaneous, sequential, who
plays ﬁrst, cooperation;
The payoﬀs to each player for every possible list of
strategy choices by the players.
A game can be represented by a matrix form or using
decision trees.
Solution(s): Nash Equilibrium: Neither player has
an incentive to change strategy, given the other
player’s choice

Complete and Incomplete Information
Complete information refers when each agent
knows the other agent’s payoﬀs and the rules of the
game. It is a central assumption of the Game Theory.
Incomplete information, also known as
asymmetric information, refers to the contrary,
where not all players know each other’s payoﬀs.

Perfect and Imperfect Information
The perfection of information is an important notion
in Game Theory when considering sequential and
simultaneous games.
Perfect information is when each player knows or
can see other player’s moves. A good example would
be chess, where each player sees the other player’s
pieces on the board.
Imperfect information is when decisions have to be
made simultaneously, and players need to balance all
possible outcomes when making a decision. A good
example is a card game where each player’s card are
hidden from the rest of the players.

Nash Equilibrium
Set of actions or decisions for which all managers are
choosing their best actions given the actions they
expect their rivals to choose
Strategic stability
No single ﬁrm can unilaterally make a diﬀerent
decision & do better.
When a unique Nash Equilibrium set of decisions
exists.
Rivals can be expected to make the decisions leading
to the Nash Equilibrium.
With multiple Nash equilibria, no way to predict the
likely outcome.
Nash equilibria can occur without dominant or
dominated strategies.

Simultaneous Decisions
Occur when managers must make individual decisions
without knowing their rivals’ decisions.
For all ﬁrms in an oligopoly to be predicting correctly
each others’ decisions:
All ﬁrms must be choosing individually best actions
given the predicted actions of their rivals, which they
can then believe are correctly predicted.
Strategically astute managers look for mutually best
decisions.

Pure Strategies: Dominant Strategy
Always provide best outcome no matter what decisions
rivals make,
When one exists, the rational decision maker always
follows its dominant strategy,
Predict rivals will follow their dominant strategies, if
they exist.
Dominant strategy equilibrium.
Exists when when all decision makers have dominant
strategies.

Prisoner’s Dilemma
2 prisoners (Bill & Jane), suspected of burglary, are
taken into custody.
There is no enough evidence to convict them of that
crime, only to convict them on the charge of possession
of stolen goods.
Prisoners are on separate interrogation rooms, which
means that they cannot communicate (imperfect
information).
Both prisoners are oﬀered the same deal and know the
consequences of each action (complete information)
Prisoners are completely aware that the other prisoner
has been oﬀered the exact same deal (common
knowledge).

Prisoner’s Dilemma- Payoﬀ matrix
2 Strategies for each player: C ∼ “Confess” and DC ∼
“Do not confess”
Players want to minimize their sentences (years in jail)
Bill
DC C
Jane
DC
A B
2 1
2 12
C
C D
12 6
1 6

Prisoner’s Dilemma-Payoﬀ Matrix
If both players choose DC, both get 2 years (Cell A)
If both players choose C, both get 6 years (Cell D)
If Bill plays DC and Jane plays C, he gets 12 years
and she gets 1 year (Cell C).
If Bill plays C and Jane plays DC, he gets 1 year and
she gets 12 years (Cell B).
Bill
DC C
Jane
DC
A B
2 1
2 12
C
C D
12 6
1 6

Prisoner’s Dilemma-Payoff Matrix: Bill’s payoffs
If Bill chooses DC (1st column), he gets either 2 years
or 12 years.. Similarly, if Bill chooses C (2nd
column), he gets either 1 year or 6 years.
Bill
DC CJane
DC A B
2 1
C
C D
12 6
C is a dominant strategy for Bill because
Payoffs of DC ( 2 & 12) bigger than payoffs of C (1 &
6). 2 >1 & 12 > 6. (DC > C)

Prisoner’s Dilemma-Payoff Matrix: Jane’s payoffs
If Jane chooses DC (1st row), she gets either 2 years or
12 years. Similarly, if Jane chooses C (2nd row), she
gets get either 1 year or 6 years.
Bill
DC CJane
DC A B
2 12
C
C D
1 6
C is a dominant strategy for Jane because
Payoffs of DC ( 2 & 12) bigger than payoffs of C (1 &
6). 2 >1 & 12 > 6. (DC > C)

Prisoner’s Dilemma-Dominant Strategy
All rivals have dominant strategies.
Nash Equilibrium is (C,C).
Both gets 6 years (Cell D).
In dominant strategy equilibrium, all are worse oﬀ
than if they had cooperated in making their decisions
(Cell A).

Prisoner’s Dilemma- Best Response Analysis
Another way to find the solution is run the best
response analysis (any player may start)
If Bill plays DC → the Jane’s best response is C
(1<2) → the Bill’s best response is C (6<12).
Bill has an incentive to change his first choice.
If Bill plays C → the Jane’s best response is C
(6<12) → the Bill’s best response is C (6<12).
Bill has no incentive to change his first choice.
Cell D is a stable solution.
The other cells are unstable because players have an
incentive to change their strategies.
The Nash Equilibrium is (C,C).

Trade War
The US is deciding whether set a tariff to the imports
coming from Japan.
Japan is deciding whether set a tariff to the imports
coming from the US.
They make their decisions separately (imperfect
information) and know the consequences of each
decisions (complete information).
It is common knowledge that they face the same
consequences. This means that imposing a tariff by
any country increases her tax collection but if the rival
imposes a tariff, her tax collection decreases through
the fall in exports.

Trade War-Payoff Matrix
2 Strategies for each player: tariff (T) and no tariff
(NT).
Players want to maximize their payoffs (in $ billions)
Japan
T NT
US
T
A B
5 4
5 9
NT
C D
9 8
4 8

Trade War-Dominant Strategy
T is a dominant strategy for Japan because
Payoffs of T (5 & 9) bigger than payoffs of NT (4 &
8). 5 >4 & 9 > 8. (T>NT)
T is a dominant strategy for US because
Payoffs of T (5 & 9) bigger than payoffs of NT (4 &
8). 5 >4 & 9 > 8. ( T>NT)
All rivals have dominant strategies.
Nash equilibrium is (T,T).
Both gets $ 5 billions (Cell A).
In dominant strategy equilibrium, all are worse off
than if they had cooperated in making their decisions.
(Cell D).

Trade War-Best Response Analysis
Any player may start.
If Japan plays NT → the US’s best response is T
(9>8) → the Japan’s best response is T (5>4).
Japan has an incentive to change her ﬁrst choice.
If Japan plays T → the US’s best response is T (5>4)
→ the Japan’s best response is T (5>4).
Japan has no incentive to change her ﬁrst choice.
Cell A is a stable solution.
There is a Nash Equilibrium at (T,T)

Multiple Equilibria
There are games that have more than one Nash
Equilibrium.
Hard to ﬁnd a unique solution in simultaneous
decisions.
Sequential decisions is the best way to ﬁnd a unique
outcome.

Battle of the sexes
It is a game of coordination.
This kind of game is where the payoﬀs of the players
are highest when they can coordinate their strategies.
Husband and wife like to spend time in either “Opera”
or “Fight”.
However, wife would rather to go to the “Opera” and
husband would rather to go to the “Fight”.
A key point is that they like to spend time together. It
is worthless to go alone to any of those meetings.
Payoﬀs are in utils (preference levels). The higher the
util, the better.

Battle of the sexes
Is there any dominant strategy for each player? Use
the best response analysis, as well.
Wife
Fight OperaHusband
Fight A B
1 0
2 0
Opera
C D
0 2
0 1

Battle of the sexes-Dominant Strategy
Wife do not have a dominant strategy because
Payoffs of “Fight” (1 & 0) are not bigger or less than
payoffs of “Opera” (0 & 2). 1 >0 & 0 < 2.
Husband do not have a dominant strategy because
Payoffs of “Fight” (2 & 0) are not bigger or less than
payoffs of “Opera” (0 & 1). 2 >0 & 0 < 1.
No player has a dominant strategy.

Battle of the sexes -Best Response Analysis
If wife plays “Fight” → the Husband’s best response is
“Fight” (2>0) → the wife’s best response is “Fight”.
(1>0).
wife has no incentive to change her ﬁrst choice.
Cell A is a stable solution.
If wife plays “Opera” → the Husband’s best response
is “Opera” (1>0) → the wife’s best response is
“Opera” (2>0).
wife has no incentive to change her ﬁrst choice.
Cell D is a stable solution.
Multiple Nash Equilibria at (“Fight”,“Fight”) and
(“Opera”,“Opera”)

Location Game
In the next matrix, is there any dominant strategy for
each player? Use the best response analysis, as well.
Payoﬀs are in utils (preference levels). The higher the
util, the better.
B
Left Right
A
Up
A B
9 8
3 1
Down
C D
0 1
0 2

Market Share Rivalry
Brander J. & Spencer B.J. (1985) Export Subsidies
and International Market Share Rivalry, Journal of
International Economics, 16, pp. 83-100
Countries often perceive themselves as being in
competition with each other for profitable
international markets.
Export subsidies can appear as attractive policy tools
because they improve the relative position of a
domestic firm in noncooperative rivalries with foreign
firms, enabling it to expand its market share and earn
greater profits.
Subsidies change the initial conditions of the game
that firms play

Market Share Rivalry-Payoﬀ Matrix
Boeing and Airbus want to enter a foreign market.
2 strategies for each player: produce (P) and do not
produce (DP).
Players want to maximize their payoﬀs (in $ billions).
Any dominant strategy?
Airbus
P DP
Boeing
P
A B
-5 0
-5 100
DP
C D
100 0
0 0

Market Share Rivalry-Dominant Strategy
Airbus do not have a dominant strategy because
Payoffs of P (-5 & 100) are not bigger or less than
payoffs of DP (0 & 0). -5 <0 & 100 > 0.
Boeing do no have a dominant strategy because
Payoffs of P (-5 & 100) are not bigger or less than
payoffs of DP (0 & 0). -5 <0 & 100 > 0.
No firm has a dominant strategy.

Market Share Rivalry -Best Response Analysis
If Airbus plays P → the Boeing’s best response is DP
(0>-5) → the Airbus’s best response is P (100>0).
Airbus has no incentive to change her ﬁrst choice
Cell C is a stable solution.
If Airbus plays DP → the Boeing’s best response is P
(100>0) → the Airbus’s best response is DP (0>-5).
Airbus has no incentive to change her ﬁrst choice
Cell B is a stable solution.
Multiple Nash Equilibria at (P,DP) and (DP,P)

European Union subsidies Airbus for $25 billions.
Use the best response analysis.
Airbus
P DP
Boeing
P
A B
20 0
-5 100
DP
C D
125 0
0 0

The US responses subsidizing Boeing for $25 billions,
as well.
Use the best response analysis.
Airbus
P DP
Boeing
P
A B
20 0
20 125
DP
C D
125 0
0 0

Pure Strategies: Dominated Strategy
Never the best strategy, so never would be chosen &
should be eliminated.
Successive elimination of dominated strategies should
continue until none remain.
Search for dominant strategies ﬁrst, then dominated
strategies
When neither form of strategic dominance exists,
employ a diﬀerent concept for making simultaneous
decisions.
Example:
2 players: Palace and Castle
They are deciding their price level for similar goods
(pizzas).
3 prices levels: high (H), medium (M), and, low (L).

Dominated Strategy
Players want to maximize their payoﬀs (in $
thousands).
For each player, ﬁnd a dominant strategy.
Palace’s price
H M L
Castle’sprice
H
1 1 0.9 1.1 0.5 1.2
M
0.9 0.4 0.4 0.8 0.35 0.5
L
1.2 0.3 0.5 0.35 0.4 0.4

Dominated Strategy
Palace’s payoffs
Palace
H M L
Castle
H
1 1.1 1.2
M 0.4 0.8 0.5L
0.3 0.35 0.4
H is a dominated strategy for Palace because
Payoffs of H ( 1, 0.4, & 0.3) less than payoffs of M
(1.1, 0.8, & 0.35), & payoffs of L (1.2, 0.5, & 0.4) .
(H < M): 1 < 1.1, 0.4 <0.8, & 0.3 < 0.35 .
(H < L): 1 < 1.2, 0.4 <0.5, & 0.3 < 0.4.

Dominated Strategy
Castle’s payoffs
Palace
H M L
Castle
H
1 0.9 0.5
M 0.9 0.4 0.35L
1.2 0.5 0.4
M is a dominated strategy for Castle because
Payoffs of M ( 0.9, 0.4, & 0.35) less than payoffs of H
(1, 0.9, & 0.5) & payoffs of L (1.2, 0.5. & 0.4) .
(M < H): 0.9 < 1, 0.4 <0.9, & 0.35 < 0.5.
(M < L): 0.9 < 1.2, 0.4 <0.5, & 0.35 < 0.4.

Dominated Strategy
Both strategies have to be eliminated, so now, there is
a 2×2 matrix
Palace
M L
Castle
H B C
1.1 1.2
0.9 0.5
L
H I
0.35 0.4
0.5 0.4
Is there any dominant strategy for each player. Find
the solution.
Using the 3×3 matrix, run a best response analysis to
corroborate the outcome of this game.

Deciding radio format
KOOL and WIRD, radio stations, are deciding their
programming formats
There are three formats:
CW: Country western ( 50% of the market).
IM: Industrial music (30% of the market).
AN: All-news (20% of the market).
Find the solution or solutions
KOOL
CW IM AN
WIRD
CW
25 25 50 30 50 20
IM
30 50 15 15 30 20
AN
20 50 20 30 10 10

Mixed Strategy
A mixed strategy is one in which a player plays his
available pure strategies with certain probabilities.
If each player in an n-player game has a ﬁnite number
of pure strategies, then there exists at least one
equilibrium in (possibly) mixed strategies.
If there are no pure strategy equilibria, there must be a
unique mixed strategy equilibrium.
You may analyze the Battle of sexes using probabilities

Employee Monitoring
Principal-Agent Problem (Moral Hazard)
Employees can work hard or shirk.
Salary: $100K unless caught shirking
Cost of eﬀort: $50K
Managers can monitor or not.
Value of employee output: $200K
Proﬁt if employee doesn’t work: $0
Cost of monitoring: $10K

Employee Monitoring-Payoﬀ Matrix
Manager
Monitor No Monitor
Employee
Work
A B
90 100
50 50
Shirk
C D
-10 -100
0 100
Best replies do not correspond.
No equilibrium in pure strategies.
What do the players do?

Skeeping Employees from Shirking
Suppose:
Employee chooses (work, shirk) with probabilities
(p,1 − p), respectively. p ∈]0, 1[.
Manager chooses (monitor, no monitor) with
probabilities (q,1 − q), respectively. q ∈]0, 1[
First, ﬁnd employee’s expected payoﬀ from each pure
strategy.
If employee works: receives 50
E(work)= 50 × q + 50 × (1 − q) = 50
If employee shirks: receives 0 or 100
E(shirk)=0 × q + 100 × (1 − q) = 100 − 100 × q

Next, calculate the best strategy for possible strategies
of the opponent
For q < 1
2
“Shirk”
E(shirk)=100 − 100 × q > 50 = E(work)
For q > 1
2
“Work”
E(shirk)=100 − 100 × q < 50 = E(work)
For q = 1
2
“Indiﬀerent”
E(shirk)=100 − 100 × q = 50 = E(work)
Employees will shirk if q < 1
2
To keep employees from shirking, must monitor at
least half of the time.

Second, ﬁnd manager’s expected payoﬀ from each pure
strategy
If manager monitors: receives 90 or -10
E(M)= 90 × p + (−10) × (1 − p) = −10 + 100 × p
If manager does not monitor: receives 100 or -100
E(NM)=100 × p + (−100) × (1 − p) = −100 + 200 × p
Next, calculate the best strategy for possible strategies
of the opponent.

For p < 9
10
“Monitor”
E(M)=−10 + 100 × p > −100 + 200 × p = E(NM)
For p > 9
10
“No monitor”
E(M)=−10 + 100 × p < −100 + 200 × p = E(NM)
For p = 9
10
“Indiﬀerent”
E(M)=−10 + 100 × p = −100 + 200 × p = E(NM)
Manager will monitor if p < 9
10
If worker puts eﬀort at least 90% of time, manager will
not monitor.

Equilibrium Payoﬀs
Mixed Strategy Nash Equilibrium occurs at p = 9
10
&
q = 1
2
1/2 1/2
Monitor No Monitor
9/10 Work 50 90 50 100
1/10 Shirk 0 -10 100 -100
Probability of Cell A = 1
2
× 9
10
= 9
20
Probability of Cell B = 1
2
× 9
10
= 9
20
Probability of Cell C = 1
2
× 1
10
= 1
20
Probability of Cell D = 1
2
× 1
10
= 1
20

Expected Nash Equilibrium Payoﬀ
Manager
90 × 9
20 + 100 × 9
20 + (−10) × 1
20 + (−100) × 1
20 = 80
Employee
50 × 9
20 + 50 × 9
20 + 0 × 1
20 + 100 × 1
20 = 50

The Overlord Game (D-day): Normandy or Calais ?
In June 1944, two potential places for invasion:
Normandy (N) & Pas de Calais (C).
Allies: where to launch the invasion.
Germans: where to concentrate defenses.
Forces need not be concentrated all in one location;
since precious lives are lost, all victories are not worth
the same, etc.

The Overlord Game (D-day) - Payoﬀ matrix
Find the Mixed Strategy Nash Equilibrium
Germans
C NAllies
C A B
80 0
20 100
N
C D
0 20
80 60

The Cuban Missile Crisis (1962)
In October 1962, USSR attempted to install
medium-range and intermediate-range nuclear-armed
ballistic missiles in Cuba that were capable of hitting a
large portion of the US.
The US considered two strategies to avoid that
attempt:
A naval blockade (B), or “quarantine” to prevent
shipment of more missiles, possibly followed by
stronger action to induce the USSR to withdraw the
missiles already installed.
An air strike (A) to wipe out the missiles already
installed.
The alternatives of USSR were:
Withdrawal (W) of their missiles.
Maintenance (M) of their missiles.

The Cuban missile crisis (1962) - Payoﬀ matrix
Find the Mixed Strategy Nash Equilibrium
USSR
W M
US
B
A B
3 4
3 1
A
C D
2 1
2 4

Repeated Games: Strategies & Commitments
Three types: commitments, threats, and, promises.
Only credible strategic moves matter.
Players (managers) announce or demonstrate to rivals
that they will bind themselves to take a particular
action or make a speciﬁc decision.
No matter what action is taken by rivals.
Threats: Explicit or tacit “If you take action A, I will
take action B, which is undesirable or costly to you”.
Promises: “If you take action A, I will take action B,
which is desirable or rewarding to you”.

Repeated Games: Cooperation or Cheating
Cooperation occurs when oligopoly firms make
individual decisions that make every firm better off
than they would be in a (noncooperative) Nash
equilibrium.
Making noncooperative decisions
Does not imply that firms have made any agreement
to cooperate.
One-time prisoners’ dilemmas
Cooperation is not strategically stable (Cell A).
No future consequences from cheating, so both firms
expect the other to cheat.
Cheating is best response for each.

Repeated Games: Pricing Dilemma
2 Players, AMD and Intel, choosing a price level.
2 Strategies for each player: “High” and “Low”
Players want to maximize their payoﬀs ($ millions
proﬁt per week)
AMD’s price
High Low
Intel’sprice
High
A:Cooperation B:Cheating
2.5 3
5 2
Low
C:Cheating D:Noncooperation
0.5 1
6 3

Repeated Games: Deciding to Cooperate
With repeated decisions, cheaters can be punished.
When credible threats of punishment in later rounds of
decision making exist
Strategically astute managers can sometimes achieve
cooperation.
Cooperate
When present value of costs of cheating exceeds
present value of benefits of cheating.
Achieved in an oligopoly market when all firms decide
not to cheat.
Cheat
When present value of benefits of cheating exceeds
present value of costs of cheating.

Present value of beneﬁts of cheating
PVBC =
B1
(1 + r)1
+
B2
(1 + r)2
+ .... +
BN
(1 + r)N
Where: Bi = ΠCheat − ΠCooperate , for all i : 1, 2, ...., N
Present value of costs of cheating
PVCC =
C1
(1 + r)N+1
+
C2
(1 + r)N+2
+ .... +
CP
(1 + r)N+P
Where: Cj = ΠCooperate − ΠNash , for all j : 1, 2, ...., P
r is the real interest rate,

AMD’s payoffs
ΠCheat = 3, ΠCooperate = 2.5, &, ΠNash = 1, so
Bi = 0.5 and Cj = 1.5.
Intel’s payoffs
ΠCheat = 6, ΠCooperate = 5, &, ΠNash = 3, so Bi = 1
and Cj = 2.
Suppose r=10% (weekly) and AMD and Intel
decided to cooperate for 24 weeks.
At the fifth week, AMD started to cheat.
Intel found it at the 12th week and started to punish
AMD (N =7, P=13).

Present value of beneﬁts of cheating
PVBC =
0.5
(1 + 0.1)1
+
0.5
(1 + 0.1)2
+ .... +
0.5
(1 + 0.1)7
= $2.43
Present value of costs of cheating
PVCC =
1.5
(1 + 0.1)8
+
1.5
(1 + 0.1)9
+ .... +
1.5
(1 + 0.1)20
= $5.47
The costs of cheating are bigger than the beneﬁts of
cheating. It is better to cooperate.
Do the same analysis for Intel.

Repeated Games: Trigger Strategies
A rival’s cheating “triggers” punishment phase
Tit-for-tat strategy
Punishes after an episode of cheating & returns to
cooperation if cheating ends.
Grim strategy
Punishment continues forever, even if cheaters return
to cooperation.

Repeated Games: Facilitating Tactics
Legal tactics designed to make cooperation more likely
Four tactics: Price matching, sale-price guarantees,
public pricing, and, price leadership.
Price matching: Firm publicly announces that it will
match any lower prices by rivals.
Usually in advertisements
Discourages noncooperative price-cutting. Eliminates
benefit to other firms from cutting prices
Sale-price guarantees: Firm promises customers
who buy an item today that they are entitled to
receive any sale price the firm might offer in some
stipulated future period.
Primary purpose is to make it costly for firms to cut
prices.

Repeated Games: Facilitating Tactics
Public pricing: Public prices facilitate quick
detection of noncooperative price cuts
Timely & authentic
Early detection reduces PVBC
Early detection increases PVCC
Early detection reduces likelihood of noncooperative
price cuts.
Price leadership: Price leader sets its price at a level
it believes will maximize total industry proﬁt
Rest of ﬁrms cooperate by setting same price
Does not require explicit agreement. Generally lawful
means of facilitating cooperative pricing

Cartels
Most extreme form of cooperative oligopoly.
Explicit collusive agreement to drive up prices by
restricting total market output
Illegal in US, Canada, Mexico, Germany, & European
Union.
Pricing schemes usually strategically unstable &
diﬃcult to maintain
Strong incentive to cheat by lowering price

OPEC
Saudi Arabia relatively large oil reserves and low
production costs made it "stronger" than the rest of
OPEC,
Even when the rest of OPEC did not obey its cartel
quota (and produced too much), it was still in the
Saudi’s interest to keep to its cartel quota given
OPEC’s dominant strategy of disobeying its quota.
The rest of OPEC does better by disobeying, no mater
what the Saudi’s do.
The Saudi’s best strategy still depends on what the
rest of OPEC does.

OPEC
Two strategies: obey cartel quota and disobey cartel
quota
Saudi Arabia (SA)
disobey obey
restofOPEC
disobey
SA low profits SA moderate profits
OPEC low profits OPEC high profits
obey
SA high profits SA moderate profits
OPEC very low profits OPEC moderate profits

Tacit Collusion
Far less extreme form of cooperation among oligopoly
ﬁrms
Cooperation occurs without any explicit agreement or
any other facilitating practices

Sequential Decisions
Sometimes a game has more than one Nash
Equilibrium and it is hard to say which is more likely
to occur.
When such a game is sequential, perhaps, one of the
Nash equilibria is more likely to occur than the other.
One firm makes its decision first, then a rival firm,
knowing the action of the first firm, makes its decision.
The best decision a manager makes today depends on
how rivals respond tomorrow.
Game tree: shows firms decisions as nodes with
branches extending from the nodes
One branch for each action that can be taken at the
node
Sequence of decisions proceeds from left to right until
final payoffs are reached

Roll-back method (or backward induction)
Method of finding Nash solution by looking ahead to
future decisions to reason back to the current best
decision.
First-mover advantage
If letting rivals know what you are doing by going first
in a sequential decision increases your payoff.
Second-mover advantage
If reacting to a decision already made by a rival
increases your payoff.
Determine whether the order of decision making can
be confer an advantage
Apply roll-back method to game trees for each
possible sequence of decisions

B
Left Right
A
Up
A B
9 8
3 1
Down C D
0 1
0 2
Player B plays ﬁrst (Node 1), Player A sees it and
makes a decision (Node 2)

B moves first
B
(3, 9)
U
(0, 0)
D
L
(1, 8)
U
(2, 1)
D
R
A
Player B’s payoffs are on the right-hand side and
player A’s payoffs are on the left hand side

Sequential Decision - Roll-Back Method
Node 2- Player A
Left-hand side: A chooses U because 3 > 0.
Right-hand side: A chooses D because 1 < 2.
Node 1- Player B
Given that we know that A will choose U (Left-hand
side) and D (Right-hand side), for the analysis, we do
not use D (Left-hand side) and U (Right-hand side)
Then, B chooses L because 9 > 1.
The Nash Equilibrium is (L,U) or (3, 9)
Do the outcome change if A decides ﬁrst?

A moves first
A
(3, 9)
L
(1, 8)
R
U
(0, 0)
L
(2, 1)
R
D
B
Player B’s payoffs are on the right-hand side and
player A’s payoffs are on the left hand side

Sequential Decision - Roll-Back Method
Node 2- Player B
Left-hand side: B chooses L because 9 > 8.
Right-hand side: B chooses R because 0 < 1.
Node 1- Player A
Given that we know that B will choose L (Left-hand
side) and R (Right-hand side), for the analysis, we
drop R (Left-hand side) and L (Right-hand side)
Then, A chooses U because 3 > 2.
The Nash Equilibrium is (L,U) or (3, 9)
The outcome does not change when A moves ﬁrst

Stackelberg Competition
von Stackelberg, H (1934) Market Structure and
Equilibrium: 1st Edition Translation into English,
Bazin, Urch & Hill, Springer 2011, XIV, 134 p, eBook
ISBN: 978-3-642-12586-7
Two ﬁrms, a large and a small ones, are deciding either
being a leader(L) or a follower(F) in the industry,
Both ﬁrms, which sell homogeneous products, and are
subject to the same demand and cost functions.
a Stackelberg duopoly is a sequential game.

Players want to maximize their payoffs (in $ millions)
Large firm
L FSmallfirm
L A B
4 2
0.5 3
F
C D
8 1
1 0.5

Large firm moves first
Largefirm
(0.5, 4)
L
(1, 8)
F
L
(3, 2)
L
(0.5, 1)
F
F
Smallfirm
Large firm’s payoffs are on the right-hand side and
small firms’s payoffs are on the left-hand side

Stackelberg Competition- Roll-Back Method
Node 2- Small firm
Left-hand side: Small firm chooses F because 0.5 < 1.
Right-hand side: Small firm chooses L because 3 >0.5.
Node 1- Large firm
Given that we know that Small firm will choose F
(Left-hand side) and L (Right-hand side), for the
analysis, we drop L (Left-hand side) and F
(Right-hand side)
Then, Large firm chooses L because 8 > 2.
The Nash Equilibrium is (L,F) or (1, 8)

Strategic Entry Deterrence
Ronnie’s Wraps is the only supplier of sandwich
food and makes a healthy proﬁt.
It currently charges a high price and makes a proﬁt of
six thousands per week.
Flash Salads is considering entering the same market.
Ronnie’s Wrap plays high price or low price.
Flash Salads plays enter to the market or stay out

The payoff matrix below defines the profit outcomes
for different possibilities.
Ronnie’s Wraps
High LowFlashSalads
Enter A B
3 1
2 -1
Stayout
C D
6 4
0 0

Complete the next decision tree and solve the game.
Ronnie’s wrap moves ﬁrst.
(, ) (, ) (, ) (, )

References
Varian, Hal R. 2010. Intermediate microeconomics: a
modern approach. New York: W.W. Norton & Co.
Gibbons, R. (1992.). A Primer in Game Theory,
Prentice Hall

Introduction to Game Theory

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