Lecture 6
Game Theory
Let’s play a game, the centipede game
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D
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2.50
1.50
3.50
3.00
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3.50
6.00
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Lecture 6
Game Theory
Strategy and Strategic Decision Making
Objective: Better decision making
Approach: Strategy
Strategic decision making is characterized by interactive pay off.
Interactive payoff means that the outcome of your decision
depends on both your actions and the action of others.
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Lecture 6
Rules to Live (or at least play) By
Rule No 1: Understand the rules
Rule No 0: Life is a game
What are the basic rules of the game?
1. There are players (stakeholders) in every game (situation).
Know as many of them as possible.
2. Know what can and what cannot be done. Identify the feasible
strategy set
3. Feasible strategy sets of players interact to form a set of
possible outcomes. Know what can and what cannot happen.
Know all (or as many as possible of) potential outcomes.
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Lecture 6
4. Know the rewards and the punishments. Each outcome has a
payoff. Know each payoff.
5. Players are rational but preferences are subjective.
6. Know the order of the play. Games can be simultaneous or
sequential.
7. Know the extent of the game. Games can be one-shots, finite
horizon repeating or indefinite repeating.
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Lecture 6
Example: as it applies to business decision-making
Dewey, Cheetham and Howe Inc. (DC&H) and Rupf & Reddy Inc (RR) are
two competing firms. DC&H and R&R both wish to launch what would
essentially be competing products.
Each has the option of either keeping their development spending at
current rates or to escalate so it can be first to market.
There are no spies, neither knows about the other’s decision until put into
effect
In a rare stroke of luck, the management of both companies are sane
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Lecture 6
If DC&H spends at the current level when R&R escalates, then DC&H will
make $3 Mil and R&R will make $2 Mil.
If DC&H spends at the current level and R&R stays, then DC&H will make
$3 Mil and R&R will make $4Mil
If DC&H escalates and R&R stays, then DC&H will make $4 Mil and R&R
will make $3 Mil
If DC&H escalates and R&R also escalates, then DC&H will make $3 Mil
and R&R will make $2 Mil
Now let us identify the central elements:
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Lecture 6
1. Players – DC&H and R&R
2. Each player has the choice to either stay (spend at current levels) or
escalate (put more money into the project)
3. There are four outcomes: when both escalate, when both stay and two
cases when one escalates and one stays.
4. Payoffs are as reflected in the table below:
5. We assume that decisions will be taken such that to maximize payoff to self.
6. We won’t know of the other player’s move so the game is simultaneous
7. Once the decision is made, that’s it, so the game is a one-shot.
Game
(Strategy)
Matrix
D&CH
Strategy
Rupf and Reddy's Strategy
Stay
Escalate
Stay
3,4
2,3
Escalate
4,3
3,2
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Lecture 6
Dominant Strategy
Should DC&H stay or escalate? How about R&R?
Game
(Strategy)
Matrix
D&CH
Strategy
Rupf and Reddy's Strategy
Stay
Escalate
Stay
3,4
2,3
Escalate
4,3
3,2
A dominant strategy is one whose payoff in any outcome, relative to
all other feasible strategies, is the highest.
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Lecture 6
Dominant Strategy
The dominant strategy is therefore to first look for a dominant strategy
How to:
identify and remove all dominated strategies
What is left is either a dominant strategy or a field of options that are clearly
not dominated.
Sometimes, removing a dominated strategy would change a previously
non-dominated strategy into a dominated one and as such a candidate for
removal
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Lecture 6
Example:
DC&H and R&R must now decide on a pricing policy for the new product.
They know now that the other party will introduce a new and similar product.
R&R will have three pricing options $1.65, $1.35 and $1.00
DC&H will also have three pricing options, $1.55, $1.30, $0.95
The payoffs are as below:
Game
(Strategy)
Matrix
D&CH
Strategy
Rupf and Reddy's Strategy
$1.00
$1.35
$1.65
$0.95
6,3
1,5
0,6
$1.30
1,7
2,8
2,6
$1.55
4,10
7,14
5,8
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Lecture 6
Game
(Strategy)
Matrix
D&CH
Strategy
Rupf and Reddy's Strategy
$1.00
$1.35
$1.65
$0.95
6,3
1,5
0,6
$1.30
1,7
2,8
2,6
$1.55
4,10
7,14
5,8
R&R will charge $1.35, and DC&H will charge $1.55
At this stage, no party needs to unilaterally change strategy, we have reached
A DOMINANT STRATEGY EQUILIBRIUM
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Lecture 6
Nash’s Equilibrium
Not many games settle in a dominant strategy equilibrium. This is because
not all games have a clear dominant strategy. Those that do are called
DOMINANCE SOLVABLE.
How do we predict behavior in a game without dominant strategies.
We need to include the future (anticipated) rational actions of others and
still arrive at a rational, stable and optimal solution.
We reach Nash Equilibrium when all players choose their best strategy
assuming that their rivals have done or will do likewise.
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Lecture 6
Note:
1. This does not mean that the game players will cooperate with each
other. It simply means that they will do the best for themselves
knowing that the competition is doing the same.
2. The essence of success becomes correctly predicting the decisions of
others.
3. Only a Nash equilibrium pair (or set) will be optimum for both (all)
players
4. There may be more than one Nash equilibrium point.
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Lecture 6
Example:
In the matrix below we have R&R and DC&H’s profit figures. We assume that
R&R entered the market first and that both firms wish to introduce new products.
Also that each can choose amongst several but must settle on only one product.
What product they will choose depends on what the competition will do.
The outcomes and payoffs are captured below:
Rupf and Reddy's Strategy
D&CH
Strategy
Product A
Product B
Product C
Product 1
4,6
9,8
6,10
Product 2
6,8
8,9
7,8
Product 3
9,8
7,7
5,5
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Lecture 6
Rupf and Reddy's Strategy
D&CH
Strategy
Product A
Product B
Product C
Product 1
4,6
9,8
6,10
Product 2
6,8
8,9
7,8
Product 3
9,8
7,7
5,5
A simple check would indicate that there is no dominated strategy for either firm.
For each strategy, we indicate the behavior of others: For example if D&CH
knew that R&R will introduce product A, what will they do?
DC&H would introduce Product 3, as it gives the highest payoff.
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Lecture 6
Rupf and Reddy's Strategy
Product A
D&CH
Strategy
Product B
Product C
Product 1
4,6
9,8 D
6,10
Product 2
6,8
8,9
7,8 D
Product 3
9,8 D
7,7
5,5
if R&R would introduce product B, then DC&H will introduce Product 1
Likewise if R&R would introduce Product C, DC&H will introduce Product 2
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Lecture 6
Game
(Strategy)
Matrix
D&CH
Strategy
Rupf and Reddy's Strategy
Product A
Product 1
4,6
Product 2
6,8
Product 3
R 9,8 D
Product B
9,8 D
R 8,9
7,7
Product C
R 6,10
7,8 D
5,5
Now doing the same analysis this time for DC&H:
The Nash equilibrium pair is when R&R introduces Product A and
D&CH introduces product 3.
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Lecture 6
Strategic Foresight
Successful game players often find that they need to make decisions now
that would be rational if what is anticipated actually happens in the future
This is called Strategic Foresight
Game theory can formally model strategic foresight through the process
of backward induction. Backward induction is using future information to
move backward in time (sequence) to arrive at a logical situation in the
present.
However, we do need to present an alternate form of game information
presentation to best utilize this approach
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Lecture 6
The Extensive Form
Game information may also be presented using what is termed a “game tree”.
Using the pricing information relative to DC&H and R&R, as presented
before we can also present the game information as below.
DC&H
$1.55
$1.30
$0.95
$1.65
R&R
$1.35
$1.00
DC&H
DC&H
$1.55
$1.30
$0.95
$1.55
$1.30
$0.95
8,5
6,2
6,0
14,7
8,2
5,1
10,4
7,1
3,6
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Lecture 6
Backward Induction
For instance R&R and DC&H wish to decide whether to expand or not. The
game tree with payoff is illustrated below
Do not
Expand
80,80
Do not
Expand
Expand
60,120
Expand DC&H
Do not
Expand
150,60
Expand
50,50
DC&H
R&R
Let us solve this game using backward induction
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Lecture 6
D
DC&H
Do not
Expand
Do not
Expand
80,80
Expand
60,120
A. If R&R expands, then DC&H will receive $50 mil if they expand
B. If R&R expands, then DC&H will receive $60 mil if they don’t expand
So R&R managers anticipate that if they
expand, DC&H will not
C
R&R
Expand DC&H
Do not
Expand
150,60
Expand
50,50
B
A
So R&R managers anticipate that if they
do not expand, then DC&H will
C. If R&R do not expand, then DC&H will receive $120 mil if they expand
D. If R&R do not expand, then DC&H will receive $80 mil if they don’t expand
So if R&R expands, they anticipate a $150 mil payoff (because R&R will not expand), and if they do
not expand, the payoff is $60 mil, so R&R will expand
Given that R&R will expand, DC&H will not
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Lecture 6
Example: The Centipede Game
A
D
B
R
d
A
r
B
R
D
d
A
r
D
B
R
r
d
1
0
2.50
1.50
3.50
3.00
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3.50
6.00
5.00
5.00
Now use backwards induction to solve the game
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Lecture 6
Threats, Commitments and Credibility
Should you believe others?
When should you believe others?
A kiss on the
hand is very
continental but
diamonds are a
girl’s best friend
How do you test for credibility?
A major use of backwards induction is to test out the credibility of threats or
commitments of your opponents.
Another dominant strategy is to ALWAYS check for credibility first
Only consider credible commitments
Always? Well , almost always
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Lecture 6
Consider the following situation:
R&R have expanded the product line and now DC&H wish to counter by
dropping the price of their product. However they are concerned that if they
dropped the price, R&R would also drop theirs. R&R are telling some common
suppliers that they would drop their price if DC&H would.
The tree below depicts the situation and the payoffs to each (values in $Million)
R&R
Maintain
Price
Maintain
Price
Drop
Price
50,30
70,20
DC&H
Drop
Price
R&R
Maintain
Price
Drop
Price
30,40
20,15
Is this threat credible?
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Lecture 6
R&R
Maintain
Price
Maintain
Price
Drop
Price
50,30
70,20
DC&H
Drop
Price
R&R
Maintain
Price
Drop
Price
30,40
20,15
If DC&H drop prices, R&R will maintain theirs (otherwise they would lose
$10mil).
If DC&H maintains prices, R&R will drop theirs (otherwise they would lose
$20mil).
DC&H should drop prices as the R&R threat is not credible
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Lecture 6
Price of Distrust
Consider the consequence
of pricing policies per sales
period
Game
(Strategy)
Matrix
DC&H
Price High
Price Low
Price High
5,5
1,20
Price Low
20,1
3,3
R&R
Playing this game once, would have both parties price low (as they cannot
afford both to price high and see the competition drop their prices). As
such they each lose $2 mil per period.
They could increase their respective profits by $2 mil each if they could
trust each other
When you are playing the game once, there is no reason to trust, but if
you are in it for the long haul, the situation is different
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Lecture 6
Repeated Games
Once there is the prospect of a future, behavior changes as new concepts
such as trust, reputation, reciprocity and revenge come into play.
Remember: Commitments must remain credible
Definite Games:
Indefinite Games:
Games that continue
for a time but is known
to end at a particular
instance.
Games that continue
without any knowledge
of whether they will
terminate or when they
will do so.
Under which circumstance is it easier to establish and maintain trust?
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Lecture 6
Definite (finite horizon) Games
Consider again the pricing policies
per sales period of DC&H and R&R
Game
(Strategy)
Matrix
DC&H
Price High
Price Low
Price High
5,5
1,20
Price Low
20,1
3,3
R&R
If the two firms cooperate and price high each receives a payoff of $5 mil.
If one defects, and prices low, it will have a windfall of $20 mil for a single
period. The other will then price low and each will receive $3 mil. So the
incremental of $15 mil will be more than eroded in 8 cycles.
In finite horizon games, as the game progresses, the impact and
importance of the future shrinks. In the last period, the Nash
equilibrium is identical to a one-shot game. Using backward induction
one can see that the equilibrium for the entire game will be forced into
one identical to a one-shot.
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Lecture 6
Indefinite (infinite horizon) Games
These are fundamentally different. The equilibrium becomes a function of
probable future behavior. These are of course much harder to predict.
The presence of a future and incomplete information are the necessary
ingredients for building
reputation
Reputation is simply the integrated history of past behavior. The past is
a good indicator of the future.
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Lecture 6
Cheating
Many of us have cheated in mathematics, let us look at the mathematics of cheating!
Imagine a one-shot game:
PN Nash equilibrium profit
In this game there are three possible profit levels:
PC Profit when cooperating
PH Profit when cheating
We can calculate the benefit from cheating as:
B=PH-PC
As presumably we cheat to get an advantage, B should always be positive.
We can calculate the cost of cheating as:
C=Pc-PN
But as the game is one-shot and there is no consequence to cheating C is always
Zero.
Absolute profit is Π=B-C=PH-PC – 0 = PH-PC which is always positive
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Lecture 6
Now Imagine a definite repeating game:
The present value of multiple rounds of such game is:
B1
B2
BN


....

( 1  r )1 ( 1  r )2
( 1  r )N
C1
C2
CP



....

( 1  r )N 1 ( 1  r )N  2
( 1  r )N P
PVbenefit _ of _ cheating 
PVCost _ of _ cheating
B j  Pcheat  Pcoperate
C j  Pcooperate  PNash
N is the number of time periods when cheating occurs undiscovered
P is the number of time periods when game continues after cheating is discovered.
Rule: Not cheating maximizes the value of a firm when the present value of
the costs of cheating is greater than the present value of the benefits from
cheating.
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Lecture 6
Finally, imagine an indefinite repeating game:
The present value of multiple rounds of such game is:
B1
B2
BN


....

( 1  r )1 ( 1  r )2
( 1  r )N
C1
C2


 ....
( 1  r )N 1 ( 1  r )N  2
PVbenefit _ of _ cheating 
PVCost _ of _ cheating
B j  Pcheat  Pcoperate
C j  Pcooperate  PNash
N is the number of time periods when cheating occurs undiscovered
Rule: Not cheating always maximizes the value of a firm because at some
stage (N) cheating will be discovered and from there on there will be a cost
that eventually will become greater than the value of benefits from cheating
(Provided of course that the game will go on for long enough)
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Lecture 6
Coordination Games
When a game has more than one Nash equilibrium, any one such
equilibrium might be selected by a given player.
By coordination they may be able to improve their odds by selecting the most
preferred equilibrium point.
There are many different types of coordination some collaborative, some
competitive. We shall investigate some here.
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Lecture 6
Matching Games
In this game, players generally have the same preferences in the outcome they
seek. Impediment may be in ability to communicate or asymmetric information.
In the game below, both 7,7 and 12,12 are Nash equilibria. But both DC&H
and R&R would no doubt prefer 12,12.
Game
(Strategy)
Matrix
R&R
DC&H
Produce for
Consumer
Market
Produce for Industrial
Market
Produce
for
Consumer
Market
0,0
7,7
Produce
for
Industrial
Market
12,12
0,0
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With collaboration and
communication (and
ensuring that neither party
will cheat, both parties can
settle on a coordinated
strategy of DC&H
producing for the
consumer market and
R&R for the industial
market.
Lecture 6
Battle of Sexes
In this game players still wish to coordinate but on different outcomes. Each
preferred payoff by one is NOT favored by the other.
If the game is repeated indefinitely, players often switch between outcomes
so both would gain. In one shot outcomes, it is impossible to predict the
outcome without good knowledge of the players reputations and styles.
Game
(Strategy)
Matrix
R&R
DC&H
Low end
product
High end product
Low end
product
0,0
11,6
High end
product
6,11
0,0
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Lecture 6
Hawks and Doves
In this game the players are locked in a conflict. If both act like hawks, there
is usually poor payoff (often loss) as a consequence of conflict. If one acts
hawkish and the other dovish, the hawk has an immediate advantage
Game
(Strategy)
Matrix
DC&H
Be a Hawk
Be a Dove
Be a Hawk
-1,-1
10,0
Be a Dove
0,10
5,5
R&R
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