Lecture 6 MGMT 7730 - © 2011 Houman Younessi 1111 A D R 0303 B d r 2.50 A D R 1.50 4.50 B d r 3.50...

Lecture 6

MGMT 7730 - © 2011 Houman Younessi

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Game Theory

Let’s play a game, the centipede game

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.

Lecture 6


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.

Rules to Live (or at least play) By

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.

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

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:

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.

Rupf and Reddy's Strategy

Stay Escalate

D&CH Strategy

Stay 3,4 2,3

Escalate 4,3 3,2

Game (Strategy)

Matrix

Lecture 6


Dominant Strategy

Should DC&H stay or escalate? How about R&R?


Stay Escalate

D&CH Strategy

Stay 3,4 2,3

Escalate 4,3 3,2

Game (Strategy)

Matrix

A dominant strategy is one whose payoff in any outcome, relative to all other feasible strategies, is the highest.

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

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:


$1.00 $1.35 $1.65

D&CH Strategy

$0.95 6,3 1,5 0,6

$1.30 1,7 2,8 2,6

$1.55 4,10 7,14 5,8

Game (Strategy)

Matrix

Lecture 6



$1.00 $1.35 $1.65

D&CH Strategy

$0.95 6,3 1,5 0,6

$1.30 1,7 2,8 2,6

$1.55 4,10 7,14 5,8

Game (Strategy)

Matrix

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

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.

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.

Lecture 6



Product A Product B Product C

D&CH Strategy

Product 1 4,6 9,8 6,10

Product 2 6,8 8,9 7,8

Product 3 9,8 7,7 5,5

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:

Lecture 6




D&CH Strategy

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.

Lecture 6




D&CH Strategy

Product 1 4,6 9,8 6,10

Product 2 6,8 8,9 7,8

Product 3 9,8 7,7 5,5D

D

D

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

Lecture 6




D&CH Strategy

Product 1 4,6 9,8 6,10

Product 2 6,8 8,9 7,8

Product 3 9,8 7,7 5,5

Game (Strategy)

Matrix

D

D

D

Now doing the same analysis this time for DC&H:

R

R

R

The Nash equilibrium pair is when R&R introduces Product A and D&CH introduces product 3.

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

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.

R&R

DC&H

DC&H

DC&H

$1.65

$1.35

$1.00

$1.55$1.30$0.95

$1.55

$1.30

$0.95$1.55

$1.30$0.95

8,5

6,2

6,0

14,7

8,2

5,110,4

7,1

3,6

Lecture 6


R&R

DC&H

DC&H

Do not Expand

Expand

Do not Expand

Expand

Do not Expand

Expand

80,80

60,120

150,60

50,50

For instance R&R and DC&H wish to decide whether to expand or not. The game tree with payoff is illustrated below

Let us solve this game using backward induction

Backward Induction

Lecture 6


R&R

DC&H

DC&H

Do not Expand

Expand

Do not Expand

Expand

Do not Expand

Expand

80,80

60,120

150,60

50,50

A. If R&R expands, then DC&H will receive $50 mil if they expand

A

B. If R&R expands, then DC&H will receive $60 mil if they don’t expand

B

So R&R managers anticipate that if they expand, DC&H will not

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

C

D

So R&R managers anticipate that if they do not expand, then DC&H will

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

Lecture 6


Example: The Centipede Game

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Now use backwards induction to solve the game

Lecture 6


Threats, Commitments and Credibility

A major use of backwards induction is to test out the credibility of threats or commitments of your opponents.

Should you believe others?

When should you believe others?

How do you test for credibility?

Another dominant strategy is to ALWAYS check for credibility first

Only consider credible commitments

Always? Well , almost always

A kiss on the hand is very continental but diamonds are a girl’s best friend

Lecture 6


DC&H

R&R

R&R

Maintain Price

Drop Price

Maintain Price

Drop Price

Maintain PriceDrop Price

50,30

70,20

30,40

20,15

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)

Is this threat credible?

Lecture 6


DC&H

R&R

R&R

Maintain Price

Drop Price

Maintain Price

Drop Price

Maintain PriceDrop Price

50,30

70,20

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

Lecture 6


Consider the consequence of pricing policies per sales period

DC&H

Price High Price Low

R&RPrice High 5,5 1,20

Price Low 20,1 3,3

Game (Strategy)

Matrix

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

Price of Distrust

Lecture 6


Repeated GamesOnce 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:

Games that continue for a time but is known to end at a particular instance.

Indefinite Games:

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?

Lecture 6


Consider again the pricing policies per sales period of DC&H and R&R

DC&H

Price High Price Low

R&RPrice High 5,5 1,20

Price Low 20,1 3,3

Game (Strategy)

Matrix

Definite (finite horizon) Games

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.

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.

Lecture 6


CheatingMany of us have cheated in mathematics, let us look at the mathematics of cheating!

Imagine a one-shot game:

In this game there are three possible profit levels:

PN Nash equilibrium profit

PC Profit when cooperating

PH Profit when cheatingWe 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

Lecture 6


Now Imagine a definite repeating game:

The present value of multiple rounds of such game is:

.discoveredis cheating aftercontinues game whenperiods time of number theis P

edundiscoveroccurs cheating whenperiods time of number theis N

PPC

PPB

r

C

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CPV

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BPV

Nashcooperatej

coperatecheatj

PNP

NNcheatingofCost

NN

cheatingofbenefit

)1(....

)1()1(

)1(....

)1()1(

22

11

__

22

11

__

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.

Lecture 6


Finally, imagine an indefinite repeating game:

The present value of multiple rounds of such game is:

edundiscoveroccurs cheating whenperiods time of number theis N

PPC

PPB

r

C

r

CPV

r

B

r

B

r

BPV

Nashcooperatej

coperatecheatj

NNcheatingofCost

NN

cheatingofbenefit

....)1()1(

)1(....

)1()1(

22

11

__

22

11

__

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)

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.

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.

DC&H

Produce for Consumer

MarketProduce for Industrial

Market

R&R

Produce for

Consumer Market 0,0 7,7

Produce for

Industrial Market 12,12 0,0

Game (Strategy)

Matrix

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 SexesIn 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.

DC&H

Low end product High end product

R&R

Low end product 0,0 11,6

High end product 6,11 0,0

Game (Strategy)

Matrix

Lecture 6


DC&H

Be a Hawk Be a Dove

R&RBe a Hawk -1,-1 10,0

Be a Dove 0,10 5,5

Game (Strategy)

Matrix

Hawks and DovesIn 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

Lecture 6 MGMT 7730 - © 2011 Houman Younessi 1111 A D R 0303 B d r 2.50 A D R 1.50 4.50 B d r 3.50...

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