3.2.2. dynamic games of complete information: backward induction and subgame perfection -...

3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection

- Credibility and Commitment -

Strategic Behavior in Business and Econ

Outline

3.1. What is a Game ?3.1.1. The elements of a Game3.1.2 The Rules of the Game: Example3.1.3. Examples of Game Situations3.1.4 Types of Games

3.2. Solution Concepts3.2.1. Static Games of complete information: Dominant

Strategies and Nash Equilibrium in pure and mixed strategies



Reminder

Dynamic games are games in which players move sequentially

In dynamic games, the players should look forward (to understand the game) and think backwards (to solve the game)

Dynamic games are better represented in tree form Backward Induction is the technique that finds only the “good” (sequentially

rational) equilibria in games with Perfect Information Subgame Perfection is the technique that finds only the “good” (sequentially

rational) equilibria in games with Imperfect Information Backward Induction and Subgame Perfection show that “incredible threats”

(or “incredible promises”) should not be taken into account• What a player knows about the choices of other players is as important as what the other players know about what you know about their choices


Credibility and Commitment

We have seen that “threats” or “promises” are not credibleif they are not sequentially rational

You don't become credible just by “talking”

To become credible, commitment is necessary


The market entry game

A firm wants to enter a new market, and its main concernis about the reaction of an incumbent company that currentlyis making a profit $100,000If the reaction is aggressive, the challenger will suffer a lossof $10,000, and the incumbent's profits will be reduced toonly $20,000 (because of the costs of the fight being $80,000). On the other hand, if the incumbent chooses to accommodate to thenew market scenario, then the two companies will share the$100,000 profit ($50,000 each)Obviously, the challenger can always choose to stay out if thatseems to be the best choice (with a profit of $0)


The game represented

(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)



(in table form)

Firm 1

Firm 2



(in table form)

Firm 1

Firm 2

There are two Nash Equilibria



(in table form)

Firm 1

Firm 2

1) Stay Out, Aggressive2) Enter, Accommodate



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

1) Stay Out, Aggressive2) Enter, Accommodate



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

1) Stay Out, AggressiveIs a Nash Equilibrium



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Given that Firm 1 is Staying Out, Firm 2 is fine with the Aggressivestrategy (Its payoff is 100,000)In fact, if Firm 1 Stays Out, anystrategy is a “Best Reply” for Firm 2



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Given that Firm 2 is Aggressive, Firm 1 is fine with the Stay Outstrategy (Its payoff is 0, which isbetter than -10,000)Given Firm 2's choice, Stay Out is a“Best Reply” for Firm 1



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Aggressive is a Best Reply to Stay Out

Stay Out is a Best Reply to Aggressive



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Firm 2 uses its strategy as a “Threat”.But the threat is incredible.If Firm 1 Enters, Firm 2 won't carryout its threat !!



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

This equilibrium won't be selected by Backward Induction



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

2) Enter, AccommodateIs the other Nash Equilibrium



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Given that Firm 1 Enters, Firm 2 is better off by choosing Accommodate (a payoff of 50,000 instead of 20,000)Accommodate is a Best Reply



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Given that Firm 2 Accommodates, Firm 1 is better off by choosing Enter (a payoff of 50,000 instead of 0)Enter is a Best Reply



(in tree form)

Firm 1

Firm 2

Enter

Aggressive

Stay out

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Accommodate is a Best Reply toEnter

Enter is a Best Reply toAccommodate



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 100,000)

(- 10,000 , 20,000)

(50,000 , 50,000)

Here there are not “incrediblethreats”

This equilibrium is the one selectedby Backward Induction


How to commit ?

Consider the following modification:

Before the game starts, Firm 2 makes an investment of $20,000 in an advertisement campaign. This investment will only be profitable in reducing the cost of being aggressive (from 80,000 to 60,000)

• If Firm 1 stays out, the profit will be 100,000 – 20,000 = 80,000 (the whole market profit minus the advertisement cost)

If Firm 1 enters and Firm 2 Accommodates, the profit will be 50,000 – 20,000 = 30,000 (half the market profit minus the advertisement cost)• If Firm 1 enters and Firm 2 is Aggressive, the profit will be 100,000 – 20,000 - 60,000 = 20,000 (the whole market profit minus the advertisement cost minus the fight cost)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 80,000)

(- 10,000 , 20,000)

(50,000 , 30,000)

The game changes

Notice !! The investment reduces (almost) ALLthe payoffs for Firm 2 !!!

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 80,000)

(- 10,000 , 20,000)

(50,000 , 30,000)

What is NOW the outcome thegame by Backward Induction ?

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 80,000)

(- 10,000 , 20,000)

(50,000 , 30,000)

Still the threat of being aggressiveis incredible !!Nothing changes with respect to the previous situation

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)


How to commit ?

Consider the following modification instead:

Before the game starts, Firm 2 makes an investment of $40,000 in an advertisement campaign. This investment will only be profitable in reducing the cost of being aggressive (from 80,000 to 45,000)

• If Firm 1 stays out, the profit will be 100,000 – 40,000 = 60,000 (the whole market profit minus the advertisement cost)

If Firm 1 enters and Firm 2 Accommodates, the profit will be 50,000 – 40,000 = 10,000 (half the market profit minus the advertisement cost)• If Firm 1 enters and Firm 2 is Aggressive, the profit will be 100,000 – 40,000 - 45,000 = 15,000 (the whole market profit minus the advertisement cost minus the fight cost)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 60,000)

(- 10,000 , 15,000)

(50,000 , 10,000)

The game changes

Notice !! The investment reduces ALL thepayoffs for Firm 2 !!!

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 60,000)

(- 10,000 , 15,000)

(50,000 , 10,000)

What is NOW the outcome thegame by Backward Induction ?

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 60,000)

(- 10,000 , 15,000)

(50,000 , 10,000)

Now the threat is credible !!

By making the right investment Firm 2 shows that it's committed to the market

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)



(in tree form)

Stay out

Firm 1

Firm 2

Enter

Aggressive

Accommodate

(0, 60,000)

(- 10,000 , 15,000)

(50,000 , 10,000)

Although Firm 2 is reducing ALL itspayoffs, at the end makes more money (60,000) than in theprevious case (50,000)

(0, 100,000)

(50,000 , 50,000)

(- 10,000 , 20,000)


Summary

Incredible threats won't be taken into account (bluffing) If you want to be credible, you must commit “Hurt yourself to help yourself” Reduce your payoffs, or your strategies, to show you are

not bluffing If done right, this is enough to convince your opponents

that your threats are credible But, once again, it is important that your opponents know

that your are committed ! This is like “burning bridges” or “throwing the steering

wheel out of the window”, it must be done with care and in the right way



- Some Games -


Military recruitment during the Civil War Cold War (revisited) Financial Crisis (“Toy” version)

Some Games


During the Civil War, recruitment was an important issue for the two sides. According to the law, if you were called to serve you could not refuse. The only way to avoid being called is that someone else is called.

(Recruitment was fully random) There was also an option to become volunteer. Being a volunteer was rewarded Being called to served had a high cost (higher than the reward) The “dilemma” was: Should I wait to see if I am not called or

should I become a volunteer to get a reward ?

Military recruitment during the Civil War(1862 - 1865)


The Game

Let us assume, for simplicity, that there are only two players in town

Player 1 chooses first and must decide between being Volunteer or Wait Then Player 2, knowing the choice by Player 1, must decide also between being Volunteer or

Wait If one of the players becomes volunteer, the game is over Otherwise, one of the players is called to serve “at random” (50% chances)

The “reward” for being volunteer is of $300. No “reward” if you are called• The “cost” of serving is of $400 (either for being volunteer or for being called to serve)

If not serving, the payoff is 0 (no reward, no cost)



The Game represented (tree form)


Wait

Player 1Volunteer

Wait

Volunteer

Player 2

Player 2 is called

Player 1

is called

Player 0

(½)

(½)

-100, 0

0, -100

-400, 0

0, -400


Expected Payoffs (for each player) if both wait


Wait

Player 1

WaitPlayer 2

Player 2 is called

Player 1

is called

Player 0

(½)

(½)

-400, 0

0, -400

E(Wait,Wait)=( ½ )·(-400) + ( ½ )·(0) = -200




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200


By Backward Induction


Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200


By Backward Induction – Final Outcome


Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

This would bethe final outcomeof the gamePlayer 1 Waitswhile Player 2goes Volunteer




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

Player 2 prefersthe sure loss of-100 rather thanthe risk of beingrecruited




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

Knowing that,Player 1 prefersto Wait




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

Player 2 is a“ReluctantVolunteer”




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

Notice thatPlayer 2 might threat Player 1with WaitingAnd then, Player 1would choose toVolunteer




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

That would be a(Nash) equilibrium.But once again, it involves anincredible threat.It won't be carriedout




Wait

Player 1Volunteer

Wait

Volunteer

Player 2

-100, 0

0, -100

-200, -200

This is the unique equilibriumthat is credibleIt exhibits a “first mover advantage”


2.1 million soldiers served in the Civil War

Only 50,000 were recruited

96 % of the soldiers were volunteers !! (reluctant ?)



Cold War (revisited) (based on “Game theory and the Cuban missile crisis”

by Steven J. Brams)

(based on “Mutually Assured Destruction” by Roy Gardner)

This is a real life (historical) version of a Game of Chicken.It goes back to the 60's, when the USA and the USSR wereengaged in a so called “cold war”One of the episodes of the “cold war” was the Cuban missile crisis

The Cuban missile crisis was precipitated by a Soviet attempt in October 1962 to install medium-range and intermediate-range nuclear-armed ballistic missiles in Cuba that were capable of hitting a large portion of the United States. The goal of the United States was immediate removal of the Soviet missiles, and U.S. policy makers seriously considered two strategies to achieve this end


A naval blockade (B), or "quarantine" as it was euphemistically called, to prevent shipment of more missiles, possibly followed by stronger action to induce the Soviet Union to withdraw the missiles already installed.

A "surgical" air strike (A) to wipe out the missiles already installed, insofar as possible, perhaps followed by an invasion of the island.

The alternatives open to Soviet policy makers were:

Withdrawal (W) of their missiles. • Maintenance (M) of their missiles.

NEW !! If both countries escalate, a FINAL DECISION mustbe taken


The environment of the game

Players: USA and USSR Strategies: For USA: Blockade (B) or Air Strike (A)

For USSR: Withdrawal (W) or Maintenance (M) (plus a FINAL DECISION)

Payoffs: (to be defined)

The Rules of the Game

Timing of moves Simultaneous AND Sequential

Nature of conflict and interaction Conflict (anti-coordination)

Information conditions Symmetric


The payoffs

This is “Game of Chicken”. Both would like the other to “swerve”. If they “stay straight”, the crash is the Nuclear War

USA USSRBest outcome (A,W) (B,M) 4Second Best (B,W) (B,W) 3Third Best (B,M) (A,W) 2Worst Outcome (A,M) (A,M) 1


Cuban Missile crisis (static version)

W M

BCompromise

3, 3

USSR wins

2 , 4 USA wins

4 , 2

Nuclear War

1 , 1 A

USA

USSRWe have two solutionsin the static version of

the Game


Cuban Missile crisis (static version)

W M

BCompromise

3, 3

USSR wins

2 , 4 USA wins

4 , 2

Nuclear War

1 , 1 A

USA

USSRLet us make the game

“sequential” and defer thefinal decision of “Nuclear

War”


Cuban Missile crisis (dynamic version)

Air Strike

USABlockade

USSR

Maintain

Withdraw3, 3

2, 4

USSR Maintain

Withdraw 4, 2

USA

USSR

The Final Decision game



Air Strike

USABlockade

USSR

Maintain

Withdraw3, 3

2, 4

USSR Maintain

Withdraw 4, 2

USA

USSR


What would bethe outcome ofthe Final DecisionGame ?



Air Strike

USABlockade

USSR

Maintain

Withdraw3, 3

2, 4

USSR Maintain

Withdraw 4, 2

USA

USSR


The Final DecisionGame has TWOequilibria1) Doomsday,Doomsday2) Retreat, Retreat



Air Strike

USABlockade

USSR

Maintain

Withdraw3, 3

2, 4

USSR Maintain

Withdraw 4, 2

Outcome of theFinal Decision game

Equilibrium of thewhole game when theoutcome of the FinalDecision (sub)game isDoomsday, Doomsday

1 , 1



Air Strike

USABlockade

USSR

Maintain

Withdraw3, 3

2, 4

USSR Maintain

Withdraw 4, 2

Outcome of theFinal Decision game

Equilibrium of thewhole game when theoutcome of the FinalDecision (sub)game isRetreat, Retreat

3 , 3


In the two equilibria, the USA strategy at the beginning is the Same: Air Strike (that is, being aggressive)

Then, in the first equilibrium the USSR Maintains the missiles while in the second Withdraws

In some sense the United States "won" by getting the Soviets to withdraw their missiles (but Premier Nikita Khrushchev of the Soviet Union at the same time extracted from President Kennedy a promise not to invade Cuba, which seems to indicate that the eventual outcome was a compromise of sorts)• According to game theory, if the USSR withdrew (first equilibrium) is because for some reason thought that the Doomsday threat was credible !!

And according to Game Theory, it is credible indeed !!


“To continue to deter in an era of strategic nuclear equivalence, it is necessary to have nuclear (as well as conventional) forces such that in considering aggression against our interests any adversary would recognize that no plausible outcome would represent a victory or any plausible definition of victory. To this end and so as to preserve the possibility of bargaining effectively to terminate the war on acceptable terms that are as favorable as practical, if deterrence fails initially, we must be capable of fighting successfully so that the adversary would not achieve his war aims and would suffer costs that are unacceptable, or in any event greater than his gains, from having initiated an attack.”

President Carter in 1980,

Presidential Directive 59, Nuclear Weapons Employment Policy


Financial Crisis (Toy version)

When we bring our money to a Bank we hope that the Bank is going to invest wisely

We expect to have our money back plus some interest earnings

But we do not know what the Bank does with our money (or, perhaps, we find out too late)



The Game

An individual (the Costumer) has $100,000 to invest and has two alternatives:

To buy a Treasury Bond (T-Bond) that has a 3% interest rate (for sure)To put the money in the Bank that offers a Certificate of Deposit (CD)

that promises an interest rate of 10% (not insured)

If the Bank gets the deposit it will invest the money in some financial derivative. Financial derivatives are very risky, but the risk can be reduced if the Bank puts a lot of effort in the search of the right investment. A good investment yields a 12% interest whereas a bad investment yields a -100% interest (total loss !)

High effort: A good investment is found with a probability of 99%• Low effort: A good investment is found with a probability of 90%

The effort has a cost: $300 for the high effort, $100 for the low effort



The Game (payoff computations)

For the Costumer

From the T-Bond: 100,000 + 3% = 103,000From the CD:

• If good Investment: 100,000 + 10% = 110,00 If bad Investment: 100,000 - 100% = 0

For the BankIf the Costumer buys the T-Bond: 0If the Costumer buys the CD: If good Investment with High effort: 100,000 * 2% - 300 = 1,700 If good Investment with Low effort: 100,000 * 2% - 100 = 1,900

If bad Investment with High effort: 100,000 * (- 100%) - 300 = -300• If bad Investment with Low effort: 100,000 * (- 100%) - 100 = -100




T-Bond

Costumer

CD

Bank

Low effort

High effort Bad Investment

Good investment

Chance

Chance

Good investment

Bad Investment

(0.99)

(0.90)

(0.01)

(0.10)

103,000 , 0

110,000 , 1700

0 , -300

110,000 , 1900

0 , -100



The Game (expected payoffs for the Bank)

E(High effort) = (0.99)·(1,700) + (0.01)·(-300) = 1,680

E(Low effort) = (0.90)·(1,900) + (0.10)·(-100) = 1,700

The Game (expected payoffs for the Costumer)

E(High effort) = (0.99)·(110,000) + (0.01)·(0) = 108,900

E(Low effort) = (0.90)·(110,000) + (0.10)·(0) = 99,000



The Game represented (simplified tree form)

T-Bond

Costumer

CD

Bank

Low effort

High effort

103,000 , 0

108,900 , 1,680

99,000 , 1,700



The Game represented (Backward Induction)

T-Bond

Costumer

CD

Bank

Low effort

High effort

103,000 , 0

108,900 , 1,680

99,000 , 1,700

The solution by Backward Induction shows that the Bank willChoose a low effort and then the Costumer will prefer theT-Bond (sure option)



The Game represented (Backward Induction)

T-Bond

Costumer

CD

Bank

Low effort

High effort

103,000 , 0

108,900 , 1,680

99,000 , 1,700

High effort would be an “incredible promise”

3.2.2. dynamic games of complete information: backward induction and subgame perfection -...

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