BUSA 4800/4810

Game Theory Lecture

Sequential Games and Credible Threats

Winter 2008

The “Mother & Child” Game

• A Child is being “BAD”

– (i.e. he is banging on the coffee table with a Hammer…)

• Moms yells “STOP!!...

• …Or I will KILL You!!”

• This is a Sequential Game

Child

good bad

Mom Mom

Kill Kill No Kill No Kill

Game Tree:

This is the EXTENDED form of the game

Child

good bad

Mom Mom

Kill Kill No Kill No Kill

C: -infinity M: -infinity

C: +1 M: -1

C: -infinity M: -infinity

C: - 1 M: +1

Mom Chooses…

Child Chooses…

With Payoffs

Introduction to Sequential Games

• Not all games are played simultaneously. In fact, many strategic situations involve sequential decision-making.

• Consider the simultaneous game of implementing a new communications system. Both Boeing and Airbus benefit by choosing the same system (scale effects, learning curve effects for airlines, etc.)

• Verify there are 2 Nash equilibria:

(alpha, alpha); (beta, beta)

(100, 50)

(40, 40)

(25, 25)

(50, 100)

Airbus

Boeing

alpha beta

alpha

beta

Simultaneous Choice in Extensive Form (game trees)

B

Alpha Beta

A

Node 1

Beta Alpha

Payoffs (100,50) (50,100)

Beta Alpha

A

We can capture the exact same game in extensive form. Note the use of an information set to capture the idea that Airbus doesn’t know whether its at Node 2 or 3 when they make their choice. In other words, Airbus must choose without knowing what Boeing has done;

this is what makes it simultaneous play.

(40,40) (25,25)

Node 2 Node 3

Information Set

2-Stage Sequential Games

B

A A

alpha beta

alpha

• Now suppose the game is played sequentially, where Boeing goes first.

• Here, Airbus knows what choice Boeing

has made; i.e., Airbus knows where it is in the game tree (nodes 2 and 3 are now in different information sets).

• To solve this game we use “backward

induction.” Boeing anticipates what Airbus will do at nodes 2 and 3, and then makes its choice knowing what Airbus will do in response.

• 2nd Stage: At node 2, Airbus will choose alpha

(50>40) At node 3, Airbus will choose beta

(100>25) • 1st Stage: Knowing how Stage2 will unfold, Boeing

will choose alpha (since 100>50) • NE consists of the strategy profile {alpha, alpha}.

beta beta alpha

(100,50) (40,40) (25,25) (50,100)

Comments

• Note that while the simultaneous game has 2 Nash equilibria, the sequential game has only 1 Nash equilibrium.

• Note also that Boeing has an advantage by virtue of choosing first. However, if Airbus made its choice first, Airbus would have an advantage, and the equilibrium would be different. In this case, Airbus would know that Boeing has an incentive to match technologies.

• Verify that the NE with Airbus choosing first is {Beta, Beta}.

• You must not conclude from this that all sequential games have first-mover advantages. They don’t. Sometimes its pays to move second (e.g., product imitation, process innovation through reverse engineering, etc.).

Credible Threats

• With Boeing choosing first, Airbus has an incentive to influence the actions of Boeing in ways favorable to itself.

• Specifically, Airbus would like Boeing to choose “beta” since 100>50.

• So how might Airbus get Boeing to choose “beta”? Suppose Airbus announces its plan (in the media) to choose “beta” no matter what Boeing does, which hopefully gives Boeing an incentive to choose “beta” as well (since 50>40).

• Is this a credible threat?

• Clearly not! If Boeing ignores Airbus’ threat and chooses alpha, Boeing knows it’s in Airbus’ interest to also adopt alpha. Thus, Airbus’ threat is cheap-talk.

Credible Threats: An Example

• A Mountain & a village

• 1000 people will climb Mountain

• At the Top of the Mountain:

– Each person wants Exactly One Beer

– Reservation Price for each is $5

• i.e. if Price < 5 Demand = 1000 beer

• if Price > 5 Demand = 0

Credible Threats: An Example

• SKIPPY decides to take beer up Mountain to sell to climbers

• Info:

– Beer in village costs $1 each

– It costs $1.50/beer to transport each way ($3/beer round trip)

– Beer at top costs $2.50

– Beer brought back costs $4.00

Credible Threats: An Example

If everything goes according to plan:

• Skippy sells 1000 beer for $5

• Skippy’s costs are $2.50

Total Revenue = $5000

Total Cost = $2500

Profit = $2500

Credible Threats: An Example

• However, MYRTLE also decides to bring 1000 Beer to the top of the mountain

• Skippy and Myrtle arrive at the same time… So do the climbers

• Questions:

– How many beer are consumed?

– How many beer are brought back to village?

– -What is the equilibrium price of BEER on the Mountain?

Credible Threats: An Example

• First, NO beer is ever brought off Mountain

• Second, only 1000 Beer are consumed

• Third, the equilibrium price is ZERO

• Now suppose that Skippy was able to get to the top of the mountain First?

Credible Threats: An Example

• If Skippy gets her beer to the top first she will have made a “Commitment”

• Myrtle knows that Skippy will giver her beer away for free rather than return it.

• If Myrtle takes beer to the top, she will incur a $2500 expense that is avoidable

• Myrtle will NOT bring beer up

Credible Threats (Boeing/Airbus cont.)

B

A A

alpha beta

alpha

• Credible threats require Airbus to restrict its own future actions (i.e., not choosing alpha) by making a binding commitment to beta. A binding commitment involves ensuring that Airbus will choose beta no matter what Boeing does. Only then will Boeing change their beliefs about what Airbus will do.

• For example, suppose Airbus signs a long-

term contract with beta company. Contract stipulates that if Airbus breaches the contract (by choosing alpha) Airbus pays beta company 20 in damages.

• Now, the payoff for {alpha, alpha} is

(100,30), and {beta, alpha) is (25,5). • Is Airbus’ threat of choosing beta credible

now? • Yes! Airbus has an incentive to choose

beta no matter what Boeing does, and this commitment is sufficient to change Boeing’s beliefs.

• The NE is now {beta, beta}=(50,100).

beta beta alpha

(100,50) (40,40) (25,25) (50,100)

30 5