1 1 deep thought ba 592 lesson i.3 sequential move theory at first i thought, if i were superman, a...

1 1

Deep ThoughtDeep Thought

BA 592 Lesson I.3 Sequential Move Theory

At first I thought, At first I thought, if I were Superman, a if I were Superman, a perfect secret identity would be “Clark perfect secret identity would be “Clark Kent, Dentist”, because you could save Kent, Dentist”, because you could save money on tooth X-rays. But then I money on tooth X-rays. But then I thought, if a patient said, “How’s my thought, if a patient said, “How’s my back tooth?” and you just looked at it back tooth?” and you just looked at it with your X-ray vision and said, “Oh it’s with your X-ray vision and said, “Oh it’s okay”, then the patient would probably okay”, then the patient would probably say, “Aren’t you going to take and X-say, “Aren’t you going to take and X-ray, stupid?” and then he probably ray, stupid?” and then he probably wouldn’t even pay his bill. --- by Jack wouldn’t even pay his bill. --- by Jack Handey.Handey.

2 2

Lesson overviewLesson overview


Chapter 3 Games with Sequential MovesChapter 3 Games with Sequential Moves

Lesson I.3 Sequential Move TheoryLesson I.3 Sequential Move Theory

Each Example Game Introduces some Game TheoryEach Example Game Introduces some Game Theory•Example 1: A Rollback SolutionExample 1: A Rollback Solution•Example 2: A Game TreeExample 2: A Game Tree•Example 3: Off the Equilibrium PathExample 3: Off the Equilibrium Path•Example 4: Multiple EquilibriaExample 4: Multiple Equilibria•Example 5: Jealous HumansExample 5: Jealous Humans•Example 6: Simple HumansExample 6: Simple HumansLesson I.4 Sequential Move ApplicationsLesson I.4 Sequential Move Applications

3 3

• Sequential moves are strategies where there is a strict order of play.

• Perfect information implies that players know everything that has happened prior to making a decision.

• Complex sequential move games are most easily represented in extensive form, using a game tree.

• Chess is a sequential-move game with perfect information.


Example 1: A Rollback SolutionExample 1: A Rollback Solution

4 4BA 592 Lesson I.3 Sequential Move Theory

Backward induction or rollbackBackward induction or rollback solves sequential move games solves sequential move games with perfect information by rolling back optimal strategies from with perfect information by rolling back optimal strategies from the end of the game to the beginningthe end of the game to the beginning..


5 5

Century Mark GameCentury Mark Game

• Played by pairs of players taking turns.Played by pairs of players taking turns.

• At each turn, each player chooses a number between 1 and 10 At each turn, each player chooses a number between 1 and 10 inclusive.inclusive.

• This choice is added to sum of all previous choices (the initial This choice is added to sum of all previous choices (the initial sum is 0).sum is 0).

• The first player to take the cumulative sum to 100 or more The first player to take the cumulative sum to 100 or more wins.wins.

How should you play the game as first player? How should you play the game as first player? Start at the end. Start at the end.

What number gets you to 100 next turn?What number gets you to 100 next turn?

BA 592 Lesson I.3 Sequential Move TheoryBA 592 Lesson I.3 Sequential Move Theory


6 6

Rollback SolutionRollback Solution

• If you bring the cumulative sum to If you bring the cumulative sum to 8989, you can take the , you can take the cumulative sum to 100 and win no matter what your opponent cumulative sum to 100 and win no matter what your opponent does. (Whatever your opponent does you can make the sum of does. (Whatever your opponent does you can make the sum of your two moves equal 11.)your two moves equal 11.)

• Hence, if you bring the cumulative sum to Hence, if you bring the cumulative sum to 7878, you can bring , you can bring the cumulative sum to 89 on your next turn (and so eventually the cumulative sum to 89 on your next turn (and so eventually win) no matter what your opponent does.win) no matter what your opponent does.

• And so on for sums And so on for sums 67, 56, 45, 34, 23, 1267, 56, 45, 34, 23, 12..

• Hence, if you play 1 first, you can bring the cumulative sum to Hence, if you play 1 first, you can bring the cumulative sum to 12 on your next turn (and so eventually win) no matter what 12 on your next turn (and so eventually win) no matter what your opponent does.your opponent does.

• That is a strategy --- a complete plan of actions no matter what That is a strategy --- a complete plan of actions no matter what your opponent does. your opponent does.




Game trees or extensive forms Game trees or extensive forms consist of nodes and branchesconsist of nodes and branches. . Nodes are connected to one another by the branches, and come in Nodes are connected to one another by the branches, and come in two types. Some nodes are decision nodes, where a player two types. Some nodes are decision nodes, where a player chooses an action. In some games, chooses an action. In some games, NatureNature is a “player”; Nature is a “player”; Nature can decide whether it rains or snows. The other nodes are can decide whether it rains or snows. The other nodes are terminal nodes, where players receive the outcomes of the actions terminal nodes, where players receive the outcomes of the actions taken by themselves and all other players.taken by themselves and all other players.

Example 2: A Game TreeExample 2: A Game Tree


Emily, Nina, and Talia Emily, Nina, and Talia all live on the same street. Each has been all live on the same street. Each has been asked to contribute to a flower garden. The quality of the garden asked to contribute to a flower garden. The quality of the garden increases with the number of contributions, but each lady also increases with the number of contributions, but each lady also prefers to not contribute. Specifically, suppose each lady gains 2 prefers to not contribute. Specifically, suppose each lady gains 2 dollars worth of happiness from each of the first two dollars worth of happiness from each of the first two contributions to the garden (including her own contribution, if contributions to the garden (including her own contribution, if any) and 0.50 dollars worth from a third contribution, but then any) and 0.50 dollars worth from a third contribution, but then looses 1 dollar if she herself contributes. looses 1 dollar if she herself contributes.

Define the game tree for this Define the game tree for this Street Garden GameStreet Garden Game, then find the , then find the rollback solution. Should Emily contribute?rollback solution. Should Emily contribute?


9 9BA 592 Lesson I.3 Sequential Move TheoryBA 592 Lesson I.3 Sequential Move Theory

3 .5 ,3 .5 ,3 .5

C o n t.

3 ,3 ,4

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

0 ,0 ,0

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ily

Street Garden Game


10 10

3 .5 ,3 .5 ,3 .5

C o n t.

3 ,3 ,4

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

0 ,0 ,0

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ily


Rolling from the end: Talia’s Strategy. (Non-optimal strategies are blacked out.)



Rolling back one from the end: Nina’s Strategy


3 .5 ,3 .5 ,3 .5

C o n t.

3 ,3 ,4

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

0 ,0 ,0

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ily


Rolling back to the beginning: Emily’s Strategy, which completes the rollback solution.


3 .5 ,3 .5 ,3 .5

C o n t.

3 ,3 ,4

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

0 ,0 ,0

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ily


Beliefs Beliefs about strategy off the equilibrium path (strategies that are never acted on) are important to keep players on the equilibrium path. Just as your belief that shooting a gun at your own head will kill you makes you decide to never shoot a gun at your own head.

Example 3: Off the Equilibrium PathExample 3: Off the Equilibrium Path


5 ,5 ,5

C o n t.

3 ,3 ,6

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

2 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ilyStreet Garden Game: alternative payoffs



Street Garden Game: Equilibrium Path


5 ,5 ,5

C o n t.

3 ,3 ,6

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

2 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ily


Emily believes that, if she contributed, then Nina would not contribute but Talia would contribute. But if, instead, Emily believed that, if she contributed, then Nina and Talia would both contribute, then Emily believes contributing gives her payoff 5, which is more than her payoff on the equilibrium path.


5 ,5 ,5

C o n t.

3 ,3 ,6

D o n 't

T a lia

C o n trib u te

3 ,4 ,3

C o n t.

1 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

C o n trib u te

4 ,3 ,3

C o n t.

2 ,1 ,2

D o n 't

T a lia

C o n trib u te

2 ,2 ,1

C o n t.

2 ,2 ,2

D o n 't

T a lia

D o n 't

N in a

D o n 't

E m ily


Rollback equilibria Rollback equilibria are are uniqueunique unless a player gets equal payoffs unless a player gets equal payoffs from two or more different actions. One method to restore a from two or more different actions. One method to restore a unique equilibrium (to be used as a prescription or prediction) is unique equilibrium (to be used as a prescription or prediction) is to question whether a game with equal payoffs is somehow to question whether a game with equal payoffs is somehow exceptional or avoidable.exceptional or avoidable.

Example 4: Multiple EquilibriaExample 4: Multiple Equilibria


Employees Employees know there is a positive gain to their continued know there is a positive gain to their continued employment, and that gain is split with their employer according employment, and that gain is split with their employer according to the employees wages. Suppose Employee A generates 100 to the employees wages. Suppose Employee A generates 100 dollars of gain by remaining employed with Employer B. dollars of gain by remaining employed with Employer B. Employee A is considering increasing his wage demands to one Employee A is considering increasing his wage demands to one of three levels. Those three levels give him either 100%, or 90%, of three levels. Those three levels give him either 100%, or 90%, or 50% of the 100 dollars of gain. or 50% of the 100 dollars of gain. Which wage should the Which wage should the employee demand? employee demand?

Define the game tree for this Define the game tree for this Bargaining GameBargaining Game, then find all the , then find all the rollback equilibria. rollback equilibria.



1 00 ,0

A cce p t

0 ,0

R e je ct

R e spo n d er

I ta ke 10 0%

9 0,10

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 0%

5 0,50

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 5 0%

P ro po serBargaining Game:Game Tree



1 00 ,0

A cce p t

0 ,0

R e je ct

R e spo n d er

I ta ke 10 0%

9 0,10

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 0%

5 0,50

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 5 0%

P ro po serBargaining Game:Partial Rollback Solution



Rollback analysis Rollback analysis was incomplete in that game because the was incomplete in that game because the responder got equal payoffs from two different actions. But that responder got equal payoffs from two different actions. But that is only because the Proposer demanded 100%. What if the is only because the Proposer demanded 100%. What if the Proposer demands 99.99%? Now there is a unique rollback Proposer demands 99.99%? Now there is a unique rollback solution with payoffs almost as high as if a demand of 100% solution with payoffs almost as high as if a demand of 100% were accepted.were accepted.



9 9.9 9 ,0 .01

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 9 .9 9%

9 0,10

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 0%

5 0,50

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 5 0%

P ro po ser

Bargaining Game:Complete Rollback Solution



Humans Humans in some sequential move games do not follow the some sequential move games do not follow the rollback solution because that solution may be felt to be too rollback solution because that solution may be felt to be too unfair.

Economic policymakers thus favor public policies whose Economic policymakers thus favor public policies whose rollback solutions seem fair enough for humans to accept. rollback solutions seem fair enough for humans to accept.

And businesspeople thus adapt strategies depending on whether And businesspeople thus adapt strategies depending on whether they are playing against jealous humans (perhaps some of their they are playing against jealous humans (perhaps some of their customers) or rational players (other businesspeople).customers) or rational players (other businesspeople).

Example 5: Jealous HumansExample 5: Jealous Humans


Shoppers Shoppers know there is a positive gain to making purchases, and know there is a positive gain to making purchases, and that gain is split with sellers according to the purchase price. that gain is split with sellers according to the purchase price. Shopper A generates 100 dollars of gain by buying from Seller B. Shopper A generates 100 dollars of gain by buying from Seller B. Buyer A is considering three alternative price offers. Those Buyer A is considering three alternative price offers. Those three offers give him either 99%, or 90%, or 50% of the 100 three offers give him either 99%, or 90%, or 50% of the 100 dollars of gain. dollars of gain. Which price should Buyer A offer? Which price should Buyer A offer?

Define the game tree for this Define the game tree for this Bargaining GameBargaining Game, then find the , then find the rollback solution. rollback solution. ..



9 9 ,1

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 9%

9 0,10

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 0%

5 0,50

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 5 0%

P ro po serBargaining Game:Game Tree


26 26

9 9 ,1

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 9%

9 0,10

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 0%

5 0,50

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 5 0%

P ro po ser


Bargaining Game:Rollback Solution



Which price should the shopper offer? In the rollback solution, Which price should the shopper offer? In the rollback solution, the shopper should offer the price that gives him 99% of the gain the shopper should offer the price that gives him 99% of the gain from trade. But humans like the seller might not follow the from trade. But humans like the seller might not follow the rollback solution because that solution is too rollback solution because that solution is too unfair. Rather, the . Rather, the shopper may have to offer a price that gives him only 90% or shopper may have to offer a price that gives him only 90% or 50% of the gain from trade. 50% of the gain from trade.



Humans Humans in some sequential move games do not follow the in some sequential move games do not follow the rollback solution because that solution is too rollback solution because that solution is too computationallycomputationally complex. complex.

Chess is a sequential move game with perfect information, so it Chess is a sequential move game with perfect information, so it has a game graph, with an estimated has a game graph, with an estimated 10120 nodes nodes describing all describing all possible board positions. There is a rollback solution, but that possible board positions. There is a rollback solution, but that solution is so computationally complex no human knows all of it. solution is so computationally complex no human knows all of it. It was, therefore, inevitable that computers eventually became It was, therefore, inevitable that computers eventually became better players than humans. In May 1997, a chess playing better players than humans. In May 1997, a chess playing machine “Deeper Blue” beat reigning champion Garry Kasparov, machine “Deeper Blue” beat reigning champion Garry Kasparov, by 3½ to 2½ in a six game match. Recent progress in computer by 3½ to 2½ in a six game match. Recent progress in computer play is software than can run on common personal computers.play is software than can run on common personal computers.

Example 6: Simple HumansExample 6: Simple Humans


Humans Humans in other sequential move games do not follow the other sequential move games do not follow the rollback solution because that solution is too rollback solution because that solution is too conceptually complex. complex.

Economic policymakers thus favor public policies whose Economic policymakers thus favor public policies whose rollback solutions are simple enough for humans to compute. rollback solutions are simple enough for humans to compute.

And businesspeople thus adapt strategies depending on whether And businesspeople thus adapt strategies depending on whether they are playing against simple humans (perhaps some of their they are playing against simple humans (perhaps some of their customers) or playing against rational players (other customers) or playing against rational players (other businesspeople).businesspeople).



Buyers and Sellers Buyers and Sellers trading over the internet risk sending money trading over the internet risk sending money or goods and not getting what was agreed upon. One solution that or goods and not getting what was agreed upon. One solution that minimizes your exposure to fraud is to trade a little at a time.minimizes your exposure to fraud is to trade a little at a time.



Suppose Albert values 6 disposable DVDs at $3 each, suppose it costs Suppose Albert values 6 disposable DVDs at $3 each, suppose it costs Blockbuster $1 to provide each DVD, and suppose Blockbuster sells Blockbuster $1 to provide each DVD, and suppose Blockbuster sells DVDs for $2 each. DVDs for $2 each. Should Blockbuster send the first DVD to Albert? Should Blockbuster send the first DVD to Albert?

• If the first DVD is sent, Albert (A) faces a decision: steal the DVD and If the first DVD is sent, Albert (A) faces a decision: steal the DVD and terminate the relationship; or, send $2 for the first DVD.terminate the relationship; or, send $2 for the first DVD.

• If the first $2 is sent, Blockbuster (B) faces a decision: take the $2 and If the first $2 is sent, Blockbuster (B) faces a decision: take the $2 and terminate the relationship; or, send the second DVD to A.terminate the relationship; or, send the second DVD to A.

• If the second DVD is sent, A faces a decision: steal the DVD and terminate If the second DVD is sent, A faces a decision: steal the DVD and terminate the relationship; or, send $2 for the second DVD.the relationship; or, send $2 for the second DVD.

• If the second $2 is sent, B faces a decision: take the $2 and terminate the If the second $2 is sent, B faces a decision: take the $2 and terminate the relationship; or, send the third DVD to A. relationship; or, send the third DVD to A.

• And so on.And so on.

• If the sixth DVD is sent, A faces a decision: steal the DVD and terminate If the sixth DVD is sent, A faces a decision: steal the DVD and terminate the relationship; or, send $2 for the sixth DVD.the relationship; or, send $2 for the sixth DVD.

Define the game tree for this Define the game tree for this Centipede Game Centipede Game (the tree looks like a centipede), (the tree looks like a centipede), then find the rollback solution.then find the rollback solution.



Centipede Game:Game Tree 3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6

S tea l 6 , pa yo ff 8 ,4 P a y 6 , p ayo ff 6 ,6

A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A



Centipede Game:A’s sixth choice


3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6


A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A


Centipede Game:B’s sixth choice


3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6


A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A


Centipede Game:A’s fifth choice 3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6


A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A



Centipede Game:B’s fifth choice 3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6


A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A



And so on, until …



Centipede Game:B’s first choice 3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6


A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A



Centipede Game:A’s first choice 3 ,-1

S tea l 1

1 ,1

T a ke 2

4 ,0

S tea l 2

2 ,2

T a ke 3

5 ,1

S tea l 3

3 ,3

T a ke 4

6 ,2

S tea l 4

4 ,4

T a ke 5

7 ,3

S tea l 5

5 ,5

T a ke 6


A

S e nd 6

B

P a y 5

A

S e nd 5

B

P a y 4

A

S e nd 4

B

P a y 3

A

S e nd 3

B

P a y 2

A

S e nd 2

B

P a y 1

A



Centipede Game: Centipede Game: Should Blockbuster send the first DVD to Should Blockbuster send the first DVD to Albert? In the rollback solution, Albert will steal the first DVD Albert? In the rollback solution, Albert will steal the first DVD and terminate the relationship. So Blockbuster should not send and terminate the relationship. So Blockbuster should not send the first DVD. the first DVD.

But humans like Albert might not follow the rollback solution But humans like Albert might not follow the rollback solution because that solution is too because that solution is too conceptually complex. Rather, Albert complex. Rather, Albert might pay for the first few DVDs, then plan to steal one of the might pay for the first few DVDs, then plan to steal one of the last DVDs. And as long as Albert pays for at least 2 DVDs last DVDs. And as long as Albert pays for at least 2 DVDs before stealing, Blockbuster makes positive profit. before stealing, Blockbuster makes positive profit.


41 41

End of Lesson I.3End of Lesson I.3

BA 592 Game BA 592 Game TheoryTheory


1 1 deep thought ba 592 lesson i.3 sequential move theory at first i thought, if i were superman, a...

Documents