Deep Thought
At first I thought, if I were
Superman, a perfect secret identity
would be “Clark Kent, Dentist”,
because you could save money on
tooth X-rays. But then I thought, if a
patient said, “How’s my back tooth?”
and you just looked at it with your Xray vision and said, “Oh it’s okay”,
then the patient would probably say,
“Aren’t you going to take and X-ray,
stupid?” and then he probably
wouldn’t even pay his bill. --- by Jack
Handey.
BA 592 Lesson I.3 Sequential Move Theory
1
Lesson overview
Chapter 3 Games with Sequential Moves
Lesson I.3 Sequential Move Theory
Each Example Game Introduces some Game Theory
• Example 1: A Rollback Solution
• Example 2: A Game Tree
• Example 3: Off the Equilibrium Path
• Example 4: Multiple Equilibria
• Example 5: Jealous Humans
• Example 6: Simple Humans
Lesson I.4 Sequential Move Applications
BA 592 Lesson I.3 Sequential Move Theory
2
Example 1: A Rollback Solution
• 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.
BA 592 Lesson I.3 Sequential Move Theory
3
Example 1: A Rollback Solution
Backward induction or rollback solves sequential move games
with perfect information by rolling back optimal strategies from
the end of the game to the beginning.
BA 592 Lesson I.3 Sequential Move Theory
4
Example 1: A Rollback Solution
Century Mark Game
• Played by pairs of players taking turns.
• At each turn, each player chooses a number between 1 and 10
inclusive.
• This choice is added to sum of all previous choices (the initial
sum is 0).
• The first player to take the cumulative sum to 100 or more
wins.
How should you play the game as first player? Start at the end.
What number gets you to 100 next turn?
BA 592 Lesson I.3 Sequential Move Theory
5
Example 1: A Rollback Solution
Rollback Solution
• If you bring the cumulative sum to 89, you can take the
cumulative sum to 100 and win no matter what your opponent
does. (Whatever your opponent does you can make the sum of
your two moves equal 11.)
• Hence, if you bring the cumulative sum to 78, you can bring
the cumulative sum to 89 on your next turn (and so eventually
win) no matter what your opponent does.
• And so on for sums 67, 56, 45, 34, 23, 12.
• 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
your opponent does.
• That is a strategy --- a complete plan of actions no matter what
your opponent does.
BA 592 Lesson I.3 Sequential Move Theory
6
Example 2: A Game Tree
Game trees or extensive forms consist of nodes and branches.
Nodes are connected to one another by the branches, and come in
two types. Some nodes are decision nodes, where a player
chooses an action. In some games, Nature is a “player”; Nature
can decide whether it rains or snows. The other nodes are
terminal nodes, where players receive the outcomes of the actions
taken by themselves and all other players.
BA 592 Lesson I.3 Sequential Move Theory
7
Example 2: A Game Tree
Emily, Nina, and Talia all live on the same street. Each has been
asked to contribute to a flower garden. The quality of the garden
increases with the number of contributions, but each lady also
prefers to not contribute. Specifically, suppose each lady gains 2
dollars worth of happiness from each of the first two
contributions to the garden (including her own contribution, if
any) and 0.50 dollars worth from a third contribution, but then
looses 1 dollar if she herself contributes.
Define the game tree for this Street Garden Game, then find the
rollback solution. Should Emily contribute?
BA 592 Lesson I.3 Sequential Move Theory
8
Example 2: A Game Tree
Street Garden Game
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
3.5,3.5,3.5
3,3,4
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
0,0,0
BA 592 Lesson I.3 Sequential Move Theory
9
Example 2: A Game Tree
Rolling from the end: Talia’s Strategy. (Non-optimal strategies
are blacked out.)
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
3.5,3.5,3.5
3,3,4
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
0,0,0
BA 592 Lesson I.3 Sequential Move Theory
10
Example 2: A Game Tree
Rolling back one from the end: Nina’s Strategy
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
3.5,3.5,3.5
3,3,4
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
0,0,0
BA 592 Lesson I.3 Sequential Move Theory
11
Example 2: A Game Tree
Rolling back to the beginning: Emily’s Strategy, which completes
the rollback solution.
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
3.5,3.5,3.5
3,3,4
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
0,0,0
BA 592 Lesson I.3 Sequential Move Theory
12
Example 3: Off the Equilibrium Path
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.
BA 592 Lesson I.3 Sequential Move Theory
13
Example 3: Off the Equilibrium Path
Street Garden Game:
alternative payoffs
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
5,5,5
3,3,6
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
2,2,2
BA 592 Lesson I.3 Sequential Move Theory
14
Example 3: Off the Equilibrium Path
Street Garden Game:
Equilibrium Path
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
5,5,5
3,3,6
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
2,2,2
BA 592 Lesson I.3 Sequential Move Theory
15
Example 3: Off the Equilibrium Path
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.
Emily
Contribute
Don't
Nina
Nina
Contribute
Don't
Contribute
Don't
Talia
Talia
Talia
Talia
Cont.
Don't
Cont.
Don't
Cont.
Don't
Cont.
Don't
5,5,5
3,3,6
3,4,3
1,2,2
4,3,3
2,1,2
2,2,1
2,2,2
BA 592 Lesson I.3 Sequential Move Theory
16
Example 4: Multiple Equilibria
Rollback equilibria are unique unless a player gets equal payoffs
from two or more different actions. One method to restore a
unique equilibrium (to be used as a prescription or prediction) is
to question whether a game with equal payoffs is somehow
exceptional or avoidable.
BA 592 Lesson I.3 Sequential Move Theory
17
Example 4: Multiple Equilibria
Employees know there is a positive gain to their continued
employment, and that gain is split with their employer according
to the employees wages. Suppose Employee A generates 100
dollars of gain by remaining employed with Employer B.
Employee A is considering increasing his wage demands to one
of three levels. Those three levels give him either 100%, or 90%,
or 50% of the 100 dollars of gain. Which wage should the
employee demand?
Define the game tree for this Bargaining Game, then find all the
rollback equilibria.
BA 592 Lesson I.3 Sequential Move Theory
18
Example 4: Multiple Equilibria
Bargaining Game:
Game Tree
Proposer
I take 100%
I take 90%
I take 50%
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
100,0
0,0
90,10
0,0
50,50
0,0
BA 592 Lesson I.3 Sequential Move Theory
19
Example 4: Multiple Equilibria
Bargaining Game:
Proposer
Partial Rollback Solution
I take 100%
I take 90%
I take 50%
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
100,0
0,0
90,10
0,0
50,50
0,0
BA 592 Lesson I.3 Sequential Move Theory
20
Example 4: Multiple Equilibria
Rollback analysis was incomplete in that game because the
responder got equal payoffs from two different actions. But that
is only because the Proposer demanded 100%. What if the
Proposer demands 99.99%? Now there is a unique rollback
solution with payoffs almost as high as if a demand of 100%
were accepted.
BA 592 Lesson I.3 Sequential Move Theory
21
Example 4: Multiple Equilibria
Bargaining Game:
Complete Rollback Solution
Proposer
I take 99.99%
I take 90%
I take 50%
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
99.99,0.01
0,0
90,10
0,0
50,50
0,0
BA 592 Lesson I.3 Sequential Move Theory
22
Example 5: Jealous Humans
Humans in some sequential move games do not follow the
rollback solution because that solution may be felt to be too
unfair.
Economic policymakers thus favor public policies whose rollback
solutions seem fair enough for humans to accept.
And businesspeople thus adapt strategies depending on whether
they are playing against jealous humans (perhaps some of their
customers) or rational players (other businesspeople).
BA 592 Lesson I.3 Sequential Move Theory
23
Example 5: Jealous Humans
Shoppers know there is a positive gain to making purchases, and
that gain is split with sellers according to the purchase price.
Shopper A generates 100 dollars of gain by buying from Seller B.
Buyer A is considering three alternative price offers. Those three
offers give him either 99%, or 90%, or 50% of the 100 dollars of
gain. Which price should Buyer A offer?
Define the game tree for this Bargaining Game, then find the
rollback solution.
.
BA 592 Lesson I.3 Sequential Move Theory
24
Example 5: Jealous Humans
Bargaining Game:
Game Tree
Proposer
I take 99%
I take 90%
I take 50%
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
99,1
0,0
90,10
0,0
50,50
0,0
BA 592 Lesson I.3 Sequential Move Theory
25
Example 5: Jealous Humans
Bargaining Game:
Rollback Solution
Proposer
I take 99%
I take 90%
I take 50%
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
99,1
0,0
90,10
0,0
50,50
0,0
BA 592 Lesson I.3 Sequential Move Theory
26
Example 5: Jealous Humans
Which price should the shopper offer? In the rollback solution,
the shopper should offer the price that gives him 99% of the gain
from trade. But humans like the seller might not follow the
rollback solution because that solution is too unfair. Rather, the
shopper may have to offer a price that gives him only 90% or
50% of the gain from trade.
BA 592 Lesson I.3 Sequential Move Theory
27
Example 6: Simple Humans
Humans in some sequential move games do not follow the
rollback solution because that solution is too computationally
complex.
Chess is a sequential move game with perfect information, so it
has a game graph, with an estimated 10120 nodes describing all
possible board positions. There is a rollback solution, but that
solution is so computationally complex no human knows all of it.
It was, therefore, inevitable that computers eventually became
better players than humans. In May 1997, a chess playing
machine “Deeper Blue” beat reigning champion Garry Kasparov,
by 3½ to 2½ in a six game match. Recent progress in computer
play is software than can run on common personal computers.
BA 592 Lesson I.3 Sequential Move Theory
28
Example 6: Simple Humans
Humans in other sequential move games do not follow the
rollback solution because that solution is too conceptually
complex.
Economic policymakers thus favor public policies whose rollback
solutions are simple enough for humans to compute.
And businesspeople thus adapt strategies depending on whether
they are playing against simple humans (perhaps some of their
customers) or playing against rational players (other
businesspeople).
BA 592 Lesson I.3 Sequential Move Theory
29
Example 6: Simple Humans
Buyers and Sellers trading over the internet risk sending money
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.
BA 592 Lesson I.3 Sequential Move Theory
30
Example 6: Simple Humans
Suppose Albert values 6 disposable DVDs at $3 each, suppose it costs
Blockbuster $1 to provide each DVD, and suppose Blockbuster sells DVDs
for $2 each. Should Blockbuster send the first DVD to Albert?
• 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.
• 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.
• If the second DVD is sent, A faces a decision: steal the DVD and terminate
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
relationship; or, send the third DVD to A.
• And so on.
• If the sixth DVD is sent, A faces a decision: steal the DVD and terminate
the relationship; or, send $2 for the sixth DVD.
Define the game tree for this Centipede Game (the tree looks like a centipede),
then find the rollback solution.
BA 592 Lesson I.3 Sequential Move Theory
31
Example 6: Simple Humans
A
Centipede Game:
Game Tree
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
32
Example 6: Simple Humans
A
Centipede Game:
A’s sixth choice
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
33
Example 6: Simple Humans
A
Centipede Game:
B’s sixth choice
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
34
Example 6: Simple Humans
A
Centipede Game:
A’s fifth choice
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
35
Example 6: Simple Humans
A
Centipede Game:
B’s fifth choice
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
36
Example 6: Simple Humans
And so on, until …
BA 592 Lesson I.3 Sequential Move Theory
37
Example 6: Simple Humans
A
Centipede Game:
B’s first choice
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
38
Example 6: Simple Humans
A
Centipede Game:
A’s first choice
Steal 1
Pay 1
3,-1
B
Take 2
Send 2
1,1
A
Steal 2
Pay 2
4,0
B
Take 3
Send 3
2,2
A
Steal 3
Pay 3
5,1
B
Take 4
Send 4
3,3
A
Steal 4
Pay 4
6,2
B
Take 5
Send 5
4,4
A
Steal 5
Pay 5
7,3
B
Take 6
Send 6
5,5
A
Steal 6, payoff 8,4
BA 592 Lesson I.3 Sequential Move Theory
Pay 6, payoff 6,6
39
Example 6: Simple Humans
Centipede Game: Should Blockbuster send the first DVD to
Albert? In the rollback solution, Albert will steal the first DVD
and terminate the relationship. So Blockbuster should not send
the first DVD.
But humans like Albert might not follow the rollback solution
because that solution is too conceptually complex. Rather, Albert
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
before stealing, Blockbuster makes positive profit.
BA 592 Lesson I.3 Sequential Move Theory
40
BA 592
Game Theory
End of Lesson I.3
BA 592 Lesson I.3 Sequential Move Theory
41