7 Section Note
October 27th
A Midterm Comments Continued
How to allocate time eciently in answering exam questions.
B Doubts about subgame perfect Nash
The Centipede Game
Two players are involved in a process that they alternately have the opportunity to
stop. Each prefers the outcome when he stops the process in any period t to that in
which the other player does so in period t + 1. However, better still is any outcome
that can result if the process is not stopped in either of these periods. After T periods,
where T is even, the process ends.
Here is a numerical example:
The name centipede comes from the shape of the diagram.
Figure 1: A Six-period Version of the Centipede Game
The game has a unique subgame perfect equilibrium; in this equilibrium each player
chooses S in every period. The outcome of this equilibrium is the same as the outcome
of every Nash equilibrium.
In the unique subgame perfect equilibrium of this game each player believes that the
other player will stop the game at the next opportunity, even after a history in which
that player has chosen to continue many times in the past. As in the subgame perfect
equilibrium of the chain-store game such a belief is not intuitively appealing; unless T
is very small it seems unlikely that player 1 would immediately choose S at the start
of the game, because if both players choose C each periods they can end up with a
huge payo rather than 1 or 0.
Again, as we have talked before, both the concepts of Nash equilibrium and subgame
perfect equilibrium are developed for us to predict the outcome of a certain game
and sometimes they both work ne, sometimes one works better the other, sometimes
neither works. Thus, we still have other concepts developed, such as the notion of
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trembling hand perfect equilibrium, as a renement of Nash Equilibrium. A trembling hand perfect equilibrium is an equilibrium that takes the possibility of o-theequilibrium play into account by assuming that the players, through a slip of the
hand or tremble, may choose unintended strategies, albeit with negligible probability.
Those of you who has interests beyond our course can look it up(you are more than
welcome to talk to me about stu beyond our course or real world applications of game
theory).
C Strategic Investment
This is an important topic since it combines game theory with a lot of real world
applications in a simple frame work. Please refer to the nice handout provided by the
Professor for detailed explanations on this topic.
Brief review of handout
Two important concepts: strategic complements(you and your opponents strategies go the same way) and strategic substitutes(you and your opponents strategies
go the opposite way).
The notion of direct eect and strategic eect. It is important to notice that the
strategic eect would disappear, if player 1s investment is not observed by player
2.
The four concepts to fully characterize the strategic investment problems. In
Top Dog the greater investment makes player 1 tough, which makes player 2
soft. Hence, the strategic eect strengthens the incentive for the investment in
the observable case. Player 1 should become big to scare the opponent. In
Lean and Hungry, it is opposite. The smaller investment makes player 1 tough,
which induces a softer choice by player 2. Hence, the strategic eect weakens
the incentive for the investment in the observable case. Player 1 should become
small to scare the opponent. You nd can similar arguments for Puppy Dog
and Fat Cat cases.
Example
Now, I am going to give a simple example that you can analyze using the framework we learned.
Consider a simple eort game. You have a course of which the grade will be curved
and there is only one other guy/girl in this class who is equally smart as you. Now
your payo in this course depends only on the your own eorts(positively) and
your classmates eorts(negatively). And we further assume that you dont like
study which means paying eort will have a cost. Intuitively, this is a strategic
complement case, since you dont want to pay eort if your classmate doesnt
pay eort and you are forced to pay more eort if your classmate does so. The
following is a simple payo function (made up) to model this simple strategic
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situation(you can come up a better one).
xj
ui = ( )100 xi
xi
We can easily solve the best response function out
BRi = 10 xj
which conrms our intuition that this is a strategic complement case.
Now imagine that you have a investment opportunity on a study machine(like
hiring a tutor?) which will make your study less costly. To be specic, your
payo function with the investment(cost you I) will be
xj
2
ui = ( 100)
xi I
xi
2
Intuitively, you will become more aggressive if you make the investment(your best
response function is now BRi = 20 xj ). But you know that this will make your
opponent also study harder which will turn out to hurt both of you two guys who
really dont like study. This means your incentive to invest in this case is smaller
if your classmate can observe whether you invest or not. But if he cannot observe
anything, you will have a higher incentive to make the investment. And along the
line with our framework, this is a Puppy Dog case. You can work out similar
algebra exercise to show the above intuition just as the homework problem. But
it is always more helpful if you have the right intuition.
Now suppose that you have an opportunity to invest on a Wii or X-Box,
which will make your study more costly(since you have better things to do other
than study). Intuitively, this investment will make you become less aggressive in
paying eorts. And you do know that your classmate will study less if he/she
knows you get an Wii which will in turn benet both of you. Thus you will
have a stronger incentive to make the investment in the observable scenario. And
this is a Fat Cat case.
D Bargaining(Finite-time)
We are still under the concept of subgame perfect equilibrium, and the time frame is
nite, thats why we can still use backward induction.
Brief review of lecture
The Professor did everything in great detail in the lecture and he will probably
post a handout on all those algebra he did in class, too. I will just talk a little
bit about two other ways to solve a dierence equation.
Some algebra issues
How to solve a simple dierence equation(rst-order linear xn+1 = xn + ).
Method one: recursive iteration, about which the Professor talked in great detail
in class.
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Method two: sum and cancel
xn =
xn1 =
2 xn2 =
.
.
.
n2 x2 =
xn =
xn1 +
(xn2 + )
2 (xn3 + )
n2 (x1 + )
(1 + + 2 + + n2 ) + n1 x1
1 n1
=
+ n1 x1
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Method three: rearrange and change to a simple one
xn = xn1 +
xn +
= (xn1
)
1
1
yn = yn1
Practice problems
I briey went over the rst practice problem. Please work through both two questions by yourselves, then compare their settings and results.
Final 2007F Q1
Final 2010F Q1
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