Download Chapter 29

• Chapter 29 Game Theory
• Key Concept: Nash equilibrium and
Subgame Perfect Nash equilibrium
(SPNE)
• Chapters 29, 30 Game Theory
• A good time to talk about game theory
since we have actually seen some types
of equilibria last time.
• Game theory is concerned with the
general analysis of strategic interaction. It
can be used to study parlor games,
political negotiation, and economic
behaviors.
• Normal form
games:
Prisoner’s
dilemma.
Defect Coop
招
不招
Defect -5,-5
招
-1,-10
Coop
不招
-2,-2
-10,-1
• This game has a strictly dominant
strategy.
• A dominant strategy is one which is
optimal no matter what the other player
does.
• Once a player has a strictly dominant
strategy, the task of predicting what other
will do (the most interesting thing in
game) becomes unnecessary.
• We may not be this lucky to have a
strictly dominant strategy all the time.
• But we might have a strictly dominated
strategy. So at least we can say we won’t
play strictly dominated strategies.
L
R
U
2, 4
6, 3
M
3, 3
3, 4
D
5, 4
4, 3
• One way to “solve” the game above is to
use iterated elimination of strictly
dominated strategies.
• So our story may go like, 1 is rational, so
1 will not play M. Given everyone knows
this, 2 is rational, 2 will not play R. Given
everyone knows this, 1 is rational, 1 will
not play U. So (D, L) becomes our
prediction. This process uses some
common knowledge of rationality.
L
R
U
2, 4
6, 3
M
3, 3
3, 4
D
5, 4
4, 3
• But, are you really a person who will play
according to this logic of iterated
elimination of strictly dominated
strategies? Will you get the highest
payoff if you do? Talk a bit about the
games we played before and the cognitive
hierarchy theory.
• Despite of this, theorists still feel that
iterated elimination does not give sharp
predictions.
Ballet
Football
Ballet
2, 1
0, 0
Football
0, 0
1,2
• In battle of sexes game, iterated
elimination does not help. A stronger
solution, called Nash equilibrium, is often
used.
• The concept of Nash equilibrium is this.
A strategy profile (so every player has a
part) is a Nash equilibrium, if given your
opponent plays his equilibrium strategy,
you play yours is a best response to that.
• In other words, players are best
responding to each other.
• Equivalently, we can say that given your
opponent is playing equilibrium, you
have no profitable deviation.
• That is exactly why we are looking at the
reaction function in Cournot competition.
• A reaction function is simply a function
that gives you the best response to every
possibly plays of your opponent. We then
intersect the two reaction functions to get
Nash equilibrium where both are best
responding to each other.
• In the battle of sexes game, there are two
pure Nash equilibria, (ballet, ballet) and
(football, football).
• Notice that in Nash equilibrium, a
player’s is checking that he is best
responding to his opponent’s play.
• In other words, implicitly we are
assuming that a player’s belief is correct,
or consistent with opponent’s play.
• So Nash equilibrium has two important
components: best responding and correct
beliefs.
• There are some interesting empirical
works supporting Nash equilibrium.
• For instance, football players in penalty
kicks.
• Yet there are also reasons sometimes
people don’t feel comfortable playing
Nash equilibrium.
• Or sometimes we can even coordinate
better than Nash.
• Look at the coordination games we
played.
left
right
top
1, 0
1, 1
bottom
-1000, 0
2, 1
• In the literature, there is a wellrecognized “problem” with Nash
equilibrium about empty threats.
• Suppose in the first class meeting I came
in and announced: “I want total
dedication from my students, so you
should drop all the other courses and just
take my course. This will guarantee you
spend all the efforts in my class. If you do
not do so, I will not let you take the final
and you will fail.”
• Now it is a sequential game. You first
decide whether you will drop other
courses and then if you do not, I decide
whether I will bar you from the final.
• My threat of not letting you take the final
seems not credible (an empty threat)
since at the final, what is the point of
baring you from exam if my purpose is to
increase your efforts in the course and
efforts are done?
• Moreover, NTU is a serious institution.
So you expect that at the final, I will not
carry out the threats and hence you will
not drop other courses. That is a NE.
• Now another Nash equilibrium looks like
this. You take my threats seriously and
you drop all other courses. Since you
drop all courses I don’t need to carry out
my threats of barring you from the exam.
So my empty threat is never tested. I am
very happy with the outcome. And you,
given your pessimistic belief, are also
doing what is the best for yourself. This
does not seem very reasonable. So we
want to rule out NE of this kind.
• This brings us to Subgame Perfect Nash
equilibrium (SPNE). A subgame is just
like a game, so let us talk a bit about
game trees.
• Drop
---- (2,5)
Don’t Drop
-----Bar (0,-5)
-----Don’t Bar (4,2)
• So for SPNE, we solve backwards.
• Talk about the sheep-lion story.
• The Stackelberg equilibrium is an SPNE.
• We are now equipped with NE and SPNE,
it seems enough so what else can theorists
do?
• There are still lots of things.
• For one, look at finite repetition of PD
game. Defect all the time is the unique
SPNE. But do you really believe in that?
• One potential answer is to appeal to
infinite horizon. Remember last time
when we talked about repeated Cournot.
With repeated interactions, as long as the
future is important enough, we might be
afraid of future punishment, which makes
current cooperation possible. This is the
result of Folk theorem.
• Another interesting direction is, do we
really believe that people are so rational?
• Can not-so-rational players play out
equilibrium? Talk a bit about traffic and
social conventions.
• Chapter 29 Game Theory
• Key Concept: Nash equilibrium and
Subgame Perfect Nash equilibrium
(SPNE)