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Nash Equilibrium

Start by explaining dominance (simpler than a nash equilibrimm)
Prisoner X
resistconfess
Prisoner 59640resist3
3		0
4
confess 4
0		 1
1
observation: people in this kind of situation will sometimes, and rationally, end up performing actions which are mutually harmful in the sense that there is a better course of actions available to them.
just think about Prisoner Y
Whatever Prisoner X does, it is always best for Prisoner Y to confess
so perhaps the notion of dominance enables us to explain why people in this kind of situation will rationally end up performing actions which are mutually harmful in the sense that there is a better course of actions available to them.

first notion : dominance

dominance weak dominance strict dominance
Prisoner X
resistconfess
Prisoner 59640resist3
3		0
4
confess 4
0		 1
1

not all games involve dominance

Gangster X
stay
home		attack
Gangster Ystay
home		2
2		1
3
attack 3
1		 0
0

No dominance! What to do?

second notion : nash equilibrium

Easier to understand if we illustrate before syaing what it is
Gangster X
stay
home		attack
Gangster Ystay
home		2
2		1
3
attack 3
1		 0
0
Here is one combination of actions. Neither ganster cannot unilaterally do better than this by changing their action ...
Consider Gangster Y’s options ...
They can move vertically, but that would only make things worse for them.
This is a nash equilibrium

A nash equilibrium for a game
is a set of actions
from which no agent can unilaterally profitably deviate

see Osborne & Rubinstein, 1994 p. 14; Dixit et al, 2014 p. 95

Let’s see another example
Gangster X
stay
home		attack
Gangster Ystay
home		2
2		1
3
attack 3
1		 0
0
There is one more nash equilibrium. Can you find it?

Why should the players not expect to settle on a non-Nash equilibrium?

Suppose they did not ...

If Gangster X expected this to happen, ...

... then they should change their action. So in fact Gangster X cannot this to happen.

A nash equilibrium for a game
is a set of actions
from which no agent can unilaterally profitably deviate

see Osborne & Rubinstein, 1994, p. 14; Dixit et al., 2014, p. 95

How does this link to our aim?

‘we wish to find the mathematically complete principles which define “rational behavior” for the participants in a social economy, and to derive from them the general characteristics of that behavior’

(Neumann, Morgenstern, Rubinstein, & Kuhn, 1953, p. 31).

An action is rational
in a noncooperative game
if it is a member of a nash equilibrium?

noncooperative game
‘Games in which joint-action agreements are enforceable are called \emph{cooperative} games; those in which such enforcement is not possible, and individual participants must be allowed to act in their own interests, are called \emph{noncooperative} games.’ (Dixit et al., 2014, p. 26)
Why is the notion of a nash equilibrium so cool? Consider:

How you should act
is a function of two things:
your preferences
and your beliefs about how others will act.

Your beliefs about how others will act
are a function of your knowledge of two things:
your beliefs about their preferences
and your beliefs about how they believe others will act.

Your beliefs about how they believe others will act ...

Consider all this complexity. The notion of a nash equilibrium cuts it out.

How you should act:

  • avoid dominated actions
  • avoid a non-Nash equilibrium
Nash equilibrium allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed.

short essay question:

What is team reasoning?

Which, if any, social interactions are better modeled by team reasoning than game theory?

plan

1. What is game theory? ✓

2a. What are its applications?

2. What are its limits? ✓

3. What is team reasoning and how might it overcome the limits? ✓

Now we have the theory, or enough of it for our purposes (dominance and Nash equilibrium).
Actually we are not going to do this quite yet ...
I wanted to skip this and get straight to the limits. But actually that makes no sense at all.
The key thing is that the limits are limits on a theory that is actually very successful—without the successes, the limits are not interesting at all because they would just show that the theory has failed.