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Nash Equilibrium

far we only considered the formalities: what games arehow they are often represented.theory is supposed to explain things. How does it do that?

‘Many events and outcomes prompt us to ask: Why did that happen? [...] For example, cutthroat competition in business is the result of the rivals being trapped in a prisoners’ dilemma’

(Dixit, Skeath, & Reiley, 2014, p. 36).

(Dixit et al., 2014, p. 36)

pd is a good target because common.

Games with the Prisoner’s Dilemma structure arise in:

bower birds (maraud/guard nests)

business organisations (product pricing)

countries (international environmental policy)

individual adult humans (suspects under arrest)

(Dixit et al., 2014, p. chapter 10)

Why is this true?
Start by explaining dominance (simpler than a nash equilibrimm)
Prisoner X
resistconfess
Prisoner Yresist3
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 Yresist3
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. I cannot unilaterally do better than this by changing my action ...
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 _cooperative_ games; those in which such enforcement is not possible, and individual participants must be allowed to act in their own interests, are called _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.