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Start by explaining dominance (simpler than a nash equilibrimm)
 
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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.
 
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just think about Prisoner Y
 
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Whatever Prisoner X does, it is always best for Prisoner Y to confess
 
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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.
 
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dominance weak dominance strict dominance
 
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Easier to understand if we illustrate before syaing what it is
 
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Here is one combination of actions. Neither ganster cannot unilaterally do better than this by changing their action ...
 
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Consider Gangster Y’s options ...
 
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They can move vertically, but that would only make things worse for them.
 
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This is a nash equilibrium
 
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A nash equilibrium for a game
is a set of actions
from which no agent can unilaterally profitably deviate
 
Let’s see another example
 
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There is one more nash equilibrium. Can you find it?
 
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Suppose they did not ...
 
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... then they should change their action. So in fact Gangster X cannot this to happen.
 
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How does this link to our aim?
 
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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)
 
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Why is the notion of a nash equilibrium so cool? Consider:
 
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Consider all this complexity. The notion of a nash equilibrium cuts it out.
 
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Nash equilibrium allows us to identify rationally optimal actions in a way that doesn’t involve working through how these beliefs might be formed.
 
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Now we have the theory, or enough of it for our purposes (dominance and Nash equilibrium).
 
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Actually we are not going to do this quite yet ...
 
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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.
 
A nash equilibrium for a game
is a set of actions
from which no agent can unilaterally profitably deviate
 
‘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)
 

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