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We'll now speak about how
we predict that people will play in a game and specifically about the
concepts of best response and Nash equilibrium.
Now, let's for one assume that you are one of the players and the other players
are going to play in the way that you know
then we can speak about what is your best thing to do and we call that the best
response
and technically speaking let's use the following notation
let's call
a-->i
the action sequence of everybody except you play write
so it's player one and so i think prayer i minus one i plus one and all the
rest
so the entire action vector we call it the action profile
a is really made up of all those a minus i and your own
uh... action
with that notation here is the definition
we will say that your best response which will call ai star
it may not be unique
but it's got to be part of the set as we call BR the best response assuming
that all the other agents play as a, a-i
if the case that
for anything you might be thinking of doing a survive
for all a survive
it's got to be the case that the u_n_ utility
for playing that in other words when you're playing
a i star
and the others are playing
a i a minus i for whatever it is
has got to be at least as great as anything else you might choose
a survive
that's the case will say that a i star is a best response
varying into the notion
and uh... that would bring us almost all the way there
the problem of course is that you don't know what the others want to
but that's okay because we'll use
the notion of best response as a building block to define what to call
the Nash equilibrium
and the Nash equilibrium briefly
is a set of actions one for each of the agents such as each is the best response to
the others
specifically
we'll look at the Nash profile here
a, a1 through an
and will say that it's a Nash equilibrium caribbean
and later on will tell you why we call it specifically a pure strategy
Nash equilibrium
if it's a case that
for every agent
that action a semi says it is the best response all the rest