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Game Theory 101 (#74): Perfect Bayesian Equilibrium - YouTube
Channel: William Spaniel
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welcome back to game theory 101 i'm
william spaniel today's topic is perfect
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Bayesian equilibrium this is the start
of a brand new solution concept so let's
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get right to it you might remember this
figure from before this lists
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equilibrium solution concepts by the
timing and information and a game we've
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started our exploration of game theory
with the simplest of games those are
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simultaneous Move games of complete
information and we saw that the
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appropriate solution concept was Nash
equilibrium then we transitioned into
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games of sequential timing where players
take turns moving and have seen previous
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moves and know what other players have
done when they make their actions and we
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saw that Nash equilibrium is
insufficient for that so we instead
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switched over to something known as sub
game perfect equilibrium then we
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switched gears and rather than changing
up the simultaneous versus sequential
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nature of a game we still looked at
simultaneous Move games but we switched
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from incomplete information to complete
information we saw that again Nash
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equilibrium was not sufficient for that
so we had to introduce Bayesian Nash
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equilibrium well the natural question is
what do you do when a game is both
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sequential and has incomplete
information we need some sort of
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combination of subgame perfect
equilibrium and Bayesian Nash
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equilibrium and that's exactly what
we're learning about today perfect
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Bayesian equilibrium so this lecture is
going to introduce the definition of
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perfect Bayesian equilibrium I'm going
to go over the important facets of that
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definition and then in later lectures
we'll actually start applying it this is
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just like what we did with Bayesian Nash
equilibrium when we get to the point
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where we're working with these really
complicated solution concepts it's very
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important to go over the specifics of
the definition before we start playing
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around with the solution concepts
themselves with those equilibria
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themselves so let's go ahead and get to
that definition a perfect Bayesian
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equilibrium or a PBE is a set of
strategies and beliefs such that the
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strategies are sequentially rational
given the players beliefs and players
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update beliefs via Bayes rule wherever
possible that's a lot to take in so let
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me go through all of the important
points of perfect Bayesian equilibrium
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one by one
to begin PBE consists of both strategies
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and beliefs that latter part is new when
you were writing down the Nash
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equilibria of a game or subgame perfect
equilibria or Bayesian Nash equilibria
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all you would do was list the strategies
there was nothing about beliefs their
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beliefs were a component of how you
would calculate a Bayesian Nash
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equilibrium but they weren't a part of
the Bayesian Nash equilibrium itself PBE
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is a whole new ballgame its strategies
and beliefs together both of them you
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need both or your answer is wrong it's
like when you go back to sub game
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perfect equilibrium and you forget to
write down the off the path of play
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strategies your answer will be wrong
with the sub game perfect equilibrium
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same thing here if you forget about the
beliefs part your answer is wrong and
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it's a common rookie mistake to forget
about the beliefs so remember last
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warning here PBE strategies and beliefs
together leave out one your answer is
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wrong you need both there's still a
question about where those strategies
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and beliefs are coming from and that's
what the rest of the definition
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addresses let's start with the
strategies strategies in a perfect
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Bayesian equilibrium are sequentially
rational think back to the difference
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between Nash equilibrium and sub-game
perfect equilibrium sub-game perfect
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equilibrium required that all threats be
credible in other words when push comes
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to shove players actually want to follow
through on the strategies listed in the
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equilibrium
well sequential rationality is buying us
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that in the definition of perfect
Bayesian equilibrium
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you'll notice that the word perfect
appears in both sub-game perfect
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equilibrium and perfect Bayesian
equilibrium and that's precisely due to
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the fact that perfection is the concept
of threats being credible so in a
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perfect Bayesian equilibrium threats
should be credible that's what the
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sequential rationality part of
strategies is here that being said
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there's an important distinction between
games of complete information and games
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with incomplete information when it
comes to the credibility of threats with
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complete information I know exactly what
I'm buying I know what my payoffs are I
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know what your payoffs
our and that's going to affect the
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credibility of my threat in a game of
incomplete information I don't know
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exactly what I'm facing you could be a
weak type you could be a strong type
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maybe I'm willing to fight a war against
a weak type but I'm not willing to fight
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a war against a strong type if that's
the case then the credibility of my
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threat depends on what I believe about
what my opponent is I have a credible
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threat to fight against a weak type I
don't have a credible threat to fight
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against a strong type and so that's why
we have sequential rationality given the
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players beliefs and that should flag to
why we care about both strategies and
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beliefs the credibility of my strategies
depends on my beliefs that takes us to
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the beliefs part how do you arrive at
these beliefs which in turn determine
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the credibility of threats well just
like with Beijing Nash equilibrium you
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start off with a prior belief here
however you update your information as
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the game progresses for example imagine
a game with a strong type in a weak type
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maybe the strong type chooses to bully
in equilibrium and the weak type always
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cowers in fear regardless of whether you
observe bullying or cowering in fear you
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have new information that you didn't
have before so you should be updating
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your prior beliefs fortunately we know
exactly how you should update prior
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beliefs in light of new information its
Bayes rule Bayes rule was literally
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created to address that exact problem so
players are going to update their
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information wherever possible via Bayes
rule but there's a caveat there it's
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that wherever possible part there might
be an action that is never taken in
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equilibrium and this is going to create
a problem and how to address your
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beliefs about being in a situation where
that strategy was pursued Bayes rule
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requires an outcome to occur with
positive probability for us to use it
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otherwise there is a divide by zero
error so in these cases where there's a
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divide by zero error we don't have a
very clear and straightforward way of
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calculating our beliefs so a large
portion of this unit is going to be
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addressing how we should be thinking
about beliefs in situations where we
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can't actually update them through
that's actually a good segue to the
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outline for the rest of this unit we're
gonna be looking at two types of games
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screening games and signaling games
signaling games have this information
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problem where it might not be easy to
update police via Bayes rule whereas
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screening games don't have that problem
it because screening games are a lot
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simpler we're going to be starting with
those in a screening game the uninformed
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actor moves first this is a lot easier
to solve for because there's no
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information conveyed and an action taken
by an uninformed actor precisely because
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that uninformed actor doesn't know
anything and so he or she can't
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communicate any sort of information in
signaling games an informed actor moves
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first and this causes the informed actor
to have to really think about what
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they're communicating with the actions
that they're taking so this is going to
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result in a bunch of different types of
equilibria pooling equilibria separating
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equilibria and semi separating or
partially pooling equilibria so if
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you've been looking at this video trying
to figure out what the differences are
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between pooling separating and semi
separating that's something that's going
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to be covered later on in this unit so I
hope you enjoyed this and I hope to see
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you next time when we start talking
about screening games take care
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