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