Hello everybody, Santiago here. I made
this video so we could dive into unit
eight a little bit more so we're gonna
explore what I mean by tension in games
by looking at the tension between
individual level and group level
incentives in the prisoner's dilemma
so some brief highlights of the unit so
far we talked about an important
assumption in game theory being that
individuals act in order to maximize
their own utility and utility remember
is this numerical measure of the outcome
of a game which could be profit
happiness reproductive success we saw
all of this earlier in the unit so we're
taking this to mean that the individuals
are selfish they're right if they act in
order to maximize their own utility
they're only looking at their personal
gain then we're calling them selfish
agents and a big question in the unit is
can selfish individuals cooperate so
let's go back to the prisoner's dilemma
the numbers in this prisoner's dilemma
are a little bit different than those
Melanie used but the game is still
essentially the same if you want to
compare this one with the other one and
see go through that process of
deliberation that each of the players
goes through to make their decision I
encourage you to do that now to make
everything even clearer in the attempt
to make things clearer I color-coded the
agents and the utilities that they
receive so player one is in blue and he
can choose between the blue cooperate
and effect player two is in green and
they can choose between the green
cooperate and effect and depending on
what they choose if player 1 chooses
cooperate player 2 chooses the fact then
they're in this cell this is the outcome
and player one gets the utility on the
Left player two gets the utility on the
right so to look at this tension let's
decompose let's rewrite the game as the
sum of cooperative and selfish forces
and to do this I'm going to use a
technique that was developed in this
paper I'm citing below strategic and
behavioral decomposition of games and
I'm adding this paper to the
supplementary materials page of the
website and I encourage you to take a
look at it unfortunately I can't go
through all the details of the paper
everything they do there's a lot of good
stuff but I gotta keep it simple keep it
short and point you where to find more
so alright let's keep going so if I
apply the decomposition to this
prisoner's dilemma game I get these
three components so if you want to make
sure pause the video add these three
matrices down here add them up and see
that you get this one and yeah so I so
let's see what's going on what are these
components telling us what do you think
they're telling us what what kind of
information is contained in each of
these components and why is it that we
can separate these information so feel
free to pause the video analyze each of
these components see what you think is
going on and because I'm about to
extenuate this first component we
interpret it to be this kind of fluff it
contains no real information it's not
really doing anything because it
contributes the same value to all the
utility's of each of the players so it
doesn't have to be the same value for
both players right we see here that both
players all have to
it could be different but I'm keeping it
simple and so this this component of the
game contains no info because since it's
the same value for all of the outcomes
well in game theory were interested in
comparing the results of different
outcomes so anyway we compare the
outcomes is gonna cancel out this common
value so saying it's just fluff there's
no real information here the authors of
the paper called us the kernel now the
second component we say contains the
game's selfish information now why is
that well so check out the other
components can you gain anything from
changing your own strategy alone no
right so check out the kernel if you
let's say player 2 is fixed playing
cooperate or defect
if you switch between cooperate out or
defect can you is there any incentive
for you to change your strategy no so
all of that information all of that
individual level incentive is contained
in this middle component in this selfish
info component which the authors called
the Nash or strategic component and the
reason they call it the Nash component
is that this component contains all the
necessary information in a game to
compute Nash equilibria which we haven't
really talked about so I won't really
get into more but right in this entire
game we can decompose it into these
three layers these three components but
to compute Nash equilibrium we only need
this it's pretty fascinating so if
you're curious if you're liking this
check out the paper study more game
theory
but anyway the next component this last
component doesn't have any selfish
information but something's going on
it's not trivial like the kernel it's
not just the same value right so what do
you think is going on here I'm calling
it left over info to keep it mysterious
we haven't figured out what this
component does so maybe pause the video
think about it a little bit here let me
blow it up
there's the leftover info component what
is happening with it okay pause the
video okay so look in this component
each agent is powerless to change their
own utility alone but what happens when
the other agent changes their strategy
so let's say okay start over let's say
your player 1 and player 2 is playing
cooperate you have a choice between
cooperate and getting a - or defect and
getting it - so if that's all you're
looking at you don't care right you're
not this isn't going to sway your
decision either way but what happens
when you play cooperate and what happens
when you play defect when you play
cooperate agent 2 is receiving a payoff
of positive - no matter what they do if
you play defect agent 2 is getting a
payoff of negative - no matter what they
do so it's like your decision is
affecting the other person so this idea
they call it externalities in X in
economics so an externality is a utility
imposed on another agent on another
person another group of people it's an
is utility imposed on others from your
actions so a simple example if I play
music really loud and it bothers my
neighbor then they're receiving a
negative externality from something I'm
doing right so maybe I'm making a
decision that gives me
a nice private payoff a utility but it
gives my neighbor a negative utility
that's the idea of externalities so this
left over information component
I'm calling the author's called the
behavioral component
I'm calling the externalities component
sorry for all the names I hope this is
not confusing yeah so I talked about
this externalities you do something the
result creates utility the result is
utility for yourself and for others so
externality is this utility that gets
imposed on others now just this
so let's look at the prisoner's dilemma
again ignoring the kernel so when I
cooperate I'm not getting the best for
myself no matter what you do right
look at that negative one if you
cooperate and I cooperate I'm getting a
negative one if you defect and I
cooperate I get a negative one remember
we're only looking at the selfish info
component so looking at the selfish info
I'd rather always defect I always get a
better individual payoff by defecting
following if I'm selfish then I'm always
gonna defect and that is shown right
here because being selfish I don't care
about a 2 or a 2 or a negative 2 and a
negative 2 right I only care about this
individual I want to maximize
individually
so following our individual interest we
choose to defect but that hurts each
other because one may defect we impose a
payoff of negative two on one another
right so here player 1 is defecting
player 2 gets a negative 2 here player 2
is defecting player 1 gets a negative 2
so following our individual interest we
hurt each other if we both cooperated
we'd supply each other with this
positive 2 utility going against that
negative one selfish utility but it'd be
better right if we look at the full game
we'd be better off both cooperating them
both defecting so the prisoner's dilemma
exemplifies this tension between
cooperating and defecting between what
does our selfish what does the
individual level pull us towards what
does it want us to do and what does the
group level want us to do right so it's
better if we cooperate but individualism
sort of unravels it and we ends up and
we end up defecting
now let me just go back so we talked
about in the previous units we talked
about tit for tat and so how does
Tiferet tat make corporation work so
Tiferet at actually changes the game one
way it changes the game as the game
becomes repeated and another way is that
the strategies change so if you play tit
for tat you're not simply playing
cooperate or defect right you're playing
this it's almost like this higher level
strategy in this repeated game and so
what this ultimately ends up doing is it
brings the externalities into the
selfish information and it makes
cooperating attractive to someone
selfish that's the only way right we're
assuming the individuals are selfish so
at the end of the day no matter how many
hoops we jump through the agents are
gonna behave selfishly so the only way
for them to cooperate at least with
these assumptions is to make cooperating
simple is to make cooperation something
a selfish agent would play so I can
barely get into that the stuff I think
is super cool and I should I suggest you
check out that cited paper they talked
about tit for tat and they show with
more detail how how things change how
things move around how some of the
externalities end up affecting a selfish
level choice which is how tit for tat
and the repeated game makes cooperation
possible so let me lastly tell you the
quote unquote rules to create the
components of these 2x2 games so you can
play around and in fact in fact I'm
gonna make an additional homework which
is going to be optional for you to
explore all of this a little bit more so
in it I'll ask you to follow these rules
and create different kinds of 2x2 games
where cooperation is aligned with
selfish incentives or it's not like we
just saw
so on all kinds of interesting examples
for you to play around with and explore
so let me tell you these rules and these
quote-unquote rules are actually
mathematical symmetries that make this
whole thing work but I'm not going to
get into the details of it but so with
The Selfish component every pair of
alternative individual choices must be
negatives of each other sorry I forgot
some words here it must be negatives of
each other anyway so here we see I put
an example player 1 chooses strategy a
and player 2 choose a strategy a so
player 1 gets a payoff of a if they
switched they'd get a payoff of negative
a and so we can see this idea repeats
throughout this selfish info component
the externality component on the other
hand remember that there's no individual
power so here if player 2 is choosing a
your player one you're just getting an M
no matter what the only way you can get
something different is if the other
agent chooses changes their strategy so
here when the other agent changes their
strategy you get the negative of what
you were getting right so player 2 by
choosing a imposes M a utility of M and
then if they changed if they played B
they impose negative N and so just to be
clear M could be three so this would be
three three negative three negative
three
but M could be negative three so then
over here we'd have negative three
negative three and then over here we'd
have negative negative 3 which would be
pot so we'd have positive 3 positive 3
and then finally the kernel was the same
value for all the utilities of each
agent so agent one would have the common
value J whatever J is an agent 2 would
have whatever value K is throughout and
so here it is here are the rules
and thanks for watching everybody
I hope this made sense if anything send
me an email post questions on the forums
I'll be happy to explain any of this a
little more and yeah thanks for watching
have a great day bye
you