as promised we're back rape so this is
actually do something he won't hear so
this is lecture 5 on game theory and
vaccinations and we are now went to
resume where we last for us and we wear
as promised we are receiving sadly where
we we're talking so when when one grades
this this vertical violence you don't
think he and bottom that part that means
if I do expression right evaluate
expressions that this would be basically
as a function of p we evaluate that it
seriously 1-1 all wear this quantity are
20 times 1-0 and we can see that these
is equal to 1 miners one of his ear
that's that's basically what we meant
their last so if the vaccine is
perceived to be sufficiently freeski dan
the relative risk exceeds the
probability of becoming infected when
the vaccination leo is 0-3 Alyse wax in
the dan dan Ashik Ali reham in that case
is to never vaccine and so this this is
our first division of we're going to use
capital letters
these star asterisk here to the Nordic
clearance so this is an extreme scenario
in which the Nash equilibrium is to know
like 0 vaccination neglect vaccination
completely ignore vaccinations and so
the vaccination the level of protection
in the population would be serious if
you think of evaluating this expression
you're basically
no i didnt know newborns in the region
and more than on newborns going to the
ER compact but going back to the islets
now examine it condition for the
scenario in which we have them makes it
clear so I serious we work out here in
which is workout expression for these
these spies of 0 is 1-1 where r is here
so if we say that we're considering a
scenario in which you can see a scenario
in which ok so maybe I should point out
that before we continue the previous
case here
the previous case for equates we have an
achingly real ways that make let's
vaccination is the condition that the
relative risks that relative risk
exceeds the priority of infection when
no one is vaccinated ok now we consider
it an arrest in a while if that relative
risk is actually less than that of
infection which is this scenario and so
we are trying to write a condition what
if I is strictly less than these dif 1-1
where are not gay so these tall error
which you may be familiar with this this
all area means if and only if so we have
the condition are relative risk last and
privilege of infection with zero
coverage even only and is happening here
is think of think of swapping these
inequalities little are
with these fraction 10 right if you say
that the one where r 02 the left side of
this inequality in now send this little
guy to the right side then you end up
with this expression ok so relative risk
less than productive infection at sea
record it even only 10 a lower bound of
the difference one miners are ok and dad
is equivalent to what do they do next
basically take these denominator or I
can you can think of multiplying both
sides of this inequality by are not and
that gives you this expression here
gives you this expression so a condition
on the scenario of makes it clear makes
Nash equilibrium condition would be that
I 0 times qty 1 miners are where artists
that will increase that exceeds one that
is exceeding 100 now
the value of the mix nasa clear in which
we are gonna steal the note here without
limitation with a star P
i series he starred what we do is that
we saw this equation we said the reality
of REEs equal to the probability is
actually at that college P start to see
are equal spies soupy star in movies we
saw we solve this equation for be start
so let's remember what expression pi is
so that expression is equal to 1-1 home
where r is the multiplier and quantity
1-3 star and what follows is we have to
a little bit of algebra here and there
is a number of ways in which these will
work out so I'm gonna continue this
equation solving trade so remember we
have these equations are equals dad and
we wouldn't do there is we went at
Seoul's when solve these equations for
peacetime ok so let me continue writing
here so we have webbing going to do is
I'm going to swap arms in our to the
right side and innocent distraction to
the left side so I'm gonna ride 140
times 10 mins P star and is equal to 1-5
are you can solve this in many many ways
is traction here and you may do the math
here in nineteen secularism steps that I
used but that's fine we will post
obtain the same equation now I can
multiply this equation by our Syrian
both sides and that would give me one on
the left side 1-1 of our 1-3 start and
here I have are 20 91 miners are ok and
next thing I can do is so I start with
this equation and dad implies this area
means implies that in place that
expression is in place these expression
and the next thing is a can take the
multiplicative inverse reckoned race the
negative 10 explaining both sides of
this equation so I have 1-2 P star
equals 21 over 20 times when miners are
going so from the previews expression
that in place the is in I can just swap
this piece star can move the peace star
to the right and this to the left so
this is gonna give me an expression 1-1
or 0-2 are equals the star so
is means that we have obtained here
expressing four beasts are great so that
expression for PCR is precisely what we
were going after that would be the
expression for the convergent least able
national clearing and you can see how
dad expression is holding for the
scenario in which that that's a
non-trivial NASA Cali right so we have
two types of nasa clearing one kind of
running out of space here so we had
first type of nasa clearing it's a are
exceeding by sub-zero and that he was
there trivia it clear which was pieced
are equal zero and then now dysentery
which power is less than the probability
that there's no vaccinations and dad
gives us these expression for this
nationally and we danced concludes were
lecture number five on game theory and
vaccinations
