today we'll go through module 4 which is
on game theory as you know
in a business many times a company makes decisions based on what the competitor does
so sometimes they take actions based on
what the competitor does and similarly a
competitor may also take some actions
related to the product or service
looking at what this company has done if
we call this company X and competitor y
so obviously X many times looks at what
options or strategies competitor Y is
choosing and accordingly they may adjust
their strategies basically this type of
situations can be modeled very easily
using concepts in game theory so game
theory is based very well suited for
this type of situations and this field
was popularized by dr. Nash who won
Nobel Prize around 1950 for developing
what is known as Nash equilibrium in
fact there was a movie made on the life
of dr. Nash called the Beautiful Mind
which is a very very interestingly made
film now in this chapter we are going to
look at situations where number of
players will be 2 we are only looking at
situations where only two players are
involved x and y and associated with
those two players and the strategies they
are using we look at payoffs and this
also we basically use a special case
where payoffs are in the form of
zero-sum so the total payoff is always
zero for example based on the strategy
by player X it may happen that the gain
by player X will be plus 5% so if there
is a 5% gain for player X that also
means that there will be a loss of 5
percent for player Y so when you add
these two the total is always 0 so these
type of situations are called
zero-sum games another situation could
be where player
X chooses a strategy which leads to let's say
10% gain in the market share and this
gain is coming basically from the
competitor who loses 10% market share so
the total is always zero the third
factor in game theory are number of
strategies players can choose
let us consider situations where these
two players x and y they choose two
strategies suppose x1 represents
the first strategy by player X and X2
represents second strategy by player X
these two strategies could be if we are
looking at say marketing maybe first
strategy is using TV for ads and second
strategy could be using radio for X
so based on this player Y also will
choose certain strategies and let's say
there are two in this table let's say
payoff is 3 when X chooses strategy 1
and y choose a strategy 1 so this table
is made with respect to player X so if
we see a positive number like 3 that
means it is a gain for X and loss for y
so let's say this value here is 5 so
again it is gain for player X this is 1
again a gain for player X and this is
negative 2 which means layer X is going
to lose two points so now if you
consider player y and the strategy is
used by player y when player y chooses
first strategy what will happen is if X
chooses first strategy Y will lose 3 and
if player X chooses second strategy Y
will lose 1 so maximum possible loss for
player y is going to be 3 so I'm going
to write maximum values in this last row
similarly maximum possible loss for
player y will be 5
and basically what player y will try to
do is to minimize the possible losses so
out of the two values player Y will
choose this one and the idea is very
simple because this payoff matrix you
can see out of
situation three situations favor X so
obviously this payoff is biased towards
player X so Y is eventually going to
lose the only thing Y can do is to
minimize that maximum loss the criteria
being used by player y in this situation
is minimax criteria which is minimizing
the maximum loss this three here is
called upper value of the game similarly
if you consider strategies used by
player X so what player X will try to do
is so if we create this last column
where we put minimum payoffs so in the
first row minimum is three so minimum
possible win for player X using strategy
one is three
similarly minimum possible gain by using
strategy two is in fact negative two so
what player X will try to do is player X
will try to maximize the minimum win so
maximum of the two is this value three
here so player X is using a strategy
called max min criteria and this value
here is called lower value of the game
now it happens in this case that the
upper value of the game is equal to
lower value of the game and in fact we
can call this is the value of the game
which is equal to three so basically
what is this number three this number
three is average or expected game
outcome if this game is played many
times like infinite number of times so
the game will converge at this value
three so both player x and y will use
strategies in such a way that they end
up with this value here
basically it means whenever player X
chooses first strategy Y will choose
first strategy similarly whenever Y
chooses second strategy obviously player
X will choose first strategy
so always they end up with three so
whenever this happens whenever both
these numbers are same we say that
equilibrium or saddle point exists
equilibrium or saddle point if you have
this so we have two situations one is
this point exists and second situation
is it does not exist
so whenever equilibrium a saddle point
in a game exists we say that we have
pure strategy games and when it does not
exist we say that we have mixed strategy
games so pure strategy obviously means
that both players will always use one
particular strategy to maximize or
optimize their situation and mixed means
they may not use the same strategy all
the time they may change strategies
so let's say a payoff matrix
looks like this
so suppose payoffs are 2 negative 4 6
and 10
now if you have to figure out whether
this is a pure strategy game or mixed
strategy game so we again apply minimax
criteria for player Y so if we find
maximum value in those columns so
maximum is 6 and 10 so we select minimum
of the two using minimax criteria
similarly if you find minimum in the
rows for player X or negative 4 & 6
so if we use maximin criteria so maximum
of these minimum values so the value is
6 so again both are equal so this is
also a pure strategy game
so this previous example also you can
see it is favoring player X because out
of 4 situations in three situations
it favors player X so let's look at this
X1 X2 y1 and y2 and suppose the
payoffs are 10 6 negative 12 and 2 so
again if you find maximum and minimum
values so you have maximum 10 and
maximum 6 here and minimum of the 2 is 6 using minimax and similarly using Maxmin
6 negative 12 so maximum of the 2
is 6 so again we have pure strategy game
so always only one solution will appear
so this game also favors player X
so maximum of the two is two so now they
do not match so obviously this is called
mixed strategy game for mixed strategy
game to find the expected value like if
the game is played for a long period of
time so what is the expected gain for
player X or what is the expected gain
for player Y to find that we have to
do this analysis in slightly different
way it's not as straightforward as we
have done in the previous examples so we
need to do some calculation to figure
out what is the expected value or the
two players so I'm going to again make
this table here slightly bigger so that
I can write few other values so
and let's say player Y plays
strategy 1 p times and strategy two
1 minus P proportion of times strategy
one is played is P and proportion of
total times when strategy 2 is used by
play y is 1 minus P so that the total is
always 1 and payoff is 4 2 1 10 and
similarly for player X let's say first
strategy is used Q times or proportion
is Q and 1 minus Q for second one so
expected gain will be so 4 times Q so we
multiply 4 and Q so basically we are
taking weighted average we are using the
weights as the proportion of times each payoff
will occur so 1 times 1 minus Q
similarly 2 times Q or this 1 plus 10
times 1 minus Q so now if we equate the
two what we are going to get here is 4 Q
plus 1 minus Q equals 2Q plus 10 1
minus Q basically we are trying to
figure out what will be the value of Q be
if you simplify this you get 4Q minus Q
is 3q plus 1 and if you simplify this
you get negative 8Q plus 10 so you
end up with if you bring negative 8Q
this side so you end up with 3q plus 8Q
equals 10 minus 1
so in fact 11Q equals nine so your Q
is equal to 9 over 11 so if Q is 9
over level so obviously 1 minus Q is
going to be 2 over 11 so basically player
X will play first strategy nine over
11 like that will be the proportion
out of all the time player X plays
strategy 1 and similarly 2 over 11 for
strategy 2
now similarly if we calculate expected
gain so now we multiply 4 with P so 4p
plus 2 times 1 minus P and then 1 times
P is P plus 10 times 1 minus P so if you
equate the two you get this if you
rearrange so 4P minus 2P will be 2 P
plus 2 equals and this becomes 10 minus
9 9 plus 2 becomes 11 and this becomes 8
so that P equals 8 over 11 similarly 1
minus P will be 3 over 11 so using
this if we want to calculate what will
be the gain for or expected gain for y
so you can actually use any one of the
two let's say I'm using the second one
so P plus 10 times 1 minus P so if you
input these values here 8 over 11 10
times because 1 minus P is 3 over 11
so 3 over 11 so if you simplify
this you will get 38 over 11 or 3.46
now expected gain for player y and
because we are getting
this 3.46 which is a positive number in
fact this is a loss for player Y
because the payoff matrix is made in
such a way that any positive value means
it is loss for player y and gain for
player X on an average player y is going
to lose 3.6 expected gain for player X
is going to be so again we can choose an
equation let's choose the first one
this one here so this equals 4Q plus 1 minus Q so this becomes 3 times
because Q is 9 over 11 plus 1 so
this will give us 27 over eleven plus 1
if you simplify this you get 38 over
11 which is same number we got
earlier and we get 3.46 these two
numbers are obviously same because this
is a zero-sum game
so whatever player X gains same thing is
lost by player y and whenever we see a
positive number for player X it means it
is gain and if we see a positive number
for player Y it is a loss
this time let's say this game favors
player Y playoff matrix is X1 X2
so again if you look at this and find
using the maxmin and minMax criteria
you will see that the two values of the
game do not match so this is also a
mixed strategy game so let's assume that
first strategy by player Y is played P
times and second strategy 1 minus P
that's the proportion and this is Q and
1 minus Q so if you calculate this value
here negative 4Q and 0 times 1 minus
Q is equal we can equate with equal to
the value here which is negative 2Q
plus negative 10 times 1 minus Q so we
are using these values here in the
payoff table and multiplying by the
proportions so if you simplify this this
whole thing becomes 0 so you are left
with negative 4Q and if you simplify
this you get negative 2q negative 10
plus 10Q if we bring all the Q's on the
other side you will end up with negative
12Q and negative 10 but negative and
negative become positive so Q equals 10
by 12 which is same as 5 by 6 so 1
minus Q obviously will be 1 by 6 now if
you do the calculation here so this will be
for the expected gain
and expected gain
so if we equate the two you get negative
4P
negative two and this becomes plus 2p
equals so zero times p becomes 0 so you
are left with negative 10 plus 10P what
you get here is negative 2p minus 10P
equals 2 minus 10 if you bring all P's
on the left and all the numbers on the
right and then we end up with negative
12P equals negative 8 but because both
are negative so we can make it positive
so we get P equals 8 by 12 or 4 times 2
is 8 and 4 times 3 is 12 so 2 by 3 1 minus P
is going to be 1 by 3 so based on the
two we can calculate expected values
so negative value for player X means it
is a loss for player X
so for player Y if we see a negative
number that means actually it is a gain
for player Y we can see both values
are same but interpretation is different
so both are negative so negative means
loss for X but gain for Y
the way this idea is used is
sometimes there are strategies which
will never be used because the other
strategies are so dominant that some
strategies may not be used anytime by
any of the two players when somebody is
working to find a solution it makes
sense to eliminate those strategies
which are dominated by other strategies 
because that will reduce the size of the
game and make things more easier to deal
with so let's say we have a situation
like this I will use example where
player X has X1 X2 X3 3 strategies for
the game and Y has 2 and let's say
payoffs look like this
so when player X will use a first
strategy obviously player y will go for
the second strategy when player X uses
second strategy player y will go for the
first strategy because that will
minimize the loss now in no case X is
going to use the third strategy because
the payoffs are very small in fact
player X will do better if player X goes
for second strategy because the payoff
will be at least 2 which is 2 times so
this strategy will never be used because
other strategies dominate this so
basically this can be deleted
so if you look carefully at this payoff
table you will find that player Y will
never choose second and third strategies
because player y can win by choosing
either first strategy or fourth strategy
so y2 will never be used similarly y3
will never be used because the other
strategies dominate second and third
strategy
so again if you look carefully you can
see that player X will never use first
strategy because the gains to be made
are much smaller compared to second and
third strategies so basically the
concept of dominance can be used to
reduce the size of the game and once the
size of the game is reduced at least
that makes things much simpler to look
at
