 Lecture 54: Axiomatic (Cooperative) Bargaining- Need and Principles/ Axioms

Hello, welcome to mooc lectures on Strategy,An Introduction to Game Theory. In this module I am going to talk about cooperative
bargaining or as it is called axiomatic bargaining. Bargaining problems represent situation in
which we have to pay attention to three things, first a surplus needs to be split as we have
learnt in the previous module.And for division there is a conflict of interest, in the sense
that giving more for one person means automatically means that giving less to the other person.There
is also a possibility of concluding a mutually beneficial agreement, if you remember the
canonical example I gave in the previous module that in the buyer and seller case, that buyer
is willing to payas much as v and seller is willing to get at least c, so v minus c needs
to be split. So, anything between this if a prize is determined
between d and c, then it would benefit both of them.No agreement may be imposed on any
individual withouthis approval that is the third thing. So, for in the previous module what we did
is called non cooperative or a strategic model. What it did?That we explicitly modeled the
process that, how this bargaining would take place. For example, we talked about one stage,
two stage, three stage or infinite stage, alternative bargaining offer. But,if we think about it in real life we face
several situation in which the bargaining process cannot be pin pointed in a exact manner,
thatfirstplayer 1 will make an offer, then player 2 will get to accept or reject, then
player 2 will get to make an offer. It is also possible that after player 2 rejects,
the player 2 makes an offer and again, player 1 rejects player 2 again gets to make an offer. So, we do not know the exact process,so we
are going to take a slightly different approach, this is called axiomatic approach in which
we abstract away from the process and consider only the set of outcome that satisfy some
reasonable properties. Nash was the one same as same economist or
mathematician, after home we have Nash equilibrium in non cooperative setting. So, this is another work from Nash, Nash proposedthis
approach and he is stated to that one states as axioms several properties that would seem
natural for the solution to haveand then, one discover that axioms actually determine
the solution uniquely. So, the most fundamental question would be
that what are those reasonable axioms, what should we take as reasonable axioms. So, let us start with a very simple example
without thinking about any game theory,without thinking about what we have learnt so far,
that two players are engaged in bargaining over one unit of good.If agreement is not
reached, thenplayers do not get anything,both the players have identical preferences, they
are in identical scenario. What do you except?What happened?What would
happen in this case?We expectthat players will agree to decide this one unit, because
it will benefit both of them. So, this is basically efficient outcome. And next that each of them obtain one half,this
is the symmetric outcome of course, you can think that it is fear and all, but we will
see how it is happening here without bringing fairness into pictures at this stage. Let us take about a general case,what happens
that let us say X is a set of possible outcome or agreement that can be reached and D represents
the disagreement outcome. So, what we can write here mathematically
is X is something like x 1 and x 2, where x 1 goes to player 1 and x 2 goes to player
2.Such that, x 1 plus x 2 is always less than or equal to 1 and D is of course,0 0 in this
case both of them get 0. Now, it is also important that we understand
the rule of utility, the sameunit of money would not benefit or would not give the same
pleasure to different people.Let us say that if you add, if you give 100 rupees to a person
who has almost to nothing, he would be very, very happy he would be very, very satisfied
that if you give 100 rupees to Bill Gates who has really high amount of money that would
not make any difference. So, rather than dealing in terms of x 1 and
x 2 we should be dealing in terms of u of x 1 that is utility that player 1 would derive
from x 1 and utility that player 2 would derive from x 2.Now, in that sense we should talk
about the utility set that would give the utility per u of x 1 for player 1 and u2 of
x 2 to player 2.Such that, x 1plus x 2 is less than or equal to 1 and D is of course,
the utility that player 1 will get from 0 and player 2 will get from 0, this is disagreement
point and capital U is the utility possibility set. So, the bargaining problem is a question ofhow
to allocate utilities among two parties, this utility how we are getting,this utility is
coming, because a surplus is getting divided between these two players. So, this is the bargaining problem, how to
divide, how to locate utilities, how to divide, so that it will give some utilities,so how to
locate utilities among two parties. And what is the bargaining solution?Bargaining
solution assigns utility outcome for every set, every utility possibility set and disagreement point. So, we are starting with this is a bargaining
problem which gives all the possibilities, all the outcome in terms of utility, here
all the outcomes are in terms of money.Here all the outcomes are, this is the set which
gives all the outcomes in terms of utility and this is the utility at disagreement point,
so this is the bargaining problem. And what is the bargaining solution? Bargaining solution is a particular assignment
that would be given to player 1 and player 2.This is small u, sorry for my poor handwriting,
this is capital U and this is small u 1 and small u 2. So, bargaining solution assigns utility outcome
for every set. So, what are those principles?What are those
axioms that we are talking about?The first important axiom is scale free. So, if you think ofutility, how do we talk
about utility, let us say when I say I prefer tea to coffee.What do I need?How much more
I preferred tea to coffee that is very difficult to determine, I am just comparing between
tea and coffee.If you remember one of the earlier modules, we are I talked about from
preference to utility I talked about that whenever we have finite choices and we have
complete and transitive preference, then we can completely rank all the outcomes and we
can assign the number in a particular order, so utility in that sense is ordinal. So,if utility is ordinal then our solution
should not dependon the scale that is being used to measure the utility. So, first principle that we are talking about
here is that bargaining solution should be scale free.What does it mean?Scale free measure
means that in concluding an international agreement for example, the solution should
not depend on the currency in which the negotiation is taking place. Let us talk about the second principle that
we had talked earlier also.If bargaining situation for both the players are exactly the same,
then an agreement should split things equally as well. Like, in the earlier example we were talking
about how to divide one between two players. So, of course,there we were talking in terms
of monitory outcome, here we are taking in terms of utility outcome. But,the notion remains the same, if situation
is the same, if a problem is symmetry, thensolution should also be the symmetric while.Let me
also do one thing to represent both the notion of symmetry and scale free in terms of a pictorial
graph.Let us say let us take a bargaining problem, this is a bargaining problem, here
this is the disagreementpoint, hereI am describing thing in terms of utility, on x axiswe have
utility of player 1 and y axis you have utility of player 2. So, let me first do what is bargaining problem,solet
us say if surplus has to be divided, then all these possible outcome can be generated. So, this is the bargaining problem, because
it has this capital U and this has this disagreement point.Now, what would be the bargaining solution?Bargaining
solutionwould be a particular outcome, may be here or may be here or may be here, soa
particular outcome is the bargaining solution, but we are concerned aboutnot just this bargaining
problem, our bargaining problem can be of this nature, what should be the outcome here
in this case. So, here we are talking about solution in
terms of a function which takes a bargaining problem and assigns a particular solution,so particular
outcome. So, bargaining solution is a function that
assigns an outcome to a bargaining problem. it is clear what is bargaining problem, now
let us talk about scale free.Let us say that if our problem is like this, here we have
utility of player 1, here we have utility of player 2. So, we are taking let us sayfor some reasons
using some axioms, we have obtained that this is the solution.Now, because we are talking
about scale free, let us say that as we had talked about that utility or utilities are
ordinal in nature. So, we canstretch it, shrink it, we just have
to maintain the order,solet us say we have stretched only on axis 1. So, the new bargaining problem is this one. So,what it says the scale free says that,
if this is the outcome recommended by bargaining solution, if we stretch it back this point
should coincide with this point, it should come back to this point,this is what a scale
free means.Third point that we talked about symmetry, what we are talking about that let
us take a symmetric problem, let us say here we have a value, here u 1, here is u 2, if
u 1 is equal to u 2, here we have d 1 comma d 2 that is d, if d 1 is equal to d 2. So, the solution should be on the 45 degree
line, here this solution we should get as at this point where both the values are equal,so that
is symmetric. The third is that there should not be any
wastage, we talked about the efficiency. The bargaining solution should exhaust all
the possible games that is, reaching to a situation in which one party cannot gain any
more utility without taking utility away from theother party.Notice, when I was talking
about the bargaining problem and I gave a particular example of this bargaining problem,
I said that the possibilities the outcome can be here, here or anywhere as long the
bargaining outcome has to be belong to this capital U. Now,no wastage see look at this point, if
this is the outcome what would happen, if we move in this direction both players would
be better half and it is possible, because these points are in utility possibility set. So, it is possible to movein this direction
which would make both of them better half. So, this cannot be the bargaining solution,
if we follow the principle of no wastage. The only outcome in this particular case which
are possible, if we follow the concept of no wastage, then what we will have, only outcome
on the boundaries would be possible.In this case what would happen?If we want to increase
the utility of any person, it will in variably mean the decreasing the utility of other person,
so no wastage is should be cleared. The forth one is alternatives not chosendo
not matter, what I mean here again let say that if we removes some of the alternativesthat
were not chosen, then it is decide in the new bargaining problem, the solution should
remain the same, let us look at graphically. So, here is the bargaining problem and the
bargaining solution recommends this particular outcome.What it saysthat let us take a new
bargaining problem, in which we have all the utility a possible utilities
except these. In that case, in the new bargaining problem
which has the boundary given by this purple color, in this case what happens, solution
should remain the same, this is also called IIA Independence of Irrelevant Alternatives.What
is the logic, that two people are bargaining over something and they consider all the choices
and they eventually reached to this particular outcome. So, if we take out some of the possibilities
which were considered in the earlier case, but now in the new problem discarded they
were any way not selected earlier.So, even in the new problemnew situation they would
not be selected, because the outcome that was selected earlier is still present,so this
is called II A. So, let us take about these are the four decide
properties that we talked about, let us talk about some of the bargaining solution proposed
by philosophers and in thesocial scientist in the earlier ages and we should check what
principle do these solution satisfy. So, let us take about egalitarian solution
first what is an egalitarian solution that it chooses an outcome giving equal utility
to each side and lying on the utility frontier. So, let us talk about all four properties,
what was the first property scale free, let us sold on for scale free.The second property
was that no wastage of course, when we are talking about that the outcomes should lie
one the utility front here what do we mean by utilityfront here that if this is the problem,
then this would be a straight line is the utility frontier. Similarly, in this case the utility frontier
this even by the boundary. So, no wastage is satisfied.Similarly, thatIIA
is also satisfied, because if equal utility if an outcome giving equal utility is present
even after applying removing some of the option that would be selected. So, IIA is satisfied if a problem is symmetric
then everyone will get the same utility as it is given in the definition. So, it is symmetric how about the scale free. So, let us take an example that would clear
whether it is scale free or not, let us saywe have to define 100 and it gets divided into
50, 50to both of the players.Now, let us say player 1 protests and says that he values
dollar or rupee twice as much as player 2.What would happen in that case? Because, the values dollar twice as much as
the other player he would get 33.33 and other would gets 66.66. Why?Because, this utility from 33.33 would
be 66.66 and while the other players utility from 66.66 we can assume it is 66.66. So, it is of course, stupid for player 1 to
argue that he values a many twice as much as player 2, but it is not scale free, because
if is coming up with new utilities scheme then it the solution is no longer the same. So, egalitarian solution does not satisfy the
scale free. The next one is utilitarian solution, as suppose
to egalitarian solution what happensin the utilitarian solution, it choose an outcome
maximizing the sum of utilities.Since, the solution lies on the utility frontier, there
is no money left on the table and it satisfies no wastage. Again here we have aalso if it delete some
of the option from the negotiation it does not change the outcome,so principle force that
is II A still holds. But,how about scale free again we will see
that there is a problem with the scale free. Let us say that player1utilities given by
2 x and player 2 is utilities given by x.What happens in thiscase?Because, the aim is to
maximize the sum of the utilities everything will be given to player 1 and nothing will
be givenand to player 2 why, let us say again you have to defined 100 rupees, if you give
it to player 1, it will translate into 200, because 2 multiplied by x if you give it to
be a player 2 it will translate into 100. But,you are attempt is to maximize the sum
of utilities,soyou will give everything to player 1, but let us rescale the utility again.Now,
let us say that player 2 utility is represented by 3x, now in this case everything will be
given to player 2 and nothing will be given to player 1,so utilitarian solution is also
not scale free. Now, we are going to talk about Nash solution,
what did the Nash propose he said the choose an allocation that maximizes the product of
the utilities. So, maximize u 1 multiplied by u 2, such that
u 1 comma u 2 belongs to theutility possibility set, it satisfies all the principle, let us
say how because if we have to, if the situation is exactly the same for both the players then
what happens, if you it will fall little bit of mathematics. But,let us pay attention u 1 if we have on
x axis u 1 or y axis u 2 are the utility of the player 2 an here utility of player 1 u
1 multiplied by u 2 this is an equation of hyperbola,so it will be like this. And what is the aim of the Nash solution to
achieve the highest hyperbola possible. So, let us
take a bargaining problem, what is the aim giventhe these are the possibilities try to
attempt maximum highest possible hyperbola of course, we can number also k 1 k 2 k 3
k 4 clearly k 4 is greater than k 3 is greater than k 2 is greater than k 1. So, it is very, very clear if we take out
some of the option, let us say if we remove these options here still the outcome would
not change as long as this outcome is present we will have this particular outcome selected
by the Nash solution. So, II A is satisfy no wastage is satisfied
the scales free is also satisfied. How? We will see let us take an example, we will
see that is So, let us take this example in which two
players would like to split 1,utility of player 1 is given by 2 x utility of player 2 x given
by x. What is the Nash solution?Nash solution would,
because the player 1is getting x then player 2 will get 1 minus x. So, x and 1 minus x since utility of player
1 is 2 x. So, what we need to do here is, maximize 2
x multiplied by 1 minus x and if youmaximize x is equal to 1 by half both the players will
get one half and one half. Let us change the utility of player let us
stretch it and let us say the utility of player 1 is 3 x, but would be the now new solution,
now here we will have to maximize 3 x multiplied by 1 minus x and if you do the maximizationagain
what do we get, first order condition that we differentiateit with respect to 2 x what
do we get 3 multiplied by 1minus x plus 3 x and here minus 1 equal to 0 and what do
we get,6 x is equal to 3. So, x is equal to half again both the players
get one half and one half,so it is also scale free. Soin fact, it is only barraging solution that
satisfies all the principle, although I discuss these four principles, but you do not have
to strict to only these four particular principles. If you think about the symmetry argument it
is kind of hard wiredin our mind that if everyone is in the same situation then the bargaining
solution. So, divide the pie equally, similarly no wastage
also make sense that if benefit has to be giant it they player should giant should that
particular benefit. So, there should it to be any dispute about
these two principles, one regardingsymmetry, secondary adding no wastage how about to other
one. The third one that we have talked about is,
scale free that is also widely excepted that itis good idea to have an outcome that is
have a solution concept that is scale free. The most problematic one is independence of irrelevant
alternative the forth one is. So, several solutions have been proposed one
notable is he solution I am not getting in to it which takes out II A and gives another
criteria and comes up with another solution concepts. So, that is it for the bargaining axiomatic
bargaining I want to close this module just by distributing, two different branches of
game theory that is non corporative and corporative game theory.Most of the things that we have discussed,so for
in this course except today that we did in bargaining everything wediscussed was non
corporative game theory.What is non corporative game theory?We assume that players possibilities
of interacting and collaborating can be fully model. We know how players move, what are their actions
available, what would be the pay off these particular combination of action would be
taken. So, it analyzes how player should strategically
behave within the rules of the game, as suppose to non corporative game theory, we have corporative
game theory in which the basic assumption is players possibilities for interacting or
collaborating are too complex to be formally modeled. Exhausted aims to allocates, allocate among
player the estimated benefit of that operation. So, these are the two major branch is to two
branches of game theory,this course is primarily devoted to non corporative game theory, but
we though it is a nice idea to just introduce the notion of corporative games. Thank you.

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