Introduction
Abdul Quadir
XLRI
December 7, 2019
What is Game Theory?
I Decision theory - (one person’s choice affects its own payoff).
I What if you take a decision or make a choice that affects the
payoff of every relevant persons.
I This situations leads to the situation of strategic thinking.
I More formally, game theory is a subject where mathematical
tools are used to model and analyze situations of interactive
decision making.
I This implies that there are multiple players having different
goals, where decision of each affects the outcome for all
decision makers.
I Caution: Knowing game theory does not guarantee winning.
I Silver lining: It provides a framework for thinking about
strategic interaction.
What is Game Theory?
I Decision theory - (one person’s choice affects its own payoff).
I What if you take a decision or make a choice that affects the
payoff of every relevant persons.
I This situations leads to the situation of strategic thinking.
I More formally, game theory is a subject where mathematical
tools are used to model and analyze situations of interactive
decision making.
I This implies that there are multiple players having different
goals, where decision of each affects the outcome for all
decision makers.
I Caution: Knowing game theory does not guarantee winning.
I Silver lining: It provides a framework for thinking about
strategic interaction.
What is Game Theory?
I Decision theory - (one person’s choice affects its own payoff).
I What if you take a decision or make a choice that affects the
payoff of every relevant persons.
I This situations leads to the situation of strategic thinking.
I More formally, game theory is a subject where mathematical
tools are used to model and analyze situations of interactive
decision making.
I This implies that there are multiple players having different
goals, where decision of each affects the outcome for all
decision makers.
I Caution: Knowing game theory does not guarantee winning.
I Silver lining: It provides a framework for thinking about
strategic interaction.
Example: Single Person Decision
I Hamid is a investor with 100 million rupees.
State
Good Bad
Bonds 10% 10%
Stocks 20% 0%
I Which one is better?
I Probability of good state is p.
I Assume that Hamid wants to maximize the amount of money
he has at the end of the year.
I Bonds: 110 million rupees.
I Stocks: Expected money holding
p × 120 + (1 − p) × 100 = 100 + 20 × p
I Invest in stocks if p > 12 .
I Invest in Bonds if p < 12 .
Example: Single Person Decision
I Hamid is a investor with 100 million rupees.
State
Good Bad
Bonds 10% 10%
Stocks 20% 0%
I Which one is better?
I Probability of good state is p.
I Assume that Hamid wants to maximize the amount of money
he has at the end of the year.
I Bonds: 110 million rupees.
I Stocks: Expected money holding
p × 120 + (1 − p) × 100 = 100 + 20 × p
I Invest in stocks if p > 12 .
I Invest in Bonds if p < 12 .
Example: Single Person Decision
I Hamid is a investor with 100 million rupees.
State
Good Bad
Bonds 10% 10%
Stocks 20% 0%
I Which one is better?
I Probability of good state is p.
I Assume that Hamid wants to maximize the amount of money
he has at the end of the year.
I Bonds: 110 million rupees.
I Stocks: Expected money holding
p × 120 + (1 − p) × 100 = 100 + 20 × p
I Invest in stocks if p > 12 .
I Invest in Bonds if p < 12 .
Example: Strategic Decision
I Hamid has two options for investing his 100 million.
I invest in bonds with certain returns of 10%.
I invest it in a risky venture
I success: return is 20%
I failure: return is 0%
I Venture is successful if and only if total investment is at least
200 million.
I There is another investor Krishna like Hamid.
I They cannot communicate and hence have to make the
investment decision without knowing the decision of each
other.
Krishna
Bonds Venture
Bonds 110, 110 110, 100
Hamid
Venture 100, 110 120, 120
Example: Strategic Decision
I Hamid has two options for investing his 100 million.
I invest in bonds with certain returns of 10%.
I invest it in a risky venture
I success: return is 20%
I failure: return is 0%
I Venture is successful if and only if total investment is at least
200 million.
I There is another investor Krishna like Hamid.
I They cannot communicate and hence have to make the
investment decision without knowing the decision of each
other.
Krishna
Bonds Venture
Bonds 110, 110 110, 100
Hamid
Venture 100, 110 120, 120
Examples of Common Games
I Driving
I Coordination
I Penalty kicks
I hunter and hunted
I Grade trap
I prisoner’s dilemma
I Group project
I free-riding
I Mean professor
I commitment
I Dating
I hidden information
Common Games Played by Businesses
I Standards adoption
I Coordination
I Audits
I hunter and hunted
I Price wars
I prisoner’s dilemma
I Pollution abatement
I free-riding
I Capacity expansion
I commitment
I External financing
I hidden information
I Spectrum allocation
I Auctions
Origins of Game Theory
I The foundations of game theory were laid down in the book
The Theory of Games and Economic Behaviour,
published in 1944.
I It was written by the mathematician Jon von Neuman and
economist Oskar Morgenstern.
I The theory was developed extensively since then.
I The Noble prize in economics has been awarded four times to
the professors working in this field.
I 1994, JohnNash, Reinhard Selten and John C Harsanyi.
I 1996, William Vickery
I 2005, Robert J Aumann and Thomas C Schelling.
I 2007,Roger Myerson, Eric Maskin, Leonard Hurwicz
Applications of Game Theory
Game theory has been applied in various fields:
1. Theoretical economics.
2. Networks.
3. Business aaplications
4. Political science.
5. Military applications.
6. Inspection.
7. Biology.
Grade Game
I Write letter α or β on a form without showing it to your
neighbour.
I We will randomly pair your form with one other form.
I Nobody will never know with whom he or she is paired.
I Grades is assigned as follows:
I If you put α and your pair put β, you will get grade A and
your pair will get C.
I If both put α, then you both will get grade B.
I If you put β and your pair put α, you will get grade C and your
pair will get A.
I If both put β, both will get grade B+.
Output Matrix
Pair
α β
α B, B A, C
Me
β C, A B+, B+
Outline of The Course
Module A: Games of Complete Information and Its Business
Applications
I Introduction to game theory - Concept of individual rationality,
decision making with certain and uncertain outcomes.
I Sequential move games, backward induction and foresight.
I Simultaneous move games - Domination, iterated elimination
of dominated strategy, Pure strategy Nash equilibrium.
I Repeated Games.
I Simultaneous move games - Mixed strategy Nash equilibrium
I Applications of Nash Equilibrium - Auctions, Buyer-seller
games, Market competition, Electoral competition.
I Commitment and Strategic moves - Credibility, threats and
promises.
I Bargaining under complete information.
Outline of The Course
Module B: Games of Incomplete Information and Its
Business Applications
I Introduction to games of incomplete information and Bayesian
Nash Equilibrium.
I Sequential move games of incomplete information and perfect
Bayesian equilibrium.
I Managing “principle-agent problems” by creating incentives -
Designing contracts.
I Auctions and Bidding
I Doing business with limited information - Limit Pricing and
predatory pricing.
I The structure of signaling contracts - Job market signaling,
Entry deterrence under incomplete information.
Textbook and Evaluation
1. Dixit, Avinash, and Susan Skeath: Games of Strategy.
W.W. Norton and Company.
2. David McAdams: Game-Changer. W.W. Norton and
Company
Evaluation
1. Quiz 1(on Module 1): 30 points
2. Quiz 2 (on Module 2): 30 points
3. End-term exam (Entire course): 40 points
Exam and quizzes will require problem solving skills.
Rational Decision Problem
The Single Person Decision Making
I Decision is ubiquitous in our day-to-day living.
I From instance, what to eat in Dhaba in evening snacks or
what to wear for a party (trivial).
I Non-trivial decision: where to invest your money?
I We provide a framework to thick about decision making
problems.
Elements of Decision Problems
A decision maker must be knowing the following three fundamental
features of a problem
1. What are his or her possible choices?
I Actions are the alternatives from which the decision maker
can choose.
2. What is the result of each of those choices?
I Outcomes are the possible consequences that can result from
any of the actions.
3. How will each result will affect his or her well-being?
I Preferences describe how the decision maker ranks the set of
possible outcomes, from most desired to least desired.
Elements of Decision Problems
A decision maker must be knowing the following three fundamental
features of a problem
1. What are his or her possible choices?
I Actions are the alternatives from which the decision maker
can choose.
2. What is the result of each of those choices?
I Outcomes are the possible consequences that can result from
any of the actions.
3. How will each result will affect his or her well-being?
I Preferences describe how the decision maker ranks the set of
possible outcomes, from most desired to least desired.
Thought Experiment
I Choosing an item for breakfast: Suppose items available are
Dosa and Paratha.
I Thus, the set of actions is A = {d, p}.
I Note that actions and outcomes are synonymous here.
I However, we treat them distinct here.
I Thus, denote the set of outcomes by X = {x, y }, where x
denotes eating dosa and y denotes eating paratha.
I Eating either one of them will provide you with some
satisfaction.
I Therefore, you will have preferences over x and y .
Preference Relation
I Suppose you prefer eating dosa over paratha.
I Then, we can say ‘x is at least as good as y ’.
I We denote the relation ‘at least as good as’ by .
I Therefore, x y is read as ‘x is at least as good as y ’.
I It is a short way to express the decision maker’s ranking of
possible outcomes.
I
I What assumptions have we made so far?
I The decision maker is able to rank any two outcomes from
the set of outcomes.
I Technically, the preference relation is complete if for any
x, y ∈ X , either x y or y x.
I It does not let you to be indecisive between any two outcomes
Preference Relation
I Suppose you prefer eating dosa over paratha.
I Then, we can say ‘x is at least as good as y ’.
I We denote the relation ‘at least as good as’ by .
I Therefore, x y is read as ‘x is at least as good as y ’.
I It is a short way to express the decision maker’s ranking of
possible outcomes.
I
I What assumptions have we made so far?
I The decision maker is able to rank any two outcomes from
the set of outcomes.
I Technically, the preference relation is complete if for any
x, y ∈ X , either x y or y x.
I It does not let you to be indecisive between any two outcomes
Preference Relation
I Suppose you prefer eating dosa over paratha.
I Then, we can say ‘x is at least as good as y ’.
I We denote the relation ‘at least as good as’ by .
I Therefore, x y is read as ‘x is at least as good as y ’.
I It is a short way to express the decision maker’s ranking of
possible outcomes.
I
I What assumptions have we made so far?
I The decision maker is able to rank any two outcomes from
the set of outcomes.
I Technically, the preference relation is complete if for any
x, y ∈ X , either x y or y x.
I It does not let you to be indecisive between any two outcomes
Preference Relation
I Suppose you prefer eating dosa over paratha.
I Then, we can say ‘x is at least as good as y ’.
I We denote the relation ‘at least as good as’ by .
I Therefore, x y is read as ‘x is at least as good as y ’.
I It is a short way to express the decision maker’s ranking of
possible outcomes.
I
I What assumptions have we made so far?
I The decision maker is able to rank any two outcomes from
the set of outcomes.
I Technically, the preference relation is complete if for any
x, y ∈ X , either x y or y x.
I It does not let you to be indecisive between any two outcomes
Rational Preferences
I If you have only two outcomes, then you can always make the
best decision.
I What if you have more than two outcomes, then what can be
your best outcome?
I For instance, X = {a, b, c}. So we can have
ab a
bc b
c a c
I We have to knit together all the elements from the set of
outcomes to find out the best outcome.
I The preference relation is transitive if for any three
outcomes x, y , z ∈ X , if x y and y z, then x z.
I Note that the preference given in table is not transitive.
I A preference relation that is complete and transitive is called a
rational preference relation.
Rational Preferences
I If you have only two outcomes, then you can always make the
best decision.
I What if you have more than two outcomes, then what can be
your best outcome?
I For instance, X = {a, b, c}. So we can have
ab a
bc b
c a c
I We have to knit together all the elements from the set of
outcomes to find out the best outcome.
I The preference relation is transitive if for any three
outcomes x, y , z ∈ X , if x y and y z, then x z.
I Note that the preference given in table is not transitive.
I A preference relation that is complete and transitive is called a
rational preference relation.
Rational Preferences
I If you have only two outcomes, then you can always make the
best decision.
I What if you have more than two outcomes, then what can be
your best outcome?
I For instance, X = {a, b, c}. So we can have
ab a
bc b
c a c
I We have to knit together all the elements from the set of
outcomes to find out the best outcome.
I The preference relation is transitive if for any three
outcomes x, y , z ∈ X , if x y and y z, then x z.
I Note that the preference given in table is not transitive.
I A preference relation that is complete and transitive is called a
rational preference relation.
Payoff Functions or Utility Functions
I Rational preferences makes the players or decision makers to
behave in a consistent and appealing way.
I Moreover, it provides a more friendlier and operational
apparatus known as payoff or utility function.
I Suppose a person wants to open a tea stall in his locality.
I He can use either low quality (l), medium quality (m) or high
quality (h) tea leaves.
I His cost for using low, medium or high quality tea leaves are
100, 150 or 200 rupees, respectively.
I His revenue for using low, medium or high quality tea leaves
are 150, 225 or 260 rupees, respectively.
I Thus, A = {l, m, h} and the outcome set is given by the
profits X = {50, 75, 60}.
I Assume higher profit is desirable, then 75 60 50.
Payoff Functions or Utility Functions
I This is a obvious profit maximizing problem.
I We have fit this problem into our framework and derived the
preference relation that is consistent with the maximizing of
profit.
I Another natural way is to consider the actions and their
associated profits.
I In fact, we can define a profit function in a obvious way:
each action a yields a profit π(a).
I Thus, we can look the profit function and choose an action
that maximizes it.
I In other words we can use the profit function to evaluate
actions and outcomes.
Payoff Functions or Utility Functions
I For this example, it is more direct to use profit to choose the
best (optimal) action.
I Can we solve any decision problem in this way?
I In other words, can we always find a function like profit
function in all decision problems?
I The answer is yes.
I A payoff function u : X → R represents the preference
relation if for any pair x, y ∈ X , u(x) ≥ u(y ) if and only if
x y.
I The above definition is an ordinal notion.
I For finite X , the answer of the above question is easy.
Theorem
If the set of outcomes X is finite then any rational preference
relation over X can be represented by a payoff function.
Payoff Functions or Utility Functions
I For this example, it is more direct to use profit to choose the
best (optimal) action.
I Can we solve any decision problem in this way?
I In other words, can we always find a function like profit
function in all decision problems?
I The answer is yes.
I A payoff function u : X → R represents the preference
relation if for any pair x, y ∈ X , u(x) ≥ u(y ) if and only if
x y.
I The above definition is an ordinal notion.
I For finite X , the answer of the above question is easy.
Theorem
If the set of outcomes X is finite then any rational preference
relation over X can be represented by a payoff function.
Payoff Functions or Utility Functions
I For this example, it is more direct to use profit to choose the
best (optimal) action.
I Can we solve any decision problem in this way?
I In other words, can we always find a function like profit
function in all decision problems?
I The answer is yes.
I A payoff function u : X → R represents the preference
relation if for any pair x, y ∈ X , u(x) ≥ u(y ) if and only if
x y.
I The above definition is an ordinal notion.
I For finite X , the answer of the above question is easy.
Theorem
If the set of outcomes X is finite then any rational preference
relation over X can be represented by a payoff function.
Decision Tree
I decision trees are very handy to represent and solve decision
problems.
I Suppose you would like to choose the best outcome from
X = {a, b, c, d}.
I Preferences are given as d a c b.
I We can consider the following payoff representation of the
preferences as v (d) = 4, v (a) = 3, v (c) = 2 and v (b) = 1.
I This could be depicted in the following decision tree:
1
b
2
c
Decision Maker a
d 3
4
Decision Tree
I decision trees are very handy to represent and solve decision
problems.
I Suppose you would like to choose the best outcome from
X = {a, b, c, d}.
I Preferences are given as d a c b.
I We can consider the following payoff representation of the
preferences as v (d) = 4, v (a) = 3, v (c) = 2 and v (b) = 1.
I This could be depicted in the following decision tree:
1
b
2
c
Decision Maker a
d 3
4
Rational Player
I We require that a decision must know the following features
of the problems:
1. A, all possible actions
2. X , all possible outcomes
3. exactly how each action affects which outcome will materialize
4. his rational preferences (payoffs) over outcomes.
I In many problems, actions and outcomes are equivalent.
I However, there are some problems they are not.
I For instance, you can either let your drunk friend derive or call
a taxi.
I The outcomes are accident or he is safe reaching home.
I But we can define a one-to-one correspondence between
action and outcome.
I Thus, we can write v (a) = u(x(a)).
I A decision maker is rational if he chooses an action a ∈ A
that maximizes his payoff. Technically, a∗ is chosen iff
v (a∗ ) ≥ v (a) for all a ∈ A.
Rational Player
I We require that a decision must know the following features
of the problems:
1. A, all possible actions
2. X , all possible outcomes
3. exactly how each action affects which outcome will materialize
4. his rational preferences (payoffs) over outcomes.
I In many problems, actions and outcomes are equivalent.
I However, there are some problems they are not.
I For instance, you can either let your drunk friend derive or call
a taxi.
I The outcomes are accident or he is safe reaching home.
I But we can define a one-to-one correspondence between
action and outcome.
I Thus, we can write v (a) = u(x(a)).
I A decision maker is rational if he chooses an action a ∈ A
that maximizes his payoff. Technically, a∗ is chosen iff
v (a∗ ) ≥ v (a) for all a ∈ A.
An Example
I Suppose you are in a party and considering in engaging in
social drinking.
I Suppose you would prefer some wine but too much will make
you sick.
I There is one litre-bottle wine.
I Thus, A = [0, 1] and a ∈ A is how much you choose to drink.
I Assume your payoff function over actions is represented by the
following function
v (a) = 2a − 4a2
I How much should you drink?
I The maximization problem is
max 2a − 4a2
a∈[0,1]
I Thus, a∗ = 0.25.
An Example
I Suppose you are in a party and considering in engaging in
social drinking.
I Suppose you would prefer some wine but too much will make
you sick.
I There is one litre-bottle wine.
I Thus, A = [0, 1] and a ∈ A is how much you choose to drink.
I Assume your payoff function over actions is represented by the
following function
v (a) = 2a − 4a2
I How much should you drink?
I The maximization problem is
max 2a − 4a2
a∈[0,1]
I Thus, a∗ = 0.25.
An Example
I Suppose you are in a party and considering in engaging in
social drinking.
I Suppose you would prefer some wine but too much will make
you sick.
I There is one litre-bottle wine.
I Thus, A = [0, 1] and a ∈ A is how much you choose to drink.
I Assume your payoff function over actions is represented by the
following function
v (a) = 2a − 4a2
I How much should you drink?
I The maximization problem is
max 2a − 4a2
a∈[0,1]
I Thus, a∗ = 0.25.
An Example
I Suppose you are in a party and considering in engaging in
social drinking.
I Suppose you would prefer some wine but too much will make
you sick.
I There is one litre-bottle wine.
I Thus, A = [0, 1] and a ∈ A is how much you choose to drink.
I Assume your payoff function over actions is represented by the
following function
v (a) = 2a − 4a2
I How much should you drink?
I The maximization problem is
max 2a − 4a2
a∈[0,1]
I Thus, a∗ = 0.25.
An Example
I Suppose you are in a party and considering in engaging in
social drinking.
I Suppose you would prefer some wine but too much will make
you sick.
I There is one litre-bottle wine.
I Thus, A = [0, 1] and a ∈ A is how much you choose to drink.
I Assume your payoff function over actions is represented by the
following function
v (a) = 2a − 4a2
I How much should you drink?
I The maximization problem is
max 2a − 4a2
a∈[0,1]
I Thus, a∗ = 0.25.
More Realistic Problem
I Consider a manager of a division in a company.
I He is thinking that should he for for research and development
(R& D) project?
I If he is not going for R&D, maybe the product will be
obsolete over time and hence there will be no taker.
I Maybe there will be constant demand for the product.
I What is he goes for R&D and the project might succeed or
fail.
I In case of success there will be great improvement of the
product and in case of failure there will be sunk cost.
I How to solve these sorts of problem where the outcome is not
certain?
I Next Class.
More Realistic Problem
I Consider a manager of a division in a company.
I He is thinking that should he for for research and development
(R& D) project?
I If he is not going for R&D, maybe the product will be
obsolete over time and hence there will be no taker.
I Maybe there will be constant demand for the product.
I What is he goes for R&D and the project might succeed or
fail.
I In case of success there will be great improvement of the
product and in case of failure there will be sunk cost.
I How to solve these sorts of problem where the outcome is not
certain?
I Next Class.
