Game Theory

Pavneet Singh

IIM Amritsar

June 20, 2025

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 1 / 64

Session Objectives

1 What is Game Theory? In what contexts can it be applied?

2 What are the elements required to build and understand models of
strategic games?
3 Terminologies- Normal Form, Strategy, Strategy Set, Strategy Profile
and Payoffs
4 An introduction to the role of Beliefs
5 The Rationality and Common Knowledge Assumptions

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 2 / 64

Introduction

Microeconomics and Game Theory- Rational choice among

interacting players
A theory of strategic interaction
Competitive decision settings – Situations where a number of
economic agents in pursuit of their respective self-interests take
actions that together affect all of their fortunes

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 3 / 64

Introduction

Microeconomics and Game Theory- Rational choice among

interacting players
A theory of strategic interaction
Competitive decision settings – Situations where a number of
economic agents in pursuit of their respective self-interests take
actions that together affect all of their fortunes
Strategic Interdependence- Your actions can affect my outcomes, and
vice versa
My optimal decision depends on what I think others will do in the
game
Key Assumption- Economic agents are rational. Is this a valid
assumption?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 3 / 64

 Pick a Number!

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 4 / 64

Game Theory in Action

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 5 / 64

Game Theory in Action

Figure: Coalition Formation

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 5 / 64

Game Theory in Action

Figure: RBI Announcements

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 6 / 64

Game Theory in Action

Figure: Formula One Pitstops

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 7 / 64

More Managerial Applications

Competitive Strategy- Market entry and exit, pricing strategies

Negotiations and Bargaining- Labor negotiations (wages and
benefits); supplier contracts
Auction design and bidding strategies
Cooperation and Alliances- Joint ventures or implicit collusion
strategies
How much to spend on advertising? How to position a brand?
Risk management and insurance decisions
Human Resource Management- Incentive schemes, hiring and
recruitment

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 8 / 64

The Course (and Logistics)

What we will do- Through relevant examples, learn the basic concepts
of game theory, and develop a thought process that can be applied to
business problems
What we will not do- Become mathematicians
Models, intuition and a word of caution
Static and dynamic games
The role of information
Simultaneous and sequential games
Exams, project, and assignments/quizzes

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 9 / 64

Elements of a Game

The list of players

Set of possible actions by players
Information available at the time of action
How actions lead to outcomes
Preferences over outcomes

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 10 / 64

 Terminology and Basic Assumptions

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 11 / 64

The Normal Form

A Tech Rivalry Game

Two smartphone manufacturers looking to enter the Indian market
Assume: Segmented market- Budget and Premium Consumers

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 12 / 64

The Normal Form

A Tech Rivalry Game

Two smartphone manufacturers looking to enter the Indian market
Assume: Segmented market- Budget and Premium Consumers
Samsung
Budget(B) Premium(P)
Budget(B) (7, 7) (10.5, 13.5)
Apple
Premium(P) (13.5, 10.5) (9, 9)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 12 / 64

The Normal Form

A Tech Rivalry Game

Two smartphone manufacturers looking to enter the Indian market
Assume: Segmented market- Budget and Premium Consumers
Samsung
Budget(B) Premium(P)
Budget(B) (7, 7) (10.5, 13.5)
Apple
Premium(P) (13.5, 10.5) (9, 9)
Does the Normal form contain all relevant information about a
strategic setting? Debatable - It works for static, one-shot games
Strategies and payoffs

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 12 / 64

Some Notations
A strategy is a complete contingent plan for a player in a game
It is a full specification of a player’s behaviour about what they would
do at each information set
A strategy space (or strategy set), Si , is a set comprising of each
of the possible strategies of player i in the game
e.g. in the tech rivalry game, S1 = {B,P}, S2 = {B,P}

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 13 / 64

Some Notations
A strategy is a complete contingent plan for a player in a game
It is a full specification of a player’s behaviour about what they would
do at each information set
A strategy space (or strategy set), Si , is a set comprising of each
of the possible strategies of player i in the game
e.g. in the tech rivalry game, S1 = {B,P}, S2 = {B,P}
Single strategies are denoted in small letters
e.g. The strategy s1 ∈ S1 , where we could have s1 = B

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 13 / 64

Some Notations
A strategy is a complete contingent plan for a player in a game
It is a full specification of a player’s behaviour about what they would
do at each information set
A strategy space (or strategy set), Si , is a set comprising of each
of the possible strategies of player i in the game
e.g. in the tech rivalry game, S1 = {B,P}, S2 = {B,P}
Single strategies are denoted in small letters
e.g. The strategy s1 ∈ S1 , where we could have s1 = B
A strategy profile is a vector of strategies, one for each player, s =
(s1 , s2 )
The set of strategy profiles, S = S1 × S2
In the tech rivalry game, S = {(B,B),(B,P),(P,B),(P,P)}

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 13 / 64

Some Notations
A strategy is a complete contingent plan for a player in a game
It is a full specification of a player’s behaviour about what they would
do at each information set
A strategy space (or strategy set), Si , is a set comprising of each
of the possible strategies of player i in the game
e.g. in the tech rivalry game, S1 = {B,P}, S2 = {B,P}
Single strategies are denoted in small letters
e.g. The strategy s1 ∈ S1 , where we could have s1 = B
A strategy profile is a vector of strategies, one for each player, s =
(s1 , s2 )
The set of strategy profiles, S = S1 × S2
In the tech rivalry game, S = {(B,B),(B,P),(P,B),(P,P)}
A game in normal form (strategic form) consists of a set of players,
strategy spaces for the players, and their payoff functions.
Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 13 / 64
Beliefs

A belief is a player’s assessment about the strategies of others in the

game
I think the other player will play...
We use probabilities to model beliefs
(p, 1-p) constitutes a probability distribution over {B,P} in the tech
rivalry game
Player 2 may not actually randomize over strategies, but Player 1 is
uncertain over Player 2’s actions
Player 1’s belief may not be accurate

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 14 / 64

Beliefs- An Example

Player 2
L M R
U (8, 1) (0, 2) (4, 0)
Player 1
C (3, 3) (1, 2) (0, 0)
D (5, 0) (2, 3) (8, 1)

Suppose P2 believes that P1 will select U with p1 = 1/2, C with

p2 = 1/4, and D with p3 = 1/4
P2’s belief, θ−2 = (1/2, 1/4, 1/4) is a probability distribution over
P1’s strategies.
Beliefs and expected payoffs

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 15 / 64

Beliefs- An Example

Player 2
L M R
U (8, 1) (0, 2) (4, 0)
Player 1
C (3, 3) (1, 2) (0, 0)
D (5, 0) (2, 3) (8, 1)

Suppose P2 believes that P1 will select U with p1 = 1/2, C with

p2 = 1/4, and D with p3 = 1/4
P2’s belief, θ−2 = (1/2, 1/4, 1/4) is a probability distribution over
P1’s strategies.
Beliefs and expected payoffs
The rationality assumption- Through some cognitive process, the
players form a belief about the strategies of others
Given this belief, they select a strategy to maximize their expected
payoff
Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 15 / 64
 The Quantification of Subjective Payoffs

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 16 / 64

Common Knowledge

Two basic assumptions- Rationality and Common Knowledge

Common Knowledge ensures that players have a shared understanding
of the entire game
A fact F is said to be common knowledge between the players if
each player knows F, each player knows that the other player knows
F, and so on..
The rules of the game are common knowledge among the players
Information asymmetry and common knowledge can co-exist!

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 17 / 64

 Strategic Tensions

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 18 / 64

Revisiting the Prisoners’ Dilemma

Player 2
C D
C (2, 2) (0, 3)
Player 1
D (3, 0) (1, 1)

The Prisoners’ dilemma highlights the first strategic tension- Conflict

between individual and group interests
How to solve the Prisoners’ Dilemma?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 19 / 64

Revisiting the Prisoners’ Dilemma

Player 2
C D
C (2, 2) (0, 3)
Player 1
D (3, 0) (1, 1)

The Prisoners’ dilemma highlights the first strategic tension- Conflict

between individual and group interests
How to solve the Prisoners’ Dilemma?
What if retribution was possible in the PD game? Could we model it
differently?
Could the players write a binding contract to commit them to C? Yes.

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 19 / 64

Prisoners’ Dilemma- Some Examples

Example- Airline Price War

Indigo
HighFare CutFare
HighFare (100, 100) (60, 150)
Spicejet
CutFare (150, 60) (70, 70)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 20 / 64

Prisoners’ Dilemma- Some Examples

Example- Airline Price War

Indigo
HighFare CutFare
HighFare (100, 100) (60, 150)
Spicejet
CutFare (150, 60) (70, 70)

Example- Coke vs Pepsi Advertising War

Coke
High(H) Low (L)
High(H) (4, 4) (7, 2)
Pepsi
Low (L) (2, 7) (5, 5)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 20 / 64

The Three Strategic Tensions

The second strategic tension- Stag-Hare Game

Player 2
S H
S (5, 5) (0, 4)
Player 1
H (4, 0) (4, 4)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 21 / 64

The Three Strategic Tensions

The second strategic tension- Stag-Hare Game

Player 2
S H
S (5, 5) (0, 4)
Player 1
H (4, 0) (4, 4)
Strategic Uncertainty and the coordination problem
How to solve strategic uncertainty?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 21 / 64

The Three Strategic Tensions

The second strategic tension- Stag-Hare Game

Player 2
S H
S (5, 5) (0, 4)
Player 1
H (4, 0) (4, 4)
Strategic Uncertainty and the coordination problem
How to solve strategic uncertainty?
Communication and social institutions

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 21 / 64

Strategic Uncertainty- An Example

Example- Coordination Game with Switching Costs

Other Users
Windows Linux
Windows (9, 9) (1, 2)
You
Linux (2, 1) (9, 9)

High switching costs lead to path dependence- Once a standard

dominates, it is hard to shift

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 22 / 64

Strategic Uncertainty- An Example

Example- Coordination Game with Switching Costs

Other Users
Windows Linux
Windows (9, 9) (1, 2)
You
Linux (2, 1) (9, 9)

High switching costs lead to path dependence- Once a standard

dominates, it is hard to shift

The third strategic tension- Inefficient coordination

The QWERTY keyboard

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 22 / 64

 Solution Concepts

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 23 / 64

Solution Concepts

A solution concept is a formal rule for predicting how a game is

likely to be played

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 24 / 64

Solution Concepts

A solution concept is a formal rule for predicting how a game is

likely to be played
Consider the following game-

Player 2
L R
U (2, 3) (5, 0)
Player 1
D (1, 0) (4, 3)

What would you do if you were Player 1?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 24 / 64

Dominant and Dominated Strategies

Dominance is the conceptual foundation of most theories of rational

behavior (along with best response)
Imagine that Player 2 has already chosen their strategy but you have
not observed it
If P1 plays rationally, they would never play D
D is dominated by U
Here, U is called the dominant strategy

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 25 / 64

Dominant and Dominated Strategies

Dominance is the conceptual foundation of most theories of rational

behavior (along with best response)
Imagine that Player 2 has already chosen their strategy but you have
not observed it
If P1 plays rationally, they would never play D
D is dominated by U
Here, U is called the dominant strategy

In the Prisoners’ Dilemma, the socially inefficient outcome turns out

to be the result of each player playing their dominant strategy

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 25 / 64

Mixed Strategies

A player plays a mixed strategy when they select a strategy

according to a probability distribution, σi ∈ ∆Si
Pure strategy (si ) vs mixed strategy (σi )
The set of mixed strategies is much larger- it includes the set of pure
strategies

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 26 / 64

Mixed Strategies- An Example

Penalty Kicks in Football

Goalkeeper
L R
L (−1, 1) (1, −1)
Penalty Taker
R (1, −1) (−1, 1)

If I try to repeatedly play a pure strategy, my opponent will figure it

out and punish me for it

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 27 / 64

Back to dominated strategies

Player 2
L R
U (4, 1) (0, 2)
Player 1
M (0, 0) (4, 0)
D (1, 3) (1, 2)

Is there any dominated strategy in Player 1 in this game?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 28 / 64

Back to dominated strategies

Player 2
L R
U (4, 1) (0, 2)
Player 1
M (0, 0) (4, 0)
D (1, 3) (1, 2)

Is there any dominated strategy in Player 1 in this game?

No pure strategy is dominant for P1
However, σ1 = (1/2,1/2,0) dominates D
A pure strategy si is dominated if there exists a strategy (pure or
mixed) σi ∈ ∆Si : ui (σi , s−i ) > ui (si , s−i ) for all strategy profiles
s−i ∈ S−i of the other players

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 28 / 64

Dominant strategies

How to check for dominance?

1 Check if a pure strategy dominates another
2 Check if a mixed strategy dominates another
To claim that a particular strategy is dominated, we need to find only
one strategy that dominates it
Strict dominance, weak dominance and strict dominance rationality

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 29 / 64

Best Response

The dominance solution does not assume anything about how one
player thinks about another player’s strategies
In most games, players have more than one undominated strategies;
in such cases, dominance does not provide a solution

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 30 / 64

Best Response

The dominance solution does not assume anything about how one
player thinks about another player’s strategies
In most games, players have more than one undominated strategies;
in such cases, dominance does not provide a solution
Suppose Player i has belief θ−i ∈ Si about other players’ strategies
Player i’s strategy, si ∈ Si , is a best response if
ui (si , θ−i ) ≥ ui (si0 , θ−i )∀si0 ∈ Si
There may be more than one best response to a given belief
In a finite game, every belief has at least one best response

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 30 / 64

Best Response- An Example

Player 2
L C R
U (2, 6) (0, 4) (4, 4)
Player 1
M (3, 3) (0, 0) (1, 5)
D (1, 1) (3, 5) (2, 3)

Denote best response of player i as BRi (θ−i )

Let θ−1 = (1/3, 1/2, 1/6)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 31 / 64

Best Response- An Example

Player 2
L C R
U (2, 6) (0, 4) (4, 4)
Player 1
M (3, 3) (0, 0) (1, 5)
D (1, 1) (3, 5) (2, 3)

Denote best response of player i as BRi (θ−i )

Let θ−1 = (1/3, 1/2, 1/6)
u1 (U, θ−1 ) = 1/3(2) + 1/2(0) + 1/6(4) = 8/6
Similarly, u1 (M, θ−1 ) = 7/6, u1 (D, θ−1 ) = 13/6
BR1 (1/3, 1/2, 1/6) = {D}

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 31 / 64

Best Response

Playing a best response to a given belief is not itself a strategic act

Forming a belief is a more important component of strategy
Success in games often hinges on whether you understand your
opponent better than they understand you
What beliefs are rational in games?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 32 / 64

Dominance and Best Response

How are dominance and best response related to each other?

Consider the following game
Player 2
L R
U (6, 3) (0, 1)
Player 1
M (2, 1) (5, 0)
D (3, 2) (3, 1)
Let Player 1 have a belief θ−1 = (p,1-p) regarding the strategies of
Player 2

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 33 / 64

Dominance and Best Response

B1 = UD1 = {U, M}

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 34 / 64

Dominance and Best Response

Let UDi be the set of strategies for player i that are not strictly
dominated, and let Bi be the set of strategies for player i that are
best responses over all possible beliefs of player i
Then, Bi = UDi in finite two-player games
More generally, Bi ⊆ UDi
Implication- To find the set of strategies that a player may adopt as a
rational response to their belief, we simply need to find the set of
undominated strategies

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 35 / 64

Dominance and Best Response

Q. How do we find the set of strategies belonging to Bi = UDi ?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 36 / 64

Dominance and Best Response

Q. How do we find the set of strategies belonging to Bi = UDi ?

1 Find the strategies that are best-responses to the simplest beliefs.
These definitely belong to Bi = UDi
2 Find the strategies that are dominated by other pure strategies. These
definitely do not belong to Bi = UDi
3 For the remaining strategies, test if they are dominated by a mixed
strategy

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 36 / 64

Solution Concepts so far

The maximin solution is a highly conservative strategy, where the

player chooses the ’best of the worst’ outcomes
Dominance is the starting point of all solution concepts involving
rational players
A rational player will never choose a strategy that is dominated by
any other (pure or mixed) strategy
The dominance solution only assumes that the players are individually
rational; it does not assume that one player might be thinking about
how the others might play the game

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 37 / 64

Dominance and best response

A player is said to play a best response when they form a belief about
how the opposing player might play the game, and choose the
strategy which maximizes their payoff as a response to their belief
about others
The best response solution concept assumes that players are actively
thinking about how others might play
For a two-player normal form game, the set of undominated strategies
available to any player is equivalent to the set of best-response
strategies to any belief that they may have about how the opposing
player will play the game

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 38 / 64

Consider the following game

Suppose you are Player 1. What would you play?

Player 2
X Y Z
A (3, 3) (0, 5) (0, 4)
Player 1
B (0, 0) (3, 1) (1, 2)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 39 / 64

Consider the following game

Suppose you are Player 1. What would you play?

Player 2
X Y Z
A (3, 3) (0, 5) (0, 4)
Player 1
B (0, 0) (3, 1) (1, 2)
Does dominance give a solution?
Does best response give a solution? Consider a belief
θ−1 = (p, q, 1 − p − q)
Is there a more rational way?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 39 / 64

Rationalizability

Key Assumption- It is common knowledge that both players are

rational!
Put yourself in Player 2’s position and think
Iterated deletion of strictly dominated strategies
Strategies that are never best responses are removed iteratively
What survives is called the set of rationalizable strategies

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 40 / 64

Rationalizability- An Example

Player 2
L C R
U (5, 1) (0, 4) (1, 0)
Player 1
M (3, 1) (0, 0) (3, 5)
D (3, 3) (4, 4) (2, 5)

Can you solve this using Rationalizability?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 41 / 64

The Hotelling (1929) Model- Location Game

Two competitors, selling the same product (e.g. cold drinks) at the
same price from a booth
Each must independently and simultaneously decide the location of
their booth, and commit to it
Nine regions, each with 50 customers, uniformly divided on a straight
line (e.g. a beach)
Profit per bottle = INR 10
The customers will go to the nearest booth to buy drinks
If the distances to both booths are equal, half of the total customers
go to each seller
Objective- Each competitor wants to maximize income. How would
they position themselves?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 42 / 64

The Location Game

P2
A1 A2 A3 A4 A5 A6 A7 A8 A9
A1 (2250, 2250) (500, 4000) (750, 3750) (1000, 3500) (1250, 3250) (1500, 3000) (1750, 2750) (2000, 2500) (2250, 2250)
A2 (4000, 500) (2250, 2250) (1000, 3500) (1250, 3250) (1500, 3000) (1750, 2750) (2000, 2500) (2250, 2250) (2500, 2000)
A3 (3750, 750) (3500, 1000)
A4 (3500, 1000) (3250, 1250)
P1 A5 (3250, 1250) (3000, 1500)
A6 (3000, 1500) (2750, 1750)
A7 (2750, 1750) (2500, 2000)
A8 (2500, 2000) (2250, 2250)
A9 (2250, 2250) (2000, 2500)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 43 / 64

Iterated Deletion- The Location Game

P2
A1 A2 A3 A4 A5 A6 A7 A8 A9
A1 (2250, 2250) (500, 4000) (750, 3750) (1000, 3500) (1250, 3250) (1500, 3000) (1750, 2750) (2000, 2500) (2250, 2250)
A2 (4000, 500) (2250, 2250) (1000, 3500) (1250, 3250) (1500, 3000) (1750, 2750) (2000, 2500) (2250, 2250) (2500, 2000)
A3 (3750, 750) (3500, 1000)
A4 (3500, 1000) (3250, 1250)
P1 A5 (3250, 1250) (3000, 1500)
A6 (3000, 1500) (2750, 1750)
A7 (2750, 1750) (2500, 2000)
A8 (2500, 2000) (2250, 2250)
A9 (2250, 2250) (2000, 2500)

u1 (2, x) > u1 (1, x) for all values of x

By symmetry, u1 (8, x) > u1 (9, x) for all values of x
Similarly, strategy A1 and A9 can be ruled out for Player 2 as well
The only rationalizable strategy left for each player at the end is A5

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 44 / 64

The Location Game

Examples- Two supermarkets in a community, product variety, median

voter theorem
Extensions-
1 The role of prices in differentiated products
2 More than two firms
3 Differentiation across more than one dimension
4 Non-uniform distribution of citizens/consumers
5 Multiple location points for the firm

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 45 / 64

The Story so far

The Dominance Solution

Key Idea- Rational players will never play a dominated strategy
Each player is only thinking about what is best for themselves, the
belief about other players’ strategies does not matter
Issue- We may not always be able to find a unique undominated
strategy for each player

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 46 / 64

The Story so far

The Dominance Solution

Key Idea- Rational players will never play a dominated strategy
Each player is only thinking about what is best for themselves, the
belief about other players’ strategies does not matter
Issue- We may not always be able to find a unique undominated
strategy for each player
Best Response
Assume- Players form a belief about others’ play before playing their
move
Key Idea- Each player tries to play the best (payoff maximizing)
response to what they believe the other player will do
Issue- What I believe about your choices may be different from what
you choose to play

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 46 / 64

The Story so far

The Rationalizability Solution

Assume- Players form beliefs about others, and best-respond to beliefs
Assume- Rationality of the players is common knowledge
Key Idea- Each player tries to find the dominated strategies of the
other player(s), and eliminates them from the other player(s)’ strategy
set as irrational, before playing their own strategy, and this process
continues iteratively

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 47 / 64

The Story so far

The Rationalizability Solution

Assume- Players form beliefs about others, and best-respond to beliefs
Assume- Rationality of the players is common knowledge
Key Idea- Each player tries to find the dominated strategies of the
other player(s), and eliminates them from the other player(s)’ strategy
set as irrational, before playing their own strategy, and this process
continues iteratively
Rationalizability does not require that the beliefs of each player be
correct, or consistent with the strategies actually played by the other
player
Q. Are beliefs always consistent with actual strategies?
No. The incoherence between beliefs and actual strategies leads to
strategic uncertainty

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 47 / 64

The Stag-Hare Game Revisited

Player 2
S H
S (5, 5) (0, 4)
Player 1
H (4, 0) (4, 4)

If I am less than 80% sure that the other player will hunt S, then I
will hunt H
Recall the devices used to coordinate behavior- Institutions, rules,
norms, communication etc.

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 48 / 64

Coordination and Rationalizability

If people have little history in common to coordinate their beliefs,

then rationalizability may be the appropriate behavioral concept to use
If coordination (or congruity) can be achieved, then Nash equilibrium
is a more appropriate solution concept
Coordination does not mean cooperation!! It only means that each
player’s beliefs are aligned with the others’ strategies

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 49 / 64

Three versions of Congruity

Repeated play may cause the game to ’settle down’

Players may meet before a game and reach an agreement on their
strategies, and actually honour the agreement
An outside mediator might recommend a strategy profile to the
players, and each player might expect that the others will follow the
mediator’s suggestions

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 50 / 64

Three versions of Congruity

Repeated play may cause the game to ’settle down’

Players may meet before a game and reach an agreement on their
strategies, and actually honour the agreement
An outside mediator might recommend a strategy profile to the
players, and each player might expect that the others will follow the
mediator’s suggestions
Nash’s insight- To achieve coordination among rational players, each
player’s chosen strategy must be a best-response to the others’
strategies

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 50 / 64

Finding the Nash Equilibrium

Player 2
L C R
U (10, 0) (0, 10) (3, 3)
Player 1
M (2, 10) (10, 2) (6, 4)
D (3, 3) (4, 6) (6, 6)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 51 / 64

Finding the Nash Equilibrium

Player 2
L C R
U (10, 0) (0, 10) (3, 3)
Player 1
M (2, 10) (10, 2) (6, 4)
D (3, 3) (4, 6) (6, 6)

One simple way of capturing the idea of capturing strategic certainty

is to assume that players have coordinated on a single strategy profile
A Nash equilibrium has no strategic uncertainty- Each player’s belief
is concentrated on the actual strategy that the other player uses

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 52 / 64

Nash Equilibrium

A strategy profile is a Nash Equilibrium if and only if si ∈ BRi (s−i )

for each player i
The strategy profile s can be stable over time only if si is a
best-response to s−i

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 53 / 64

Rationalizability and Nash Equilibrium- An Example

Consider the following Hotelling Game with two modifications-

1 There are six regions with 50 consumers in each region
2 If the two firms locate in the same region, they incur an additional loss
of INR 100 each
The Normal form of the game would look as below-

P2
A1 A2 A3 A4 A5 A6
A1 (1400, 1400) (500, 2500) (750, 2250) (1000, 2000) (1250, 1750) (1500, 1500)
A2 (2500, 500) (1400, 1400) (1000, 2000) (1250, 1750) (1500, 1500) (1750, 1250)
A3 (2250, 750) (2000, 1000) (1400, 1400) (1500, 1500) (1750, 1250) (2000, 1000)
A4 (2000, 1000) (1750, 1250) (1500, 1500) (1400, 1400) (2000, 1000) (2250, 750)
P1 A5 (1750, 1250) (1500, 1500) (1250, 1750) (1000, 2000) (1400, 1400) (2500, 500)
A6 (1500, 1500) (1250, 1750) (1000, 2000) (750, 2250) (500, 2500) (1400, 1400)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 54 / 64

Rationalizability and Nash Equilibrium- An Example

The rationalizability solution concept in the above game would yield

four potential strategy profiles- {A3, A3}, {A3, A4}, {A4, A3}, and
{A4, A4}
The strategies A1, A2, A5 and A6 will be iteratively deleted for both
players
The Nash Equilibrium solution goes one step further as it assumes
that rational players will be able to coordinate their rational beliefs
Hence, if Player 1 decides to set up shop in A3, Player 2 would want
to set up shop in A4 to avoid the extra loss of INR 100
The Nash Equilibrium solution predicts two potential strategy profiles-
{A3, A4} and {A4, A3}

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 55 / 64

Rationalizability and Nash Equilibrium

Player 2
L R
U (0, 0) (1, 1)
Player 1
D (1, 1) (1, 1)

Q. How many Nash equilibria does this game have?

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 56 / 64

Rationalizability and Nash Equilibrium

Player 2
L R
U (0, 0) (1, 1)
Player 1
D (1, 1) (1, 1)

This game has three Nash equilibria, as shown

However, if we delete weakly dominated strategies using
rationalizability, we might end up deleting some Nash equilibria

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 57 / 64

Rationalizability and Nash Equilibrium

If iterated deletion of strictly dominated strategies leaves only one

strategy for each player, that strategy profile is the Nash equilibrium
of the model
If iterated deletion of both strictly and weakly dominated strategies
leaves only one strategy for each player, that strategy profile is also a
Nash equilibrium
But, deletion of weakly dominated strategies might delete some Nash
equilibria from the game
Hence, the Nash equilibrium outcome in the second method may not
be the only Nash equilibrium of the game
Changing the order of deletion with weakly dominated strategies may
lead to a different Nash equilibrium outcome, in case the game has
multiple Nash equilibria

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 58 / 64

Finding the NE- The Prisoners’ Dilemma

Player 2
C D
C (2, 2) (0, 3)
Player 1
D (3, 0) (1, 1)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 59 / 64

Finding the NE- The Prisoners’ Dilemma

Player 2
C D
C (2, 2) (0, 3)
Player 1
D (3, 0) (1, 1)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 60 / 64

Finding the NE- Battle of the Sexes

Wife
Opera Movie
Opera (2, 1) (0, 0)
Husband
Movie (0, 0) (1, 2)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 61 / 64

Finding the NE- Battle of the Sexes

Wife
Opera Movie
Opera (2, 1) (0, 0)
Husband
Movie (0, 0) (1, 2)

A game might have multiple Nash equilibria

The players can coordinate with each other and arrive at a unique
solution

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 62 / 64

Finding the NE- Matching Pennies

Player 2
Heads Tails
Heads (1, −1) (−1, 1)
Player 1
Tails (−1, 1) (1, −1)

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 63 / 64

Finding the NE- Matching Pennies

Player 2
Heads Tails
Heads (1, −1) (−1, 1)
Player 1
Tails (−1, 1) (1, −1)

A game might not have even a single Nash equilibrium

A zero-sum game of pure conflict would not have any Nash
equilibrium

Pavneet Singh (IIM Amritsar) Sessions 1 - 5 June 20, 2025 64 / 64