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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
GAME THEORY & APPLICATIONS
J. Ajith Kumar, TAPMI, Manipal
Pure strategy, Nash Equilibrium and Dominance
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 4
INTRODUCTION
E.g. cricket, where batsman has options of multiplestrokes, and bowler has option of multiple types of ball.
Simplistically speaking, the number of runs scored by
the batsman can be thought of as a function of the
pairing between the type of ball bowled and type ofstroke played.
Analysis of a game can also be thought of as solving a
decision tree with more than one agent, with each
agent trying to maximize her returns.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
Game theoretic analyses can help us understand realitybetter and
explain current reality why things are the way they
are.
prescribe actions what should be done to shapereality in some desirable manner.
HOW GAME THEORY IS USEFUL
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
* Simultaneous game
a model that assumes playersdont get to see each othersactions before they
perform their own actions.
SIMULTANEOUS vs. SEQUENTIAL GAMES
* Sequential game a model that accommodates the
time dimension, and assumes players get to see whatthe other players did before performing their own
actions.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
A game is typically characterized bya) the context,
b) the players,
c) actions (strategies) available to the players,
d) the payoff functions of the players , and
e) whether the game is simultaneous or sequential.
CHARACTERISTICS OF A GAME
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 8
Context PlayersActions
availablePayoffs
How the game is
played*
Pricing in the
telecom
industry
Telecomfirms
Different
levels of price(e.g. high,
medium, low)
Revenues
Profits
Market share
Simultaneous
(over multiple
rounds)
Writing an
Examination
Invigilators,
Students
Invigilators:
(strict,
lenient),
Students:
(cheat, dont
cheat)
Invigilators:
how well the
exam isconducted.
Students:
performance
in the exam
Simultaneous /
sequential
EXERCISE (10 minutes)
List real life situations that you can model as games. See
examples below.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 9
Theory of Rational Choice(Choice Theory, Rational Actor Theory)
Decision makers are rational actors. While making a
decision, they choose the alternative(s) that maximizes
their utility under given circumstances. They also have
adequate knowledge and ability to estimate their utilities
and to use them to discriminate between alternatives.
Axioms of Rationality
For rationality to be possible, we assume these to be true:
1. Reflexivity (trivial)2. Completeness
3. Transitivity
4. Continuity (sufficient but not necessary)
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
App Development Project Game
10
Each student is given a slip in which s/he indicates
her/his choice after reading the problem and thinking
over it. No one should talk to each other.
ReadApp Development Project
in Exercise Set A
How many marked Work hardand how many Hardly
Work? Why did you mark what you did?
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
App Development Project Game
Player 1s utilities are:
u1(a1 = WH | a2 = WH) = 30
u1(a1 = HW | a2 = WH) = 70
u1(a1 = WH | a2 = HW) = 10
u1(a1 = HW | a2 = HW) = 20
Player 2s utilities will be similar.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 12
App Development Project Game
PARTNER
WH HW
YOU
WH
30, 30 10, 70
HW 70, 10 20, 20
Strategic form or normal form representation
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
App Development Project Game
What is the best action for you? Why?
What is the best action for your partner? Why?
We see that regardless of what your partner does, you
get higher payoffs if you chose HW. Thus HW is the
best action for you.
By the same reasoning as above, HW is the best action
for your partner too.
If both players are rational, then what will the
outcome be?
(HW, HW): also called the Nash Equilibrium of this
game.
13
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 14
NASH EQUILIBRIUM
A Nash Equilibrium is an action profile in which no
player can improve his payoff by changing his action,given that the other playersactions remain the same
as in that action profile.
An action profile is a Nash equilibrium if and only if
each players action is his best response to the actionsof all other players in that action profile.
In other words, iffui(ai, a-i) ui (ai, a-i) for all i, ai, is
satisfied, then (ai
, a-i
) is a Nash Equilibrium.
Examine if (HW, HW) fulfils the above condition.
Examine if any of the other three action profiles in this game
fulfil the condition and can be called a Nash Equilibrium.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 15
NASH EQUILIBRIUM
Important Caveat:
An NE is an action profile, not a payoff profile.
To say that (HW, HW) is a NE correct
But to say that (20, 20) is an NE is incorrect.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 16
STRICT vs. NON-STRICT NASH EQUILIBRIUM
If in a NE, ui(ai, a-i) > ui (ai, a-i) for all i, ai,
then (ai, a-i) is said to be a Strict Nash Equilibrium.
If not, then it is a nonstrict NE.
Player 2
H T
Player1
H 1, 2 3, 0
T 2, 1 3, 2
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 17
NASH EQUILIBRIUM: A SOLUTION CONCEPT
1. Indicates possible actions that players might take in agiven situation.
2. Indicates possible stable steady statethat a system
might reach after a period of time, over several trials
of the game.
3. An NE can be thought of as a stable social norm. If
everyone else adheres to it, no individual would wish
to deviate from it.
4. Give us a language to discuss what can be done to
change the system behavior (alter playersbehaviors,
alter the equilibrium). Mechanism Design.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 18
If the players had chosen WH instead of HW, then
both would have got higher payoffs. Yet, why is (HW,
HW) the rational outcome of this game?
App Development Project Game
Lack of adequate trust, understanding? If I choose
WH, how do I know for sure that my partner will alsochoose WH? He might choose HW. If he does that
then my payoff will be just 10 points. On the other
hand, I can be assured of getting at least 20 points if I
choose HW and might even get 70 if my partner
chooses WH.
This reasoning implicitly assumes that each player
thinks purely selfishly.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 19
Lesson: When everyone acts rationally, it can lead tosuboptimal outcomes for everyone. The combined
effect of individual rationality can sometimes be
collective irrationality.
App Development Project Game
Can control and coordination mechanisms help?
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 20
They did not understand the question correctly.
They did not calculate all the payoffs correctly.
...other reasons?
Our model predicts that HWwould be the rational
choice. Why then did everyone not choose HW?
Lesson: Bounded Rationality(Herbert Simon)the
NE may not always be reached in all real-life
situations.
Lesson: To be perfectly selfish, a person needs tohave complete knowledge and intelligence...!
Otherwise, it may not be possible. Can we say the
same of being perfectly unselfish?
App Development Project Game
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
IMPORTANT LESSONS
21
Lesson: The payoffs that each player gets are afunction of her actions in combination with the actions
of all other players in the game.
Lesson: The payoffs need not always be in terms of atangible measure such as money, market share etc...
Utilityis a more general conceptualization of payoff.
Covers intangibles like satisfaction, happiness,
frustration etc... Utility: assessed by the modeller andis thus, subjective in nature.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 22
Action: decision choice available to a player. More
generally, also referred to as pure strategy. Actions are
mutually exclusive and collectively exhaustive. In this
model, a player can play only one action at a time and
not a combination of actions. Often denoted as ai.
Action set: defined for each player in the game, as a
listing of all possible actions that the player can choose.
Denoted by Ai. In the above game, A1 = {WH, HW} and
A2 = {WH, HW}.In this example, equal numberand identicalactions
available to all players (i.e. two) but not necessary, in
general.
Some Terminology
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES
Some Terminology
23
Action profile: a distinct combination of actions of all
the players, in which for each player, one action islisted, and collectively, the actions are listed in a
standard order. In a two player game, the row players
action is listed first.
An action (strategy) profile is also referred to as anoutcome.
A 2 x 2 game (two players, two actions each) has four
possible action profiles. Likewise a m x n xp game will
have mnp possible action (pure strategy) profiles.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 24
Utility (payoff) profile: Corresponding to each strategy
profile is an utility profile: the distinct combination ofutilities received by all players if the corresponding
strategy profile is played.
Some Terminology
For each player, one utility value is listed, and
collectively, the utilities are listed in the same order inwhich actions are listed in the action profile.
In a two player game, the row players utility is listed
first.
A 2 x 2 game has four possible utility profiles. E.g. (30,
30), (70, 10), (10, 70), (20, 20). Likewise a m x n xp
game will have mnp possible utility profiles.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 25
The NE represents a possible outcome
But is it the best, or optimal outcome?
What would be an optimal outcome,
when there are multiple agents?
How do we determine optimality,
when there are trade-offs?
WHAT IS THE BEST OUTCOME?
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 26
PARETO OPTIMALITY: ANOTHER SOLUTION
CONCEPT
Pareto Domination:
Action profile a Pareto dominates action profile a if for
all i N, ui (a) ui (a), and there exists somej N for
which uj(a) > u
j(a).
Pareto Optimal:
Action profile a is Pareto optimal (or, Pareto efficient) if
there does not exist another action profile a S, which
Pareto dominates a.
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apProf. J. Ajith Kumar, TAPMI, Manipal GAME THEORY NOTES 27
PARETO OPTIMALITY: ANOTHER SOLUTION
CONCEPT
Player 2
H T
Player
1 H1, 2 3, 0
T 2, 1 3, 2
Shyam
Run Stand
Ram R
un
1, 1 2, 2
Stan
d3, 3 4, 4
Ashwin
R Q
Murali M 3, 2 2, 3
L 2, 1 1, 2
India
Fight Dont
Pakistan
Fig
ht
5, 4 5, 4
Don
t5, 4 5, 4
Find the NE(s) as well as the Pareto optimal solution(s)