1. An Introduction to Game Theory
Speaker
Abhinav Srivastava
M.Tech IInd Year
School of Information Technology
Indian Institute of Technology, Kharagpur
11/18/2018
An Introduction to Game Theory

2. An Introduction to Game Theory
Contents
History
Introduction
Definitions related to game theory
What is a game?
Types of Games
Static games of complete information
Dominant Strategy
Zero-Sum Game
Nash Equilibrium
References
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An Introduction to Game Theory

3. An Introduction to Game Theory
History
Game Theory is an interdisciplinary approach to the study of human behavior.
It was founded in the 1920s by John von Neumann.
In 1994 Nobel prize in Economics awarded to three researchers.
HARSANYI, JOHN C., U.S.A., University of California, Berkeley, CA, b (in Budapest, Hungary);
NASH, JOHN F., U.S.A., Princeton University, NJ, b. 1928; and
SELTEN, REINHARD, Germany, Rheinische Friedrich-Wilhelms-Universit,t, Bonn, Germany, b. 1930:
“Games” are a metaphor for wide range of human interactions.
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An Introduction to Game Theory

4. An Introduction to Game Theory
Game Theory (GT) can be regarded as
A multi-agent decision problem.
Many people contending for limited rewards/payoffs.
Moves on which payoffs depends.
Follow certain rules while making the moves.
Each player is supposed to behave rationally.
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An Introduction to Game Theory

5. Definitions Related to GT
Game Theory: Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave strategically.
Player: Each participant (interested party) is called a player.
Strategy: A strategy of a player is the predetermined rule by which a player decides his course of action from his own list of actions during the game.
Rationality: It implies that each player tries to maximize his/her payoff irrespective to what other players are doing.
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An Introduction to Game Theory

6. Definitions of GT (Contd…)
Rule: These are instructions that each player follow. Each player can safely assume that others are following these instructions also.
Outcome: It is the result of the game.
Payoff: This is the amount of benefit a player derives if a particular outcome happens.
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An Introduction to Game Theory

7. An Introduction to Game Theory
What is a Game?
A game has the following
Set of players D = { Pi | 1 <= i <= n}
Set of rules R
Set of Strategies Si for each player Pi
Set of Outcomes O
Pay off ui (o) for each player i and for each outcome o e O
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An Introduction to Game Theory

8. An Introduction to Game Theory
Coin Matching Game
Coin Matching Game : Two players choose independently either Head or Tail and report it to a central authority. If both choose the same side of the coin , player 1 wins, otherwise 2 wins.
This game has the following :-
Set of Players: P={P1,P2} (The two players  who are choosing either Head or Tail.)
Set of Rules: R (Each player can choose either Head or Tail. Player 1 wins if both selections are the same otherwise player 2 wins.)
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An Introduction to Game Theory

9. Coin Matching Game (Contd…)
Set of Strategies: Si for each player Pi (For example S1 = { H, T}  and S2 = {H,T}  are the strategies of the two players.)
Set of Outcomes: O {Loss, Win} for both players (This is a function of the strategy profile selected. In our example S1 x S2 = {H,H),(H,T),(T,H),(T,T)} is the strategy profile.)
Pay off: ui (o)  for each player i and for each outcome o e O. (This is the amount of benefit a player derives if a particular outcome happens.)
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An Introduction to Game Theory

10. Coin Matching Game (Contd…)
Player 2
Head
Tail
1, -1
-1, 1
Head
Player 1
Tail
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An Introduction to Game Theory

11. An Introduction to Game Theory
Types of Games
There are four types of Games:
Static Games of Complete Information
Dynamic Games of Complete Information
Static Games of Incomplete Information
Dynamic Games of Incomplete Information
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An Introduction to Game Theory

12. Static games of complete information
Simultaneous-move
Each player chooses his/her strategy without knowledge of others’ choices.
Complete information
Each player’s strategies and payoff function are common knowledge among all the players.
Assumptions on the players
Rationality
Players aim to maximize their payoffs
Players are perfect calculators
Each player knows that other players are rational
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An Introduction to Game Theory

13. Static games of complete information
The players cooperate?
No only non-cooperative games
Represented as normal-form or strategic form.
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An Introduction to Game Theory

14. The Prisoner’s Dilemma
Two burglars, Bob and Al, are captured and separated by the police.
Each has to choose whether or not to confess and implicate the other.
If neither confesses, they both serve one year for carrying a concealed weapon.
If each confesses and implicates the other, they both get 10 years.
If one confesses and the other does not, the confessor goes free, and the other gets 20 years.
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An Introduction to Game Theory

15. The Prisoners’ DilemmaAl
Confess
Don’t Confess
Bob
10 / 10
0 / 20
20 / 0
1 /1
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An Introduction to Game Theory

16. An Introduction to Game Theory
Dominant Strategies
The prisoners have fallen into a “dominant strategy equilibrium”
DEFINITION: Dominant Strategy
Evaluate the strategies.
For each combination, choose the one that gives the best payoff.
If the same strategy is chosen for each different combination, that strategy is called a “dominant strategy” for that player in that game
DEFINITION: Dominant Strategy Equilibrium
If, in a game, each player has a dominant strategy, and each player plays the dominant strategy, then that combination of (dominant) strategies and the corresponding payoffs constitute the dominant strategy equilibrium for that game.
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An Introduction to Game Theory

17. An Introduction to Game Theory
Traffic Lights
There are two players in this game: Player I and Player II.
Player I is the commuter and All other people at the  intersection (signal) can be considered as the second player in the game.
When a commuter arrives and faces a red light he/she has two options:
Wait for light to turn Green
Jump the Red light
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An Introduction to Game Theory

18. An Introduction to Game Theory
Traffic Lights
If the commuter obeys and others also obey he will have to suffer delay of  'd' that is the time required for the red light to turn green.
If he disobeys but others obey his delay is 0.
If he obeys but others disobey let additional delay is D ( due to congestion ) over 'd' .
If  all disobey total delay is D.
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An Introduction to Game Theory

19. An Introduction to Game Theory
Traffic Lights
Player II
Obey
Disobey
Player I d
d+D D
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An Introduction to Game Theory

20. Information Technology Example
Players
Company considering a new internal computer system
A supplier who is considering producing it
Choices
To install an advanced system with more features
To install a proven system with less functionality
Payoffs
Net payment of the user to the supplier
Assumptions
A more advanced system really does supply more functionality
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An Introduction to Game Theory

21. An Introduction to Game Theory
IT Example
User
Advanced
Proven
Supplier
20 / 20
0 / 0
5 /5
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An Introduction to Game Theory

22. An Introduction to Game Theory
Complications
There are no dominant strategies.
The best strategy depends on what the other player chooses!
Need a new concept of game-equilibrium
Nash Equilibrium
Occurs when each participant chooses the best strategy given the strategy chosen by the other participant
Advanced/Advanced
Proven/Proven
Can be more than one Nash equilibrium
This is considered a cooperative game
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An Introduction to Game Theory

23. An Introduction to Game Theory
Zero-Sum Game
It was discovered by Von Neumann.
A zero-sum game is a game in which one player’s winnings equal to the other player’s losses.
If there is even one strategy set for which the sum differs from zero, then the game is not zero sum.
In a zero-sum game, the interest of the players are directly opposed, with no common interest at all.
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An Introduction to Game Theory

24. An Introduction to Game Theory
Bottled Water Game
Players: Evian, Perrier
Each company has a fixed cost of $5000 per period, regardless of sales
They are competing for the same market, and each must chose a high price ($2/bottle), and a low price ($1/bottle)
At $2, 5000 bottles can be sold for $10,000
At $1, bottles can be sold for $10,000
If both companies charge the same price, they split the sales evenly between them
If one company charges a higher price, the company with the lower price sells the whole amount and the higher price sells nothing
Payoffs are profits – revenue minus the $5000 fixed cost
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An Introduction to Game Theory

25. An Introduction to Game Theory
Bottled Water Game
Perrier $1 $2
Evian
$0, $0
$5000, -$5000
-$5000, $5000
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An Introduction to Game Theory

26. An Introduction to Game Theory
The Maximin Criterion
There is a clear concept of a solution for the zero-sum game.
Each player chooses the strategy that will maximize the minimum payoff.
The pair of strategies and payoffs such that each player maximizes her minimum payoffs is the “solution of the game”.
Both choose $1 prices
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An Introduction to Game Theory

27. Non-constant Sum Games
Games that are not zero-sum or constant sum are called Non-constant sum games.
These are in some sense natural games.
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An Introduction to Game Theory

28. An Introduction to Game Theory
Widgets Game
Let’s sell widgets
Set pricing to $1, $2, or $3 per widget
Payoffs are profits, allowing for costs
General idea is that company with lower price gets more customers, and more profits, within limits.
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An Introduction to Game Theory

29. An Introduction to Game Theory
Widgets Game
ACME Widgets $1 $2 $3
Widgeon Widgets
0, 0
50, -10
40, -20
-10, 50
20, 20
90, 10
-20, 40
10, 90
50, 50
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An Introduction to Game Theory

30. An Introduction to Game Theory
Nash Equilibrium
On the name after Nobel Laureate and mathematician John Nash.
Nash, a student of Tucker’s contributed several key concepts to game theory around 1950.
Nash Equilibrium is the most widely used “solution concept” in game theory.
If there is a set of strategies with the property that no player can benefit by changing their strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.
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An Introduction to Game Theory

31. An Introduction to Game Theory
Nash Equilibrium
Maximin is a Nash Equilibrium.
Dominant Strategy is also a Nash Equilibrium.
Nash Equilibrium is an extension of the concepts of dominant strategy and the maximin solution for zero-sum games.
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An Introduction to Game Theory

32. Limitations of Nash Equilibrium
Can there be more than one Nash-Equilibrium in the same game?
What if there are more than one?
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An Introduction to Game Theory

33. Multiple Nash Equilibrium
There are two cola companies, Pepsi and Coke.
Each own a vending machine in the dormitory. 
Each must decide how to stock its machine. 
They can fill the machine with diet soda, regular soda, or a combination of the two. 
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An Introduction to Game Theory

34. An Introduction to Game Theory
The Cola Wars
Coca-Cola
Diet
Both
Classic
Pepsi
25, 25
50, 30
50, 20
30, 50
15, 15
30, 20
20, 50
20, 30
10, 10
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35. An Introduction to Game Theory
Game Over
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36. An Introduction to Game Theory
Application of GT
Game theory has applications
Economics
International relations
Political science
Military strategy
Operations research
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An Introduction to Game Theory

37. An Introduction to Game Theory
References
Formal Theory for Political Science by Andrew Kydd
Game Theory: An Introductory Sketch by Roger McCain
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38. An Introduction to Game Theory
Questions?
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An Introduction to Game Theory

39. An Introduction to Game Theory
Thank You
11/18/2018
An Introduction to Game Theory