The document discusses various concepts in game theory including the Prisoner's Dilemma, Battle of the Sexes game, Nash Equilibrium, mixed strategies, dynamic games, auctions, and applications in computer science. It provides examples and explanations of these game theory concepts and how they apply to situations involving strategic decision making between multiple players.

1. SARANI SAHABHATTACHARYA, HSS
ARNAB BHATTACHARYA, CSE
07 JAN, 2009
Game Theory and its
Applications

2. Prisoner’s Dilemma
Two suspects arrested for a crime
Prisoners decide whether to confess or not to confess
If both confess, both sentenced to 3 months of jail
If both do not confess, then both will be sentenced to
1 month of jail
If one confesses and the other does not, then the
confessor gets freed (0 months of jail) and the non-
confessor sentenced to 9 months of jail
What should each prisoner do?
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Game Theory

3. Battle of Sexes
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A couple deciding how to spend the evening
Wife would like to go for a movie
Husband would like to go for a cricket match
Both however want to spend the time together
Scope for strategic interaction

4. Games
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Normal Form representation – Payoff Matrix
Confess Not Confess
Confess -3,-3 0,-9
Not Confess -9,0 -1,-1
Movie Cricket
Movie 2,1 0,0
Cricket 0,0 1,2
Prisoner 1
Prisoner 2
Wife
Husband

5. Nash equilibrium
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Each player’s predicted strategy is the best response
to the predicted strategies of other players
No incentive to deviate unilaterally
Strategically stable or self-enforcing
Confess Not Confess
Confess -3,-3 0,-9
Not Confess -9,0 -1,-1
Prisoner 1
Prisoner 2

6. Mixed strategies
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A probability distribution over the pure strategies of
the game
Rock-paper-scissors game
 Each player simultaneously forms his or her hand into the
shape of either a rock, a piece of paper, or a pair of scissors
 Rule: rock beats (breaks) scissors, scissors beats (cuts) paper,
and paper beats (covers) rock
No pure strategy Nash equilibrium
One mixed strategy Nash equilibrium – each player
plays rock, paper and scissors each with 1/3
probability

7. Nash’s Theorem
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Existence
 Any finite game will have at least one Nash equilibrium
possibly involving mixed strategies
Finding a Nash equilibrium is not easy
 Not efficient from an algorithmic point of view

8. Dynamic games
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Sequential moves
 One player moves
 Second player observes and then moves
Examples
 Industrial Organization – a new entering firm in the market
versus an incumbent firm; a leader-follower game in quantity
competition
 Sequential bargaining game - two players bargain over the
division of a pie of size 1 ; the players alternate in making offers
 Game Tree

9. Game tree example: Bargaining
0
1
A
0
1
B
0
1
A
B B
A
x1
(x1,1-x1)
Y
N
x2
x3
(x3,1-x3)
(x2,1-x2)
(0,0)
Y
Y
N
N
Period 1:
A offers x1.
B responds.
Period 2:
B offers x2.
A responds.
Period 3:
A offers x3.
B responds.

10. Economic applications of game theory
The study of oligopolies (industries containing only
a few firms)
The study of cartels, e.g., OPEC
The study of externalities, e.g., using a common
resource such as a fishery
The study of military strategies
The study of international negotiations
Bargaining

11. Auctions
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Games of incomplete information
First Price Sealed Bid Auction
 Buyers simultaneously submit their bids
 Buyers’ valuations of the good unknown to each other
 Highest Bidder wins and gets the good at the amount he bid
 Nash Equilibrium: Each person would bid less than what the good
is worth to you
Second Price Sealed Bid Auction
 Same rules
 Exception – Winner pays the second highest bid and gets the good
 Nash equilibrium: Each person exactly bids the good’s valuation

12. Second-price auction
Suppose you value an item at 100
You should bid 100 for the item
If you bid 90
 Someone bids more than 100: you lose anyway
 Someone bids less than 90: you win anyway and pay second-price
 Someone bids 95: you lose; you could have won by paying 95
If you bid 110
 Someone bids more than 11o: you lose anyway
 Someone bids less than 100: you win anyway and pay second-price
 Someone bids 105: you win; but you pay 105, i.e., 5 more than what
you value
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13. Mechanism design
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How to set up a game to achieve a certain outcome?
 Structure of the game
 Payoffs
 Players may have private information
Example
 To design an efficient trade, i.e., an item is sold only when
buyer values it as least as seller
 Second-price (or second-bid) auction
Arrow’s impossibility theorem
 No social choice mechanism is desirable
Akin to algorithms in computer science

14. Inefficiency of Nash equilibrium
Can we quantify the inefficiency?
Does restriction of player behaviors help?
Distributed systems
 Does centralized servers help much?
Price of anarchy
 Ratio of payoff of optimal outcome to that of worst possible
Nash equilibrium
In the Prisoner’s Dilemma example, it is 3
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15. Network example
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Simple network from s to t with two links
 Delay (or cost) of transmission is C(x)
Total amount of data to be transmitted is 1
Optimal: ½ is sent through lower link
 Total cost = 3/4
Game theory solution (selfish routing)
 Each bit will be transmitted using the lower link
 Not optimal: total cost = 1
Price of anarchy is, therefore, 4/3
C(x) = 1
C(x) = x

16. Do high-speed links always help?
½ of the data will take route s-u-t, and ½ s-v-t
Total delay is 3/2
Add another zero-delay link from u to v
All data will now switch to s-u-v-t route
Total delay now becomes 2
Adding the link actually makes situation worse
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C(x) = x
C(x) = 1
C(x) = 1
C(x) = x
C(x) = x
C(x) = 1
C(x) = 1
C(x) = x
C(x) = 0

17. Other computer science applications
Internet
Routing
Job scheduling
Competition in client-server systems
Peer-to-peer systems
Cryptology
Network security
Sensor networks
Game programming
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18. Bidding up to 50
Two-person game
Start with a number from 1-4
You can add 1-4 to your opponent’s number and bid
that
The first person to bid 50 (or more) wins
Example
 3, 5, 8, 12, 15, 19, 22, 25, 27, 30, 33, 34, 38, 40, 41, 43, 46, 50
Game theory tells us that person 2 always has a
winning strategy
 Bid 5, 10, 15, …, 50
Easy to train a computer to win
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19. Game programming
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Counting game does not depend on opponent’s choice
Tic-tac-toe, chess, etc. depend on opponent’s moves
You want a move that has the best chance of winning
However, chances of winning depend on opponent’s
subsequent moves
You choose a move where the worst-case winning
chance (opponent’s best play) is the best: “max-min”
Minmax principle says that this strategy is equal to
opponent’s min-max strategy
 The worse your opponent’s best move is, the better is your move

20. Chess programming
How to find the max-min move?
Evaluate all possible scenarios
For chess, number of such possibilities is enormous
 Beyond the reach of computers
How to even systematically track all such moves?
 Game tree
How to evaluate a move?
 Are two pawns better than a knight?
Heuristics
 Approximate but reasonable answers
 Too much deep analysis may lead to defeat
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21. Conclusions
Mimics most real-life situations well
Solving may not be efficient
Applications are in almost all fields
Big assumption: players being rational
 Can you think of “unrational” game theory?
Thank you!
Discussion
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