mechanical engineering

1. S A R A N I S A H A B H A T T A C H A R Y A , H S S
A R N A B B H A T T A C H A R Y A , C S E
0 7 J A N , 2 0 0 9
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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