Transcript Game Theory and its Applications

Game Theory and its
Applications
SARANI SAHABHATTACHARYA, HSS
ARNAB BHATTACHARYA, CSE
07 JAN, 2009
Prisoner’s Dilemma
2
 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 nonconfessor sentenced to 9 months of jail
 What should each prisoner do?
Game Theory
Jan 07, 2009
Battle of Sexes
3
 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
Game Theory
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Games
4
 Normal Form representation – Payoff Matrix
Prisoner 2
Prisoner 1
Confess
Not Confess
Confess
-3,-3
0,-9
Not Confess
-9,0
-1,-1
Husband
Wife
Game Theory
Movie
Cricket
Movie
2,1
0,0
Cricket
0,0
1,2
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Nash equilibrium
5
 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
Prisoner 2
Prisoner 1
Game Theory
Confess
Not Confess
Confess
-3,-3
0,-9
Not Confess
-9,0
-1,-1
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Mixed strategies
6
 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
Game Theory
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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
Game Theory
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Dynamic games
8
 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
Game Theory
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Game tree example: Bargaining
Period 2:
B offers x2.
A responds.
(x1,1-x1)
1
(x3,1-x3)
1
1
Y
Y
x3
x1
B
A
N
N
B
B
x2 A
A
N
Y
0
Period 1:
A offers x1.
B responds.
0
(x2,1-x2)
0
Period 3:
A offers x3.
B responds.
(0,0)
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
Auctions
11
 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
Game Theory
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Second-price auction
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 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
Game Theory
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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
Game Theory
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Inefficiency of Nash equilibrium
14
 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
Game Theory
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Network example
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C(x) = 1
C(x) = x
 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
Game Theory
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Do high-speed links always help?
16
C(x) = x
C(x) = 1
C(x) = 1
C(x) = x
C(x) = x
C(x) = 0
C(x) = 1
C(x) = x
C(x) = 1
 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
Game Theory
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Other computer science applications
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 Internet
 Routing
 Job scheduling
 Competition in client-server systems
 Peer-to-peer systems
 Cryptology
 Network security
 Sensor networks
 Game programming
Game Theory
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Bidding up to 50
 Two-person game
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 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
Game Theory
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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
Game Theory
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Chess programming
20
 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
Game Theory
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Conclusions
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 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
Game Theory
Jan 07, 2009