sarani sahabhattacharya, hss arnab bhattacharya, cse 07 jan, 2009 game theory and its applications

SARANI SAHABHATTACHARYA, HSSARNAB BHATTACHARYA, CSE

07 JAN, 2009

Game Theory and its Applications

Prisoner’s Dilemma

Two suspects arrested for a crimePrisoners decide whether to confess or not to

confessIf both confess, both sentenced to 3 months of

jailIf both do not confess, then both will be

sentenced to 1 month of jailIf 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?

Jan 07, 2009

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Game Theory

Battle of Sexes

Jan 07, 2009Game Theory

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A couple deciding how to spend the eveningWife would like to go for a movieHusband would like to go for a cricket matchBoth however want to spend the time

togetherScope for strategic interaction

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

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 unilaterallyStrategically stable or self-enforcing

Confess Not Confess

Confess -3,-3 0,-9

Not Confess -9,0 -1,-1

Prisoner 1

Prisoner 2

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 equilibriumOne mixed strategy Nash equilibrium – each

player plays rock, paper and scissors each with 1/3 probability

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

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

Game tree example: Bargaining

0

1

A

0

1

B

0

1

AB B

A

x1

(x1,1-x1)

Y

Nx2

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.

Economic applications of game theory

The study of oligopolies (industries containing only a few firms)

The study of cartels, e.g., OPECThe study of externalities, e.g., using a

common resource such as a fisheryThe study of military strategiesThe study of international negotiationsBargaining

Auctions


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Games of incomplete informationFirst 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

Second-price auction

Suppose you value an item at 100You should bid 100 for the itemIf 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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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

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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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 1Optimal: 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

Do high-speed links always help?

½ of the data will take route s-u-t, and ½ s-v-tTotal delay is 3/2Add another zero-delay link from u to vAll data will now switch to s-u-v-t routeTotal delay now becomes 2Adding 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

Other computer science applications

InternetRoutingJob schedulingCompetition in client-server systemsPeer-to-peer systemsCryptologyNetwork securitySensor networksGame programming


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Bidding up to 50

Two-person gameStart with a number from 1-4You can add 1-4 to your opponent’s number and

bid thatThe first person to bid 50 (or more) winsExample

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 winJan 07, 2009Game Theory

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Game programming


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Counting game does not depend on opponent’s choiceTic-tac-toe, chess, etc. depend on opponent’s movesYou want a move that has the best chance of winningHowever, chances of winning depend on opponent’s

subsequent movesYou 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

Chess programming

How to find the max-min move?Evaluate all possible scenariosFor 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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Conclusions

Mimics most real-life situations wellSolving may not be efficientApplications are in almost all fieldsBig assumption: players being rational

Can you think of “unrational” game theory?

Thank you!Discussion


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sarani sahabhattacharya, hss arnab bhattacharya, cse 07 jan, 2009 game theory and its applications

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