Game Theory and its Applications

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Game Theory and its Applications Sarani SahaBhattacharya, HSS Arnab Bhattacharya, CSE 07 Jan, 2009 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

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### Game Theory and its Applications

Sarani SahaBhattacharya, HSS

Arnab Bhattacharya, CSE

07 Jan, 2009

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?

Game Theory

Battle of Sexes
• 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

Games
• Normal Form representation – Payoff Matrix

Prisoner 2

Prisoner 1

Husband

Wife

Game Theory

Nash equilibrium
• 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

Mixed strategies
• 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

Nash’s Theorem
• 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

Dynamic games
• 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

Game tree example: Bargaining

Period 2:B offers x2.

A responds.

(x1,1-x1)

(x3,1-x3)

1

1

1

Y

Y

x3

x1

N

(0,0)

B

B

N

x2

A

B

A

A

N

Y

0

0

0

Period 1:A offers x1.

B responds.

Period 3:A offers x3.

B responds.

(x2,1-x2)

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
• 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

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

Game Theory

Mechanism design
• 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

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

Game Theory

Network example
• 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

s

t

C(x) = x

Game Theory

• ½ of the data will take route s-u-t, and ½ s-v-t
• Total delay is 3/2
• All data will now switch to s-u-v-t route
• Total delay now becomes 2

u

u

C(x) = x

C(x) = x

C(x) = 1

C(x) = 1

C(x) = 0

s

t

s

t

C(x) = 1

C(x) = 1

C(x) = x

C(x) = x

v

v

Game Theory

Other computer science applications
• Internet
• Routing
• Job scheduling
• Competition in client-server systems
• Peer-to-peer systems
• Cryptology
• Network security
• Sensor networks
• Game programming

Game Theory

Bidding up to 50
• Two-person game
• 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

Game programming
• 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

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
• Too much deep analysis may lead to defeat

Game Theory

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

Game Theory