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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

Game Theory and its Applications

Sarani SahaBhattacharya, HSS

Arnab Bhattacharya, CSE

07 Jan, 2009

prisoner s dilemma
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
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
Games
  • Normal Form representation – Payoff Matrix

Prisoner 2

Prisoner 1

Husband

Wife

Game Theory

nash equilibrium
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
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
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
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
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
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
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
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
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
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
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

do high speed links always help
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

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
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
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

Game Theory

game programming
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
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

Game Theory

conclusions
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