GameTheory_popular.ppt in the operations reearch
- 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? Jan 07, 2009 2 Game Theory
- 3. Battle of Sexes Jan 07, 2009 Game Theory 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
- 4. Games Jan 07, 2009 Game Theory 4 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 Jan 07, 2009 Game Theory 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 Confess Not Confess Confess -3,-3 0,-9 Not Confess -9,0 -1,-1 Prisoner 1 Prisoner 2
- 6. Mixed strategies Jan 07, 2009 Game Theory 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
- 7. Nash’s Theorem Jan 07, 2009 Game Theory 7 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 Jan 07, 2009 Game Theory 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
- 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 Jan 07, 2009 Game Theory 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
- 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 Jan 07, 2009 Game Theory 12
- 13. Mechanism design Jan 07, 2009 Game Theory 13 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 Jan 07, 2009 Game Theory 14
- 15. Network example Jan 07, 2009 Game Theory 15 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 Jan 07, 2009 Game Theory 16 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 Jan 07, 2009 Game Theory 17
- 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 Jan 07, 2009 Game Theory 18
- 19. Game programming Jan 07, 2009 Game Theory 19 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 Jan 07, 2009 Game Theory 20
- 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 Jan 07, 2009 Game Theory 21