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GameTheory_popular.ppt

25. Mar 2023
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GameTheory_popular.ppt

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
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