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Game Theory and its Applications

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

  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? Game Theory

  3. 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

  4. Games • Normal Form representation – Payoff Matrix Prisoner 2 Prisoner 1 Husband Wife Game Theory

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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)

  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 • 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

  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 Game Theory

  13. 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

  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 Game Theory

  15. 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

  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 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

  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 Game Theory

  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 Game Theory

  19. 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

  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 Game Theory

  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 Game Theory