SARANI SAHABHATTACHARYA, HSS ARNAB BHATTACHARYA, CSE 07 JAN, 2009 Game Theory and its Applications

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, Game Theory

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

Games Jan 07, 2009 Game Theory 4 Normal Form representation – Payoff Matrix ConfessNot Confess Confess-3,-30,-9 Not Confess-9,0-1,-1 MovieCricket Movie2,10,0 Cricket0,01,2 Prisoner 1 Prisoner 2 Wife Husband

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 ConfessNot Confess Confess-3,-30,-9 Not Confess-9,0-1,-1 Prisoner 1 Prisoner 2

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

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

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

Game tree example: Bargaining 0 1 A 0 1 B 0 1 A B B A x1x1 (x 1,1-x 1 ) Y N x2x2 x3x3 (x 3,1-x 3 ) (x 2,1-x 2 ) (0,0) Y Y N N Period 1: A offers x 1. B responds. Period 2: B offers x 2. A responds. Period 3: A offers x 3. B responds.

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 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

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

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

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

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

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) = x C(x) = 1 C(x) = x C(x) = 0

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

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

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

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

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