Game theory and its applicationsPresentation Transcript
GAME THEORY AND ITS APPLICATION S Yen 2
CONTENThat is game theorypplicationsefinitionsssumptionsame theory Models with examples(Zero sum game, Prisoner’s Dilemma etc…)
WHAT IS GAME THEORY John von Neumannheory Oskar Morgenstern research of and is associated with the in early 1940st deals with Bargaining/Decision analysis.concerned with strategic behavior
GAME THEORY APPLICATIONSn Economics decision makingargainingilitary strategiesomputer networking, network security
GAME THEORY DEFINITIONTheory of rational behavior for interactive decision problems. In agame, several agents strive to maximize their (expected) utility indexby choosing particular courses of action, and each agents final utilitypayoffs depend on the profile of courses of action chosen by allagents. The interactive situation, specified by the set of participants,the possible courses of action of each agent, and the set of all possibleutility payoffs, is called a game; the agents playing a game are calledthe players.
DEFINITIONSEg : Two companies are the only manufactures of a particularproduct, they compete each other for market share
DEFINITIONSDefinition: Maximin strategy – If wedetermine the least possible payoff for eachstrategy, and choose the strategy for which thisminimum payoff is largest, we have themaximin strategy.
FURTHER DEFINITIONSDefinition: Constant-sum and non constant-sum game– If the payoffs to all players add up to the sameconstant, regardless which strategies they choose, thenwe have a constant-sum game. The constant may bezero or any other number, so zero-sum games are aclass of constant-sum games. If the payoff does not addup to a constant, but varies depending on whichstrategies are chosen, then we have a non-constant sumgame
GAME THEORY :ASSUMPTIONS(1) Each decision maker has available to him two ormore well-specified choices or sequences of choices.(2) Every possible combination of plays available to theplayers leads to a well-defined end-state (win, loss, ordraw) that terminates the game.(3) A specified payoff for each player is associated witheach end-state.
GAME THEORY :ASSUMPTIONS(4) Each decision maker has perfect knowledge of thegame and of his opposition.(5) All decision makers are rational; that is, each player,given two alternatives, will select the one that yields himthe greater payoff.
GAME THEORY MODELSTwo-person, zero-sum games.Two-person, constant-sum game.Two-Person Non constant-Sum Games
TWO PERSON ZERO-SUM AND CONSTANT-SUM GAMESTwo-person zero-sum and constant-sum games are playedaccording to the following basic assumption:Each player chooses a strategy that enables him/her to do the besthe/she can, given that his/her opponent knows the strategy he/she isfollowing.A two-person zero-sum game has a saddle point if and only if Max (row minimum) = min (column maximum) all all rows columns
TWO PERSON ZERO-SUM GAME Company B Payoff Matrix to company A Maximin strategy Company AStrategies a1,b1- increase advertising Mini max strategy a2,b2-provide discounts a3,b3-Extend Warranty Example from text book
ZERO-SUM GAME• Game theory assumes that the decision maker and the opponent are rational, and that they subscribe to the maximin criterion as the decision rule for selecting their strategy• This is often reasonable if when the other player is an opponent out to maximize his/her own gains, e.g. competitor for the same customers.• Consider: Player 1 with three strategies S1, S2, and S3 and Player 2 with four strategies OP1, OP2, OP3, and OP4
ZERO-SUM GAME Player 2 OP1 OP2 OP3 OP4 Row Minima S1 12 3 9 8 3 Player 1 S2 5 4 6 5 4 maximin S3 3 0 6 7 0 Column 12 4 9 8 maxima minimax• Using the maximin criterion, player 1 records the row minima and selects the maximum of these (S2)• Player 1’s gain is player 2’s loss. Player 2 records the column maxima and select the minimum of these (OP2) example
ZERO-SUM GAME Player 2 OP1 OP2 OP3 OP4 Row Minima S1 12 3 9 8 3 Player 1 S2 5 4 6 5 4 maximin S3 3 0 6 7 0 Column 12 4 9 8 maxima minimax• The value 4 achieved by both players is called the value of the game• The intersection of S2 and OP2 is called a saddle point. A game with a saddle point is also called a game with an equilibrium solution.• At the saddle point, neither player can improve their payoff by switching strategies example
TWO PERSON NON CONSTANT SUM GAME• Most game-theoretic models of business situations are not constant-sum games, because it is unusual for business competitors to be in total conflict• As in two-person zero-sum game, a choice of strategy by each player is an equilibrium point if neither player can benefit from a unilateral change in strategy
PRISONER’S DILEMMAwo suspects arrested for a crimerisoners decide whether to confess or not to confessf both confess, both sentenced to 3 months of jailf both do not confess, then both will be sentenced to 1 month of jailf one confesses and the other does not, then the confessor gets freed
ormal Form representation – Payoff Matrix Prisoner 2 Confess Not ConfessPrisoner 1 Confess -3,-3 0,-9 Not Confess -9,0 -1,-1
ach player’s predicted strategy is the best response to the predicted strategies of other players o incentive to deviate unilaterally Prisoner 2 Confess Not ConfessPrisoner 1 player can benefit from a unilateral change in strategy either Confess -3,-3 0,-9 Not Confess -9,0 -1,-1 trategically stable or self-enforcing
NASH EQUILIBRIUM Prisoner 2 Confess Not ConfessPrisoner 1 Confess -3,-3 0,-9 Not Confess -9,0 -1,-1
MIXED STRATEGIESprobability distribution over the pure strategies of the gameock-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) rocko pure strategy Nash equilibriumne mixed strategy Nash equilibrium – each player plays rock, paper and
FURTHER STUDIESWill Game Theory give us the optimum or best solution/decision?When there is n number of players and n number ofstrategies need to go for Leaner programming.
Multi agent Systems: Algorithmic, Game-Theoretic, and Logical Foundationsby Yoav Shoham, Kevin Leyton-BrownPublisher: Cambridge University Press 2008ISBN/ASIN: 0521899435ISBN-13: 9780521899437Number of pages: 532Description:Multiagent systems consist of multiple autonomousentities having different information and/or diverginginterests. This comprehensive introduction to the fieldoffers a computer science perspective, but also drawson ideas from game theory, economics, operationsresearch, logic, philosophy and linguistics. It will serveas a reference for researchers in each of these fields,and be used as a text for advanced undergraduate andgraduate courses.
Game Theory for Applied Economists Robert GibbonsThis book introduces one of the most powerfultools of modern economics to a wide audience:
Strategy: An Introduction to Game Theory,2nd EditionJoel WatsonBook Description October 16, 2007 | ISBN-10: 0393929345Publication Date:| ISBN-13: 978-0393929348 | Edition: 2Strategy, Second Edition, is a thorough revisionand update of one of the most successful GameTheory texts available.Known for its accurate and simple-yet-thoroughpresentation, Joel Watson has refined his text tomake it even more student friendly. Highlights ofthe revision include the addition of Guided (orSolved) Exercises and a significant expansion ofthe material for political economists and politicalscientists.