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game theory in economics

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- 1. GAME THEORY

2. Game theory and Strategic behaviour under oligopolyGame theory is a technique used to analyse situationswhere individuals or organisations have conflictingobjectives. Three important concepts are often used ingame theory. (i)Players (ii)Strategy and (iii) Payoffs.Players are the decision makers (mangers). A strategy isa course of action to change price, develop newproducts, adopt new advertising, etc. The payoff is theoutcome of the strategy. The players always try tooptimise their strategy. This theory was first developedby Von Neuman (mathematician)and OskarMorgenstern(economist). It explains the strategicinteraction among the Oligopoly firms. Each playerneeds to adopt dominant strategy to maximise profit. 3. Types of games Constant sum of game: In this case the total benefit of theplayers given each strategy, is a constant and the players have toshare the profit Zero sum game: In this game the total benefit, given eachstrategy, is equal to zero. Under this game the gain of one playeris the loss by the other player. Positive sum games: The total benefit of the players addedtogether, given each strategy, is more than zero(+ve). Negative sum game: When the total benefit , given the strategy,is less than zero (-Ve) Co operative game: The games where the strategies of theplayers are coordinated or joint action.Q.14.2 4. Role of InterdependenceThe essence of game is the interdependence ofplayer strategies.It may be sequential game or a simultaneous game.In a sequential game, each player moves insuccession, and each player is aware of all priormoves.A simultaneous game is one in which all playersmake decisions (or select a strategy) withoutknowledge of the strategies that are being chosen byother players. Even though the decisions may bemade at different points of time, the decisions aremade simultaneously. Simultaneous games aresolved using the concept of Nash equilibrium. 5. The nature of Problems faced by the Oligopoly firms isbest explained by the Prisoners Dilemma game.Let two persons are involved with some illegal activitiessay; match fixing; were arrested and kept separately sothat they cannot communicate each other. Four possibleoptions were kept before them:(i) If both confess, each one will get 5 years ofimprisonment;(ii) If Both deny, each one will be put in jail for 1 year;(iii) If A confesses and B denies, A will go free and B willget 10 years of imprisonment ;(iv) If B confesses and A denies, B will go free and A willget 10 years of imprisonment . 6. Prisoners Dilemma: The Pay-off matrix With much ofIndividualsStartegies Do not confessDo not confessIndividual AIndividual BConfessConfess 5,5 0,1010,0 1,1uncertainty, no oneknows the action of eachother, there is dilemmain taking a decision. Thedominant strategy for Ais to confess. Thedominant strategy for B isalso is to confess. Eachone will end up with 5years of imprisonment.It refers to a situation inwhich each individualfirm adopts its dominantstrategy and earnsmaximum profits. 7. Payoff Matrix for an advertising gameIndividualsStartegies Do not advertiseDo not advertiseIndividual BAdvertiseIndividual AAdvertise 4,3 5,12,5 3,2Firm As profit is always greater if it advertises than not advertising regardless of what Bdoes and the dominant strategy for A is to advertise. The same is the case for B also.This is the final equilibrium as it is the optimal choice of both the players. 8. In oligopoly, the business firm chooses itsstrategies to achieve equilibrium. There areactions, reactions and interactions to increasetheir prices to achieve the optimum profit. To analyse this type of situation, an Americanmathematician (John Nash)developed atechnique which is known as Nash equilibrium. Itis defined as a situation where each playerchooses his/her optimal strategy, given thestrategy chosen by other players. A game mayhave more than one Nash equilibrium. 9. Simple Two Persons, Zero Sum GameASSUMPTIONS Each player knows bothhis and his opponentsalternatives Preferences of allplayers are known Single period game Sum of payoffs are zero An Equilibrium (orNash Equilibrium) - ifnone of the participantscan improve theirpayoffPLAYER 2PLAYER 1abc d1, -1 3, -3-2, 2 0, 0Player 1 is the first number ineach pair. We will get to {a,c}which is an Equilibrium 10. Two Person Game, Non-Zero Sum Game:ASSUMPTIONS Each player can invade the territory ofthe other (no guard)or Guard his ownterritory Paks payoff is given first. Inida always ranks Guard above noguard, so India has a Dominant Strategy Knowing what India will do, Pak decidesto Guard as well. An Equilibrium--none of the participantscan improve their payoffIndiaGuardPAKNo guardGuard no guardBetter, 1st Worst, 4thWorse, 2nd Best, 3rdWe will get to {Guard, Guard}which is an Equilibrium 11. Unstable Games:No Equilibrium Is Found Suppose KIM thinksthat the solution isgoing to be: {b, c} Then, KIM has anincentive to switch tostrategy-a Then JOHN has anincentive to switch tostrategy-d, etc., etc.Johnc dKIMab3, - 3 1, - 12, - 2 4, - 4There is no, single stable equilibriumEach player may elect a randomstrategy 12. Dominant strategy and domonated startegyPLAYER 2aPLAYER 1bc d1, -1 3, -3-2, 2 0, 0 For Player 1, strategy (a) is adominant strategy - anaction that maximizes thedecision makers welfareindependent of the actionsof the other players. Also, strategy (b) is adominated strategy,which is worst regardlessof what others do Player 2 also has adominant strategy of (c). Dominant strategies makegames easy to solve.With dominant strategies of (a)for Player 1 and ( c) for Player2, the solution will be {a, c},which is an Equilibrium. 13. IndividualsStartegies Do not advertiseDo not advertiseIndividual BAdvertiseIndividual AAdvertise4,3 5,12,5 8,2In real world the one player or both of them may not have a dominant strategy as shownin the matrix. Here firm B has the dominant strategy is to advertise whether firm Aadvertises or not, because the payoffs for B remain the same. Firm A has no dominant strategy now. Because if B advertises, A earns a profit of 4 if itadvertises and 2 if it does not. Thus if B advertises, firm A should advertise.A earns a profit of 5 if it advertises and 8 if it does not advertise. Here the assumption isthat A uses an expensive advertisement and it adds to its cost than revenue. It would shiftthe burden of increased cost to its consumers and get the larger share. It indicates that if Bdoes not advertise, it is better for A not to advertise and get the larger share of the market.Hence A does not have a dominant strategy. 14. IndividualsStartegies Do not advertiseDo not advertiseIndividual BAdvertiseIndividual AAdvertise4,3 5,12,5 8,6B gets better payoff by advertising than not advertising. But if A does not advertise,B will not advertise as advertisement leads to lower payoffs to B than not advertisingwhen A does not advertise. Hence the decision of B to advertise or not depends onwhether A advertises or not. In other words, B does not have a dominant strategyon his own. B gets a share of 6 if it does not advertise when A also does notadvertise.No one can choose a dominant strategy independently of the other firm. When eachplayer chooses its optimal strategy given the strategy of the other player, we willhave a Nash equilibrium. Hence a dominant strategy equilibrium is always a Nashequilibrium, but a Nash equilibrium is not necessarily a dominant strategy.Problem 14.3 for assignment 15. Every dominant strategy equilibriumis also a Nash equilibrium. Nash equilibrium can exist wherethere is no dominant strategyequilibrium. In Nash bargaining, two competitorsor players "bargain" over some itemof value. In a simultaneous-move,one-shot game, the players haveonly one chance to reach anagreement. 16. A. Yes, the dominant strategy for firm A is up. If firm B chooses left,the highest payoff of $5 million can be achieved if Firm A chooses up.On the other hand, if firm B chooses right, the highest payoff of $7.5million can be achieved if firm A again chooses up. No matter what firmB chooses, the highest payoff results for firm A occurs if A chooses up.Therefore, up is a dominant strategy for firm A.B. No, there is no dominant strategy for firm B. If firm A chooses up,the highest payoff of $10 million can be achieved if firm B chooses left.On the other hand, if firm A chooses down the highest payoff of $5million can be achieved if firm B chooses right. Therefore, there is nodominant strategy for firm B. The profit-maximizing choice by firm Bdepends upon the choice made by firm A.