- 1. Game TheoryAbu Bashar
2. What is game theory? Game theory is the study of how optimalstrategies are formulated in conflict. It is concerned with the requirement of decisionmaking in situations where two or more rationalopponents are involved under conditions ofcompetition and conflictinginterests inanticipation of certain outcomes over a period oftime. 3. Contd. In a competitive environment the strategies taken bythe opponent organizations or individuals candramatically affect the outcome of a particulardecision by an organization. In the automobile industry, for example, thestrategies of competitors to introduce certain modelswith certain features can dramatically affect theprofitability of other carmakers. So in order to make important decisions in business,it is necessary to consider what other organizationsor individuals are doing or might do. Game theory is a way to consider the impact of thestrategies of one, on the strategies and outcomes ofthe other. 4. What is a GAME??? A GAME refers to a situation in which two or moreplayers are competing. It involves the players (decision makers) who havedifferent goals or objectives. They are in a situation in which there may be anumber of possible outcomes with different values tothem. Although they might have some control that wouldinfluence the outcome, they do not have completecontrol over others. Unions striking against the company management,players in a chess game, firm striving for larger shareof market are a few illustrations that can be viewed as 5. GAME THEORY MODELS Models in the theory of games can be classifieddepending upon the following factors: Number of Players Total Pay Off and Strategy 6. Number of players It is the number of competitive decision makers, involved in the game. A game involving two players is referred to as a Two-person game. However if the number of players is more ( say n>2 ) then the game is called an n-person game. 7. Total Payoff It is the sum of gains and losses from the game thatare available to the players. If in a game sum of the gains to one player is exactlyequal to the sum of losses to another player, so thatthe sum of the gains and losses equals zero then thegame is said to be a zero-sum game. There are also games in which the sum of theplayers gains and losses does not equal zero, andthese games are denoted as non-zero-sum games. 8. Strategy The strategy for a player is the set of alternativecourses of action that he will take for every payoff(outcome) that might arise. It is assumed that the players know the rulesgoverning the choices in advance. The different outcomes resulting from the choicesare also known to the players in advance and areexpressed in terms of the numerical values ( e.g.money, market share percentage etc. ) 9. Strategy Contd.. Strategy may be of two types: (a) Pure strategy If the players select the same strategy each time, thenit is referred as pure strategy. In this case each playerknows exactly what the opponent is going to do and theobjective of the players is to maximize gains or tominimize losses. (b) Mixed Strategy When the players use a combination of strategies withsome fixed probabilities and each player kept guessingas to which course of action is to be selected by theother player at a particular occasion then this is knownas mixed strategy. 10. TWO PERSON ZERO SUM GAME A game which involves only two players, say player Aand player B, and where the gains made by oneequals the loss incurred by the other is called a twoperson zero sum game. For example, If two chess players agree that at the end of the gamethe loser would pay Rs 50 to the winner then it wouldmean that the sum of the gains and losses equalszero. So it is a two person zero sum game. 11. Pay Off Matrix If Player A has m strategies represented as A1, A2, -------- , Am and player B has n strategies represented byB1, B2, ------- ,Bn. Then the total number of possible outcomes is m x n. Here it is assumed that each player knows not only hisown list of possible courses of action but also those ofhis opponent. It is assumed that player A is always a gainer whereasplayer B a loser. Let aij be the payoff which player Agains from player B if player A chooses strategy i andplayer B chooses strategy j. 12. By convention, the rows of the payoff Matrix denoteplayer As strategies and the columns denote playerBs strategies. Since player A is assumed to be the gainer always sohe wishes to gain a payoff aij as large as possibleand B tries to minimize the same. 13. Now consider a simple game Suppose that there are two lighting fixture stores, Xand Y. The respective market shares have beenstable until now, but the situation then changes. The owner of store X has developed two distinctadvertising strategies, one using radio spots and theother newspaper advertisements. Upon hearing this,the owner of store Y also proceeds to prepare radioand newspaper advertisements. 14. Here a positive number in the payoff matrix means that Xwins and Y loses. A negative number means that Y wins and X loses. Thisgame favors competitor X, since all values are positiveexcept one. If the game had favored player Y, the values in the tablewould have been negative. So the game is biased against Y. However since Y must play the game, he or she will play to 15. From this game the outcomes of eachplayer 16. ASSUMPTIONS OF THE GAME Each player has to choose from a finite number ofpossible strategies. The strategies for each player may or may not be thesame. Player A always tries to maximize his gains andplayers B tries to minimize the losses. The decision by both the players is taken individuallyprior to the play without any communication betweenthem. The decisions are made and announcedsimultaneously so that neither player has anadvantage resulting from direct knowledge of theother players decision. 17. MINIMAX AND MAXIMIN PRINCIPLE The selection of an optimal strategy by each playerwithout the knowledge of the competitors strategy isthe basic problem in playing games. The objective of the study is to know how theseplayers must select their respective strategies so thatthey could optimize their payoff. Such a decision making criterion is referred to as theminimax maximin principle. 18. For player A Minimum value in each row represents the least gain tohim if he chooses that particular strategy. These are written in the matrix by row minima. He will then select the strategy that gives maximum gainamong the row minimum values. This choice of player A is called the maximin criterionand the corresponding gain is called the maximin valueof the game. Similarly, for player B who is assumed to be the loser, the maximum value ineach column represents the maximum loss to him if hechooses his particular strategy. These are written as column maxima. He will select thatstrategy which gives minimum loss among the columnmaximum values. This choice of player B is called theminimax criterion, and the corresponding loss is the 19. Saddle Point If the maximin value equals the minimax value,then the game is said to have a saddle point andthe corresponding strategies are called optimalstrategies. The amount of payoff at an equilibrium point isknown as the VALUE of the game. A game mayhave more than one saddle point or no saddlepoint. 20. Example of minimax maximin principle consider a two person zero sum game with thegiven payoff matrix for player A. 21. Suppose that player A starts the game knowing thatwhatever strategy he adopts, B will select that particularcounter strategy which will minimize the payoff to A. Thus if A selects the strategy A1 then B will select B2 ashis corresponds to the minimum payoff to Acorresponding to A1. Similarly, if A chooses the strategyA2, he may gain 8 or 6 depending upon the choice of B. If A chooses A2 then A can guarantee a gain of at leastmin {8,6} = 6 irrespective of the choice of B. Obviously Awould prefer to maximize his minimum assured gains. In this example the selection of strategy A2 gives themaximum of the minimum gains to A. This gain is called as maximin value of the game and thecorresponding strategy as maximum strategy. 22. RULES FOR DETERMINING A SADDLEPOINT STEP 1: Select the minimum element of each row ofthe payoff matrix and mark them as (O) . This is rowminima of the respective row. STEP 2: Select the greatest element of each columnof the payoff matrix and mark them as ( ) . This iscolumn maxima of the respective column. STEP 3: If there appears an element in the payoffmatrix marked with (O) and ( )both, then thiselement represents the value of the game and itsposition is a saddle point of the payoff matrix. 23. Example 24. Example 25. What would be the optimum strategy of the game given below. 26. Games without saddle point 27. Example Solve the following game and determine the value of the game:Clearly, the pay-off matrix does not possess anysaddle point. The two players, therefore, use mixedstrategies. Let 28. The optimum mixed strategies for player X and Y are determined by 29. But... Did you observe one thingthat it was applicable toonly 2 x 2 payoffmatrices? So let us implement it toother matricesusingdominance and study theimportance of DOMINANCE 30. PRINCIPLES OF DOMINANCE In a game, sometimes a strategy available to a player mightbe found to be preferable to some other strategy / strategies.Such a strategy is said to dominate the other one(s). The rules of dominance are used to reduce the size of thepayoff matrix. These rules help in deleting certain rows and/or columns ofthe payoff matrix, which are of lower priority to at least one ofthe remaining rows, and/or columns in terms of payoffs toboth the players. Rows / columns once deleted will never be used fordetermining the optimal strategy for both the players. This concept of domination is very usefully employed insimplifying the two person zero sum games without saddlepoint. In general the following rules are used to reduce the size of 31. Contd. The RULES Rule 1: If all the elements in a row ( say ithrow ) of a pay off matrix are less than orequal to the corresponding elements of theother row ( say jth row ) then the player A willnever choose the ith strategy then we say ithstrategy is dominated by jth strategy and willdelete the ith row. Rule 2: If all the elements in a column ( sayrth column ) of a payoff matrix are greaterthan or equal to the corresponding elementsof the other column ( say sth column ) thenthe player B will never choose the rth strategy 32. EXAMPLE So the reduced matrix will be 33. Calculate Saddle point?? If saddle point exist then we have value for thegames. If not calculate????????????? The optimal values for Players A and B are 34. There is no saddle point. Row 1 is dominated by row 2 and column 1 is dominated by column 2 giving the reduced 2 x 2 matrix 35. So the reduced 2x2 matrix would beFind out the value of the game 36. GRAPHICALSOLUTIONTOsolve the zerosum games which donot possess a saddlepoint using 37. Solution of 2 x n and m x 2 Games consider the solution of games where eitherof the players has only two strategiesavailable: When the player A, for example, has only 2strategies to choose from and the player Bhas n, the game shall be of the order 2 x n,whereas in case B has only two strategiesavailable to him and A has m strategies, thegame shall be a m x 2 game. The problem may originally be a 2 x n or a mx 2 game or a problem might have been 38. Graphical method Contd.. In either case, we can use graphical method tosolve the problem. By using the graphical approach, it is aimed toreduce a game to the order of 2 x 2 by identifyingand eliminating the dominated strategies, andthen solve it by the analytical method used forsolving such games. The resultant solution is also the solution to theoriginal problem. Although the game value and the optimal strategycan be read off from the graph, we generallyadopt the analytical method (for 2 x 2 games) toget the answer. 39. The lines are marked B1, B2, B3 and B4 and theyrepresent the respective strategies. For each value of p1, the height of the lines at thatpoint denotes the pay-offs of each of Bs strategiesagainst (p1, 1 p1) for A. A is concerned with his least pay-off when heplays a particular strategy, which is represented bythe lowest of the four lines at that point, andwishes to choose p1 so as to maximize thisminimum pay-off. This is at K in the figure where the lower envelope(represented by the shaded region), the lowest ofthe lines at point, is the highest. This point lies atthe intersection of the lines representing strategies 40. Thankyou Very
