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Decisiontree&game theory
- 1. Decision making and GameTheory
- 2. Decisions aremade at every level of management to ensure organizational or business goals are achieved. Decision-making involves the selection of a course of action from among two or more possible alternatives in order to arrive at a solution for a given problem”. Steps to Good Decision Making 1.Define problem and influencing factors. 2. Establish decision criteria. 3. Select decision-making tool (model). 4. Identify and evaluate alternatives using decision-making tool (model). 5. Select best alternative. 6. Implement decision. 7. Evaluate the outcome.
- 3. Terminologies Decision Maker:The decision maker is charged with responsibility of making the decision. The acts or decision: The alternative decisions or strategies that are available to decision maker. Events: Occurrences which affect the achievement of the objectives, these are also called as outcomes or state of nature. Pay off table: It is the outcome in quantitative form if the decision maker adopts a particular strategy under a particular state of nature
- 4. Types of DecisionMaking 1) Decision making under certainty The outcome of a decision alternative is known (i.e., there is only one state of nature.) 2) Decision making under risk The outcome of a decision alternative is not known, but its probability is known. 3) Decision making under uncertainty The outcome of a decision alternative is not known, and even its probability is not known.
- 5. Decision underCertainty: The decisions may be taken when the problems are under certainty i.e., where a complete knowledge about the nature of future conditions is known. For example, we know that if we toss an unbiased coin, one of two equally likely outcomes (i.e., either head or tail) occur, and the probability of each outcome is predetermined. Decision making under uncertainty. In this case, each course of action has several possible consequences and the decision maker does not know the probability of each of them. It is therefore a scenario poor in information, as opposed to the previous case. The decision is complicated because past experiences do not make it possible to predict the future and there are many uncontrollable variables. Decision making under risk. This scenario presents an intermediate situation between the two two previous ones: each alternative, strategy or course of action has several possible consequences, but the decision maker knows the probability of each of them. Although the choice choice will not be as easy as in the case of decisions under certainty, it will be possible to apply a a decision-making model that facilitates it.
- 6. Decision Making UnderUncertainty 1.Maximax Criterion /optimism 2. Maximin Criterion/ pessimism 3. Minimax Criterion /regret 4. Laplace Criterion (Equally likelihood/Rationality/Bayes) 5. Hurwicz alpha Criterion ( Rationality or Realism)
- 7. Criterion 1: Maximax(Optimistic) An adventurous and aggressive decision maker may choose the act that would result in the maximum payoff possible Step 1 - Pick maximum payoff of each alternative. Step 2 - Pick maximum of those maximums in Step 1; its corresponding alternative is the decision. This is viewed as an optimistic approach, “Best of bests”.
- 8. States of NatureDecisions A1 A2 A3 S1 220 180 100 S2 160 190 180 S3 140 170 200 maximum 220 190 200 Max(220,190,200)=220 is the maximax i.e maximum of maximums Ex1: What is the decision taken by the decision maker for the following payoff matrix under maximax criterion? States of Nature Decisions A1 A2 A3 S1 220 180 100 S2 160 190 180 S3 140 170 200
- 9. (Payoffs) State ofNature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 State of Nature Maxim um payoff Action Oil Dry Drill for oil 700 -100 700 Sell the land 90 90 90 Ex2: What is the decision taken by the decision maker for the following payoff matrix under maximax criterion? Max(700,90)=700,Hence the decision taken by decision maker is to drill for oil
- 10. Criterion 2: Maximin(Pessimistic) This is also called Waldian criterion. This criterion of decision making stands for choice between alternative courses of action assuming pessimistic view of nature (payoff matrix) • Step 1 - Pick minimum payoff of each alternative • Step 2 - Pick the maximum of those minimums in Step 1, its corresponding alternative is the decision This is viewed as a pessimistic approach, “Best of worsts”
- 11. State of NatureDecision A1 A2 A3 S1 -1 4 10 S2 3 -2 6 S3 18 14 -3 minimum -1 -2 -3 Maximum(-1,-2,-3)=-1 i.e. maximum of minimum decisions, The decision taken by decision maker is A1 Ex1: What is the decision taken by the decision maker for the following payoff matrix under maximin criterion? State of Nature Decision A1 A2 A3 S1 -1 4 10 S2 3 -2 6 S3 18 14 -3
- 12. For eachaction, find minimum payoff over all states of nature Then choose the action with the maximum of these minimum payoffs State of Nature Minim um payoff Action Oil Dry Drill for oil 700 -100 -100 Sell the land 90 90 90 (Payoffs) State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 Ex2: Ex2: What is the decision taken by the decision maker for the following payoff matrix under maximin criterion? Max(-100,90)=90,Hence the decision chosen by decision maker is to sell the land
- 13. Criterion 3: Minimaxdecision criterion Application of the minimax criterion requires a table of losses or table of regret instead of gains. Regret is amount you give up due to not picking the best alternative in a given state of nature. Regret = Opportunity cost = Opportunity loss (cost matrix) • Step 1 - Construct a ‘regret table’, • Step 2 - Pick maximum regret of each row in regret table, • Step 3 - Pick minimum of those maximums in Step 2, its corresponding alternative is the decision
- 14. State of NatureDecisions A1 A2 A3 S1 0 4 10 S2 3 0 6 S3 18 14 0 maximum 18 14 10 Minimum(18,14,10)=10 i.e.)minimum of maximum decisions,Decision made by decision maker is A3 Ex1: What is the decision taken by the decision maker for the following Regret matrix under minimax criterion? State of Nature Decisions A1 A2 A3 S1 0 4 10 S2 3 0 6 S3 18 14 0
- 15. For eachaction, find maximum regret over all states of nature Then choose the action with the minimum of these maximum regrets (Regrets) State of Nature Max Regret Action Oil Dry Drill for oil 700 -100 700 Sell the land 90 90 90 (Payoffs) State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 Ex2: What is the decision taken by the decision maker for the following Regret matrix under minimax criterion? Min(700,90)=90,Hence the decision taken is to sell the land
- 16. Criterion 4: EquallyLikely or Laplace criterion The decision maker makes a simple assumption that each state of nature is equally likely to occur & compute average payoff for each. Choose decision with highest average payoff. Also known as Laplace criterion • Step 1 - Calculate the average payoff for each alternative. • Step 2 - The alternative with highest average if the decision.
- 17. Events Acts A1 A2A3 E1 20 12 25 E2 25 15 30 E3 30 20 22 All the three outcomes has equal probability of occurrence i.e.) 1/3=0.34,find expected payoff for each alternative or acts A1->0.34(20+25+30)=25.5 A2->0.34(12+15+20)=15.98 A3->0.34(25+30+22)=26.18 A3 has maximum payoff, hence it is the optimal choice. Ex1: What is the decision taken by the decision maker for the following payoff matrix under laplace criterion?
- 18. Criterion 5: Hurwicz(Realism) This method is a combination of Maximin and Maximax criterion. Also known as criterion of rationality. neither too optimistic nor too pessimistic • Step 1 - Calculate Hurwicz value for each alternative • Step 2 - Pick the alternative of largest Hurwicz value as the decision. 𝐻𝑢𝑟𝑤𝑖𝑐𝑧 𝑣𝑎𝑙𝑢𝑒 𝑎𝑙𝑡𝑒𝑟𝑛𝑎𝑡𝑖𝑣𝑒 = 𝑚𝑎𝑥 𝛼 + (𝑚𝑖𝑛)(1 − 𝛼) where 𝛼 is called coefficient of realism
- 19. Events Acts A1 A2A3 E1 20 12 25 E2 25 15 30 E3 30 20 22 Let ᾱ=0.6 For A1, maximum payoff=30,minimum payoff=20,Hence decision D1=(30*0.6)+20*(1-0.6)=26 For A2,D2=(20*0.6)+12*(1-0.6)=16.8 For A3,D3=(30*0.6)+22*(1-0.6)=26.8 D3 is maximum, hence the optimal decision is A3 Ex1: What is the decision taken by the decision maker for the following payoff matrix under Hurwicz criterion?
- 20. Ex2:The followingpayoff matrix gives the payoff of different strategies S1,S2,S3 against conditions N1,N2,N3 and N4 Conditions strategies S1 S2 S3 N1 4000 20000 20000 N2 -100 5000 15000 N3 6000 400 -2000 N4 18000 0 1000 Indicate the decision under 1)pessimistic 2)optimistic 3)Equal probability 1)Pessimistic : max((-100,0,-2000))=0,S2 is the strategy 2)Optimistic : max(18000,20000,20000)=20000,either S2 or s3 can be chosen 3)Equal probability value: For S1:1/4(400-100+6000+18000)=6975 For S2:1/4(20000+5000+400+0)=6350 For s3:1/4(20000+15000-2000+1000)=8500, Hence S3 is the optimal decision
- 21. Decision under risk Risk implies a degree of uncertainty and an inability to fully control the outcomes or consequences of such an action. Risk or the elimination of risk is an effort that managers employ. However, in some instances the elimination of one risk may increase some other risks. Effective handling of a risk requires its assessment and its subsequent impact on the decision process. The decision process allows the decision-maker to evaluate alternative strategies prior to making any decision. The process is as follows: 1. The problem is defined and all feasible alternatives are considered. The possible outcomes for each alternative are evaluated. 2. Outcomes are discussed based on their monetary payoffs or net gain in reference to assets or time. 3. Various uncertainties are quantified in terms of probabilities. 4. The quality of the optimal strategy depends upon the quality of the judgments. The decision-maker should identify and examine the sensitivity of the optimal strategy with respect to the crucial factors.
- 22. Expected Monetary Value(EMV) To calculate the EMV in project risk management: 1. Assign a probability of occurrence for the risk. 2. Assign monetary value of the impact of the risk when it occurs. 3. Multiply Step 1 and Step 2.
- 23. Ex1:A1,A2,A3 arethe acts and S1,S2,S3 are the state of nature for the following payoff matrix, Also known that p(S1)=0.5, P(S2)=0.4 and P(S3)=0.1, what is the decision taken by decision maker? State of nature Decision A1 A2 A3 S1 30 25 22 S2 20 35 20 S3 40 30 35 EMV for A1=0.5*30+0.4*20+0.1*40=15+8+4=27 EMV for A2=0.5*25+0.4*35+0.1*30=12.5+14+3=29.5 EMV for A3=0.5*22+0.4*20+0.1*35=11+8+3.5=22.5 Highest EMV is for A3,hence the decision chosen is A3
- 24. Ex2: Giventhe following pay off of three acts A1,A2 and A3 and states of nature S1,S2 and S3, the probabilities of state of nature are respectively 0.1,0.7,0.2.calculate and tabulate the EMV and conclude which of the acts can be chosen as the best. States of Nature Acts A1 A2 A3 S1 25 -10 -125 S2 200 440 400 S3 650 740 750 Solution: Tabulating the EMV, EMV is maximum for A2,hence A2 is chosen act Act A1 Prob*payoff Act A2 Prob*payoff Act A3 Prob*payoff 0.1*25=2.5 0.1*-10=-1 0.1*-125=-12.5 0.7*200=280 0.7*440=308 0.7*400=280 0.2*650=130 0.2*740=148 0.2*750=150 EMV for A1 =412.5 EMV for A2=455 EMV for A3=417.5
- 25. DECISION MAKING DecisionMaking Under RiskCalculate Expected Values (Ei) Well Drilling Example-Decision Making Under Risk N1:Dry Hole N2 :Small Well N3:Big Well P1=0.6 P2=0.3 P3=0.1 A1:Don’t Drill $0 $0 $0 A2:Drill Alone $-500,000 $300,000 $9,300,000 Alternative State of Nature / Probability A3:Farm Out $0 $125,000 $1,250,000 Expected Value E1=0.6*0+0.3*0+ 0.1*0 $0 E2=0.6*(-500,000)+0.3*(300,000)+ 0.1*(9,300,000) $720,000 E3=0.6*0+0.3*(125,000)+ 0.1*(1,250,000) $162,000 $720,000 A2 is the solution if you are willing to risk $500,000
- 26. n j=1 (pjOij) Ei= E1=0.999*(-200)+0.001*(-200) E1=$-200 E2=0.999*0+0.001*(-100,000)E2=$-100 Example of Decision Making Under Risk Not Fire in your house P1 =0.999 P2=0.001 Insure house $-200 $-200 Do not Insure house 0 $-100,000 Alternatives Fire in your house Would you insure your house or not?
- 27. Decision trees ADecision tree is graphical representation of the decision process indicating decisions alternatives,states of nature, probabilities attached to the states of nature and conditional benefits and losses. It consist of network of nodes and branches. • Root Node: A root node compiles the whole sample, it is then divided into multiple sets which comprise of homogeneous variables. • Decision Node: That sub-node which diverges into further possibilities, can be denoted as a decision node. • Terminal Node: The final node showing the outcome which cannot be categorized any further, is termed as a value or terminal node. • Branch: A branch denotes the various alternatives available with the decision tree maker. • Splitting: The division of the available option (depicted by a node or sub-node) into multiple sub-nodes is termed as splitting.
- 28. Steps in Decisiontree analysis identity all the options you have to complete your project. Predict potential outcomes i.e evaluate the results that each option will bring Analyse the results:determine the different outcomes are. Optimise your decisions: determining which option will best fit your project
- 29. Ex1:Consider thefollowing payoff table States of Nature Probability Alternatives A1 A2 S1 0.6 4000 10000 S2 0.4 2000 -5000 EVM of decision alternative A1=4000*0.6+2000*0.4=3200 EMV of decision alternative A2=10000*0.6-5000*0.4=4000,alternative A2 yields higher EMV,hence A2 is the best decision
- 30. Ex2:Construct thedecision tree for following State of Nature Decision Expand Maintain status quo Sell now Good competit ion Probability=0.7 800000 1300000 320000 Poor competit ion Probability=0.3 500000 -150000 320000 Solution:EMV for Expand=800000*0.7+500000*0.3=710000 EMV for Status quo=1300000*0.7+150000*0.3=865000 EMV for sell=320000*0.7+320000*0.3=320000 Status quo gives maximum EMV hence is the best option
- 32. DECISION MAKING DecisionMaking Under RiskCalculate Expected Values (Ei) Decision Trees No Fire: Fire: No Fire: Fire: Decision node Ai Chance node Nj Outcome (Oij) Probability (Pj) Expected Value Ei (-200) (0.999) x = x = (-199.8) (-200) (0.001) x = (-0.2) (0) (0.999) x = (0) (-100,000) (0.001) x = (-100) + + = $-200 =$-100 Mathematical solution is identical, visual representation is different
- 33. Advantages of Decisiontrees Easy to read and interpret One of the advantages of decision trees is that their outputs are easy to without even requiring statistical knowledge. • Important insights can be generated based on experts describing a situation (its alternatives, probabilities, and costs) and their preferences for outcomes. • Help determine worst, best and expected values for different scenarios. • Can be combined with other decision techniques. • Depicts Most Suitable Project/Solution: It is an effective means of picking out the most appropriate project or solution after examining all the possibilities.
- 34. Game Theory Theoryof games is decision theory with general features of competitive situations. It is helpful when two or more individuals have conflicting objectives when trying to make decision. Properties of a Game There are finite numbers of competitors called ‘players’ Each player has a finite number of possible courses of action called ‘strategies’ All the strategies and their effects are known to the players but player does not know which strategy is to be chosen Strategy can be classified as: 1. Pure strategy 2. Mixed Strategy
- 35. Basic Terms usedin game theory Player –Competitor(individual or group or organization) Strategy – Alternate course of action(choices) Pure strategy – Using same strategy each time (deterministic) Mixed strategy – Using the course of action depending on some fixed probability. Optimum strategy – The choice that puts the player in the most preferred position irrespective of his competitors strategy.
- 36. Two person zero– sum game Definition: Only 2 persons are involved in the game and the gain made by one player is equal to the loss of the other. As the name implies, these games involve only two players .They are called zero-sum games because one player wins whatever the other one loses, so that the sum of their net winnings is zero. In general, a two-person game is characterized by The strategies of player 1. The strategies of player 2. The pay-off table.
- 37. Terms used Pay off matrix: The representation of gains and losses resulting from different actions of the competitors is represented in the form of a matrix Payoff is the outcome of playing the game.. Value of game: It is the expected outcome of the player when all the players of the game follow their optimum strategy. Fair game: Value of the game is zero. Total payoff to the two players at the end of each play is zero
- 38. Formulation of Twoperson zero – sum game payoff table a11 a12 ……… a1n a21 a22 …........ a2n . . am1 am2 ………. amn B1 B2 ……… Bn A1 A2 . . Am
- 39. Formulation of Twoperson zero – sum game A1,A2,…..,Am are the strategies of player A ,row designates the course of actions available for A B1,B2,…...,Bn are the strategies of player B, column designates the course of actions available for B aij is the payoff to player A (by B) when the player A plays strategy Ai and B plays Bj (aij is –ve means B got |aij| from A)
- 40. Ex1.Consider thegame of the odds and evens. This game consists of two players A,B, each player simultaneously showing either of one finger or two fingers. If the number of fingers matches, so that the total number for both players is even, then the player taking evens (say A) wins Rs.1 from B (the player taking odds). Else, if the number does not match, A pays Rs.1 to B. Thus the payoff matrix to player A is the following table:
- 41. Optimum Solution Agame can be solved by using the following three methods, based on the nature of the problem. Saddle point concept/Max-min and Min max principle Dominance rule Graphical method.
- 42. Min- Max andMax-Min principle Max –Min : A row(winning) payer will select the maximum out of the minimum gains. Min- Max : A column(loosing) player will always try to minimize his maximum losses. Saddle point: If the max-min and min-max values are same then the game has a saddle point and is the intersection point of both the values. A game is fair if maximin value=minimax value=0 A game is said to be determinable if maximin=minimax≠0
- 43. 8 6 28 8 9 4 (SP) 5 7 5 3 5 B1 B2 B3 B4 A1 A2 A3 Max Row min Col 8 9 4 8 2 4 3 min max max min
- 44. Solution The solutionof the game is based on the principle of securing the best of the worst for each player. If the player A plays strategy 1, then whatever strategy B plays, A will get at least 2. Similarly, if A plays strategy 2, then whatever B plays, will get at least 4. and if A plays strategy 3, then he will get at least 3 whatever B plays. Thus to maximize his minimum returns, he should play strategy 2.
- 45. Solution (cont..) Nowif B plays strategy 1, then whatever A plays, he will lose a maximum of 8. Similarly for strategies 2,3,4. (These are the maximum of the respective columns). Thus to minimize this maximum loss, B should play strategy 3. and 4 = max (row minima) = min (column maxima) is called the value of the game. 4 is called the saddle-point. 6/19/2021 45
- 46. Ex 2:Solvethe game whose payoff is given by solution B1 B2 B3 Row minima A1 1 3 4 1 A2 0 -4 -3 -4 A3 1 5 -1 -1 Column maxima 1 5 4 Maximin=max(1,-4,-1)=1 Minimax=min(1,5,4)=1 Minimax=maximin, Hence saddle point exists i.e 1 and the optimal strategy is (A1,B1) B1 B2 B3 A1 1 3 4 A2 0 -4 -3 A3 1 5 -1
- 47. Ex3: Determinewhich of the following two person zero sum games are strictly determinable and fair. Given the optimum strategy for each player in the case of strictly determinable games. B1 B2 A1 -5 2 A2 -7 -4 Solution: B1 B2 Row minima A1 -5 2 -5 A2 -7 -4 -7 Column maxima -5 2 Maximin=max(-5,-7)=-5 Minimax=min(-5,7)=-5 Maximin=minimax=-5,saddle point exist at(A1,B1)and is not equal to 0,hence game is strictly determinable
- 48. Ex4: Determinewhich of the following two person zero sum games are strictly determinable and fair. Given the optimum strategy for each player in the case of strictly determinable games. B1 B2 A1 0 2 A2 -1 4 Solution B1 B2 Row minima A1 0 2 0 A2 -1 4 -1 Column maxima 0 4 Maximin=max(0,-1)=-0 Minimax=min(0,4)=-0 Maximin=minimax=0,saddle point exist at(A1,B1)and is equal to 0,hence game is fair
- 49. Games without saddlepoint(mixed strategy) No pure strategy or no saddle point exists. The optimal mix for each player may be determined by assigning each strategy a probability of it being chosen. Thus these mixed strategies are probabilistic combinations of available better strategies and these games hence called Probabilistic games. The probabilistic mixed strategy games without saddle points are commonly solved by any of the following methods Analytical Method Graphical Method Simplex Method 6/19/2021 49
- 50. Analytical Method A2x2 game without saddle point can be solved using following formula. The optimum mixed strategies is given by Sa and Sb,where A1 A2 P1 P2 SA= B1 B2 q1 q2 Sb= Sb= Where p1,p2,q1,q2 are as follows and V is the value of the game B1 B2 A1 a11 a12 A2 a21 a22
- 51. Ex1:Solve thefollowing game and determine its value Player Y strategy1 Strategy 2 Row minima Player X strategy 1 4 1 1 Strategy 2 2 3 2 Column maxima 4 3 Maximin=max(1,2)=2 Minimax=min(4,3)=3 Maximin ≠ minimax, saddle point does not exist, hence apply mixed strategy
- 52. Conclusion: Out of4 trials player X will use first strategy once and second strategy thrice Out of 2 trials player Y will use first strategy once and second strategy once
- 53. Ex2:Two playerA and B match coins. If the coins match, then A wins two units of value, if the coin do not match, then B win 2 units of value. Determine the optimum strategies for the players and the value of the game. Solution: The pay off matrix for player A Since maxmin=-2 and minmax=2. i.e maxmin is not equal to minmax, therefore there is no unique saddle point. Games with no saddle point should be solved using mixed strategy
- 55. Ex3:Consider amodified form of " matching biased coins" game problem. The matching player is paid Rs. 8.00 if the two coins turn both heads and Rs. 1.00 if the coins turn both tails. The non-matching player is paid Rs. 3.00 when the two coins do not match. Given the choice of being the matching or non- matching player, which one would you choose and what would be your strategy? Solution: The pay off matrix for matching player Since maxmin=-3 and minmax=1. i.e maxmin is not equal to minmax, therefore there is no unique saddle point. Games with no saddle point should be solved using mixed strategy.
- 57. Dominance Rule Definition:A strategy is dominated by a second strategy if the second strategy is always at least as good (and sometimes better) regardless of what the opponent does. Such a dominated strategy can be eliminated from further consideration. dominance method Steps (Rule) Step-1: 1. If all the elements of Column-i are greater than or equal to the corresponding elements of any other Column-j, then the Column-i is dominated by the Column-j and it is removed from the matrix. eg. If Column-2 ≥ Column-4, then remove Column-2 Step-2: 1. If all the elements of a Row-i are less than or equal to the corresponding elements of any other Row-j, then the Row-i is dominated by the Row-j and it is removed from the matrix. eg. If Row-3 ≤ Row-4, then remove Row-3 Step-3: Again repeat Step-1 & Step-2, if any Row or Column is dominated, otherwise stop the procedure.
- 58. Ex1: Solvethe game given below in Table after reducing it to 2 × 2 game Solution: Reduce the matrix by using the dominance property. In the given matrix for player A, all the elements in Row 3 are less than the adjacent elements of Row 2. Strategy 3 will not be selected by player A, because it gives less profit for player A. Row 3 is dominated by Row 2. Hence delete Row 3, as shown in table.
- 59. For PlayerB, Column 3 is dominated by column 1 (Here the dominance is opposite because Player B selects the minimum loss). Hence delete Column 3. We get the reduced 2 × 2 matrix as shown below in table. Now, solve the 2 × 2 matrix, using the maximin criteria as shown below in table. There is no saddle point and the game has a mixed strategy.
- 61. Ex2:Solve thefollowing game ayer APlayer B B1 B2 B3 B4 A1 3 5 4 2 A2 5 6 2 4 A3 2 1 4 0 A4 3 3 5 2 Solution: Use Dominance rule to reduce the size of the payoff matrix to 2X2 Row-3 ≤ Row-4, so remove Row-3
- 62. Column-2 ≥Column-4, so remove Column-2 Column-1 ≥ Column-3, so remove Column-1
- 63. Row-1 ≤Row-3, so remove Row-1 Check for the saddle point B3 B4 Row minima A2 2 4 2 A4 5 2 2 Column maxima 5 4 Maximin is not equal to minimax,2≠4,Hence no saddle point exist apply mixed strategy formula to find p1,p2 and q1,q2
- 64. Player A A1A2 A3 A4 0 3/5 0 2/5 Player B B1 B2 B3 B4 0 0 2/5 3/5 The optimal strategy for the players is
- 65. Ex3:Find Solutionof game theory problem using dominance method layer APlayer B B1 B2 B3 A1 1 7 2 A2 6 2 7 A3 5 1 6 Solution:Use Dominance rule to reduce the size of the payoff matrix row-3 ≤ row-2, so remove row-3 column-3 ≥ column-1, so remove column-3
- 66. Check forsaddle point B1 B2 Row minima A1 1 7 1 A2 6 2 2 Column maxima 6 7 maximin≠minimax,2≠6,saddle point does not exist hence use mixed strategy formulae P1=(2-6)/(3-13)=-4/-10=2/5 P2=1-p1=1-(2/5)=3/5 Q1=(2-7)/)3-13)=-5/-10=1/2 Q2=1-q1=1-(1/2)=1/2 Hence the optimal strategy is Player A A1 A2 A3 2/5 3/5 0 Player B B1 B2 B3 1/2 1/2 0