Subject Name OPERATIONS RESEARCH Subject Code 10 CS

Subject Name: OPERATIONS RESEARCH Subject Code: 10 CS 661 Prepared By: Sindhuja K Department: CSE 2/25/2021

UNIT VII- Game Theory, Decision Analysis 2/25/2021 2

Objective • Competition in business, military operations, advertising about a product , marketing etc. , • It is essential to guess the activities or actions of his opponent or competitor. 2/25/2021 3

Game Theory • Game theory is a mathematical theory that deals with the general features of competitive situations like these in a formal, abstract way. It places particular emphasis on the decision-making processes. • Game theory is a decision theory in where ones choice of action is determined after talking into account all possible alternatives available to an opponent playing in a same field. 2/25/2021 4

Basic Terms used in 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. 2/25/2021 5

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. 2/25/2021 6

Two person zero – sum game • 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. – 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. 2/25/2021 7

Formulation of Two person zero – sum game B 1 2/25/2021 B 2 ……… Bn A 1 a 12 ……… a 1 n A 2 a 21 a 22 …. . . . a 2 n . . Am am 1 am 2 ………. amn 8

Formulation of Two person zero – sum game • A 1, A 2, …. . , Am are the strategies of player A • B 1, B 2, …. . . , Bn are the strategies of player 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) 2/25/2021 9

Example Consider the game 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: 2/25/2021 10

Optimum Solution • A game 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 • Games without saddle point – Dominance rule – Graphical method. • A primary objective of game theory is the development of rational criteria for selecting a strategy. Two key assumptions are made: – Both players are rational – Both players choose their strategies solely to promote their own welfare 2/25/2021 11

Min- Max and Max-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. 2/25/2021 12

B 1 B 2 B 3 B 4 Row min A 1 8 6 2 8 2 A 2 8 9 4 (SP) 5 4 A 3 7 5 3 5 3 Col 8 9 4 8 Max min max 2/25/2021 max min 13

Solution • The solution of 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. 2/25/2021 14

Solution (cont. . ) • Now if 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. 2/25/2021 15

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. • The following rules of dominance is used reduce the sixe of the matrix – – 2/25/2021 Row dominance Column dominance Modified row dominance- Average of rows Modified column dominance- Average of columns 16

Example • Thus in our example (below), for player A, strategy A 3 is dominated by the strategy A 2 and so can be eliminated. • Eliminating the strategy A 3 , we get the B 1 B 2 B 3 B 4 2/25/2021 A 1 8 6 2 8 A 2 8 9 4 5 A 3 7 5 3 5 17

Cont. . • following reduced payoff matrix: • Now , for player B, strategies B 1, B 2, and B 4 are dominated by the strategy B 3. • Eliminating the strategies B 1 , B 2, and B 4 we get the reduced payoff matrix: B 1 B 2 B 3 B 4 2/25/2021 A 1 8 6 2 8 A 2 8 9 4 5 18

Cont. . • following reduced payoff matrix: • Now , for player A, strategy A 1 is dominated by the strategy A 2. • Eliminating the strategy A 1 we thus see that A should always play A 2 and B always B 3 and the value of the game is 4 as before. B 3 2/25/2021 A 1 2 A 2 4 19

Example The following game gives A’s payoff. Determine p, q that will make the entry (2, 2) a saddle point. B 1 B 2 B 3 A 1 1 q 6 A 2 p 5 10 A 3 6 2 3 Col max(p, 6) max(q, 5) 10 Row min(1, q) min(p, 5) 2 Since (2, 2) must be a saddle point, 2/25/2021 20

Example Specify the range for the value of the game in the following case assuming that the payoff is for player A. B 1 B 2 B 3 Row min A 1 3 6 1 1 A 2 5 2 3 2 A 3 4 2 Col max 5 6 -5 -5 3 Thus max( row min) <= min (column max) The game has no saddle point. Thus the value of the game lies between 2 and 3. 2/25/2021 21

Games without saddle point(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 2/25/2021 22

Analytical Method • A 2 x 2 game without saddle point can be solved using following formula. 2/25/2021 23

Formula 2/25/2021 24

Example • Solve the following game and determine its value 2/25/2021 25

2/25/2021 26

Graphical Method : Solution of 2 x n and m x 2 Games • 2 x n and m x 2 Games : When the player A, for example, has only 2 strategies to choose from and the player B has n, the game shall be of the order 2 x n, whereas in case B has only two strategies available to him and A has m strategies, the game shall be a m x 2 game. 2/25/2021 27

Example • Solve the following using graphical method. 2/25/2021 28

Algorithm for solving 2 x n matrix games • Draw two vertical axes 1 unit apart. The two lines are x 1 = 0, x 1 = 1 • Take the points of the first row in the payoff matrix on the vertical line x 1 = 1 and the points of the second row in the payoff matrix on the vertical line x 1 = 0. • The point a 1 j on axis x 1 = 1 is then joined to the point a 2 j on the axis x 1 = 0 to give a straight line. Draw ‘n’ straight lines for j=1, 2… n and determine the highest point of the lower envelope obtained. This will be the maximin point. • The two or more lines passing through the maximin point determines the required 2 x 2 payoff matrix. This in turn gives the optimum solution by making use of analytical method. 2/25/2021 29

Example • Solve using graphical method 2/25/2021 30

Solution 2/25/2021 31

Cont. . • V = 66/13 • SA = (4/13, 9 /13) • SB = (0, 10/13, 3 /13 2/25/2021 32

Algorithm for solving m x 2 matrix games • Draw two vertical axes 1 unit apart. The two lines are x 1 =0, x 1 = 1 • Take the points of the first row in the payoff matrix on the vertical line x 1 = 1 and the points of the second row in the payoff matrix on the vertical line x 1 = 0. • The point a 1 j on axis x 1 = 1 is then joined to the point a 2 j on the axis x 1 = 0 to give a straight line. Draw ‘n’ straight lines for j=1, 2… n and determine the lowest point of the upper envelope obtained. This will be the minimax point. • The two or more lines passing through the minimax point determines the required 2 x 2 payoff matrix. This in turn gives the optimum solution by making use of analytical method. 2/25/2021 33

Example • Solve by graphical method 2/25/2021 34

Solution 2/25/2021 35

Cont. . • V = 3/9 = 1/3 • SA = (0, 5 /9, 4/9, 0) • SB = (3/9, 6 /9) 2/25/2021 36

Decision Analysis • Decision making without experimentation • Decision making criteria • Decision making with experimentation • Expected value of experimentation • Decision trees • Utility theory 2/25/2021 37

Decision Making without Experimentation • • • Goferbroke Company owns a tract of land that may contain oil Consulting geologist: “ 1 chance in 4 of oil” Offer for purchase from another company: $90 k Can also hold the land drill for oil with cost $100 k If oil, expected revenue $800 k, if not, nothing Payoff Alternative Oil Dry 1 in 4 3 in 4 Drill for oil Sell the land Chance 2/25/2021 38

Notation and Terminology • Actions: {a 1, a 2, …} – The set of actions the decision maker must choose from – Example: • States of nature: { 1, 2, . . . } – Possible outcomes of the uncertain event. – Example: • Payoff/Loss Function: L(ai, k) – The payoff/loss incurred by taking action ai when state k occurs. – Example: • Prior distribution: – Distribution representing the relative likelihood of the possible states of nature. • Prior probabilities: P( = k) – Probabilities (provided by prior distribution) for various states of nature. – Example: 2/25/2021 39

Decision Making Criteria • Can “optimize” the decision with respect to several criteria – – 2/25/2021 Maximin payoff Minimax regret Maximum likelihood Bayes’ decision rule (expected value) 40

Maximin Payoff Criterion • For each action, find minimum payoff over all states of nature • Then choose the action with the maximum of these minimum payoffs State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 2/25/2021 Min Payoff 41

Minimax Regret Criterion • For each action, find maximum regret over all states of nature • Then choose the action with the minimum of these maximum regrets (Payoffs) State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 (Regrets) State of Nature Action Oil Dry Max Regret Drill for oil Sell the land 2/25/2021 42

Maximum Likelihood Criterion • Identify the most likely state of nature • Then choose the action with the maximum payoff under that state of nature State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 0. 25 0. 75 Prior probability 2/25/2021 43

Bayes’ Decision Rule (Expected Value Criterion) • For each action, find expectation of payoff over all states of nature • Then choose the action with the maximum of these expected payoffs State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 0. 25 0. 75 Prior probability 2/25/2021 Expected Payoff 44

Sensitivity Analysis with Bayes’ Decision Rule • What is the minimum probability of oil such that we choose to drill the land under Bayes’ decision rule? State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 p 1 -p Prior probability 2/25/2021 Expected Payoff 45

Decision Making with Experimentation State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 0. 25 0. 75 Prior probability • Option available to conduct a detailed seismic survey to obtain a better estimate of oil probability • Costs $30 k • Possible findings: – Unfavorable seismic soundings (USS), oil is fairly unlikely – Favorable seismic soundings (FSS), oil is fairly likely 2/25/2021 46

Posterior Probabilities • Do experiments to get better information and improve estimates for the probabilities of states of nature. These improved estimates are called posterior probabilities. • Experimental Outcomes: {x 1, x 2, …} Example: • Cost of experiment: Example: • Posterior Distribution: P( = k | X = xj) 2/25/2021 47

Goferbroke Example (cont’d) • Based on past experience: • If there is oil, then • the probability that seismic survey findings is USS = 0. 4 = P(USS | oil) • the probability that seismic survey findings is FSS = 0. 6 = P(FSS | oil) • If there is no oil, then • the probability that seismic survey findings is USS = 0. 8 = P(USS | dry) • the probability that seismic survey findings is FSS = 0. 2 = P(FSS | dry) 2/25/2021 48

Bayes’ Theorem • Calculate posterior probabilities using Bayes’ theorem: Given P(X = xj | = k), find P( = k | X = xj) 2/25/2021 49

Goferbroke Example (cont’d) Optimal policies • If finding is USS: State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 Expected Payoff Posterior probability • If finding is FSS: State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 Expected Payoff Posterior probability 2/25/2021 50

The Value of Experimentation • Do we need to perform the experiment? – As evidenced by the experimental data, the experimental outcome is not always “correct”. We sometimes have imperfect information. • 2 ways to access value of information • Expected value of perfect information (EVPI) – What is the value of having a crystal ball that can identify true state of nature? • Expected value of experimentation (EVE) – Is the experiment worth the cost? 2/25/2021 51

Expected Value of Perfect Information • Suppose we know the true state of nature. Then we will pick the optimal action given this true state of nature. State of Nature Action Oil Dry Drill for oil 700 -100 Sell the land 90 90 0. 25 0. 75 Prior probability • E[PI] = expected payoff with perfect information = 2/25/2021 52

Expected Value of Experimentation • We are interested in the value of the experiment. If the value is greater than the cost, then it is worthwhile to do the experiment. • Expected Value of Experimentation: EVE = E[EI] – E[OI] where E[EI] is expected value with experimental information. 2/25/2021 53

Decision Tree • Tool to display decision problem and relevant computations • A decision tree consists of 3 types of nodes: 1. Decision nodes - commonly represented by squares 2. Chance nodes - represented by circles 3. End nodes - represented by triangles/ellipses • A decision tree has only burst nodes (splitting paths) but no sink nodes (converging paths) on a decision tree branch 2/25/2021 54

Decision Tree Example 2/25/2021 55

Analysis Using Decision Trees • Start at the right side of tree and move left a column at a time. For each column, if chance fork, go to (2). If decision fork, go to (3). • At each chance fork, calculate its expected value. Record this value in bold next to the fork. This value is also the expected value for branch leading into that fork. • At each decision fork, compare expected value and choose alternative of branch with best value. Record choice by putting slash marks through each rejected branch. • Comments: • • 2/25/2021 This is a backward induction procedure. For any decision tree, such a procedure always leads to an optimal solution. 56

Decision Tree Advantages • • Are simple to understand interpret Have value even with little hard data Possible scenarios can be added Worst, best and expected values can be determined for different scenarios • Use a white box model. If a given result is provided by a model • Can be combined with other decision techniques • e. g. Net Present Value calculations 2/25/2021 57