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QUANTITATIVE BUSINESS ANALYSIS Simple Decision Tree Analysis and Utility Theory Oliver Yu, Ph

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QUANTITATIVE BUSINESS ANALYSIS Simple Decision Tree Analysis and Utility Theory Oliver Yu, Ph.D. Oliver Yu © 2021 DT&UT-1 OBJECTIVE AND BACKGROUND Objective In addition to AHP, we will present another approach for quantifying values of a decision-maker by using mathematical axioms and decision-tree analysis. Background This approach has a long history dating back to Daniel Bernoulli in the 18th century, but it was first formalized by the great mathematician, John von Neumann, in the 1950s. In this approach, utility is defined as a relative quantitative measure of the decision-maker’s values (degree of happiness, preference, desirability, etc.). It is relative because, like temperature with the Fahrenheit and Celsius measures and altitude with the English and metric measures, the values of a decision-maker can be represented by different but internally consistent measures of utilities. Furthermore, the decision-tree analysis is both a way to estimate utilities and a way to apply utilities for making choices. Oliver Yu © 2021 DT&UT-2 DECISION TREE WITH UNCERTAINTY: FRAMEWORK A Decision Tree A tree representation of the decision process over time as • a series of decision time points (decision or choice nodes conventionally in square shape) at which the decision-maker has full control, with available choices as emanating branches • interspersed by a series of uncertainty time points (probability or chance nodes conventionally in round shape) at which the decision maker has no control, with probable outcomes as emanating branches Basic Elements of the Decision-Making Process Value: Ultimate outcome of each series of branches Choices: branches at each decision node Relationships: The tree structure Oliver Yu © 2021 DT&UT-3 DECISION TREE WITH UNCERTAINTY: A SIMPLE GRAPHICAL EXAMPLE Probability p Outcome Value A, Best or Highest utility Probability Node Decision Node Choice 1, Uncertainty Choice, 2 Certainty Oliver Yu © 2021 Probability 1-p Outcome Value C, Worst or Lowest utility Outcome Value B, In-Between utility DT&UT-4 DECISION TREE WITH UNCERTAINTY: ANALYSIS For a decision tree with uncertainty, we compare the expected value of each choice and select the one with the greatest expected value. Specifically, in the above example of a simple decision tree, Expected value of Choice 1: p (Outcome Value A) + (1-p) (Outcome Value C) Expected value of Choice 2: Outcome Value B The best choice is the one with the greatest expected value or utility. Note: This simple decision tree can also be used as a mechanism for estimating the decision-maker’s relative degree of preference for a choice that lies between the best and worst choices Oliver Yu © 2021 DT&UT-5 UTILITY: DEFINITION Utility is a measure of decision-maker’s relative degree of happiness, preference, desirability, importance, benefit, or other value of the outcome of an Choice. In resource allocation, the degree of preference, or utility, U(x), of the outcome from allocating resource x to an investment is generally a nonlinear, often concave, function. A concave utility function generally represents diminishing incremental increases in the relative degree of happiness, preference, desirability, importance, benefit, or value of the allocation. If time factor is considered, then utility generally decreases with time. This phenomenon is often called the time preference of utility. Such a decrease in utility or preference is generally caused by the perceived risk of future returns, which usually increases with time. This decrease is often represented in a simplified manner as the discount rate. Oliver Yu © 2021 DT&UT-6 UTILITY: AXIOMS 1. Rankability and Completeness Let U be the utility of an Choice, then for a choice between Choices A and B, a decision-maker either prefers A to B, i.e., u(A)>u(B); or B to A, i.e., u(A)u(B)>u(C) implies u(A)>u(C). 3. Computability If a lottery L has probability p for A and 1-p for B, then the utility of L will be u(L) = (p) u(A) + (1-p) u(B). 4. Substitutability If u(A)=u(B), then A and B are substitutable for the decision-maker. 5. Continuity and Certainty Equivalent If a decision-maker prefers A to B and B to C, then there exists a lottery L with probability p for A and 1-p for C such that the u(L) = u(B), and B is the certainty equivalent of the lottery L. Oliver Yu © 2021 DT&UT-7 UTILITY IS A RELATIVE MEASURE OF PREFERENCE Similar to temperature, utility is a relative measure, in the sense that we can develop a complete utility measurement system by defining a base value and a measuring unit. For example, there are two common measuring systems for temperature, Celsius and Fahrenheit. I can also develop a Yu system for measuring temperature, in reference to the two common systems as follows: Temperature (degree) Celsius Fahrenheit Yu Water Freezing Water Boiling Normal Body 0 100 37 32 212 98 -100 1000 307 Conversion formulas: F = 32+C(212-32)/100; Y = -100+C(1000-100)/100. Oliver Yu © 2021 DT&UT-8 A SIMPLE DECISION TREE FOR UTILITY ESTIMATION Probability p* U(Best Outcome) = U(A) Lottery L U(Worst Outcome) = U(C) Probability 1-p* U(Outcome in Between) = U(B) Certainty Equivalent Oliver Yu © 2021 U(A) > U(B) = p*U(A) + (1-p*)U(C) = U(L) > U(C), where is p* is the indifference probability between B and the Lottery. DT&UT-9 UTILITY: ESTIMATION PROCESS As a relative measure of value, a decision-maker’s utility can be quantified through the simple decision tree analysis of two choices. Using money as an example. 1. Assuming that utility as the measure for the value of money is a non-decreasing function of the quantity of money for a decisionmaker, set an arbitrarily high utility, say 100, for a large amount of money, say $1 million; and an arbitrarily low utility, say 0, for a small amount of money, say $0. 2. Develop a simple decision tree with two choices: • A lottery, L, with a probability p of getting the best outcome (A) of winning $1 million and probability 1-p of getting the worst outcome (C) of $0. • A certainty choice of getting the in-between outcome (B) of x amount of money lying between $0 and $1 million with certainty. Then by Utility Axioms 3 and 5, U(L) = pU(A) + (1-p)U(C). Oliver Yu © 2021 DT&UT-10 UTILITY ESTIMATION - Concluded 3. For this artificial simple decision tree, if p = 1, then clearly the decision-maker will choose the lottery L; if p = 0, then clearly the decision-maker will choose the certainty choice with outcome B. By Utility Axiom 5, as we vary p continuously from 1 to 0, there exists a probability value p* at which the decision-maker will choose L if the probability p of getting the best outcome A is greater than p*, and choose the certainty Choice if the probability p of getting the best outcome A in the gamble is less than p*, and will be indifferent between L and the certainty Choice if p = p*. 4. By Utility Axiom 4, if the probability of getting the best outcome A in the lottery is p*, then the utility of B, U(B) = U(L) = p*U(A) + (1-p*)U(C), and B is the certainty equivalent of L with p = p*. Oliver Yu © 2021 DT&UT-11 UTILITY: RISK ATTITUDES This simple decision tree also can be used to determine the risk attitudes of the decision-maker. Again using money as an example: If U(x) is proportional to x, then the decision-maker is risk neutral as the value of money is proportional to the amount of money. If U(x) is more than proportional to x, then the decision-maker is viewed as risk avoiding, as the utility of having x for sure is preferred to the utility of a lottery with higher expected payoff. On the other hand, if U(x) is less than proportional to x, then the decision-maker is viewed as risk preferring, as the utility of having x for sure is less than the utility of a lottery with lower expected payoff. Risk preference is totally subjective, depending on the decisionmaker’s personality, the size of decision-maker’s assets, and the amount at stake. Oliver Yu © 2021 DT&UT-12 UTILITY: DIFFERENT RISK ATTITUDES U Risk Avoiding Risk Neutral Risk Preferring $ Oliver Yu © 2021 DT&UT-13 UTILITY: A SIMPLE DETERMINATION OF RISK ATTITUDE FOR MONEY A simple way to determine whether a decision-maker’s risk attitude for money is to compare the utility function with a risk neutral person’s utility function that has the same utility for the maximum amount of money, M, as well as the same utility for the minimum amount of money, m (for example, both persons have utility 100 for M=$1 million and 0 for m=$0). Let x be an amount of money between M and m, then the risk-neutral person’s utility for x, Un(x), is (x-m)/(M-m)[U(M)-U(m)] or (x/1 million)(100) for the example. Now, let U(x) be the utility of the decision-maker for x, then If U(x) > Un(x), the decision-maker is risk avoiding; If U(x) = Un(x), the decision-maker is risk neutral; If U(x) < Un(x), the decision-maker is risk preferring. Oliver Yu © 2021 DT&UT-14 UTILITY: RISK PREMIUM For a risk-averse decision-maker, if a lottery, L, with expected monetary value E(L), then the monetary value of the certainty equivalent, x, for the lottery is necessarily lower than E(L), because of the uncertainty in the lottery. E(L) - x is then the risk premium of the lottery. In other word, because of the risk involved in the lottery, the decision-maker has discounted the monetary value of the lottery to the amount represented by the risk premium. In dealing with a lottery of potential loss, the reverse will be true; i.e., a risk-averse decision-maker would have a certainty equivalent x higher than the expected loss E(L) from the lottery. Using insurance as an example, the lottery for the insured has a expected loss E(L) less than the insurance premium x charged by the insurer. In this case, the risk premium is x-E(L), i.e., the portion of the premium that the insured pays in excess of the expected payout by the insurer in order to avoid the risk of a large loss without insurance coverage. Oliver Yu © 2021 DT&UT-15 UTILITY: STYLIZED UTILITY FUNCTION For analytical simplicity, utility functions for money are often stylized. For example, the utility function for money of a riskavoiding person may be stylized to be proportional to the root function of the amount of money, while that of a risk-preferring person may be proportional to the power function of the amount of money. Example: The utility function for money U(x) of a risk-avoiding person is stylized to be proportional to the square root of the amount of money x, i.e., U(x) = c x0.5 . If the person sets U($1 million) = 1,00 and U($0) = 0, then the proportionality constant c can be determined to be 10, because U(1,000,000) = c (1,000,000)0.5 = c (1,000) = 100. Now, U(x) can be estimated for any x. Oliver Yu © 2021 DT&UT-16 PRINCIPLE OF INSURANCE PRICING Insurers generally set premiums based on the following principle: The maximum insurance premium an insured person is willing to pay is at an amount that the person is indifferent between having and not having the insurance; i.e., when the two choices have same utility. Application: If a person has net monetary asset A, including an asset under risk B, and a utility function for money U(x). During a year, assume that B has a probability p of being totally destroyed in an accident, and probability 1-p of being not harmed at all. Then the maximum insurance premium IP the person would be willing to pay to fully insure B can be obtained by equating the utility of the asset after subtracting IP, i.e., U(A-IP), and the expected utility of the assets without insurance, i.e., p U(A-B) + (1-p) U(A). Oliver Yu © 2021 DT&UT-17 INSURANCE PRICING EXAMPLE A person has a net asset A=$40,000, including a car valued at B=$7,600. Through demographic profiling, the utility function for money of this person U(x) for x amount of money is stylized to be proportional to the square root of money; i.e., U(x) = c x0.5. By setting U($10000) = 100 and U(0) = 0, we will have c = 1, and can then estimate U(x) for any amount of money. Accident statistics for the person’s demographic segment has shown that the car has a probability p= 0.05 of being totaled during a year. The insurer can estimate the maximum insurance premium, IP, the person would be willing to pay to have the car fully insured by equating the utilities of the person for having and not having the insurance as follows: Utility of full insurance = (40000-IP) = (40000-IP)0.5 = Utility of no insurance = 0.05 U(car is totaled) + 0.95 U(car not damaged) = 0.05U(40000-7600)+0.95U(40000)=0.05(1800)+0.95(2000) = 199 or 40000-IP = (199)2 = 39601 or IP = $399. Risk premium = IP – Insurance expected payout = 399 – [0.05 (7600) + 0.95 (0)] = $19, which is the insurer’s gross profit. Oliver Yu © 2021 DT&UT-18 RECONCILIATION WITH ANALYTIC HIERARCHY PROCESS For computers A, B, and C, assume that a decision-maker prefers A the most and C the least. Using AHP, we can estimate the values or degrees of preference, VA, VB, and VC of the three computers respectively. Clearly VA>VC>VC. By setting U(A)=VA and U(C)=VC, we can then apply the utility decision tree to estimate, U(B), the utility of B. Since utility also measures the relative preference, U(B) should equal VB. What if U(B) does not equal VB, how can we reconcile the difference? In this case, because of the hierarchical structure and comparison precision, AHP generally produces more accurate and consistent reflections of the decision-maker’s preferences. Furthermore, since, VB is between VA and VC, U(B) can always be made equal to VB by setting p for getting A in the gamble to be (VB-VC)/(VA-VC). However, if U(B) obtained directly from the Utility Theory approach is very different from VB, then the decision-maker should review the AHP process to gain a deeper insight about the difference. Oliver Yu © 2021 DT&UT-19 LINEAR PROGRAMMING: OUTLINE ? Key characteristics and applications ? Decision framework ? LP formulation ? Graphical solution and computer solvers ? Sensitivity analysis and shadow price ? LP of a two-person zero-sum game ? Free downloadable textbook at King Library: Bruce R. Feiring, Linear Programming: An Introduction,1986 ? Free downloadable LP solver: QM for Window free download ? Another free download: https://linear-program-solver.soft112.com ? Third free source: Solver on Excel Oliver Yu © 2021 LP-1 LINEAR PROGRAMMING: Key Characteristics & Major Applications Key Characteristics ? Proportionality: Linearity and continuity ? Non-negativity Some Major Applications Application Activity Constraints Production mix Diet problem Portfolio selection 2-person 0-sum game Transportation problem Assignment problem Project crashing Blending problem Labor planning Truck loading Media selection Market research Production scheduling Production level Ingredient amount Investment level Strategy probability Transporting amount Assignment Time reduced Ingredient amount Time/level of assignment Loading quantity Media exposure Sample size Time/level of production Resources, demand requirements Nutrition/caloric requirements Budget, risk requirements Minimax requirements Availability, demand requirements Labor, job requirements Time precedence Availability, mix requirements Labor, schedule requirements Capacity Budget, exposure requirements Budget, minimum sample sizes Schedule requirements Oliver Yu © 2021 LP-2 LINEAR FROGRAMMING FORMULATION The Production Mix Example A furniture company makes only Tables (T) and Chairs (C). Each table produces $60 net profit and each chair produces $40 net profit. Each table takes 4 hours of carpenter’s time, 2 hours of painter’s time, and 0 hour of metal worker’s time; each chair takes 3 hours of carpenter’s time, 1 hour of painter’s time, and 2 hours of metal worker’s time. Each week, 240 hours of carpenter’s time, 100 hours of painter’s time, and 120 hours of metal worker’s time are available. Find the production mix of tables and chairs that will maximize the weekly net profit P. Maximize V = Profit = 60T + 40C 4T + 3C < 240 (Carpenter’s time) 2T + 1C < 100 (Painter’s time) 0T + 2C < 120 (Metal-worker’s time) T, C > 0 Oliver Yu © 2021 LP-3 LINEAR FROGRAMMING FORMULATION The Diet Problem Example A company produces pound-size package of dog food that is made only from Beef (B) and Grain (G). Each package must have 10 units of vitamin 1 and 9 units of vitamin 2. A pound of beef costs 90 cents and contains 12 units of vitamin 1 and 10 units of vitamin 2. A pound of grain costs 60 cents and contains 9 units of vitamin 1 and 6 units of vitamin 2. Find the combination of beef and grain for the package that will minimize the cost C while meeting the vitamin requirements. Minimize V = Cost = 90B + 60G 12B + 9G > 10 (Vitamin 1) 10B + 6G > 9 (Vitamin 2) B+ Oliver Yu © 2021 G= 1 B, G > 0 LP-4 LINEAR FROGRAMMING FORMULATION The Portfolio Selection Example A person wants to invest up to $10,000 into stocks: a high tech company (T) with an expected annual return of 12% and a risk index of 8; and a regulated power company (P) with an expected annual return of 6% and a risk index of 2. To limit risk, the combined portfolio risk must be no more than 6 and the proportion of investment in T must be less than 70%. Find the portfolio that will maximize the annual return R while meeting the risk limitations. Maximize V = Return = 12T + 6P T + P < 10 8T + 2P < 6 (T+P) 2T -4P < 0 T < 0.7 (T+P) Oliver Yu © 2021 0.3T-0.7P < 0 T, P > 0 Invest all $10,000 T+ P = 10 8T+2P < 60 T 0 LP-5 Combined portfolio risk index It’s the weighted average based on the proportions invested in the stocks. For example, if you invest 30% in T with risk index 8 and 70% in P with risk index 2, then the portfolio risk index is 0.3 (8) + 0.7(2) = 3.8. In general, if you invest T/(T+P) in T and P/(T+P) in P, then the portfolio risk is 8T/(T+P) + 2P/(T+P). If the portfolio risk index is required to be less than 6, then we can transform this constraint into a linear constraint as follows: (8T+2P)/(T+P) 0 or equivalently such that Maximize EP EP-Si pi Pij < 0, for all j Si pi =1 pi > 0 Similarly, for a mixed strategy T of Y, with strategy j having probability qj, let EP(T) be the maximum expected loss in face of all possible strategies of X. Then T* that minimizes the value of EP(T) is the Optimal Mixed Strategy for Y, and the Linear Program is: Minimize EP such that Sj qj Pij < EP, for all i Sj qj = 1 qj > 0 or equivalently such that MinimizeEP EP-Sj qj Pij > 0, for all i Sj qj =1 qj > 0 Note that pi, qj, and EP are the activities. Oliver Yu © 2021 LP-12 GAME THEORY: OUTLINE ? Quantitative Methods for Simple Group Decisions: TwoPerson Zero-Sum Game – A simple examples and key characteristics – Decision framework – A conservative optimization approach: the Minimax approach – Dominated strategy and irreducible matrix – Pure-strategy solution and a saddle point – Games without a saddle point with square irreducible matrix – Games without a saddle point with rectangle irreducible matrix Oliver Yu © 2021 GT-1 TWO-PERSON ZERO-SUM GAMES: A PLAYER’S GAIN IS THE OPPONENT’S LOSS The payoff matrix for Row-player X and Column-player Y: Rowplayer X1 X2 X3 . Xm Y1 P11 -P11 P21 -P21 P31 -P31 Column-player (Opponent) Y Y2 Y3 …. P12 -P12 P13 -P13 P22 -P22 P23 -P23 P32 -P32 P33 -P33 Yn P1n -P1n P2n -P2n P3n -P3n Pm1 -Pm1 Pm2 -Pm2 Pmn -Pmn Pm3 -Pm3 Therefore, one matrix is sufficient to represent both players. Example X1 X2 Oliver Yu © 2021 Y1 3 -3 1 -1 Y2 5 -5 -2 +2 GT-2 TWO-PERSON ZERO-SUM GAME: A Simple Example A simple 2-person 0-sum game: the game of rock, scissors, and paper. Each player has three possible pure strategies: Rock, Scissors, Paper. The following is the payoff matrix from the Row-player point of view: Rock Row-Player Scissors Paper Column-Player (Opponent) Rock Scissor Paper 0 +1 -1 -1 +1 0 -1 +1 0 For the Row-player, there is no pure strategy that always beats the opponent. However, it is intuitive that a mixed strategy is to play randomly with 1/3 chance for each strategy, which will achieve a stable result of a tied game, i.e., a value of the game of 0, no matter what the opponent does. Oliver Yu © 2021 GT-3 DECISION FRAMEWORK FOR A TWOPERSON ZERO-SUM GAME For each player: ? Value: Expected payoff, where the player’s gain is the opponent’s loss and the sum of the gains or losses is zero ? Alternatives: Various pure or mixed strategies ? Relationships: Payoff matrices ? Optimization: A strategy that maximizes the minimum expected gain, or equivalently minimizes the maximum expected loss to the opponent This Minimax approach is a conservative optimization strategy as each player looks for the worst among the best or best among the worst. Oliver Yu © 2021 GT-4 DOMINATED STRATEGIES AND IRREDUCIBLE MATRIX For a player, a dominated strategy is a strategy that is always (not most of the time or on the average) worse or no better than one (not all) other strategy of the player. A dominated strategy should be eliminated, because the player has no advantage in playing this strategy. To solve the game, we first eliminate in turn all dominated strategies of each player; the order of elimination is not important. When all dominated strategies have been eliminated, the remaining is an irreducible matrix. Example X1 X2 Y1 3 1 Dominated strategy of X eliminated: X1 3 -3 Irreducible matrix Oliver Yu © 2021 X1 Y2 5 -2 dominated strategy of X 5 -5 dominated strategy of Y 3 -3 GT-5 PURE STRATEGY SOLUTION If the irreducible matrix is a single value, i.e., a single strategy is left for each player; in this case, these single strategies are respectively the optimal strategies for the players, and they are called Pure Strategy Solutions. A Saddle Point then exists at the intersection of these Pure Strategies, and the best expected payoff for each player is the respective Value of the Game. Example Y1 3 1 Y2 5 -2 Player Best strategy X X1 Y Y1 Value of the Game +3 -3 X1 X2 Solution: Oliver Yu © 2021 Saddle-point GT-6 MIXED STRATEGY SOLUTION FOR A SQUARE IRREDUCIBLE MATRIX A Mixed Strategy is one that strategy i is played with probability pi with Spi = 1; a Pure Strategy is a special case where a strategy is played with probability 1. If the irreducible matrix is not a single value but a square matrix, then an optimal mixed strategy exists for each player is to equalize the expected payoffs for all opponent’s strategies. For 2x2 irreducible matrix, it can be shown that p1 and p2 = 1-p1 for X and q1 and q2 = 1-q1 for Y obtained below are optimal mixed strategy probabilities: p1 P11 + (1-p1) P21 = p1 P12 + (1-p1) P22 or p1 = (P22 - P21)/(P11 + P22 - P12 - P21) Example X1 X2 Y1 4 1 Y2 2 10 q1 P11 + (1-q1) P12 = q1 P21 + (1-q1) P22 or q1 = (P22 - P12)/(P11 + P22 - P12 -P21) p1 = (10-1)/(4+10-2-1) = 9/11 q1 = (10-2)/(4+10-2-1) = 8/11 Value of the game for X = 4x8/11 + 2x3/11 = 1x8/11 + 10x3/11 = 4x9/11 + 1x2/11 = 2x9/11 + 10x2/11 = 38/11 Oliver Yu © 2021 GT-7 MORE EXAMPLES OF ZERO-SUM GAME Game with Saddle Point X1 Player X X2 X3 Y1 -3 -2 -3 Player Y Y2 Y3 Game without Saddle Point Player Y Y1 Y2 Y3 2 1 -3 X1 Player X X2 X3 Value of the game for X is -2 5 6 -7 -2 0 -5 4 -3 1 5 6 -6 Reduce to a 2x2 game of X1, X2 & Y1, Y2 Value of the game for X is -2/3 Caution: If a pure strategy game is solved as a mixed strategy game, then one of the probabilities of one of the player’s strategies will be either greater than 1 or negative, which is not acceptable. Oliver Yu © 2021 GT-8 SOLUTION FOR RECTANGULAR IRREDUCIBLE PAYOFF MATRIX Solution procedure: 1. Decompose the irreducible rectangular payoff matrix for the player with more pure strategies to all possible largest square payoff sub-matrices, 2. Solve each square payoff sub-matrix. 3. The player with more strategies will choose the square payoff sub-matrix with the largest expected payoff by dismissing strategies not in this matrix, and the player with less strategies is forced to play this matrix even though not desirable. Example: Y1 Y2 Y3 X1 1 3 5 X2 4 2 1 Y has the choice Value of the Game for Y: Solution: Oliver Yu © 2021 Sub-matrices Y1 Y2 Y1 Y3 Y2 Y3 1 3 1 5 3 5 4 2 4 1 2 1 Mixed strategy Mixed strategy Pure strategy Y2 (1/4)(-1)+(3/4)(-3) (4/7)(-1)+(3/7)(-5) =-5/2 = -2.5 best =-19/7 = -2.67 -3 Player X Y Best strategy ½ X1, ½ X2 ¼ Y1, ¾ Y2 Value of the Game +2.5 -2.5 GT-9 NON-ZERO-SUM GAMES Generic Payoff Matrix (P for Player A and Q for Player B) Player B Strategy 1 (Cooperate) Strategy 2 (Defect) Player A Strategy 1 (Cooperate) P11, Q11 P12, Q12 Strategy 2 (Defect) P21, Q21 P22, Q22 Non-Zero Sum: Symmetric Games: Pij + Qij not equal 0. P11 = Q11 = a; P12 = Q21 = b; P21 = Q12 = c; P22 = Q22 = d. Prisoners' Dilemma: c > a > d > b (e.g., 3,3 0,5 5,0 1,1) (1,1) is a stable strategy, but (3,3) is a better strategy; that's the dilemma. Win-Win Cooperation: a > c > b > d (e.g., 3,3 1,2 2,1 0,0) Deadlock: c > d > a > b (e.g., 1,1 0,3 3,0 2,2) In negotiation, we want to persuade the opponent to change the payoff matrix from that of the Deadlock or Prisoner’s Dilemma to that of a Win-Win Cooperation. Oliver Yu © 2021 GT-10 e. Project and term paper: Based on methods and tools you learned in this course, with the explicit exception of Multifactor Evaluation Process, conduct a project for a realistic application of quantitative analysis to your business or personal decisions. The term paper for this project should include: (1) A one-page description of your problem and its importance (10%). (2) A one-page discussion of the solution procedure and the rationale (10%) borlupos (3) One or more pages of presentation of your quantitative analysis and results (40%) (4) One or more pages of discussion of your insights, critique, and possible expansion for the application (40%). The term paper should be submitted as a MSWord file with 12-point font and double-spaced on an 8.5x11 format on the day of the final exam. It will be subjectively evaluated by the instructor on the basis of Originality and relevance of the problem - Clarity and accuracy of the problem formulation Correctness of the results Depth of insights from the experience and creativity in the expansion of the application. Please note if you submit a decision problem almost identical to a homework problem, you will get at most 50 points. To get higher points, your problem must be more realistic and sophisticated than a homework problem.