Quantitative Methods for Business - Utility and Game Theory

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Jan 9, 2024

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Quantitative Methods for Business Chapter 5 – Utility and Game Theory Chapter 5 Exercises 7. Suppose that the point spread for a particular sporting event is 10 points and that with this spread you are convinced you would have a 0.60 probability of winning a bet on your team. However, the local bookie will accept only a $1000 bet. Assuming that such bets are legal, would you bet on your team? (Disregard any commission charged by the bookie.) Remember that you must pay losses out of your own pocket. Your payoff table is as follows: a. What decision does the expected value approach recommend? Compute the expected value using the formula: EV (d) = pv 1 + (1-p) v 2 Here, p 1 and p 2 are probabilities, and v 1 and v 2 are the best and the worst payoffs. EV (bet) = 0.60 x 1000 + 0.40 x -1000 1 | P a g e

= 600 – 400 = $200 EV (do not bet) = 0.60 (0) + 0.40 (0) = $0 The expected value for the bet is $200, and to bet is the most suitable option, Therefore the recommended decision is to bet, d 1 b. What is your indifference probability for the $0 payoff? (Although this choice isn’t easy, be as realistic as possible. It is required for an analysis that reflects your attitude toward risk.) Compute the expected value using the formula: EV (d) = pv 1 + (1-p) v 2 Here, p 1 and p 2 are the probability values, and v 1 and v 2 are the best and the worst payoffs EV (bet) = 1000p – 1000(1-p) EV (do not bet) = p x 0 + (1-p) x 0 = 0 Set both the expected value equal to the probability. 1000p – 1000 (1-p) = p x 0 + (1-p) x 0 1000p – 1000 + 1000p = 0 2000p = 1000 P = 0.5 As a result, the indifference probabilities are equal to 0.5 for failure or success. 2 | P a g e

c. What decision would you make based on the expected utility approach? In this case are you a risk taker or a risk avoider? As obtained in part (b), the indifference probabilities are equal to 0.5. Assign utility 10 for the best payoffs and 0 for the worst payoffs. Compute the utility value for the percentage viewing audience by using the indifference probability. Bet or do not bet Indifference probability Utility value 1000 0 10 -1000 0 0 0 0.50 10 x 0.50 = 5 Compute the expected utility by adding the value of multiplications of the utility value, and the probability of a win or a loss. EU (win) = 0.6 x 10 + 0.4 x 0 = 6 + 0 = 6 EU (loss) = 0.40 x 9 + 0.6 x 9 = 3.6 + 5.4 = 9 The expected utility of a loss is greater than the expected utility of a win. Consequently, the decision (Do not bet” should be selected. Refer to part (a), the recommendation was also to bet on the basic of expected value. Consequently, the decision is a risk avoider. Therefore, “Do not bet” decision should be selected. Risk Avoider. 3 | P a g e

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d. Would other individuals assess the same utility values you do? Explain. No. If the utility value is dependent on the indifference probability, the utility values will be different for other people. The utility values will be different for different individuals as the utility value depends on the indifference probability. e. If your decision in part (c) was to place the bet, repeat the analysis assuming a minimum bet of $10,000. As obtained in part (b), the indifference probabilities are 0.5. Assign utility 10 for the best payoffs and 0 for the worst payoffs. Compute the utility value for the percentage viewing audience using the indifference probability. Bet or do not bet Indifference probability Utility value 1000 0 10 -1000 0 0 0 0.50 10 x 0.50 = 5 Compute the expected utility by adding the value of the multiplications of the utility value and the probability of a win or a loss. EU (win) = 0.6 x 10 + 4 x 0 = 6 + 0 = 6 EU (loss) = 0.40 x 9 + 0.6 x 9 = 3.6 + 5.4 = 9 4 | P a g e

The expected utility of a loss is greater than the expected utility of a win. Consequently, the decision for a ‘do not bet’ should be selected. 9. A new product has the following profit projections and associated probabilities: a. Use the expected value approach to decide whether to market the new product. Compute the expected value using the formula: EV (Profit) = ∑ i = 1 n p i v i Here, p 1 and p 2 are probability values and v 1 and v 2 are payoffs EV (Profit) = ∑ i = 1 n p i v i = (0.10 x 150000 + 0.25 x 100000 + 0.20 x 50000 + 0.15 x 0 + 0.20 x -50000 + 0.10 x 100000) = 15000 + 25000 + 10000 + 0 – 10000 – 10000 = 30000 The expected value for profit is positive. So the recommendation is to launch a new model. The expected value for profit is $30,000. So, it is recommended to begin with a new model. b. Because of the high dollar values involved, especially the possibility of a $100,000 loss, the marketing vice president has expressed some concern about the use of the expected value approach. As a consequence, if a utility analysis is performed, what is the appropriate lottery? 5 | P a g e

In case of profit, the best profit is $150,000 and the maximum loss is $100000. So, lottery will be p, with profit of $150000 and 1-p will be the probability for -$100000 Therefore, p is probability for profit, which is $150,000, and 1-p is the probability for loss of $100,000 c. Assume that the following indifference probabilities are assigned. Do the utilities reflect the behavior of a risk taker or a risk avoider? The utilities reflect the behavior of a risk avoider. Assign utility 10 for best playoffs and 0 for worst playoffs. Compute utility value for profit using the indifference probability. Profit Indifference Probability Utility Value 150,000 0 10 100,000 0.95 10 x 0.95 = 9.5 50,000 0.70 10 x 0.70 = 7 0 0.50 10 x 0.50 = 5 -50,000 0.25 10 x 0.25 = 2.5 -100,000 0 0 Compute the expected value in terms of utility by summing the product of utility value and probability for profit. EV (Profit) = 0.10 x 10 + 0.25 x 9.5 + 0.20 x 7 + 0.15 x 5 6 | P a g e

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= 1 + 2.375 + 1.4 + 0.75 = 5.525 EV (Loss) = 0.2 x 2.5 + 0.10 x 0 = 0.5 The expected value in terms of utility of loss is less than the expected value in terms of utility of profit. So, the new product should be launched. Refer part (a). Using the expected value approach, profit is $30,000. So, the recommendation is to launch the new product. The expected value in terms of utility also recommended to launch the new product. Therefore, the utility reflects risk avoider behavior. d. Use expected utility to make a recommended decision. Assign utility 10 for best payoffs and 0 for worst payoffs. Compute utility value for profit using indifference probability. Profit Indifference Probability Utility Value 150,000 0 10 100,000 0.95 10 x 0.95 = 9.5 50,000 0.70 10 x 0.70 = 7 0 0.50 10 x 0.50 = 5 -50,000 0.25 10 x 0.25 = 2.5 -100,000 0 0 Compute expected utility for market and do not market. EU (market) = (do not market) = EU ($0) 7 | P a g e

= 5.0 EU (market) = (0.10 x 10 + 0.25 x 9.5 + 0.20 x 7 + 0.15 x 5 + 0.20 x 2.5 + 0.10 x 0.0) = 1 + 2.375 + 1.4 + 0.75 + 0.5 + 0.0 = 6.025 The expected utility of loss is less than the expected utility of profit. Therefore, the new product should be launched. e. Should the decision maker feel comfortable with the final decision recommended by the analysis? Yes. Refer to subpart (a). The expected value approach profit is $30,000. So the recommendation is to launch new product and refer subpart (c). The expected utility also recommended launching new model. Therefore, the utility reflects risk avoider behavior, and decision-makers will be good to launch new model. 15. In a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B’s bill. If the bills match, Player B wins Player A’s bill. a. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. In the game theory, the table represents two persons’ zero-sum game. For individual A, assume r 1 represents for $1, and r 2 represents $5 invoice. For individual B, assume c 1 represents $1, and c 2 for $5 invoice. As a result, the required tabular representations and stated as following: C 1 C 2 r 1 -1 5 r 2 1 -5 8 | P a g e

In this case, each negative value at the intersection of (r 1 , c 1 ) and (r 2 , c 2 ) represents a loss of $1 and $5, respectively, and each positive value at the intersection of (r 1 , c 2 ) and (r 2 , c 1 ) represents a profit of $5 and $1, respectively. b. Is there a pure strategy? Why or why not? From part (a), the tabular representation from the zero sum G was obtained as: C 1 C 2 r 1 -1 5 r 2 1 -5 Determine the row minimum and the column maximum, and assign in the next row and the next column, respectively. C 1 C 2 Minimum r 1 -1 5 -1 r 2 1 -5 -5 Maximu m 1 5 Determine the maximum among the row minimums and the minimum among the column maximums and represent it with (). C 1 C 2 Maximum of minimums r 1 -1 5 (-1) r 2 1 -5 -5 Minimum among maximums (1) 5 9 | P a g e

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Observe the maximum of the row minimum is not equal to the minimum of the column maximum. As a result, the given G cannot be won by pure strategy. Therefore, No, there is no pure strategy. The maximum of the row minimum is not equal to the minimum of the column maximum. c. Determine the optimal strategies and the value of this game. Does the game favor one player over the other? From part (a), the tabular representation for the zero-sum G was obtained as follows: C 1 C 2 r 1 -1 5 r 2 1 -5 For Individual A, assume p is the probability for strategy r 1 and (1-p) is the strategy for r 2 . Determine the expected values of c 1 and c 2 using the above table. E (c 1 ) = -1p + 1(1-p) E (c 2 ) = 5p – 5(1-p) Individual A cannot change its strategy for decreasing the expected value. Equate both the expected value equations obtained above to calculate the corresponding value of p. -1p + 1(1-p) = 5p – 5(1-p) -p + 1-p = 5p – 5 + 5p -2p – 10p = -1 -5 -12p = -6 P = 0.50 Substitute the probability for p = 0.50 to calculate the probability for r 2 using the formula: r 2 = 1 – p 10 | P a g e

r 2 = 1 - 0.50 = 0.50 For the individual A, assume q is the probability for strategy c 1 and (1 – q) is the strategy for c 2 . Determine the expected values of r 1 and r 2 using the table given above. E (r 1 ) = -1q + 5(1 – q) E (r 2 ) = 1q – 5(1 – q) Individual B cannot change its strategy for decreasing the expected value. Equate both the expected values obtained above to calculate the corresponding value of q. -1q + 5(1 – q) = 1q – 5(1 – q) -q + 5 – 5q = q – 5 + 5q -6q - 6q = -5 – 5 -12q = -10 q = 5/6 Substitute the probability for q = 5/6 to calculate the probability for c 2 using the formula: c 2 = 1 – q c 2 = 1 - 5/6 = 1/6 Substitute p = 0.50 and 1 - p = 0.50 to calculate the value of G for the Individual A using the formula: E (A) = -1(p) + 1(1 – p) E (A) = -1(0.5) + 1(0.5) = -0.5 + 0.5 = 0 11 | P a g e

The expected value for Individual A is 0. AAs a result, in this case G does not prefer one individual over the other. Therefore P (r 1 ) = 0.50 P (r 2 ) = 0.50 P (c 1 ) = 5/6 P (c 2 ) = 1/6 Value (A) = 0 No. The game does not favor one player over the other, d. Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy. From part (c), the obtained expectation for individual A can be stated as: E (r 1 ) = -1q + 5(1 - q) E (r 2 ) = 1q - 5(1 - q) If Individual B formulates the given strategy, q = 0.50. Substitute q = 0.50 to calculate the expected value, E (r 1 ) using the above formula. E (r 1 ) = -1 (0.50) + 5 (1 – 0.50) = -0.50 + 2.50 = 2 Substitute 1 – q = 0.50 to calculate the value of E (r 2 ) using the formula: 12 | P a g e

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E (r 2 ) = 1q – 5 (1 – q) E (r 2 ) = 1 (0.50) – 5 (1- 0.50) = 0.50 – 2.50 = -2 The expected values of r 1 is greater than that of r 2 . As a result, if individual B continues to use this strategy Individual A should use the strategy r 1 . In this case, Individual B may also reciprocate in the same manner. The most appropriate approach for individual A is to follow the optimal strategy obtained in part (c). Therefore, in this case, Individual A should use the strategy r 1 . It is because the optimal strategy obtained in part (c), gives a uniform appropriate approach for the win considering all the factors for the individual A. 13 | P a g e

Quantitative Methods for Business - Utility and Game Theory

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