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Name: Trương Hoài Anh Facebook: Quasar Hoaianh Website: qm-for-business-class-of-mr-Huy8.webnode.vn Class Pass: qmiu

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**Chapter 1: Introduction to Quantitative Analysis**

Describe the quantitative analysis approach Understand the application of quantitative analysis in a real situation Describe the use of modeling in quantitative analysis Use computers and spreadsheet models to perform quantitative analysis Discuss possible problems in using quantitative analysis Perform a break-even analysis

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Chapter 2: Probability Basic Definitions: Events, Sample Space, and Probabilities Basic Rules for Probability Conditional Probability Independence of Events Combinatorial Concepts The Law of Total Probability and Bayes’ Theorem Random variables

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Module 1: Game theory Two-person means there are two competing players in the game. Zero-sum means the gain (or loss) for one player is equal to the corresponding loss (or gain) for the other player. The gain and loss balance out so that there is a zero-sum for the game. What one player wins, the other player loses.

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**Module 1: Game theory Payoff table 4 steps**

Row minimum Maximin Column maximum Minimax Pure vs Mixed vs Dominated strategies Expected value (EV)/saddle point

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**Example 1 Player II Player I b1 b2 a1 19 20 a2 5 -4 Row minimum 19 -4**

Best Strategy For Player II Player II Player I b1 b2 a1 19 20 a2 5 -4 Row minimum 19 -4 Column maximum 19 20 Maximin Payoff Best Strategy For Player I Minimax Payoff

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Example 2 Player I holds a black Ace and a red 8. Player II holds a red 2 and a black 7. The players simultaneously choose a card to play. If the chosen cards are of the same color, Player I wins. Player II wins if the cards are of different colors. The amount won is a number of dollars equal to the number on the winner’s card (Ace counts as 1.) Establish the payoff table Find the value of the game and the optimal mixed strategies of the players

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**Player II Player I Red 2 Black 7 Row minimum b1 b2 Black Ace a1 Red 8**

Column maximum -2 -2 1 -7 8 -7 8 1

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**Expected value Player I EV for q: -2q+8(1-q)=q-7(1-q) => q=15/18**

Red 2 Black 7 EV Player I b1 (p) b2 (1-p) Black Ace a1 (q) Red 8 a2 (1-q) -2 1 8 -7 -2p + (1-p) 8p -7(1-p) -2q + 8(1-q) q -7(1-q) EV for q: -2q+8(1-q)=q-7(1-q) => q=15/18 EV for p: -2p+(1-p)=8p-7(1-p) => p=4/9 EV: -2(4/9)+(1-4/9)= -1/3

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**Dominated strategy 2 players, zero-sum**

At least one player has more than 2 options Solution Payoff 4 steps Pure strategy or mixed strategy Pure => EV Mixed => elimination => EV

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**Example 3 Row minimum Y1 Y2 X1 13 X2 6 8 X3 12 14 6 12 Column maximum**

X2 6 8 X3 12 14 6 12 Column maximum 13 14

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**Example 3 Row minimum Y1 Y2 X1 13 X2 6 8 X3 12 14 12 Column maximum 13**

X2 6 8 X3 12 14 12 Column maximum 13 14

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M4.15 ST Co. and FF Co. are both vying for more share of the market. If ST does no advertising, it will not lose any share of market if FF does nothing. It will lose 2% of market if FF invests $10,000, and it will lose 5% if FF invests $20,000 in advertising. On the other hand, if ST invests $15,000 it will gain 3% if FF does nothing; gain 1% if FF invests $10,000; and lose 1% if FF invests $20,000 in advertising. Develop a payoff table Find the value of the game

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**M4.15 (sol.) ST Co. Do nothing $10,000 $20,000 Row minimum b1 b2 b3 a1**

-2 -5 $15,000 a2 3 1 -1 Column maximum -1

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