MGT 2255 Test 2 Formula Sheet PDF

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ch.3 “Decision Analysis” Formula Sheet 1. Decision Making without Probabilities

Optimistic criterion Under an optimistic criterion, the “best” payoff for each decision alternative is determined, with the “best” of those selected as the preferred decision alternative under the optimistic approach. Pessimistic (Conservative) criterion Under a conservative approach, the “worst” payoff for each decision alternative is determined, with the “best” of those selected as the preferred decision alternative. Minimax Regret criterion Under the minimax regret approach, we choose the decision alternative that minimizes the maximum possible regret, also referred to as opportunity loss. The Laplace (equally likely) solution is computed averaging the payoffs for each alternative and choosing the best.

2. Decision Making with Probabilities

Expected Monetary value (EMV) approach: Choose the decision alternative having the best expected value, we first calculate the EMV for each decision alternative:

EMV (d i )=∑ P( S j )(Pij )=∑ P (S j )(payoff from choosing d i should S j occur ) j

j

Specifically, the expected return for each decision is calculated by summing the products of the payoff under each state of nature and the probability of the respective state of nature occurring. Expected opportunity loss (EOL) approach The expected opportunity loss is the expected value of the regret for each decision alternative. Thus we calculate the regrets for each alternative and calculate the expected opportunity loss or regret based upon the weight (probability) of each state of nature occurring:

EOL(d i )=∑ P ( S j )( R ij )=∑ P(S j )(Regret from choosing d i should S j occur ) j

j

Expected Value of Perfect Information (EVPI) EVPI Calculation Step 1: Determine the optimal value corresponding to each state of nature. (referred to as EV with PI). Thus we take the best possible payoff for each state of nature and multiply by the prior probability of that state of nature and sum them together.

∑ P( S j )⋅¿

¿ EV with PI = (best possible payoff under S j ) Step 2: Compute the expected monetary value of the optimal decision without additional information. Step 3: Subtract the EV of the optimal decision from the amount determined in step (1). j

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EVPI (EV of PI) = │EV with PI – EV of our optimal decision (EMV without PI)│ We take the absolute value of this difference because in a minimization problem, the EV of our optimal decision will be greater than the ‘EV with PI’, while EVPI is always expressed as a positive value. Expected Value of Sample Information (EVSI) The Expected Value of Sample Information (EVSI) is the additional expected profit possible through knowledge of the sample or survey information. • EVSI (EV of SI) = │(EV with SI +cost of survey) – EMV without any additional information│ EVSI calculation Step 1: Determine the optimal decision and its expected return (you need to subtract the cost of survey from the original payoff) for the possible outcomes of the sample using the posterior probabilities for the states of nature, resulting in EV with SI. Step 2: Compute the EMV of the optimal decision (obtained without using the sample information). Step 3: Find (EV with SI +cost of survey)=amount of step 1+cost of survey. Subtract the EMV of the optimal decision from the amount determined in step (3). EVSI (EV of SI) = │(EV with SI +cost of survey) – EMV without any additional information│= =Amount of Step 3-Amount of Step 2: Efficiency of sample information =

EVSI EVPI , converted to a %

Bayes’ Theorem and Posterior Probabilities In some situations, the prior (initial) probability of a state of nature S j is modified in light of “new information” by relying on Bayes Theorem. That is, knowledge of sample/survey information, for example, can be used to revise the probability estimates for the states of nature. Bayes’ Theorem: Given mutually exclusive events S 1, S2, S3, …, Sn (our States of Nature) whose probabilities sum to 1, and given an event E (for us, this typically represents our sample information outcomes, for example a favorable or unfavorable analysis) with a non-zero probability:

P(S j|E )=

P( E∩S j ) P( E)

P( E|S j ) P( S j )

=

∑ P( E|S j ) P(Sj)

. Computing Posterior Probabilities using tabular approach 2 3 4=2*3 5=(4)/(sum(marginal probability)) Prior Conditional Joint Posterior: P(S1) P (E | S1) P(S1)* P (E | S1) P(S1|E) P(S2) P (E | S2) P(S2)* P (E | S 2) P(S2|E) Sum of these joint probabilities (marginal) j

1 State of Nature S1 S2

3...