You have decided to open a restaurant in Huntington. The followingtable gives the payoffs based on two states of nature, favorable and unfavorable demand. You believe that the probability of a favorable market is 0.8. You can open any one of three different sizes of restaurant.

Open	Favorable	Unfavorable
Large Restaurant	55,000	-30,000
Medium Restaurant	40,000	-5,000
Small Restaurant	20,000	15,000

1. What should you do? Show your work.

Adding notes below:

Module 12 - Decision Theory

Decision analysis is used to find the optimal solution when one has multiple alternatives and different states of nature (future events).

Analysis should include risk analysis that provides probability information about favorable and unfavorable outcomes.

The problem consists of different decision alternatives, different states of nature, and payoffs associated with combination of alternative and states of nature.

Example - A company has land that can be used to build a condominium complex. The options are to build a 30 unit, 60 unit, or 90 unit complex. There are two states of nature: high demand and low demand. The following table gives the payoffs associated with each combination of decision and state of nature.

Payoff Table

States of Nature

Decision Alternative	Strong Demand, s1	Weak Demand, s2
Small Complex, d1	8	7
Medium Complex, d2	14	5
Large Complex, de	20	-9

Without having any probabilities, there are three different ways to analyze the table.

1. Optimistic Approach - the decision-maker believes that the best that can happen will happen. Therefore, choose the decision with the highest payoff.

20 is the highest payoff, so choose Large Complex

1. Conservative Approach - the minimum payoff is list and then the decision-maker chooses the highest of these values. See the table above. All of the values are for Weak Demand.

Decision Alternative	Weak Demand, s2
Small Complex, d1	7
Medium Complex, d2	5
Large Complex, de	-9

Choose Small Complex with a payoff of 7. That is the highest of all the numbers.

1. Minimax Regret Approach - need to calculate the regret for each option. Regret is the difference between each payoff and the largest payoff for that state of nature. For example, if there is strong demand, then you should build a large complex with payoff of 20. If you built a small complex, then you make less money (8) then you could have. This is your regret. In this case, that would be 20 - 8 = 12. The table below shows the regret for each option.

Regret Table

States of Nature

Decision Alternative	Strong Demand, s1	Weak Demand, s2
Small Complex, d1	20-8 = 12	7 - 7 = 0
Medium Complex, d2	20 - 14 = 6	7 - 5 = 2
Large Complex, de	20 - 20 = 0	7- (-9) = 16

For each decision list the maximum regret, then choose the decision with the minimum of these values.

Decision Alternative	Maximum Regret
Small Complex, d1	12
Medium Complex, d2	6
Large Complex, de	16

Choose Medium Complex with regret of 6.

Decision Making with Probabilities

Expected Value (EV) Approach is used where you have probabilities for the occurrence of each state of nature.

The expected return for each decision is calculated by the summing the products of the payoff for each state times the probability for that that state.

The decision yielding the best expected return in chosen.

The expected value (EV) of decision alternative d_i is defined as:

EV(d_i) =

Where N = the number of states of nature

P(s_j) = the probability of state of nature s_j

V_ij = the payoff corresponding to decision alternative d_i and state of nature s_j

Decision trees can help with examining these types of problems .A decision tree is a representation of the problem over time.

Each decision tree has two types of nodes:

Round nodes - states of nature

Square nodes - decision alternatives.

The branches leaving the round nodes represent different states of nature.

The branches leaving the square nodes represent the different decision alternatives.

At the end of each limb of a tree are the payoffs attained.

Lines from node 2, 3, and 4 represent the various states of nature. The top line represents s₁, strong demand, with a probability of 0.8. The bottom line represents s₂, weak demand, with probability of 0.2

EMV (small) = .8($8mil) + .2 ($7 mil) = $6.4 + $1.4 = $7.8 mil

EMV (medium) = .8($14mil) + .2 ($5 mil) = $11.2 + $1.0 = $12.2 mil

EMV (large) = .8($20mil) + .2 ($-9 mil) = $16.0 + $-1.8 = $14.2 mil

Choose the one with the largest payoff. That is EMV (large) with a value of $14.2.