Decision Analysis

MGMT 306

Alex L. Wang

Purdue University

Decision Analsis Using Payoff Tables

Without probabilities

Decision Analysis

Decision analysis is used to determine a strategy when a decision maker is faced with several decision alternatives and uncertainty
A decision problem is characterized by
- decision alternatives
- states of nature
- resulting payoffs

Payoff table

Payoff: Outcome (a quantity) of a decision alternative and a particular state of nature
- For example: profit, cost, time, distance

Inventory Problem

A department store wants to decide on how much to order of a toy for the Christmas season
The manufacturer sells toys in lots at a price of $2,000 per lot
The store considers ordering 1, 2, or 3 lots
The demand can be high (3 lots), medium (2 lots) or low (1 lot)
The department store will sell each lot for $4000 if there is sufficient demand

Construct a payoff table indicating the department store’s profit for each decision alternative and state of nature

Inventory Problem Payoff Table

		Demand
		1 lot	2 lots	3 lots
Order quantity	1 lot	2000	2000	2000
	2 lots	0	4000	4000
	3 lots	-2000	2000	6000

Decision making criteria

Three commonly used criteria for decision making under uncertainty (without probabilities):
- Optimistic approach
- Conservative approach
- Minimax regret approach

Optimistic Approach

We will be optimistic and assume that nature will help us
- Largest profit
- Lowest cost
Finding the best decision in a payoff table:
For each decision (row), calculate the best payoff for that row
Pick the row with the best optimistic payoff

Optimistic Approach Payoff Table

		Demand			Optimistic Payoff
		1 lot	2 lots	3 lots	Optimistic Payoff
Order quantity	1 lot	2000	2000	2000	2000
	2 lots	0	4000	4000	4000
	3 lots	-2000	2000	6000	6000

The best decision under the optimistic approach is to order 3 lots of toys for an optimistic payoff of $6,000

Conservative Approach

We will be pessimistic and assume that nature will work against us
- Lowerst profit
- Highest cost
Finding the best decision in a payoff table:
For each decision (row), calculate the worst payoff for that row
Pick the row with the best conservative payoff

Pessimistic Approach Payoff Table

		Demand			Pessimistic Payoff
		1 lot	2 lots	3 lots	Pessimistic Payoff
Order quantity	1 lot	2000	2000	2000	2000
	2 lots	0	4000	4000	0
	3 lots	-2000	2000	6000	-2000

The best decision under the pessimistic approach is to order 1 lot of toys for a conservative payoff of $2,000

Minimax Regret Approach

We want to minimize our worst-case regret
What is regret?
- Imagine we make a decision and then a state of nature is revealed
- After the state of nature is revealed, we may regret making our original decision
- The regret is the difference between our payoff and the best possible payoff for that fixed state of nature
For each decision (row), the worst-case/maximum regret is the maximum regret along the row
Pick the decision (row) with the minimum maximum regret

Regret table

		Demand			Regret			Max. regret
		1 lot	2 lots	3 lots	1 lot	2 lots	3 lots	Max. regret
Order quantity	1 lot	2000	2000	2000	0	2000	4000	4000
	2 lots	0	4000	4000	2000	0	2000	2000
	3 lots	-2000	2000	6000	4000	2000	0	4000

The best decision under the minimax regret approach is to order 2 lot of toys for a maximum/worst-case regret of $2,000

Practice Problem

Consider the following payoff table (profits)

		States of nature
		$s_1$	$s_2$	$s_3$
Decision alternatives	$d_1$	4	4	-2
	$d_2$	0	3	-1
	$d_3$	1	5	3

What is the best decision and payoff in the:
- optimistic approach?
- conservative approach?
- minimax regret approach?

Practice Problem - Optimistic

		States of nature			Optimistic payoff
		$s_1$	$s_2$	$s_3$	Optimistic payoff
Decision alternatives	$d_1$	4	4	-2	4
	$d_2$	0	3	-1	3
	$d_3$	1	5	3	5

Under the optimistic approach, the best decision is $d_3$ with optimistic payoff 5

Practice Problem - Pessimistic

		States of nature			Pessimistic payoff
		$s_1$	$s_2$	$s_3$	Pessimistic payoff
Decision alternatives	$d_1$	4	4	-2	-2
	$d_2$	0	3	-1	-1
	$d_3$	1	5	3	1

Under the pessimistic approach, the best decision is $d_3$ with pessimistic payoff 1

Practice Problem - Minimax Regret

		States of nature			Regret			Max Regret
		$s_1$	$s_2$	$s_3$	$s_1$	$s_2$	$s_3$	Max Regret
Decision alternatives	$d_1$	4	4	-2	0	1	5	5
	$d_2$	0	3	-1	3	2	4	4
	$d_3$	1	5	3	3	0	0	3

Under the optimistic approach, the best decision is $d_3$ with worst-case/maximum regret 3

Decision Analsis Using Payoff Tables

With probabilities

Expected Value (EV) Approach

If probabilities for the different states of nature are known, can use the Expected Value (EV) approach
- For example, suppose in the department store problem, we know

Demand	Probability
1 lot	10%
2 lots	50%
3 lots	40%

The expected value of each decision is the expected payoff
Pick the row with the best expected payoff

EV Approach Payoff Table

		Demand			Expected value
		1 lot	2 lots	3 lots
Probability		10%	50%	40%
Order quantity	1 lot	2000	2000	2000	2000
	2 lots	0	4000	4000	3600
	3 lots	-2000	2000	6000	3200

Under the EV Approach, the best decision is 2 lots with an expected profit of $3,600

\[\begin{aligned} \textup{EV} = 3,600 \end{aligned}\]

Perfect Information

It is possible to do better if have perfect information
For example, state of nature is still

Demand	Probability
1 lot	10%
2 lots	50%
3 lots	40%

However, we are told in advance which of the three scenarios will occur

Computing the Expected Value with Perfect Information

We know ahead of time which state of nature will occur

		Demand
		1 lot	2 lots	3 lots
Probability		10%	50%	40%
Order quantity	1 lot	2000	2000	2000
	2 lots	0	4000	4000
	3 lots	-2000	2000	6000
Payoff with perfect information		2000	4000	6000

The expected value with perfect information is \[\begin{aligned} \textup{EVwPI} = (0.1)\times 2000 + (0.5) \times 4000 + (0.4) \times 6000 = 4600 \end{aligned}\]

Expected Value of Perfect Information

Compare

\[\begin{aligned} \textup{EV} &= 3600\\ \textup{EVwPI} &= 4600 \end{aligned}\]

The difference is called the Expected Value of Perfect Information (EVPI)
Measures expected increase in payoff due to perfect information
How much should we pay for accurate forecasting?

EV and EVPI Practice problem

You are at an auction bidding on a gently-used dining table set
The seller is holding a second-price auction for this item, where you and one other bidder will place bids of either $20, $30, $40, $50
The seller will sell the dining table set to the highest bidder at the price of the second-highest bid
Since you contacted the seller first, you are considered the “highest bidder” in the case of ties
The table is worth $50 to you
Construct a payoff table for this problem

EV and EVPI Practice problem (continued)

Assume the other bidder has the following probability over their bid

Demand	Probability
$20	40%
$30	25%
$40	25%
$50	10%

Compute the best choice under the EV approach and its expected value
Compute the Expected Value with Perfect Information (EVwPI)
Compute the Expected Value of Perfect Information

		States of nature
		\(s_1\)	\(s_2\)	\(s_3\)
Decision alternatives	\(d_1\)	4	4	-2
	\(d_2\)	0	3	-1
	\(d_3\)	1	5	3

		States of nature			Regret			Max Regret
		\(s_1\)	\(s_2\)	\(s_3\)	\(s_1\)	\(s_2\)	\(s_3\)	Max Regret
Decision alternatives	\(d_1\)	4	4	-2	0	1	5	5
	\(d_2\)	0	3	-1	3	2	4	4
	\(d_3\)	1	5	3	3	0	0	3