Last updated 1996 Sep 05 Accounting 503

Decision Theory

Environment | Criteria | Decision Trees

Environment

States of Nature | Decisions | Payoffs | Utility

Top of Page

States of Nature

certainty, the deterministic case
uncertainty, without probabilities
risk, with probabilities

Payoffs

payoff tables, for discrete states and decisions
payoff formulas

Utility is usually a nonlinear function of monetary payoff. For example, as wealth increases the marginal utility of another dollar decreases. Also, a loss and a gain might have equal monetary value but different utility.

Human Perception

events of very low probability, for example, infants suffocating under plastic bags

Decision Criteria

Alternatives

Maximize Profit
Minimize Cost

Maximizing Expected Value

Required Environment

knowledge of payoffs
knowledge of probabilities
favorable odds
ability to cumulate gains with losses
- play many times
- don't bet the whole company
A story: In the dark ages of Europe an alchemist conjured up the devil. The devil proposed drawing a random card from a deck. Somehow the archangel Gabriel was prevailed upon to guarantee the deck is fair. If the card is the ace of spades the alchemist forfeits his soul forthwith. Otherwise, the alchemist receives power, riches, and the adoration of all humanity, with no claim on his soul at the end of a long life.

Value of Perfect Information and Expected Regret.

Criteria Independent of Probability

Maximax, the Gambler's Criterion
Maximin, the Widows' and Orphans' Criterion
Minimax Regret, the "If only I had ..." Criterion

Expected Value

Relation to Expected Regret

Decision Trees

Sequences of Decisions

states of nature
payoffs
probabilities

can all be dependent on prior decisions.

Procedure

Construct the geometry of the tree
1. identify initial decision alternatives
2. for each alternalternative decision identify the possible states of nature
3. for each state of nature go back to the first step
Assign probabilities wherever there are alternative states of nature.
Assign payoffs to the final branch ends of the tree
Starting with the final branch ends compute expected values of the preceding branches. This is called folding back.
Repeat the preceding step back to the root of the root.
Reverse direction, away from the root, always taking the branch with the highest expected value.

Conditional Scenarios

Example:

Expected Value

Credit Card Usage at a Parking Garage, article in Management Accounting

Payoff Tables

Perishable product (Newsboy Problem)

Payoff Formulas

Lakeside Resort

Winds from surrounding regions make the weather at a lakeside resort unpredictable on a day to day basis. In turn, sales of fresh fruit punch are a function of the 3pm temperature, approximately 5 + T liters, where T is the temperature in degrees Celsius in the range 25 to 40. The punch must be ordered in five liter increments at a cost of $.50 per liter. The selling price is $1 per liter. Apply various decision criteria.

Decision Trees

Rain or shine

The probability of rain is 0.2. Taking your umbrella when the sun shines has a cost of 8. Not taking your umbrella when it rains has a cost of 10. Guessing right has a cost of zero.

Draw both the payoff table and the decision tree.

Suppose nature acts on your decision and changes the probability of rain. The decision tree formulation can show the change.

Inspect and repair as needed

You're about to drive to Montreal. If your car's engine is out of tune, your gas cost will increase by $100. Having the engine tested will cost $20. If it's out of tune repairs will cost $60. Before testing the probability is 30% that the engine is out of tune. What should you do?

Perishable Product Example

Stock fresh each day. Throw out unsold product at day's end.

Accounting Data
amount	per unit	alternately
sales	$5	revenue
cost of goods sold	$3	variable cost
gross margin	$2	contribution margin

Payoff Table
In terms of gross margin per day
Decision, Stock Level Selected 30 20 10

State of Nature, Demand

30 60 40 20

20 10 40 20

10 -40 -10 20

**Payoff Table**
In terms of gross margin per day
Decision, Stock Level Selected	30	20	10
State of Nature, Demand
30	60	40	20
20	10	40	20
10	-40	-10	20

Regret Table
For each state of nature it's the best payoff minus the payoff for the decision you made.
Decision, Stock Level Selected 30 20 10

State of Nature, Demand

30 0 20 40

20 30 0 20

10 60 30 0

**Regret Table**
For each state of nature it's the best payoff minus the payoff for the decision you made.
Decision, Stock Level Selected	30	20	10
State of Nature, Demand
30	0	20	40
20	30	0	20
10	60	30	0

Criterion Stock Level Selected

maximax 30

maximin 10

minimax regret 20

Criterion	Stock Level Selected
maximax	30
maximin	10
minimax regret	20

Expected Value The decision depends on the probabilities assigned.

Case 1

Demand 30 20 10 Stock 30 20 10 Perfect Info

Probability 0.3 0.6 0.1 Expected Value 20 35 20 9

Case 2

Demand 30 20 10 Stock 30 20 10 Perfect Info

Probability 0.33 0.34 0.33 Expected Value 10 23.5 20 16.5

Case 3

Demand 30 20 10 Stock 30 20 10 Perfect Info

Probability 0.1 0.1 0.8 Expected Value -25 0 20 6

Case 1
Demand	30	20	10	Stock	30	20	10	Perfect Info
Probability	0.3	0.6	0.1	Expected Value	20	35	20	9
Case 2
Demand	30	20	10	Stock	30	20	10	Perfect Info
Probability	0.33	0.34	0.33	Expected Value	10	23.5	20	16.5
Case 3
Demand	30	20	10	Stock	30	20	10	Perfect Info
Probability	0.1	0.1	0.8	Expected Value	-25	0	20	6

References

Michael S. Luehfing, Driving Out Inefficiency, Management Accounting, March, 1993, pages 33 to 36.

Tom Peters, A private opinion on worker drug testing, Tribune Media Service, October 17, 1994.

Top of Page | ACCT 503 Syllabus | Warren Home Page