DECISION TABLES AND SCENARIO ANALYSIS
2.4 DECISION TABLES AND SCENARIO ANALYSIS
Decision tables and trees are simple mechanisms for structuring some decision problems involving uncertainty and solving them. It requires that an objective, the problem’s states and probabilities be given. To construct a payoff table we proceed as follows:
r Identify the alternative courses of action, mutually exclusive, and collectively exhaustive, which are variables (at least two) we can control directly.
r Consider all possible and relevant states of a problem. Each state represents one and only one potential event; each state may itself be defined in terms of multi-
ple other states, however; states represent events which are mutually exclusive; they are collectively exhaustive; one and only one state will actually result.
r Assign to each state a probability of occurrence. This probability should be based on the information we have regarding the problem and, since states are
mutually exclusive and collectively exhaustive, these probabilities (summed over all states) should be equal to one.
All conditional (payoff or cost) consequences are then assembled in a table for- mat – see Table 2.2 where [c ij ] are the conditional costs of alternative i if event
j occurs.
Table 2.2 The payoff table. States
... n Probabilities
p 1 p 2 p 3 ...
... p n
A 1 c 11 c 12 c 13 ...
... c 1n
A 2 c 21 c 22 c 23 ...
... c 2n
... c 3n ...
A 3 c 31 c 32 c 33 ...
... ... A m
c m 1 c m 2 c m 3 c mn
32 MAKING ECONOMIC DECISIONS UNDER UNCERTAINTY
For example, for a credit manager, what are the relevant states to consider when a customer comes in and demands a loan? Simply grant the loan (state 1) or not (state 2). If the loan is not reimbursed on time (and reimbursement delays are introduced) there may be other ways to express these states. For example,
a first state would stand for no delay, a second, would stand for a one-period delay, a third state for a two-period delay, and so on. The entries in the table tell us what will be the conditional payoffs (or costs) associated with each action. The sample Table 2.2 specifies n states 1, 2, 3, 4, . . . n and m alternatives. When alternative A i , i = 1, 2, . . . , m is taken and say state j occurs with probability
p j , j = 1, 2, . . . , n, then the cost (or payoff) is c ij (or π ij ). Thus, in such a decision problem, there are:
(1) n potential, mutually exclusive and exhaustive states, (2) m alternative actions, one of which only can be selected, (3) nm conditional consequences we should be able to define.
If we use an expected cost (or expected payoff) criterion, then the decision selected would be the one yielding the least expected cost (or, equivalently, the largest expected payoff). The expected monetary cost of alternative i is then:
while the least cost alternative k selected is:
k ∈ Min i ∈[1,..n] {EMC i }
Problem
Cash management consists of managing the short-term flow of funds in order to meet a potential need or demand for cash. Cash is kept primarily because of its need in the future. Assume, for example, that an investor has the following needs for money:
0.05 0.25 0.50 0.15 0.05 (1) What are the potential courses of action?
(2) What are the problem states? And their probabilities? (3) What are the conditional costs if the bank rate is 20 % yearly?
2.4.1 The opportunity loss table
Say that action i has been selected and event j occurs and thus payoff π ij is gained. If we were equipped with this knowledge prior to making a decision, it is possible that another decision would bring greater profits. Assume such knowledge and let the maximum payoff, based on the best decision be
Max j [π ij ]
33 Table 2.3 Opportunity loss table. States
EMV , EOL , EPPI , EVPI
... n Probabilities
The difference between this maximum payoff and the payoff obtained by taking any other decision is called the opportunity loss, denoted by:
l ij = Max j [π ij ]−π ij
The opportunity loss table is therefore a matrix as given in Table 2.3. Thus, the opportunity loss is the difference between the costs or profits actually realized and the costs or profits which would have been realized if the decision had been the best one possible. A project might seem like a good investment, but it means that we have lost the opportunity to do something else that might be more profitable. This loss may be likened to the additional income a trader would have realized had he been an inside trader, benefiting from information regarding stock prices before they reach the market! As a result, we can verify that the difference between the expected profits of any two acts is equal in magnitude but opposite in sign to the difference between their expected losses. By the same token, the difference between the expected costs of any two acts is equal in magnitude and identical in sign to the difference between their expected opportunity losses. With these definitions on hand we can also state that: the cost of uncertainty is the expected opportunity loss of the best possible decision under a given probability distribution.