• The Hong Kong University of Science and Technology

• ISOM 390

Decision Tree

Decision Making under Uncertainty: Recognize all possible states of the world, If possible, estimate the probability for each state

Evaluate outcomes (in each state) of each action; Rank actions; Collect information only if it is worthy

Example: Marketing a Movie

Maximax: Best of the best = Maximum possible result à Maximize the most optimal result;

Maximin: Best of the worst; Minimin: Minimum of the minimum; Laplace (Equal Likelihood): Best/ Maximum of the average

Minimax Regret: Least of the worst regret = Loss because you didn’t choose the best option; Regret = The most can get – Actual Get

The appropriate criterion depends on the “risk” attitude, personality, and philosophy of the decision maker

Decision Making with Probabilities: Not probability in the sense of frequencies; Best guesses, or subjective probability

E.g. The management assesses that the probability (P) for this new production to be a hit is 30%; ↓Decision Tree without Information↓

Event = Not your choice; But depends on the state of the world

Decision Point = Make Choice; Need to tell Optimal Choice

Expected Profit of “Action” = 0.3 x 25m + 0.7 x -10m = 0.5m

Sensitivity Analysis (P is unknown)

Expected-value: best of the weighted average

Action is the best choice if 35p-10 > 20p-5 and 35p-10 > 5 à p > 3/7

Comedy is the best choice if 20p-5 > 35p-10 and 20p-5 > 5

TV network is the best choice if 5 ≥ 35p-10 and 5 ≥ 20p-5 à p < 3/7 Example: Surgery

A patient is deciding whether and when to have a surgery. If she does not have a

surgery, her payoff is 40. If she has a surgery now, her payoff is as follows:

She can also postpone the decision by five years. By then possible advancements in

technology may improve the surgery outcome. Her payoff of having a surgery in five

years is on the next slide. Five Years Later:

Information: Any piece of knowledge able to alter your beliefs about the uncertainty that you are facing; e.g. forecasts, market surveys,

medical tests; Information is valuable only if it changes your decision. Allan H. Murphy: Forecasts possess no intrinsic value. They

acquire value through their ability to influence the decisions made by users of the forecasts

Reliability of Information

Real information is often incomplete and unreliable. Prob [X happens if the information predicts so] < 100% (Forecast Error)

Problem: Not always possible to assess reliability of the information. à No forecast is 100% reliable (Misleading information)

Perfect information à Predict with 100% accuracy à Best information we can get; No noise information exists.

Provide an upper bound for value of any information; Determine maximum information search; Same cost

Example: Marketing a Movie

Suppose that a crystal ball can perfectly foretell the box office result before the marketing strategy is determined

If the ball says “success”, choose “Action” and get $25m; If the ball says “failure”, choose TV network and get $5m

Expected profits with the crystal ball = 30% x 25 + 70% x 5 = $11m; 30% = chance the ball says success,

70% = chance the ball says failure; Decision Tree without Information

Expected value of perfect information (EVPI)

= Expected profit with perfect info - Expected profit without information

Value of the crystal ball = Expected value of perfect information

= $11M - $5M = $6M

EVPI = Upper bound for value of information, if cost of acquiring info > EVPI, then not worthy

If cost of acquiring information is < EVPI; Cost may change according to the information accuracy

When facing multiple uncertainties, EVPI may help decide which one is most worthy of further study

Imperfect Information from a Single Source (Bayesian updating)

Prior probabilities are initial assessment of the uncertainty before new information

Posterior probabilities are updated assessment after new information

Example: Marketing a Movie: Historically, favorable forecasts have been obtained for 70% of all successful films, while unfavorable

forecasts have resulted from 80% of box-office failures. The cost of obtaining the forecast is $500,000.

à Prior: Prob(S) = 0.3, Prob(F) = 0.7; Reliability: Prob(+|S) = 0.7, Prob(-|F) = 0.8

Posterior = Prob (S | +) = Prob (+ , S) / Prob (+) = Prob (+ | S) x Prob (S) / [ Prob (+ | S) Prob (S) + Prob (+ | F) x Prob (F)

= (0.7 x 0.3) / (0.7 x 0.3 + 0.2 x 0.7) = 0.6

Expected Value of Imperfect Information (EVII) = Expected profit with (costless) imperfect info - Expected profit without information

Cost of acquiring information is not included in the calculation of EVII; Information is worth acquiring if its value > cost.

Imperfect Information from Two Independent Sources

Example: The Party Problem: Anna has two weather detectors, “alpha” and “beta”, with reliability of 80% and 90% respectively. Weather

forecasts from the two detectors are independent of each other à Sequence of inspection does not affect the posterior probabilities.

P(A"S", B"S", S) = P(A"S" , B"S" | S) x P(S)

= P(A"S" | B"S", S) x P(B"S" | S) x P(S) = P(A"S" | S) x P(B"S" | S) x P(S)

= 80% x 90% x 40% = 0.288 Prob(Sunny) = 40%

P(S | A"S", B"S") = P(A"S", B"S", S) / P(A"S", B"S") = 0.288 / 0.3 = 0.96

Joint Value of Two Forecasts

EV of the two detectors > EV of the Alpha detector alone

EV of the two detectors < EV of the Alpha alone + Beta alone

In general, expected value of multiple forecasts can be

<, =, or > sum of expected value of individual forecasts

Have 2 strong signal may substitutable to each other

à Reduce the benefit of the first information

Have 1 strong signal can be enough to change decision

à Remember only info that change your decision is valuable

Exchangeability Marginal value of the detector fluctuating

When n becomes sufficiently large, the joint value of n

independent detectors approaches the expected value of perfect information. It applies even if the detectors are individually valueless

The More Information à Total value increases in n; Marginal value fluctuates; Cost also increases in n

Optimal information search: Expected return from information (value - cost) is usually maximized at some intermediate n

Can Detectors be Complementary?

Consider a detector with 52% reliability, Prob[S|”S”]=0.42, Prob[S|”R”]=0.38

Because within the same "Porch" range; Decision will not change à Value = 0

Two detectors

Prob[S|2 S indications]=0.44, Prob[S|1 S indication]=0.4, Prob[S|0 S indication]=0.36

Fall to indoor range à Decision may change à Expected value of information