2.3. Value of Information: Decision Trees and Backward Induction

2.1 Basic concepts: Preferences, and utility 2.2 Choice under uncertainty: Lotteries and risk aversion 2.3 Value of information: Decision trees and backward induction Outline

Complex decision scenarios usually involve both the choice of an alternative (action) and the influence of uncertain events (lotteries) on the final outcome of the decisions In many cases, reducing the “degree” of the uncertainty will help the decision process. In this sense, we will see that Information is a valuable element

Decision Trees Decision Trees (as we have already seen) are useful diagrammatic representations of decision problems that help in the search for the best decision Usually, a Decision Tree consists of a sequence of action branches (decisions) and chance branches (lotteries) that represent the problem at hand Solving a Decision Tree means computing a value for each action branch (Expected Value, Expected Utility) and finally decide what is the best action based on these values.

Example: The garden apples oranges apricots E(apples) = 0.25 x x 40 = 55 E(apricots) = 0.25 x x 60 = 65 E(oranges) = 0.25 x (-20) x 140 = 100 E(u(apples)) = 0.25 x x 40 2 = 3700 E(u(oranges)) = 0.25 x -(20) x = 14,600 E(u(apricots)) = 0.25 x x 60 2 = 4300 Risk Neutral: Risk Lover: u(x)=x u(x)=x 2

The method of “solving the tree” moving from right to left is called Backward Induction (or tree rollback) In this example, it is clear that knowing before hand what the weather is going to be like would be a valuable information for taking the best decision

apples oranges apricots apples oranges apricots Snows Does not Snow

apples oranges apricots apples oranges apricots Snows Does not Snow

What is the value of information ? If we do not have any information regarding the weather in the future, our best choice is to plant Oranges as it is the alternative that has the highest Expected Value E(Oranges) = 100 But if we had full information about the weather, that is, if we knew before hand if it is going to snow or not, our best choice would be: Apples if it is going to snow Value = 100 Oranges if it is not going to snow Value = 140

What is the value of information ? Since the ”real” probability of snowing is of 0.25, the probability that you are informed that it is going to snow is also of Hence, your decision Apples will be taken with probability 0.25 Oranges will be taken with probability 0.75 Therefore, the Expected Value using this information is E(Information) = 0.25 x x 140 = 130

What is the value of information ? Thus, Expected revenue without infomation E(Oranges) = 100 Expected revenue with information E(Information) = 130 The information has a value of 30

Nevertheless, having full information is rarely the case In most situations, some partial information (for instance, a weather forecast) is all we can get This partial information, though, can be useful to obtain a more accurate measure (updated probabilities) of the uncertain events

Asian Import Company (i) Asian Import Company is a small Chicago based firm specialized on the import and distribution of Asian collectibles. AIC is about to close a deal with a large Chinese company consisting of the acquisition a large collection of Ming porcelain for reselling in the U.S. market. Such operation would have a cost of $500,000 for AIC and would produce a revenue of $800,000 (net profit of $300,000) However, the U.S. Government is currently negotiating with China over a special trade agreement for “arts and crafts” items. If the negotiations are successful, AIC will be allowed to freely import the Ming collection at zero cost, but if an agreement is not reached, AIC will be forced to pay a 50% tariff on the sale of any imported item. Before hand, without any specific information, the chances that the negotiation is successful are 50%.

(For simplicity, we will assume that AIC is risk neutral, that is, AIC uses Expected Value (Expected Profit in this case) as the decision criterion) Should AIC buy the Ming collection ?

Buy Don't buy Agreement Disagreement (0.5) $0 $8 - $5 = $3 $4 - $5 = -$1 E(Profit) = 0.5 x x (-1) = $1

Should AIC buy the Ming collection ? At this point the decision is clear. Since the Expected Profit is positive ($100,000), AIC should purchase the Ming collection

Asian Import Company (ii) Reducing the risk After “cheap talking “ to an analyst, AIC realize that they could wait until the uncertainty is resolved. The problem is that then it might be too late and that AIC looses the Chinese deal to a competitor firm ! Based on previous experiences, AIC knows that the chances that this happens are of 70 % Should AIC wait ?

Buy Don't buy Agreement Disagreement (0.5) $0 $3 -$1 E(Profit) = 0.5 x x (-1) = $1 Wait Agreement Disagreement (0.5) $0 Still Available Not available (0.3) (0.7) $3 $0

Backward Induction Solving the tree by Backward Induction consists of: Start at the top-rightmost end of the tree For each set of chance branches, find the corresponding Expected Value (or Expected Utility) Chop off those branches and replace them by the computed Expected Value (or Expected Utility)

Backward Induction (continued) For each set of choice branches, find the best choice Chop off those branches and replace them by the value that corresponds to that best choice Proceed left wise in the same way until all chance branches Are removed

Buy Don't buy Agreement Disagreement (0.5) $0 $3 -$1 E(Profit) = 0.5 x x (-1) = $1 Wait Agreement Disagreement (0.5) $0 Still Available Not available (0.3) (0.7) $3 $0 Remove these chance branches Replace them by the Expected Value

Buy Don't buy $0 $1 Wait Agreement Disagreement (0.5) $0 Still Available Not available (0.3) (0.7) $3 $0

Buy Don't buy $0 $1 Wait Agreement Disagreement (0.5) $0 Still Available Not available (0.3) (0,.7) $3 $0 E(Profit) = = 0.3 x x 0 = $0.9

Buy Don't buy $0 $1 Wait Agreement Disagreement (0.5) $0 $0.9

Buy Don't buy $0 $1 Wait Agreement Disagreement (0.5) $0 $0.9 E(Profit) = 0.5 x x 0 = $0.45

Buy Don't buy $0 $1 Wait $0.45

Should AIC wait ? Clearly not. The expected profit of waiting is of only $ (because of the high chances of loosing the deal with the Chinese firm). Buying the collection without knowing the outcome of the negotiations is a better choice ($ expected profit)

Asian Import Company (iii) Full information Suppose now that AIC has the opportunity to access a source of full information. That is, AIC could know right away whether the trade agreement is going to be signed or not. How much would AIC pay for such information ?

Buy Don't buy $0 $1 Wait $0.45 Agreement Disagreement (0.5) Buy Don't buy $3 $0 Buy Don't buy -$1 $0 Full Information

using Backward Induction

Buy Don't buy $0 $1 Wait $0.45 Agreement Disagreement (0.5) Buy Don't buy $3 $0 Buy Don't buy -$1 $0 Full Information

Buy Don't buy $0 $1 Wait $0.45 Agreement Disagreement (0.5) $3 $0 Full Information

Buy Don't buy $0 $1 Wait $0.45 Agreement Disagreement (0.5) $3 $0 E(Profit) = 0.5 x x 0 = $1.5 Full Information

Buy Don't buy $0 $1 Wait $0.45 $1.5 Full Information

How much would AIC pay for such information ? Expected gain without infomation (best option) E(Buy) = $100,000 Expected gain with information E(Full Information) = $150,000 Thus, the information has a value of $50,000

Asian Import Company (iv) Partial information Suppose, finally, that AIC can hire an analyst that has good connections with the Washington bureaucracy. For a fee he can tell you what we knows on the status of the negotiations.. This analyst is known to be very successful in similar situations. Every time an agreement was reached he had predicted it 90% of the times. However, he hasn't been so reliable when the negotiations were not successful. In such cases, he only called it right 60% of the time. How much would AIC pay now for such information ?

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy Agreement (??) Disagreement (??) Agreement (??) Disagreement (??) $0 $3 -$1 Full Information

Disagreement (0.5) This is what we know... Predicts Success Predicts Failure (0.9) (0.1) (0.3) (0.45) (0.05) (0.2) Path probabilites Agreement (0.5) Predicts Success Predicts Failure (0.4) (0.6)

Predicts Failure (B) This is what we need to know... Agreement Disagreement (C) (D) (0.3) (0.45) (0.05) (0.2) Path probabilites Predicts Success (A) Agreement Disagreement (E) (F) ”Flipped Probabilites”

A = = 0.65 B = = 0.35 A X C = X C = 0.45 C = 0.69 A X D = X D = 0.20 D = 0.31 B X E = X E = 0.05 E = 0.14 B X F = X F = 0.30 F = 0.86 Computation of ”Flipped Probabilites”

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy Agreement (C) Disagreement (D) Agreement (E) Disagreement (F) (A) (B) $0 $3 -$1 Full Information

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy Agreement (0.69) Disagreement (0.31) Agreement (0.14) Disagreement (0.86) (0.65) (0.35) $0 $3 -$1 Full Information

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy Agreement (0.69) Disagreement (0.31) Agreement (0.14) Disagreement (0.86) (0.65) (0.35) $0 $3 -$1 Full Information E(Buy) = 0.69 x x -1 = $1.76

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy Agreement (0.14) Disagreement (0.86) (0.65) (0.35) $0 $3 -$1 Full Information $1.76

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy Agreement (0.14) Disagreement (0.86) (0.65) (0.35) $0 $3 -$1 Full Information $1.76 E(Buy) = 0.14 x x -1 = -$0.44

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy (0.65) (0.35) $0 Full Information $1.76 -$0.44

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure Buy Don't buy Don't buy (0.65) (0.35) $0 Full Information $1.76 -$0.44

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure (0.65) (0.35) $0 Full Information $1.76

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Predicts Success Predicts Failure (0.65) (0.35) $0 Full Information $1.76 E(Partial Information) = = 0.65 x x 0 = $1. 144

Buy Don't buy $0 $1 Wait $0.45 $1.5 Partial Information Full Information $1. 144

How much would AIC pay for such partial information ? Expected gain without infomation E(Buy) = $100,000 Expected gain with partial information E(Partial Information) = $114,400 Thus, the information has a value of $14,400

Summary Complex decision scenarios usually involve both the choice of an alternative (action) and the influence of uncertain events (lotteries) on the final outcome of the decisions Information on these uncertain events is, hence, valuable Full Information is seldom available Partial information, though, can be useful to obtain a more accurate measure (updated probabilities) of the uncertain events Decision Trees are useful diagrammatic representations of decision problems that help in the search for the best decision using Backward Induction