Abstract

The decision tree is widely used to represent and solve decision problems. Because it involves the explicit use of mathematics, it tends to be used for technical audiences. This paper presents an alternative way of representing and solving the decision tree for nontechnical audiences.

A common step in quantifying a decision tree is to ask subjects to quantify their beliefs about an event using a probability wheel. This procedure implies that subjects do have geometrical intuitions about uncertainties. Our new approach to decision analysis builds on this insight.

Instead of representing the tree with lines, we represent the various levels of a decision tree as concentric rings. Specific events are represented as segments in those rings with the size of the segment being proportional to the probability of the event occurring. Payoffs are represented by coloring various segments. The standard expected value calculation of decision analysis corresponds to mixing the colors from outer segments to color inner segments.

This new way of representing and solving a decision tree is completely visual and eliminates the need for any mathematics.

Introduction

The classical approach to modeling a problem involving risk (Bernstein, 1996) uses the decision tree (Skinner, 1999; Chernoff, 1987, DeGroot, 1970; Lindley, 1991; Berger, 1985; Baron and Brown, 1991; Pratt, Raiffa and Schlaiffer, 1964, Savage, 1954; LaValle, 1968, Clemen & Reilly, 2000). Creating a decision tree involves structuring, quantification and computation. There are a number of difficulties involved in all three steps:

1. Problem structuring using the decision tree becomes very cumbersome for all but very small problems
2. The explicit use of numbers in decision analysis discourages mathematically unsophisticated users from applying the technique. The use of numbers can also lead to a sense of artificial precision.
3. The expected value computation is likewise non-intuitive for non-mathematical audiences.

This paper will present Decision Rings as a more compact and totally visual alternative to the standard decision tree which eliminates the need for any quantification. While Decision Rings can be created by hand, they can also be created and solved using Microsoft Excel's doughnut chart capability¹.

Before introducing our innovation, we discuss each step in conventional decision tree analysis and its potential limitations.

1. The Conventional Decision Tree

1.1 Structuring the Decision Tree:

Consider the problem of deciding whether or not to fund an R&D project. If we fund the project, then the project will either be technically successful or unsuccessful. Given the project is technically successful, the project either leads to high profit or to low profit. A decision tree would represent this problem as follows:

Figure 1

This decision tree has three levels where

1. the first level corresponds to the decision about whether to fund. (Hence there are two outcomes at this level.)
2. the second level corresponds to the immediate implications of that decision, i.e. the project being funded and successful, the project being funded and failing, the project not being funded. (Hence there are three outcomes at this level.)
3. the third level corresponds to the immediate implications of the second level outcomes, i.e., the project being successful and having high payoff, the project being successful and having low payoff, the project not being successful, the project not being funded. (Hence there are four outcomes at this level.)

Note that the number of possible outcomes in a given level generally increases as we move toward later and later levels in the tree.

This creates the `bushy mess' problem. As Raiffa (1967) noted:

Trees that exhibit the structure of real problems have a nasty habit of getting rapidly out of control---branches seem to proliferate everywhere and the tree never seems to stop growing…the further we look ahead and the more refined our analysis becomes, the more complex the tree becomes, and if we carry matters to an extreme, the tree begins to resemble a gigantic bush"(pg 239).

To avoid this problem, it's common to `prune' the tree in advance by including only those nodes which, based on sensitivity analysis, impact the decision the most. But there can be a danger in oversimplifying a problem. For example recent work on real options (Amram & Kulatilaka, 1998) led Howard (1966) to warn against the common practice of ignoring downstream decisions in decision risk analyses².

1.2 Quantifying the Problem

Once the tree is constructed, the probability of each uncertainty and the desirability of each consequence is assessed. To assess the probability of each event, decision analysts often proceed by presenting a subject with a choice between the following two bets:

Bet 1: You win if the R&D project is successful. Otherwise you lose

Bet 2: Consider the following probability wheel.

Figure 2

You win if a dart, randomly hurled at the wheel, lands in the white region. You lose if it lands in the black region

The subject is asked to specify which bet they prefer. Suppose the subject prefers to bet on the white region occurring. You then ask the subject how they would choose between

Bet 1 and the following adjusted bet:

Bet 2': Consider the following adjusted probability wheel

Figure 3

You win if a dart, randomly hurled at the wheel, lands in the white region. You lose if it lands in the black region

The subject is asked to specify which bet they prefer. Suppose the subject prefers to bet on the white region occurring. Then the analyst presents the subject with another altered version of Bet 2'. Eventually the analyst finds a version of Bet 2 such that the subject is indifferent between Bet 1 and this version of Bet 2. The probability assigned to the event in Bet 1, success of the R&D project, is set equal to the probability of winning Bet 2.

Suppose this procedure leads to a probability of 0.2 for the R&D project being successful and a probability of 0.5 for the R&D project having a high payoff, given it is successful.

The decision analyst now assigns numerical payoffs describing the desirability of various consequences. To do so, payoffs are ranked from best to worst. In this case, high profit from a successful R&D project is the best payoff, low profit from a successful R&D project is the next best payoff, not funding the R&D is the next best payoff and funding the R&D and having it fail is the worst payoff. The best payoff is assigned a utility of 100 and the worst payoff is assigned a utility of 0. To quantify the other payoffs, we again present the subject with the following choices:

Choice 1: You're guaranteed that the R&D project is successful but you get a low payoff

Choice 2: You throw a dart at the following chart:

Figure 4

If the dart hits white, then you get the best payoff, the R&D project is successful and you get a high payoff. If the dart hits black, then you get the worst payoff, you fund the R&D and it fails.

If the subject indicates that they prefer betting on the chart, then we revise the chart to get the chart in bet 2' and repeat choices 1 and 2. We continue until we identify a scenario in which the subject is indifferent between choice 1 and choice 2 using a variant of the chart in bet 2. When the subject is indifferent, the utility of the consequence in choice 1, the R&D project is successful and we get the low payoff, is set equal to the probability of winning the gamble presented in the altered version of choice 2.

Suppose that this procedure leads the analyst to attach a payoff of 100 at the end of the branched labelled `high profit', a payoff of 20 to the branch labelled `low profit', a payoff of 0 to the branch labelled `project failure' and a payoff of 10 to the branch labeled `No Funding.' This leads to the tree

Figure 5

This procedure makes the plausible presumption that the individual can describe his uncertainty about an event geometrically, i.e., in terms of the probability of a random dart hitting a sector on a circle. It then uses our knowledge of the numerical probability of hitting a sector on a circle to assign a number to the events. Hence this procedure involves translating a subject's geometrical intuition about probability into a numerical value. In this paper, we will present a new approach to decision trees that leverages the subject's geometrical intuition about uncertainty but avoids making any translation of this geometric intuition into numerical values.

1.3 Computation

Decision analysis then assigns an intermediary payoff to each node of the tree based on the possible payoffs arising immediately after that node. If the node represents an uncertainty, the intermediary payoff is the expectation of the possible payoffs occurring after that node. If the node represents a decision, the intermediary payoff is the maximum of the possible payoffs occurring after that node. Thus the intermediary payoff attached to the `uncertainty' node `Profit' is 50% of 100 plus 50% of 20 = 60. The intermediary payoff attached to `the uncertainty node, Project', is 20% of the value assigned to `Profit' (or 60) plus 80% of the value assigned to project failure(or 0) which equals 12. Finally the payoff attached to the decision node, `Fund' is the maximum of the payoff attached to project(or 12) and the payoff attached to no funding(or 10), which equals 12.

Figure 6

Thus the decision tree represents quantitative information by directly writing numbers under arcs or at the end of branches. While not a drawback to the mathematical modeler, many clients of decision analysis are uncomfortable with numbers. Hence the need to represent quantitative information directly is a second limitation of the classical decision tree. While we might be tempted to accept this limitation as inevitable, the successes of Tufte (1992, 1997) and others have demonstrated that quantitative information can frequently be communicated visually.

2. Structuring Problems with Decision Rings

2.1 Structuring the Problem

As our discussion of quantification emphasizes, decision analysis presumes that subjects have some intuition about the probability of a randomly thrown dart hitting various sectors on a probability wheel. We will build on this assumption by showing how the probability wheel concept can be used as the basis for decision rings. This section will focus on structuring a problem and assigning probabilities using decision rings. Our third section will focus on assigning utilities and solving a decision problem using decision rings. (As will be apparent, this circular representation can be easily created using the doughnut chart feature in Microsoft Excel.)

Decision rings, like the classical decision tree, models the problem beginning with the first stage, then the second stage, etc. It starts by drawing a small circle.

Figure 7

To represent the first stage of the decision problem, we now draw a ring around this circle. In our example, this corresponds to a decision which has two possible choices. We will assign 50% of the ring to the first choice and 50% of the ring to the second choice. This involves splitting the ring into two segments, one segment corresponding to the first choice(funding the R&D) and one segment corresponding to the second choice(not funding the R&D.) This gives the following representation:

Figure 8

This represents a one-stage decision tree.

To represent the second stage, we draw a second ring around this first ring. We then extend whatever cuts were drawn in the first inner ring into this second ring. In our example, this causes the second inner ring to be divided into two segments. We now focus on the segment which is adjacent to the `Fund' portion of the first ring.

Our decision tree indicates that the fund decision is following by an uncertainty about whether or not the project will be successful or not. The two possible outcomes are `project success' and `project failure'. The probability of `project success' was 20% and the probability of `project failure' was 80%. We now cut this segment into two pieces---one corresponding to project success and one corresponding to project failure. As before, we will later extend this cut to all rings containing this second ring. Of the total area of the original segment, 20% is assigned to the subsegment corresponding to `project success' and 80% to the subsegment corresponding to `project failure'.

We now focus on the portion of the second ring adjacent to `Don't Fund'. In this case, there is no uncertainty. Hence we don't cut the ring adjacent to `Don't Fund'.

Figure 9

Note that outcomes of decisions were labeled using bold italics while outcomes of an uncertainty were labeled in standard font.

We now turn to the third and last stage in our example. As before, we represent this stage by drawing a third ring about the second ring. Since we extend the cuts in the second ring into the third ring, this third ring will have already been cut into three pieces. We focus first on that portion of the third ring which is adjacent to `project success'. The third stage of the decision tree indicates that there is one uncertainty following project success---which has two possible outcomes `High' or `Low'. Both are equally likely. As before, we now cut this portion of the third ring into two equal parts, with one part labeled `High' and the other part labeled `Low.' We then move to the portion of the ring adjacent to `Project Failure.' In this case, there is no uncertainty following project failure so that, again, there is no need to split this segment further. We then move to that portion adjacent to the ring which is adjacent to the decision `Don't Fund.' As before, there are no uncertainties. Hence there is no reason to cut this ring. This gives us the following representation:

Figure 10

In summary, we structure a problem using decision rings as follows:

1. If the decision problem involves n stages, then draw n+1 concentric circles.
2. Focus first on the first stage . If this stage involves a decision with m possible choices, then split the first ring---and all rings containing it---into m equal parts. If this stage involves an uncertainty with m possible outcomes, then split the first ring-- -and all rings containing it---into m parts with the area of each part proportional to the probability of the associated outcome occurring.
3. Now turn to the second stage . This second stage has already been cut into segments based on what was done in the previous step. If the segment corresponds to the occurrence of a decision with m* possible outcomes, then split the segment---and all segments immediately above it---into m* equal portions. If the segment corresponds to the occurrence of an event with m* possible outcomes, then split the segment---and all segments immediately above it---into m* parts with the area proportional to the probability of each outcome occurring.
4. Repeat this procedure with each successive stage .

The next section focuses on quantification of this ring---which corresponds to choosing how rings will be colored.

2.2 Quantification and Computation

Our circular tree has already represented the probabilistic information in the decision tree using the lengths associated with segments. Hence there is no need to write down probabilities with decision rings.

The other piece of quantitative information used in a decision tree is the payoffs associated with various branches. To represent this information on our tree rings, we---following standard decision analysis practice---rank the payoffs from best to worst. We then associate colors (or shadings) with each payoff. The best payoff is colored `white' and the worst payoff is colored `black.' (Obviously the reader can alter these conventions freely.) Intermediate payoffs are assigned intermediate shades of gray in proportion to their relative value.

There are several ways to do this. If there is direct quantitative information on the payoff, then we could use standard color schemes to translate a numerical value into a shade of color. If there isn't direct quantitative information on the payoff, we might have the client directly assign the brightest color to the best outcome, the darkest color to the worst outcome and intermediate colors to intermediate outcomes. This would represent an application of an established psychophysical technique, cross-modality matching (Stevens and Galanter, 1957). Hence decision rings open up the possibility of quantifying relative value by using colors and never using numbers.

The classical decision tree assigns payoffs to the endpoints of the decision tree. In decision rings, we color the segments in the outer ring according to their payoffs. Hence we would first modify the colors assigned to the outer ring. This would give us

Figure 11

After assigning payoffs to the endpoints on the tree (i.e. to the last stage), classical decision theory then solves the decision tree by proceeding to the next to the last stage and, for each node, in that stage

1. Writing a number on that node which---if the node is followed by an uncertainty---is the expected value of the endpoints arising from that node
2. Writing a number on that node which---if the node is followed by a decision---is the maximum value associated with any of the endpoints arising from that node

This is known as `folding back' the decision tree.

As we now show, decision rings provide an alternate way of `folding back' the decision tree which makes no explicit use of calculations.

Suppose we have colored the segments in the outer ring. We now go the next outer ring. To color a segment in this ring, we look at the color of the segments that are immediately above this ring. Suppose there is only one segment above that segment and it has been colored `black'. Then we color that segment `black.' Suppose that there are several segments above the segment in question and that these segments represent the different outcomes of an uncertainty. Then we color the segment in question by proportionately `mixing' the colors of these segments. Hence if a segment lies below two equally sized segments, one of which is `white' and the other of which is `gray', then we color the segment `light gray.' In our example, this specifies how to color the second ring

Figure 12

We repeat the procedure in order to color the first ring. Note, for example, that one segment lies below two segments, one of which is `light gray' and the other of which is `black.' The length of the light gray segment is one quarter of the length of the `black' segment. When we mix the colors, we get `gray'. Hence we color the segment `gray'. As before, if a segment lies below a segment which is completely `dark gray', then we color that segment `dark gray'.

Figure 13

When we get to the center, we look at the colors of the segments that surround the center. In our example, this segment is a decision segment. Hence the color of the center will be the lighter of the two colors present in the decision segment. Hence we color the center `gray'. This gives

Thus the value of this decision tree is gray---which corresponds to a payoff of 3³.

We have `solved' the decision tree without explicitly using any numbers! Note that in examining this circle, the human eye can immediately discern a small sliver of `white' and light brown against a background of dark colors. This immediately communicates the fact that R&D represents a small chance of a very good outcome and a large chance of a poor outcome.

3. Benefits of Decision Rings

3.1 Elimination of the Need for Quantification

Assessing an individual's subjective probability for an event is potentially rather tedious and involves comparison of the event with the sizes of various sectors in a single-stage doughnut chart. Decision rings skip the quantification step by directly representing the probability of an event through an appropriately sized sector in a multi-stage doughnut chart. In our tree, probabilities are represented as the lengths of arcs and payoffs as colors. Unlike the decision tree, there is no reason to write down numbers. This can be a tremendous asset with nontechnical audiences. In addition, it gives us the option of valuing consequences in terms of colors, as opposed to numbers. This creates the possibility of doing decision analysis without explicitly making use of any numbers. This might be helpful in those cases where clients feel that the value issues are too sensitive to be quantified in numbers.

3.2 Elimination of Artificial Precision

Individual impressions are usually quite imprecise. Because a number can be made infinitely precise, subjects can be misled into thinking that a unit difference in expected value between two outcomes implies a real difference in value between the payoffs. Since our approach avoids introducing numbers, it avoids introducing artificial precision. Hence if a subject cannot easily distinguish between the colors assigned to two different decisions, then the implication is that the subject, in practice, will not be able to distinguish between the practical value of those two decisions.

3.3 Elimination of Clutter

The standard decision tree begins with one node on the lefthandside of the tree and ends up with a possibly very large number of nodes on the righthandside. Spatially, this means that some of the space on the lefthandside of the diagram is wasted while much of the space on the righthandside looks cluttered. This arises because the decision tree allocates the same amount of space to the tenth stage of a tree as it allocates to the first stage of the tree.

In our representation, there is no waste of space. The lefthandside of the tree is placed in the tiny center of the circle. The more cluttered righthandside of the tree is now spread across the outer ring of the circle. If our classical decision tree is W units along the horizontal and W units along the vertical, decision rings represents this by a circle with a radius of (W/2) and a circumference of p W. Thus it more than triples the width of the space available for representing the last layer of the tree.

3.4 Allowing for More Decisions and Uncertainties

One important side-effect of reducing clutter is reducing the need to limit the number of decisions and uncertainties going into a decision tree. We can now visually represent much more complex problems with decision rings than is currently possible with a classical decision tree.

3.5 Allowing for an Arbitrary Number of Possible Outcomes

Many uncertainties have an infinite number of possible outcomes. In decision analysis, it's common to approximate that infinite range of outcomes by two or three representative outcomes(a pessimistic, optimistic and most likely outcome.) This reflects the fact that decision trees are best applied when each decision and each uncertainty only has a discrete number of possible outcomes.

But it's very simple to replace a white segment and a black segment in decision rings by a continuous band of color ranging from white to black. Hence one can use decision rings when uncertain outcomes have an infinite number of possible consequences just as easily as when they have a discrete number of outcomes.

3.6 Presenting Expected Value and Risk Profiles

In many applications of decision analysis, decision makers often focus too heavily on the overall expected value of the decision and not on the uncertain distribution of the payoffs from that decision. To counteract this overemphasis, it's also common to report the risk profile which reports the distribution of payoffs associated with each decision.

But in decision rings, the expected value is represented by the color of the innermost circle while the risk profile is described by the distribution of colors along the outmost ring. Hence both pieces of information are represented simultaneously. Indeed, the diagram represents the risk profiles associated with each of the various decisions in the decision tree. Because the diagram represents all this information in a single visual format, there's a much better chance that it will be properly factored into the decision.

3.7 Value of Information:

The value of information represents how much the value of a decision problem increases if a specific uncertainties could be resolved before a decision is made. In decision rings, this corresponds to simply moving a segment(or ring) representing an uncertainty into the interior of the circle. The value of such information corresponds to the amount by which the color of the inner circle lightens. Hence the value of information can be computed visually.

4. Summary

Most tools of operations research involve translating a subject's intuition about a problem situation into a mathematical model which is then analyzed. The results of the model are then translated back into intuitive terms. In some contexts, translating intuitions into mathematical terms leads to distortions, e.g., artificial precision, oversimplification of the decision problem, oversimplification in the range of possible outcomes associated with uncertainties.

This paper presents a way of reformulating the decision tool of operations research in a way that eliminates the need to create a mathematical model. The representation is completely visual. All calculations are done geometrically. Hence it can potentially allow for more realistic decision models which are simultaneously more communicable to nontechnical audiences.

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