Decision Flexibility

The development of new methods and rep­ resentations for temporal decision-making requires a principled basis for characterizing and measuring the flexibility of decision strategies in the face of uncertainty. Our goal in this paper is to provide a framework not a theory for observing how decision policies behave in the face of informational perturbations, to gain clues as to how they might behave in the face of unanticipated, possibly unartic­ ulated uncertainties. To this end, we find it beneficial to distinguish between two types of uncertainty: "Small World" uncertainty and "Large World" uncertainty. The first type can be resolved by posing an unambiguous question to a "clairvoyant," and is anchored on some well-defined aspect of a decision frame. The second type is more troublesome, yet it is often of greater interest when we address the issue of flexibility; this type of uncertainty can be resolved only by consulting a "psychic." We next observe that one approach to flexibility frequently used in the economics literature is already implicitly accounted for in the Maximum Expected Utility (MEU) princi­ ple from decision theory. Though simple, the obser­ vation establishes the context for a more illuminating notion of flexibility, what we term flex­ ibility with respect to information revelation. We show how to perform flexibility analysis of a static (i.e., single period) decision problem using a simple example, and we observe that the most flexible alter­ native thus identified is not necessarily the MEU alternative. We extend our analysis for a dynamic (i.e., multi-period) model, and we demonstrate how to calculate the value of flexibility for decision strat­ egies that allow downstream revision of an upstream commitment decision. 1.0 Flexibility, Uncertainty, and Information Researchers in decision-making under uncertainty have recognized the importance of developing models and methods that more adequately address the notion of time: How do decisions taken now shape or constrain decision opportunities confronted in the future? Whether we are building decision analyses to clarify human-oriented pol­ icy decisions or algorithms to approximate intelligent agency in a robot, the question poses a significant chal­ lenge for any methodology which claims to provide a complete, normative basis for action. Planning systems, for example, need rigorous methods for judging whether a particular plan is more "brittle" or less "flexible" than another, as well as sensible metrics for quantifying those features precisely. Decision consultants, meanwhile, increasingly focus on building strategies that create and sustain value for a decision maker or an organization over time, strategies that are somehow robust in the face of rap­ idly changing circumstances and unanticipated outcomes. 78 Chavez and Shachter The reason we care about notions such as flexibility, brit­ tleness, and robustness in decision-making is that the world is uncertain. If the world were perfectly determinis­ tic -or even approximately so we could simply order our actions to respond to future events using propositional logic. For the many problem areas where such reasoning falls short, we model uncertainty using the axiomatic framework of probability theory. Utility theory provides a normative basis for action using the probabilities assessed by a decision maker. Flexibility enters the discussion because we would like our actions to accommodate uncer­ tain outcomes, and even better, to respond to uncertainties that we perhaps do not explicitly consider at the time we make a decision. As the mechanism through which we reduce uncertainty, information ought to figure promi­ nently in any analysis of flexibility: given more informa­ tion, we might do otherwise and thereby achieve higher value; the revelation of information downstream of a deci­ sion often creates or constrains opportunities to respond effectively to our upstream commitments. Practicing decision analysts use a variety of sensitivity techniques to determine which sources of uncertainty weigh most heavily in the identification of optimal courses of action (see, for example, [Morgan and Henrion, 1990]). Deterministic perturbation, proximal analysis, and rank­ order correlation are useful tools for measuring the relative importance of different uncertainties in a decision model. Value of information is the most powerful approach to sensitivity analysis because it measures not just whether uncertainty in an input variable could affect the output value, but rather whether reducing uncertainty in the input variable could change the recommended decision. Recent research provides efficient techniques for estimating infor­ mation value in very large models [Chavez and Henrion, 1994]. All such sensitivity methods, however, occupy a separate phase of the decision analysis cycle or a special function of a decision support system, in both cases usually at what we might call the back end. They tell a decision maker which uncertainties matter, but they do not provide a nec­ essary loop back to the front end: How should a decision maker use sensitivity measures to gain insight into the rec­ ommended action, or to craft better strategies? How should a decision maker use sensitivity results on vari­ ables to identify strengths and weaknesses in decisions? So far, sensitivity methods provide clues, but no comprehensive basis, for measuring the relative robustness or flexibility of competing plans. In this paper, we study the issue of flexibility using deci­ sion analysis. Our goal is to provide a framework not a theory -for observing and measuring how decision poli­ cies respond to informational perturbations, to gain clues as to how they might respond to unanticipated, possibly unarticulated uncertainties. In Section 2, we distinguish between two central types of uncertainty, in an effort to delineate a more precise sense in which flexibility analysis is possible. In Section 3, we present and motivate a defini­ tion of flexibility developed in the economics literature, and we show how decision theory implicitly accommo­ dates it. In simple terms, decision theory delivers one type of flexibility -what we call flexibility with respect to values on outcomes (F vo>"for free ." Though simple, the analysis sets the stage for a more illuminating notion of flexibility which explicitly takes information and uncer­ tainty into account: flexibility with respect to informa­ tion revelation (F1a). In Section 4, we demonstrate how to measure this type of flexibility in a static (i.e., single period) decision model, using a canonical problem from decision analysis. We observe that the most flexible action thus identified is not necessarily the MEU (Maximum Expected Utility) alternative. In Section 5, we extend the analysis for a dynamic model which allows a downstream revision of an upstream commitment decision. 2.0 Distinctions about Uncertainty We can think of flexibility generally as "the ability to adapt to changing circumstances" [Mandelbaum, 1978]. Changing circumstances are just the outcomes of random variables: for example, the interest rate suddenly dips, your car's electrical system suddenly blows a fuse, or your apartment is damaged in an earthquake. Adapting to changing circumstances requires that the actions you take now allow you to respond effectively to new discoveries or new situations in the future. As observed in [Ghemawat, 1991], " ... a strategic option has flexibility value not because it is a sure thing but to the extent that it is an abun­ dant store of potentially valuable revision possibilities." For example, if I am attempting to decide what car to buy, I might buy a Jeep instead of a Cadillac if I anticipate hav­ ing to drive along treacherous mountain roads frequently in the future. If an agent is attempting to decide what resources to gather from its environment, it needs to determine which resources will best ensure its long-term sur­ vival given reasonable expectations about future states of the world. We find it useful to fix ideas as follows: "Flexibility is the ability to achieve greater value given the revelation of missing but knowable information down­ stream." 2.1 Small Worlds, Large Worlds, Clairvoyants, and Psychics Missing but knowable information can assume several forms. In addressing issues of flexibility, we have found it useful to distinguish between two broad categories of information or uncertainty: "Small World" and "Large World."1 We use the term "Small World" to bound the type of uncertainty to which it corresponds: Small World uncertainty can be resolved by observing an outcome on a variable which is clearly and explicitly defined by a deci­ sion-making agent. Decision analysts often use the con­ cept of the clairvoyant to pose and resolve information questions. The clairvoyant is a thought experiment (similar to Maxwell's Demon, for example), a hypothetical person who can answer questions relevant to a decision problem. For example, if you are betting on coin flips, then the clair­ voyant can tell you the outcome of the next flip. The clair­ voyant cannot tell you whether you should consider betting on horses rather than coin flips. Nor can the clair­ voyant offer information, e.g., he cannot suddenly tell you that the next coin flip will turn up something besides heads or tails. He can only answer unambiguous questions of fact. Another type of missing but knowable information relates to those alternatives, uncertainties, or preferences that we have not explicitly defined or articulated. This type of information corresponds to Large World uncertainty. For example, knowing that the next coin flip will cause your opponent to suffer a heart attack at the gambling table cer­ tainly counts as missing, knowable information, but it probably is not anything we explicitly include in our bet­ ting analysis. In general, Large World uncertainties can be resolved only by posing open-ended questions to a psy­ chic: for example, "Will anything strange happen when I next flip this coin?" Hypothetical answer: "Yes, your opponent will suffer a heart attack while it is in the air." 1. For treatment of a similar distinction, see [Laskey, 1992a,b]. Decision Flexibility 79 Large World uncertainties are those uncertainties that we have not already specified, even though their outcomes could significantly affect the value we achieve. Missing but knowable information provided by a clairvoy­ ant corresponds exactly to the class of Small World uncer­ tainties. Missing but knowable information provided by a psychic corresponds exactly to the class of Large World uncertainties. 2.2 Missionaries and cannibals revisited The Large World/Small World distinction is key because it draws the line between what's possible and what's infeasi­ ble for flexibility analysis. We worry about flexibility in decision-making because we want our decision strategies to be responsive to a range of uncertain outcomes. But such strategies can be responsive only with respect to uncertainty that we deliberately, explicitly articulate. It is perhaps useful to think of an analogue to the well­ known AI Frame Problem in this context: Suppose you are charged with the task of transporting a group of missionar­ ies safely between opposite sides of a river. Cannibals lurk in the bushes at both banks. A decision-analytic treatment of the problem would take account of the risks of encoun­ tering cannibals at different locations along the banks, given rustling movements in the bushes, say, and the dis­ utility of getting shot by a cannibal's arrow. In implement­ ing a decision strategy for transporting the missionaries, you might attempt to take action that flexibly accommo­ dates all manner of bizarre risks. For example, you might worry about the possibility of an oar snapping in half or the possibility of a sudden, violent thunderstorm erupting while you are crossing the river. Yet it is clearly impossibleand certainly impractical­ to include all such remotely relevant, low-probability uncertainties in your analysis. Suppose for the moment that you construct a plan flexible enough to handle a snap­ ping oar or a violent thunderstorm, and you begin to cross the river. In the middle of the river, your boat pops a leak and sinks; the missionaries drown and die, while the can­ nibals at the banks wail and wring their hands at their loss. Regardless of how flexible you thought your plan was, it certainly was not flexible enough to accommodate this bizarre circumstance. 80 Chavez and Shachter It is important to observe that sometimes the unanticipated event can be beneficial. For example, the cannibals might be away for the wedding of their chieftain's daughter in a nearby village. Yet this is probably outside the realm of anything a reasonable person would include in a plan for transporting missionaries. Trying to accommodate all pos­ sible uncertainty particularly Large World uncertainty -in flexibility analysis is infeasible, and probably inco­ herent in any case. The difficulty with this is that people often evaluate actions retroactively with respect to out­ comes on Large World uncertainties. For example, if the boat pops a leak and sinks, then you are accused of imple­ menting a ''bad" plan to save the missionaries. If you are a decision analyst, then you respond that a leak in the boat was not even in the realm of discourse at the time you for­ mulated the model; the event was vaguely relevant to the decision, but only marginally more relevant than a host of other variables which appear natural to eliminate from the analysis. The best we can doand this is the thrust of the framework we propose in this paper-is to analyze flexi­ bility with special attention to Small World uncertainty, in an effort to gain clues as to how strategies thus analyzed will behave in the face of Large World uncertainty. 3.0 Flexibility on Outcomes We define a decision problem as a triple [O(X), O(D), v(D,X)], where X is a vector of state variables [X1, ... .Xn1• n (X) is the space of outcomes over X, and D is a single decision with m alternatives [d1 , ... ,dm1· We will use capital letters to denote a variable or decision, and lower-case let­ ters to denote a corresponding value or alternative, respec­ tively: e.g., D denotes the decision, and d1 denotes a particular alternative for it; X is a state variable, and x denotes a particular value for X. We will also use O(X) to denote the set of possible values for X, where X can be a state variable or a decision: e.g., de n (D) and x e n (X) . The value function v(d,x) specifies the pay­ off/value/utility when action dis taken and outcome x e 0 (X) obtains. X may be defined probabilistically. If X is a random vari­ able, then P{XI�} denotes a probability mass or probabil­ ity density assignment on X, conditional on �, our prior state of knowledge. We will use E[v(D,X)I�] to denote the expected value of v(D,X). We will also use Ex[v(D,X)II�] to denote the same measure, subscripting by X to indicate that the expectation is taken with respect to X. Stigler [1939] presents a useful and intuitive notion of flexibility, which has been studied and extended by a num­ ber of other researchers (see, e.g., Marschak and Nelson [1962], Jones and Ostroy [1984], Epstein [1980], and Merkhofer [1975]). Stigler characterizes the flexibility of two alternative plants using the second derivative of their total cost curves: a less flexible Plant A has a second deriv­ ative that is strictly greater than the second derivative cor­ responding to a more flexible Plant B, for all possible output values X. Figure I demonstrates the idea. FIGURE 1. Stigler's approach to flexibility.