A Smooth Model of Decision Making under Ambiguity

We propose and characterize a model of preferences over acts such that the decision maker prefers act f to act g if and only if 𝔼_μφ(𝔼_πu○f) 𝔼_μφ(𝔼_πu○g), where 𝔼 is the expectation operator, u is a von Neumann–Morgenstern utility function, φis an increasing transformation, and μis a subjective probability over the set Πof probability measures πthat the decision maker thinks are relevant given his subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective beliefs, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes toward pure risk are characterized by the shape of u, as usual, while attitudes toward ambiguity are characterized by the shape of φ. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures Π. One advantage of this model is that the well‐developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's paradox). The maxmin expected utility model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as a limiting case of our model with infinite ambiguity aversion. Two illustrative portfolio choice examples are offered.     

Vector Expected Utility and Attitudes Toward Variation

This paper proposes a model of decision under ambiguity deemed vector expected utility, or VEU. In this model, an uncertain prospect, or Savage act, is assessed according to (a) a baseline expected‐utility evaluation, and (b) an adjustment that reflects the individual's perception of ambiguity and her attitudes toward it. The adjustment is itself a function of the act's exposure to distinct sources of ambiguity, as well as its variability. The key elements of the VEU model are a baseline probability and a collection of random variables, or adjustment factors, which represent acts exposed to distinct ambiguity sources and also reflect complementarities among ambiguous events. The adjustment to the baseline expected‐utility evaluation of an act is a function of the covariance of its utility profile with each adjustment factor, which reflects exposure to the corresponding ambiguity source. A behavioral characterization of the VEU model is provided. Furthermore, an updating rule for VEU preferences is proposed and characterized. The suggested updating rule facilitates the analysis of sophisticated dynamic choice with VEU preferences.     

Partial Ambiguity

We extend Ellsberg's two‐urn paradox and propose three symmetric forms of partial ambiguity by limiting the possible compositions in a deck of 100 red and black cards in three ways. Interval ambiguity involves a symmetric range of 50 − n to 50 + n red cards. Complementarily, disjoint ambiguity arises from two nonintersecting intervals of 0 to n and 100 − n to 100 red cards. Two‐point ambiguity involves n or 100 − n red cards. We investigate experimentally attitudes towards partial ambiguity and the corresponding compound lotteries in which the possible compositions are drawn with equal objective probabilities. This yields three key findings: distinct attitudes towards the three forms of partial ambiguity, significant association across attitudes towards partial ambiguity and compound risk, and source preference between two‐point ambiguity and two‐point compound risk. Our findings help discriminate among models of ambiguity in the literature.     

Perceived Ambiguity and Relevant Measures

We axiomatize preferences that can be represented by a monotonic aggregation of subjective expected utilities generated by a utility function and some set of i.i.d. probability measures over a product state space, S^∞. For such preferences, we define relevant measures, show that they are treated as if they were the only marginals possibly governing the state space, and connect them with the measures appearing in the aforementioned representation. These results allow us to interpret relevant measures as reflecting part of perceived ambiguity, meaning subjective uncertainty about probabilities over states. Under mild conditions, we show that increases or decreases in ambiguity aversion cannot affect the relevant measures. This property, necessary for the conclusion that these measures reflect only perceived ambiguity, distinguishes the set of relevant measures from the leading alternative in the literature. We apply our findings to a number of well‐known models of ambiguity‐sensitive preferences. For each model, we identify the set of relevant measures and the implications of comparative ambiguity aversion.     

Risk Attitude,Ambiguity Intolerance and Decision Making: An Exploratory Investigation*

Although recent research has identified attitudes towards ambiguity and risk to be important determinants of choice behavior [8] [18], no prior work jointly assessed the roles of both attitudes. We conducted a laboratory experiment using a real decision scenario and conducted exploratory analyses of the relationship between attitudes towards risk and ambiguity and the decision taken by the subjects. The results support the prediction that attitudes towards both risk and ambiguity affect choice behavior. Our exploratory analyses indicate interesting avenues for future research, including an examination of the decision process itself.     

PREFERENCE MAPPING OF PERCEIVED INFORMATION STRUCTURE: IMPLICATIONS FOR DECISION SUPPORT SYSTEMS DESIGN

In a recent issue of this journal, Watkins [13] presented an approach for discovery of decision-maker perceptions of the complexity (dimensionality) of information items that might be supplied by a decision support system. Through use of multidimensional scaling and cluster analysis, relatively homogeneous groups of decision makers, sharing common perceptions of various information items, were formed. This prior research was referred to as a first step in suggesting that information reports could be tailored to groups of decision makers classified on the basis of common perceptions of information. The current research extends the prior study by evaluating decision maker preferences for information in a variety of decision-making scenarios in relation to the previously identified perceptions of the information. Based on the results of the study, conclusions are made which suggest that the tailoring of information to groups of decision makers should be based on both perceptions and preferences for information. Even so, it is demonstrated that the decision tasks have an impact on the preferences for information which may affect the attempt to tailor information to groups of decision makers.     

Risk,Ambiguity, and Decision Choice: Some Additional Evidence*

This paper extends prior research by jointly assessing the roles of risk attitude and tolerance for ambiguity in predicting choice. An experiment examined the effects of these variables on decisions made in four different scenarios. The four scenarios (treatment combinations) were generated by manipulating risk and ambiguity into two levels (high and low). The context was defined in terms of a sample size selection problem. The second issue explored was the effect of attitudes toward risk and ambiguity on decision confidence. The results indicate that (1) both risk attitude and ambiguity intolerance determined choice behavior, (2) the roles of these individual attitudes depend on the levels of the two treatment variables of risk and ambiguity, (3) the presence of ambiguity accentuates the perception of risk in individual subjects, and (4) decision makers who are less risk averse, and have more tolerance for ambiguity, display greater confidence in their choice. The paper discusses some of the managerial implications of the results.     

Alpha as Ambiguity: Robust Mean‐Variance Portfolio Analysis

We derive the analogue of the classic Arrow–Pratt approximation of the certainty equivalent under model uncertainty as described by the smooth model of decision making under ambiguity of Klibanoff, Marinacci, and Mukerji (2005). We study its scope by deriving a tractable mean‐variance model adjusted for ambiguity and solving the corresponding portfolio allocation problem. In the problem with a risk‐free asset, a risky asset, and an ambiguous asset, we find that portfolio rebalancing in response to higher ambiguity aversion only depends on the ambiguous asset's alpha, setting the performance of the risky asset as benchmark. In particular, a positive alpha corresponds to a long position in the ambiguous asset, a negative alpha corresponds to a short position in the ambiguous asset, and greater ambiguity aversion reduces optimal exposure to ambiguity. The analytical tractability of the enhanced Arrow–Pratt approximation renders our model especially well suited for calibration exercises aimed at exploring the consequences of model uncertainty on equilibrium asset prices.     

Ellsberg Revisited: An Experimental Study

An extension to Ellsberg's experiment demonstrates that attitudes to ambiguity and compound objective lotteries are tightly associated. The sample is decomposed into three main groups: subjective expected utility subjects, who reduce compound objective lotteries and are ambiguity neutral, and two groups that exhibit different forms of association between preferences over compound lotteries and ambiguity, corresponding to alternative theoretical models that account for ambiguity averse or seeking behavior.     

Expected Uncertain Utility Theory

We introduce and analyze expected uncertain utility (EUU) theory. A prior and an interval utility characterize an EUU decision maker. The decision maker transforms each uncertain prospect into an interval‐valued prospect that assigns an interval of prizes to each state. She then ranks prospects according to their expected interval utilities. We define uncertainty aversion for EUU, use the EUU model to address the Ellsberg Paradox and other ambiguity evidence, and relate EUU theory to existing models.     

THE EFFECTS OF TWO NORMATIVE STRUCTURAL INTERVENTIONS ON ESTABLISHED AND AD HOC GROUPS: IMPLICATIONS FOR IMPROVING DECISION MAKING EFFECTIVENESS1

This paper focuses on a direct comparison of consensual, nominal, and conventional decision making techniques in established and ad hoc groups. The impact of the structural interventions on group decision quality and group attitudes is examined, and the appropriateness of the techniques in various situations is discussed.     

Finding the anti-block vital edge of a shortest path between two nodes

Let P _G (s,t) denote a shortest path between two nodes s and t in an undirected graph G with nonnegative edge weights. A detour at a node u∈P _G (s,t)=(s,…,u,v,…,t) is defined as a shortest path P _G−e (u,t) from u to t which does not make use of (u,v). In this paper, we focus on the problem of finding an edge e=(u,v)∈P _G (s,t) whose removal produces a detour at node u such that the ratio of the length of P _G−e (u,t) to the length of P _G (u,t) is maximum. We define such an edge as an anti-block vital edge (AVE for short), and show that this problem can be solved in O(mn) time, where n and m denote the number of nodes and edges in the graph, respectively. Some applications of the AVE for two special traffic networks are shown. This research is supported by NSF of China under Grants 70471035, 70525004, 701210001 and 60736027, and PSF of China under Grant 20060401003.     

Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u _ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector [^(u)]\hat{u} for the arc set A such that a given feasible flow [^(x)]\hat{x} is optimal with respect to the modified capacities. Among all capacity vectors [^(u)]\hat{u} satisfying this condition, we would like to find one with minimum ||[^(u)]-u||\|\hat{u}-u\| value. We consider two distance measures for ||[^(u)]-u||\|\hat{u}-u\| , rectilinear (L ₁) and Chebyshev (L _∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP\mathcal{NP} -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.     

Which trees have a differentiating-paired dominating set?

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. The set S is called a differentiating-paired dominating set if for every pair of distinct vertices u and v in V(G), N[u]∩S≠N[v]∩S, where N[u] denotes the set consisting of u and all vertices adjacent to u. In this paper, we provide a constructive characterization of trees that do not have a differentiating-paired dominating set.     

The Microeconomics of Efficient Group Behavior: Identification1

Consider a group consisting of S members facing a common budget constraint p'ξ=1: any demand vector belonging to the budget set can be (privately or publicly) consumed by the members. Although the intragroup decision process is not known, it is assumed to generate Pareto‐efficient outcomes; neither individual consumptions nor intragroup transfers are observable. The paper analyzes when, to what extent, and under which conditions it is possible to recover the underlying structure—individual preferences and the decision process—from the group's aggregate behavior. We show that the general version of the model is not identified. However, a simple exclusion assumption (whereby each member does not consume at least one good) is sufficient to guarantee generic identifiability of the welfare‐relevant structural concepts.     

Lower bounds and a tabu search algorithm for the minimum deficiency problem

An edge coloring of a graph G=(V,E) is a function c:E→ℕ that assigns a color c(e) to each edge e∈E such that c(e)≠c(e′) whenever e and e′ have a common endpoint. Denoting S _v (G,c) the set of colors assigned to the edges incident to a vertex v∈V, and D _v (G,c) the minimum number of integers which must be added to S _v (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum ∑ _v∈V D _v (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.     

Illustration of a Method to Incorporate Preference Uncertainty in Benefit–Cost Analysis

Benefit–cost analysis is widely used to evaluate alternative courses of action that are designed to achieve policy objectives. Although many analyses take uncertainty into account, they typically only consider uncertainty about cost estimates and physical states of the world, whereas uncertainty about individual preferences, thus the benefit of policy intervention, is ignored. Here, we propose a strategy to integrate individual uncertainty about preferences into benefit–cost analysis using societal preference intervals, which are ranges of values over which it is unclear whether society as a whole should accept or reject an option. To illustrate the method, we use preferences for implementing a smart grid technology to sustain critical electricity demand during a 24‐hour regional power blackout on a hot summer weekend. Preferences were elicited from a convenience sample of residents in Allegheny County, Pennsylvania. This illustrative example shows that uncertainty in individual preferences, when aggregated to form societal preference intervals, can substantially change society's decision. We conclude with a discussion of where preference uncertainty comes from, how it might be reduced, and why incorporating unresolved preference uncertainty into benefit–cost analyses can be important.     

A Note on Decoding of Superimposed Codes

A superimposed code with general distance D can be used to construct a non-adaptive pooling design. It can then be used to identify a few unknown positives from a large set of items by associating naturally an outcome vector u. A simple method for decoding the outcome vector u is given whenever there are at most errors occuring in the outcome vector u. Moreover, another simple method of detecting whether there is any error occuring in the outcome vector u is also given whenever there are at most D – 1 errors in u. Our method is a generalization of the classical result of Kautz and Singleton (Nonadaptive binary superimposed codes, IEEE Trans. Inform. Theory, vol. 10, pp. 363–377, 1964).