Fast and frugal heuristics for portfolio decisions with positive project interactions
Introduction
Portfolio decisions involve selecting a subset of alternatives or “projects” that together maximize some measure of value, subject to resource constraints [1]. Examples include capital investment [2,3], R&D project selection [[4], [5], [6], [7], [8]], maintenance planning [9], and windfarm location [10]. This paper considers portfolio problems in which benefits and costs are not necessarily additive: some projects may interact with one another.
Exact solutions to this problem require that all project interactions are assessed, and the time and effort involved in this can be considerable. As the starting point for this paper we take the view that in some problems project interactions can only be assessed by consulting a human decision maker or expert, and that sometimes the number of interactions will be too large for the assessment of all of them to be feasible. The purpose of this paper is to propose several heuristics that limit the number of assessments that are made and thus may be suitable for portfolio decision problems in which the complete assessment of interactions is not an option. We evaluate these heuristics in terms of how many assessments they save, and how close their portfolio values are to the theoretical optimal value that would be achieved if all interactions were known and exact methods used. We also use a behavioral laboratory experiment to provide evidence of behaviour that is consistent with using some of the proposed heuristics.
We draw a distinction between our heuristics and those developed in the optimization literature, where the problem above has been extensively studied for decades, either in its interaction-free version as the standard knapsack problem or, with some restrictions (value interactions involving pairs of projects only) as the quadratic knapsack problem. Exact algorithms (pseudo-polynomial in the standard case), efficient approximations, and numerous computational heuristics have been developed for both problems [11]. These require all interactions to be assessed upfront and their goal is to limit the amount of computation time required to solve the problem. This is important when the number of projects is very large, but less relevant when projects number in the tens or hundreds, as is typically the case for portfolio problems in which decision support is provided (see e.g. applications reported in Salo et al. [1]). In these cases using a computational heuristic is inappropriate – if all interactions can be assessed then an exact method should be used. The heuristics we propose address a different kind of time- and effort-saving to computational heuristics – time and effort in assessment – and are in the tradition of so-called fast and frugal heuristics [12] or psychological heuristics [13], which use limited information and process this information in computationally simple ways e.g. elimination-by-aspects Tversky [14], take-the-best [15]. These heuristics are typically not normative, but invoke bounded rationality arguments to argue for both potential prescriptive use (if environments in which cases good performance is obtained are known) and descriptive plausibility [15]. Different heuristics may of course vary in the degree to which they emphasise prescriptive or descriptive aspects [16,17].
Our heuristics construct a portfolio by iteratively adding a project that is best in a greedy (i.e. locally optimal) sense. Sharing this common structure, we collectively call them the add-the-best family of heuristics. For example, in a computationally demanding version of add-the-best, the “best” project is the one whose selection leads to the largest immediate increase in portfolio value, including the value added by project interactions. In computationally simpler heuristics, a best project is again one which leads to the largest immediate increase in portfolio value, but this is now calculated without considering interactions. Add-the-best heuristics are conceptually closely related to single-cue heuristics that make decisions using a single piece of information; in cases where this single piece of information does not discriminate among the projects, the heuristic decides randomly [18].
The primary goal of our paper is to extend fast and frugal heuristics, which have been extensively studied in traditional choice problems, to portfolio decision making involving project interactions. We find that, in contrast to choice problems, where simple heuristics often perform unexpectedly well (e.g. [17,18]), it is much harder to strike a balance between frugal information use and good performance in portfolio problems. Our main contribution is to develop two heuristics called Added Value and Unit Value with Synergy that achieve this balance, returning portfolios that are competitive with those obtained by exact methods while limiting the number of assessments to potentially manageable levels. These heuristics combine descriptive plausibility with effort-accuracy trade-offs that make them potentially attractive for prescriptive use in cases where complete assessement of interactions is not feasible.
Section snippets
Portfolio decision making
Stummer and Heidenberger [19] describe the formulation of the portfolio decision problem with interactions, whose goal is to decide which projects to select from a set of candidates {P1, …, PJ}, so as to maximize the overall value of the portfolio subject to budget and any other constraints. Interactions between projects are modelled by defining interaction subsets containing those projects making up interaction k = 1, …, K. A set is defined for each subset of projects whose total value or
Proposed fast and frugal portfolio heuristics
In this section we propose a family of fast and frugal heuristics for selecting portfolios. A numerical example illustrating each heuristic is given in Appendix A. The heuristics are frugal in that they do not use all of the available information, and fast because they integrate the information in simple ways to decide which project to include next, and when to stop. All except one use a single well-defined criterion in adding projects to the portfolio, extending single-cue heuristics developed
Analytical results on information requirements
Exact methods require the assessment of all m-way interactions up to order M. Assuming that M is somehow known, this equates to interactions. While many of these interactions could easily be ruled out by statements such as “project X does not interact with any other project”, the number of interactions provides a useful baseline for comparison with heuristics.
How much information do the add-the-best heuristics use? Let P(s) denote the s-th project added, and denote the set of J − s
Simulation-based comparison of heuristic and optimal portfolios
In previous sections we proposed a number of fast and frugal heuristics for portfolio selection, and showed that these have relatively low information requirements. In this section we evaluate the ability of these heuristics to achieve overall portfolio values comparable with those obtained by optimal portfolios. Our simulation structure consists of (a) generating a number of projects and their individual values and costs, (b) creating interdependencies between the projects, (c) defining the
Task description
We presented 75 participants with two versions of a simple portfolio selection task (the same one used in the numerical illustration in Appendix A). One version of the task was exactly the same as the example (Task 2); in the other version no project interactions were present (Task 1). Participants saw tasks in random order, were students from the African Institute of Mathematics and the University of the Western Cape, and were paid approximately $4 for their participation. Data collection
Conclusions and further research
Portfolio decisions are an important and increasingly studied class of decision problem, with optimization models developed for a variety of settings (e.g. [1,10,24]). We see two gaps in this literature. Firstly, portfolio optimization typically means that one has to assess all project interactions. The effort involved in this can be considerable and, even in a prescriptive setting, it is reasonable that decision makers might want to limit this. There is currently relatively little guidance
Acknowledgements
ID is supported in part by funding from the National Research Foundation of South Africa (Grant ID 90782, 105782).
Ian N. Durbach is adjunct associate professor in the Centre for Statistics in Ecology, the Environment and Conservation, Department of Statistical Sciences, University of Cape Town, South Africa, and research fellow in the Centre for Research into Ecological and Environmental Modelling, School of Mathematics and Statisics, University of St Andrews, UK. His research area investigates simplified approaches to decision support.
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Ian N. Durbach is adjunct associate professor in the Centre for Statistics in Ecology, the Environment and Conservation, Department of Statistical Sciences, University of Cape Town, South Africa, and research fellow in the Centre for Research into Ecological and Environmental Modelling, School of Mathematics and Statisics, University of St Andrews, UK. His research area investigates simplified approaches to decision support.
Sim'on Algorta is a postdoctoral research fellow in the Faculty of Mechanical Engineering and Transport Systems, Technische Universit¨at Berlin, Germany. He did his PhD at the Max Planck Institute Centre for Adapative Behaviour and Cognition. His research interests include sequential decision making and bounded rationality.
Dieudonn'e Kabongo Kantu a PhD student in the Department of Statistical Sciences, University of Cape Town, supported by the African Institute for Mathematical Sciences. He is also a statistician at Ipsos Laboratories, Cape Town. His dissertation topic is on simplified approaches to portfolio decision making.
Konstantinos V. Katsikopoulos is associate professor of behavioral operations at the Centre for Operational Research, Management Science and Information Systems, University of Southampton Business School, UK, and was previously at the Max Planck Institute Centre for Adapative Behaviour and Cognition. His research integrates standard decision theory with simpler boundedly rational models.
Özgür Şimşek is senior lecturer in the Department of Computer Science, University of Bath, UK, and was previously at the Max Planck Institute Centre for Adapative Behaviour and Cognition. Her research interests include machine learning, bounded rationality, and reinforcement learning.