Incorporation of uncertainty and risk attitude into multi-criteria forest planning
P. Leskinen 1 , J. Viitanen 2 , A. Kangas 3 and J. Kangas 4
1
Finnish Forest Research Institute, Joensuu Research Centre, P.O. Box 68, FIN-80101
Joensuu, Finland. e-mail: pekka.leskinen@metla.fi
2
Finnish Forest Research Institute, Joensuu Research Centre, P.O. Box 68, FIN-80101
Joensuu, Finland. e-mail: jari.viitanen@metla.fi
3
University of Helsinki, Faculty of Agriculture and Forestry, Department of Forest
Resource Management, P.O. Box 27, FIN-00014 University of Helsinki, Finland. e-mail:
annika.kangas@helsinki.fi
4
UPM Forest, P.O. Box 32, FIN-37601 Valkeakoski, Finland. e-mail: jyrki.kangas@upmkymmene.com
_______________________________________________________________________
Introduction
In multi-criteria forest planning, the performance of forest planning alternatives are evaluated
with respect to several, generally conflicting decision criteria. The Analytic Hierarchy Process
(AHP) (Saaty, 1977) gives an example of decision-support tools that can be used to solve
multi-criteria planning problems. It is based on pairwise comparison of forest plans on ratio
scale and estimation of ratio scale utility measures from the pairwise comparisons data.
Unfortunately, the calculations in the AHP are deterministic and possibilities to incorporate
uncertainties are limited. Stochastic Multiobjective Acceptability Analysis (SMAA) (Lahdelma
et al., 1998) is another method, which has been developed to also incorporate uncertain or
inaccurate data. This is a clear advantage, because the uncertainties are evident in the
assessment of decision maker's subjective preferences.
In addition to SMAA, another possibility to incorporate uncertainty is to analyse pairwise
comparisons data through regression models (e.g. Alho et al., 1996, Alho & Kangas, 1997,
Leskinen & Kangas, 1998, Alho et al., 2001, Leskinen et al., 2003, Leskinen et al., 2004).
The regression approach has turned out to be a flexible tool that enables versatile
possibilities in analysing uncertainties in the planning calculations.
Forest planning with uncertainty also reflects the forest owner’s attitude towards risk (e.g.
Pratt, 1964, Arrow, 1974). Forest owners are usually risk averse, i.e. high level of uncertainty
must be compensated by high expected outcome. Risk-seeking is also possible in some
special cases, i.e. a decrease in the expected outcome is acceptable if simultaneously the
level of uncertainty increases. Risk-neutral forest owner makes decisions based on expected
utility only.
This presentation evaluates alternatives to incorporate uncertainty into the regression
analysis of pairwise comparisons data so that also attitude towards risk can be taken into
account.
Regression model to analyse preference data
Let rij be the relative value of forest plan i compared to forest plan j evaluated by the
forest owner with respect to a single decision criterion such as timber harvesting income. For
example, rij = 2 / 1 means that the value of forest plan i is two times higher than the value of
forest plan j. Assume that rij = (vi / v j )exp(ε ij ), where vi and v j are the true and unknown
values of forest plans i and j, and ε ij measures the uncertainty or error with which the true
values are obtained. Then, by defining y ij = log (rij ), the regression model for the pairwise
comparisons data gets the form (Alho & Kangas, 1997)
yij = α i − α j + ε ij ,
where α i = log(vi ),
(1)
and the residuals
ε ij
are uncorrelated with
E (ε ij ) = 0 and
Var(ε ij ) = σ 2 . The parameters of model (1) can be estimated by ordinary least squares
(OLS). The model is used repeatedly in multi-criteria forest planning (e.g. Alho & Kangas,
1997, Leskinen, 2001) so that one model is constructed for the evaluation of decision
alternatives with respect to each decision criterion. Moreover, one additional regression
model is required to assess the importance of the criteria.
Uncertainty and risk attitude
Hypothesis testing
Among others, classical statistical inference provides hypothesis tests as one tool to
incorporate uncertainty into decision-making so that the null hypothesis H 0 : U i = U j , i.e.
the overall utility of forest plans i and j are equal, is tested against H A : U i ≠ U j , for
example. If the p-value of the test is small enough, the null hypothesis is rejected and it is
concluded that forest plans i and j are not equal. However, the hypothesis test study the
utility values of alternative forest plans and uncertainty is incorporated into the test as
uncertainty concerning the estimation of utilities from the pairwise comparisons data. This
indicates that risk attitude is not explicitly included in the hypothesis testing.
Indices derived from pairwise probabilities
Pairwise probabilities that forest plans beat each other (Alho & Kangas, 1997) can be utilised
in choosing the best alternative by summarising the information into different indices. One
possibility is to utilise the rules from pairwise voting in the form of Simpson score (e.g. Martin
et al., 1996). In the case of pairwise probabilities, it would be the minimum of pairwise
winning probabilities for one alternative and it can be expressed as
Simpson k = min( p (a k , a l )), l = 1,..., n, k ≠ l ,
(2)
where p (a k , a l ) is the probability of alternative k beating alternative l and n is the number
of alternatives. On the other hand, for example the Positive Outranking Index (POI) is
calculated as the mean of the observed pairwise probabilities of alternative k beating each of
the other alternatives l , i.e.
POI k =
1 n
∑ p(ak , al ) .
n − 1 l =1
(3)
k ≠l
The above scores differ with respect to risk aversion and all of them are more risk averse
than, for example, selecting the alternative having largest probability for the first rank. The
advantage is that they summarise a large number of pairwise probabilities into single
numerical measures.
Mean-variance utility
A common and widely accepted method is to reduce the decision rule to contain only the
expected value and the variance of the outcome (Markowitz, 1959) so that both the expected
value and the variance are used directly as decision criteria. If the forest owner's behaviour
can be characterised in this way, then a utility function for mean and variance will be able to
rank choices analogously to the standard von Neumann and Morgenstern’s (1944) utility
function. In other words, the utility can be expressed as U ( Mean, Std ) and e.g. the marginal
utilities
U ' Mean ( Mean, Std ) > 0, U ' Std ( Mean, Std ) < 0
(4)
imply that the higher the mean, the higher the utility, while a higher standard deviation
decreases the utility of the forest owner.
Discussion
With hypothesis testing, the incorporation of risk attitude is not straightforward, because
uncertainty measures concerning alternative forest plans have only a secondary role in the
analysis. The indices based on the pairwise winning probabilities have the advantage that
they summarise a large amount of information into single numerical measures. Different
indices measure the risk differently, but the risk attitude is hidden behind the expected
performance of decision alternatives. The advantage of using mean-variance utility is that
uncertainty and expected performance of forest plans are both measured separately. As a
consequence, the mean and the standard deviation can be presented in a single plot that
illustrates the performance of the forest plans in these two dimensions. Because the meanstandard deviation plot is easily interpreted, it can be given to the forest owner directly as a
final decision support to solve the planning problem.
The mean-variance-utility has also a problem when compared to the weighted average of
percentiles of the utility distributions or the rank probabilities (see Leskinen et al., 2005 for
the discussion of weighted averages in incorporation of the risk attitude). Consider a risk
seeking forest owner with two forest plans so that plan A has a small expected utility and a
large variance and plan B has a large expected utility and a small variance. Assume also
that plan B has larger minimum utility value than plan A's maximum value is. Then,
depending on the degree of risk seeking, it appears to be possible that the forest owner
chooses plan A, although it will produce smaller utility to the forest owner than plan B with
certainty. However, this is irrational and the theory of stochastic dominance prevents it by
comparing cumulative distribution functions of the random outcomes (Rothschield & Stiglitz,
1970). However, this requires that additional information would be given to the decision
maker besides the mean and the variance, and the analysis may eventually resemble the
weighted average analysis. It remains to be an empirical question when additional
information is needed and when and how it can be reliably assessed from the forest owner.
Reference
Alho, J.M.; Kangas, J. and Kolehmainen, O. (1996) Uncertainty in expert predictions of the
ecological consequences of forest plans. Applied Statistics 45: 1-14.
Alho, J.M. and Kangas, J. (1997) Analyzing uncertainties in experts’ opinions of forest plan
performance. Forest Science 43: 521-528.
Alho, J.M.; Kolehmainen, O. and Leskinen, P. (2001) Regression methods for pairwise
comparisons data. Schmoldt, D.L.; Kangas, J.; Mendoza, G.A. and Pesonen, M. (Eds.). The
Analytic Hierarchy Process in Natural Resource and Environmental Decision Making. Kluwer
Academic Publishers, 235-251.
Arrow, K. (1974) Essays in the theory of risk bearing. Amsterdam, North Holland.
Lahdelma, R.; Hokkanen, J. and Salminen, P. (1998) SMAA - Stochastic Multiobjective
Acceptability Analysis. European Journal of Operational Research 106: 137-143.
Leskinen, P. and Kangas, J. (1998) Analysing uncertainties of interval judgment data in
multiple-criteria evaluation of forest plans. Silva Fennica 32: 363-372.
Leskinen, P. (2001) Statistical methods for measuring preferences. University of Joensuu,
Publications in Social Sciences 48.
Leskinen, P.; Kangas, J. and Pasanen, A.-M. (2003) Assessing ecological values with
dependent explanatory variables in multi-criteria forest ecosystem management. Ecological
Modelling 170: 1-12.
Leskinen, P.; Kangas, A.S. and Kangas, J. (2004) Rank-based modelling of preferences in
multi-criteria decision making. European Journal of Operational Research 158: 721-733.
Leskinen, P.; Viitanen, J.; Kangas, A. and Kangas, J. (2005) Alternatives to incorporate
uncertainty and risk attitude in multi-criteria evaluation of forest plans. Manuscript.
Markowitz, H. (1959) Portfolio selection. Wiley, New York.
Martin, W.E.; Schields, D.J.; Tolwinski, B. and Kent, B. (1996) An application of social
choice theory to U.S.D.A. Forest Service decision making. Journal of Policy Modelling 18:
603-621.
Pratt, J. (1964) Risk aversion in the small and in the large. Econometrica 32: 122-136.
Rothschield, M. and Stiglitz, J. (1970) Increasing risk I: a definition. Journal of Economic
Theory 2: 225-243.
Saaty, T.L. (1977) A scaling method for priorities in hierarchical structures. Journal of
Mathematical Psychology 15: 234-281.
von Neumann, J. and Morgenstern, O. (1944) Theory of games and economic behavior.
Princeton N.J., Princeton University Press.