Selecting a portfolio of actions with incomplete and action

Selecting a portfolio of actions with incomplete and
action-dependent scenario probabilities
Eeva Vilkkumaa∗, Juuso Liesiö, Ahti Salo
Systems Analysis Laboratory
Department of Mathematics and Systems Analysis
Aalto University School of Science
P.O. Box 11100, 00076 Aalto, Finland
Abstract
In order to deal with major changes in the operational environment, organizations can use
scenario planning to (i) build scenarios that characterize different future states of this environment, (ii) assign probabilities to these scenarios, (iii) evaluate the performance of alternative actions across the scenarios, and (iv) select those actions that are expected to
perform best. In this paper, we develop a portfolio model to support the selection of such
actions when (i) information about the scenario probabilities is possibly incomplete and (ii)
some actions can affect these scenario probabilities. This model helps select action portfolios
which are resilient in that they perform relatively well in view of all available probability
information, and proactive in that the actions they contain can help steer the future towards
the desired direction. The model is illustrated with an example based on a reported case
study for the selection of R&D projects at a high-tech company.
Keywords: decision analysis, portfolio selection, scenarios, incomplete probabilities
1. Introduction
The ability to respond to major changes in the operational environment is important
to organizations such as businesses, nation states, and international institutions. These organizations take strategic actions to mitigate threats and to seize opportunities offered by
the changing environment. Traditional strategic planning approaches for selecting a combination or portfolio of such actions build on forecasts based on trend extrapolation. Such
approaches are, however, inadequate in highly uncertain, intensive and complex environments (Meristö, 1989; Mobasheri et al., 1989; Bunn and Salo, 1993; Chermack et al., 2001;
Corresponding author. Tel.: +358-50-412 8066; Fax: +358-9-863 2048.
Email addresses: [email protected] (Eeva Vilkkumaa), [email protected] (Juuso
Liesiö), [email protected] (Ahti Salo)
∗
Preprint submitted to European Journal of Operations Research
March 5, 2014
Varum and Melo, 2010). Consequently, strategic planning in organizations has increasingly
been complemented and even replaced by scenario planning, which, instead of focusing on
the future that is perceived as the most likely, considers a set of plausible futures, called
scenarios (Schnaars and Berenson, 1986; Schoemaker, 1995; Peterson et al., 2003). Scenarios
draw the decision-maker’s (DM’s) attention to uncertainties and help her build resilient action portfolios that perform relatively well across different operational environments (Wilson,
2000; Lempert et al., 2006).
A conventional approach to scenario-based action portfolio selection is to assess the probability of the scenarios, to evaluate the impacts of the actions in each scenario, and, finally, to
select those actions that, in light of the available information, have the highest expected performance across the scenarios (e.g., Poland, 1999). Yet, this approach poses some challenges.
First, it may be difficult to estimate precisely how probable the different scenarios are – for
instance, to say that the probability of a major legislative change that shapes the operational
environment is precisely 40%. Second, the actions may affect the scenario probabilities: for
instance, investments in an operating system by major smartphone manufacturers may increase the probability that this system becomes the industry standard (Toppila et al., 2011).
Failing to account for the impacts of such proactive actions may lead to suboptimal decisions.
In this paper, we develop a scenario model to support the selection of portfolios consisting
of actions when (i) information about the scenario probabilities is incomplete, and (ii) the
selected actions may affect these probabilities. The incomplete probability information is
modeled by bounding the set of feasible probabilities through constraints that may depend
on which actions are selected. Dominance relations are employed to identify portfolios which
are not outperformed by any other portfolio given any feasible scenario probabilities. There
is no need to carry out exhaustive pairwise comparisons among all possible action portfolios
because (i) the set of feasible portfolios is partitioned with respect to those actions that affect
the scenario probabilities, and (ii) a multi-objective zero-one linear programming (MOZOLP)
problem is formulated within the partitioned sets to identify those non-dominated portfolios
that satisfy the resource and other feasibility constraints.
In particular, our model provides recommendations for choosing action portfolios that are
(i) resilient across the range of future scenarios in light of the incomplete information about
scenario probabilities, and (ii) proactive in that they help steer the course of the change
by influencing these probabilities. The resulting decision recommendations help prioritize
individual actions by dividing them into three categories: (i) core actions that should be
selected (included in all non-dominated portfolios), (ii) exterior actions that should not be
selected (not included in any non-dominated portfolios), and (iii) borderline actions (included
in some non-dominated portfolios but not all).
2
The rest of the paper is structured as follows. Section 2 discusses earlier literature
on scenario-based action portfolio selection. Section 3 introduces the model framework.
Section 4 presents an illustrative example. Section 5 discusses the implementation of the
model and the accommodation of risk preferences, and Section 6 concludes.
2. Earlier approaches to scenario-based strategy development
Scenario planning emerged in the aftermath of World War II as a method for military
planning. Later, it was extended to support social forecasting, public policy, and strategic
management (Bunn and Salo, 1993; Van der Heijden, 1996; Varum and Melo, 2010). The
early scenario planning methodologies can be divided into three schools. First, there is the
intuitive logics school comprising of qualitative methods for developing scenarios and evaluating strategies against these scenarios (Bunn and Salo, 1993; Bradfield et al., 2005). Second,
the probabilistic modified trends school generates scenarios by asking experts to provide subjective probability estimates about the occurrence of unprecedented events. Trend-impact
analysis (TIA) uses these probabilities and the expected impacts of the events to perturb
trends extrapolated from historical data. Cross-impact analysis (CIA) incorporates an additional layer of complexity in that also the probabilities of the events conditioned on the
occurrence or non-occurrence of some other events are elicited (Godet, 1987; Bradfield et
al., 2005). The third school, La Prospective (Godet, 2000), can be viewed as an elaborate,
complex and somewhat mechanistic blending of the intuitive logics and the probabilistic
modified trend methodologies (Bradfield et al., 2005).
These early methodologies have been criticized for not providing sufficient support for
the evaluation of different strategies across the range of scenarios (Wilson, 2000; Goodwin and Wright, 2001). The comparison of strategies without the help of formal methods
is particularly difficult if the DM has multiple objectives (Kahneman and Tversky, 1982;
Goodwin and Wright, 2001). In consequence, several approaches have been developed to
integrate scenarios within a Multi-Criteria Decision Analysis (MCDA) framework (Stewart,
1997, 2005; Wright and Goodwin, 1999; Belton and Stewart, 2002; Montibeller et al., 2006).
These approaches help make trade-offs between possibly conflicting objectives and make it
possible to compare strategies across all scenarios – for instance, based on the total multiattribute value (e.g., Goodwin and Wright, 2001; Karvetski and Lambert, 2012), or the regret
of each strategy in every scenario; here, regret is defined as the difference between the value
of the strategy in the given scenario and that of the best-performing strategy in the same
scenario (Lempert et al., 2006; Ram et al., 2011).
Recently, methods of Portfolio Decision Analysis (PDA; see Salo et al., 2011 for an
overview) have been developed to support scenario-based strategy selection (Toppila et al.,
3
2011; Liesiö and Salo, 2012). In these methods, strategies are modeled as combinations or
portfolios of possibly interdependent actions (e.g., strategic research agendas or R&D portfolios). For instance, actions can be mutually exclusive (e.g., only one of actions A and B can
be selected) or there can be synergies between some actions (e.g., if both action A and action
B are selected, their combined impact is larger than the sum of their individual impacts).
Typically, PDA methods assign probabilities to scenarios. Toppila et al. (2011), for instance,
use decision trees to determine the optimal allocation of resources to standardization and
technology development activities in a telecommunications company. Specifically, they develop four scenarios which correspond to the four combinations defined by (i) successful
vs. unsuccessful standardization, and (ii) wide vs. limited adoption of technology. Liesiö
and Salo (2012) use scenario probabilities to generate robust recommendations for portfolios of investment projects in the presence of incomplete information about the DM’s utility
function and scenario probabilities.
Some authors have argued against the use of scenario probabilities, for instance because of
the psychological biases associated with subjective probability estimation, including overconfidence (Tversky and Kahneman, 1974; Hogarth and Makridakis, 1981; Goodwin and Wright,
2001). The use of probabilities has also been criticized for filtering out important information
about vulnerabilities and opportunities, and for forcing stakeholder consensus (Karvetski and
Lambert, 2012); moreover, probability estimation has been seen as tantamount to forecasting (Mobasheri et al., 1989). Bunn and Salo (1993), however, point out that if scenario
analysis is to support strategic choices, then some judgement about the relative likelihood of
scenarios is implicit even in those methods that deliberately attempt to avoid assessing these
likelihoods. It is, therefore, better to make this important aspect explicit by introducing as
much information about subjective probabilities as can be elicited with reasonable effort.
For instance, these probabilities can be characterized through feasible sets instead of point
estimates (cf. Liesiö and Salo, 2012). Such information can be elicited even through ordinal
statements such as ‘scenario 1 is more probable than scenario 2’, whereafter the resulting information can be harnessed when developing decision recommendations. Moreover, decision
recommendations can be generated even if the DM does not express any statements about
the scenario probabilities.
In this paper we develop a scenario-based portfolio model that helps generate recommendations for resilient and proactive action portfolios. In our model, a resilient action portfolio
is defined as one which is not outperformed by any other feasible portfolio, given the estimated impacts of the actions across the scenarios and the incomplete information about the
probabilities of these scenarios. This definition parallels that by Wilson (2000) who, in the
context of business strategy development, defines resilience as ‘the ability to deal with wide
4
variations in business conditions’ when these variations are captured by scenarios about the
future business environment. Woods (2005, 2006) suggests a broader definition of resilience
as ‘a system capability to create foresight, to recognize, to anticipate, and to defend against
the changing shape of risk before adverse consequences occur’. In our approach, this broader
definition can be taken to refer to the entire process of (i) building scenarios to characterize
the future operational environment, (ii) developing a list of possible actions that help mitigate threats and seize opportunities offered by the changing environment, and (iii) selecting
an action portfolio which performs relatively well across the scenarios.
We define a proactive action portfolio as one that accounts for the effect that the actions
may have on scenario probabilities. Indeed, it is often recognized that some futures are
more desirable than others, and that the DM’s actions can significantly affect the odds of
reaching these desirable futures (Schoemaker, 1995; Godet and Roubelat, 1996; Godet, 2000;
Peterson et al., 2003; Robinson, 2003; Porter et al., 2004; Palomo et al., 2011). Such impacts
are explicit in decision tree modeling; however, to our knowledge, this is the first paper to
present a decision analytic model that accommodates both incomplete and action-dependent
information about scenario probabilities.
3. Model framework for action portfolio selection
3.1. Portfolio selection with complete information
Consider a DM who wants to select a portfolio consisting of a subset of m actions.
The impacts of these actions are evaluated in n scenarios s1 , . . . , sn . The impact of action j in scenario si , denoted by Xj (si ), can represent, for instance, the net present
cash flow of the action in scenario si , or the cardinal multi-attribute value of the action, as obtained from conventional MAVT analysis (see, e.g., Dyer and Sarin, 1979).
These impacts form a vector-valued random variable X : {s1 , . . . , sn } → Rm such that
X(si ) = [X1 (si ), . . . , Xm (si )]T . The matrix consisting of the actions’ impacts in the n scenarios is denoted by X = [X(s1 ), . . . , X(sn )].
An action portfolio is a subset of the m available actions, represented by a binary row
vector z = [z1 , . . . , zm ] ∈ {0, 1}m where zj = 1 if and only if action j is included in the
portfolio. The set of feasible portfolios ZF ⊆ {0, 1}m can be restricted by different kinds of
constraints (e.g., availability of resources, action interdependencies, requirements of balance;
see Stummer and Heidenberger, 2003; Liesiö et al., 2008). Such constraints are modeled
through linear inequalities so that
ZF = {z ∈ {0, 1}m | Az T ≤ B}.
5
(1)
The coefficients of the feasibility constraints are contained in matrix A ∈ Rq×m and vector
B ∈ Rq×1 , where q is the number of constraints.
The value of portfolio z in a given scenario si is the sum of the impacts of the actions
contained in it
m
X
zj Xj (si ) = zX(si ).
(2)
j=1
When the action portfolio is selected, it is not known which scenario will be realized. A risk
neutral DM would, therefore, seek to maximize the expected value of the selected portfolio.
The portfolio that maximizes this expected value can be obtained from the zero-one linear
programming (ZOLP) problem
max Ep [zX] = max zXT p,
z∈ZF
z∈ZF
(3)
where p = [p1 , . . . , pn ]T is the vector of the probabilities such that pi is the probability of
P
scenario si . By definition, p is in the n-dimensional simplex Δn = {p ∈ [0, 1]n | ni=1 pi = 1}.
The decision tree corresponding to the ZOLP problem (3) is shown in Figure 1.
Figure 1: Decision tree for portfolio selection with scenario probabilities which do not depend on
the selected actions.
3.2. Action-dependent scenario probabilities
In some decision contexts, scenario probabilities depend on which actions are selected.
For instance, future climate conditions depend on which actions are adopted to reduce CO2
emissions (Hamin and Gurran, 2009; Moss et al., 2010). Therefore, we relax the assumption
that the probability vector p = [p1 . . . , pn ] is constant and, instead, assume that it may
6
depend on the selected portfolio. Whereas the probabilities of the n scenarios may depend
on which portfolio of actions is selected, we assume that the contents of these scenarios are
fixed. This is in line with the usual definition of scenarios as descriptions of the external
operational environment in which the organization acts (Coates, 2000; Ram et al., 2011).
Technically, we assume that the set of feasible portfolios ZF is partitioned into K disjoint
k
k
sets Z 1 . . . , Z K (∪K
k=1 Z = ZF ) such that if portfolio z ∈ Z is selected, then the scenario
probability vector is [Pk,1 , . . . , Pk,n ] ∈ Δn . In particular, this partition is made with respect
to those actions that affect the scenario probabilities. For instance, assume that the selection
of action j = 1 affects the scenario probabilities in one way, and the selection of both of
actions j = 2 and j = 3 affects them in some other way. Then, the set of portfolios ZF is
partitioned into four sets Z 1 , . . . , Z 4 which correspond to those portfolios which (i) include
at most one of actions j = 2 and j = 3 but do not include action j = 1, (ii) include action
j = 1 but neither action j = 2 nor j = 3, (iii) contain both actions j = 2 and j = 3 but not
action j = 1, and (iv) contain all three actions j = 1, 2, 3. That is,
z ∈ ZF |z1 = 0 ∧ (z2 = 0 ∨ z3 = 0) ,
= z ∈ ZF |z1 = 1 ∧ (z2 = 0 ∨ z3 = 0) ,
= z ∈ ZF |z1 = 0 ∧ z2 = z3 = 1) ,
= z ∈ ZF |z1 = z2 = z3 = 1) .
Z1 =
Z2
Z3
Z4
(4)
(5)
(6)
(7)
Given K portfolio sets Z k , K × n scenario probabilities need to be estimated. These
estimates can be represented by the matrix P ∈ ΔK×n such that

P =

Scenario s1
. . . Scenario sn
P1,1
..
.
...
..
.
P1,n
..
.
PK,1
...
PK,n



Portfolios z ∈ Z 1
P
Portfolios z ∈ Z K
P
..
.
= 1,
i
(8)
..
.
i
= 1.
The row vector of this matrix corresponding to those scenario probabilities which result from
selecting portfolio z ∈ Z k is denoted by
p(z) = [Pk,1 , . . . , Pk,n ],
where k is such that z ∈ Z k .
(9)
The portfolio that maximizes the expected portfolio value can be determined by solving the
problem
max zEp(z) [X] = max zXT pT (z).
(10)
z∈ZF
z∈ZF
7
The decision tree for this problem is shown in Figure 2.
Figure 2: Decision tree for portfolio selection with action-dependent scenario probabilities.
3.3. Incomplete information
Due to elicitation costs and time constraints, it may be difficult to obtain estimates for
the K × n probabilities needed to form matrix P in (8). Moreover, if these probabilities
are elicited from many experts, the experts may provide different estimates which cannot
be readily aggregated into a single probability vector. Thus, it is instructive to admit incomplete probability information that covers the full range of stated probability estimates,
and to examine which decision recommendations are compatible with such information (cf.
e.g., White et al., 1982; Hazen, 1986; Walley, 1991; Moskowitz et al., 1993; Liesiö and Salo,
2012).
We model incomplete probability information by set inclusion. That is, instead of a single
scenario probability matrix P , we consider a set of feasible probability matrices P ∈ ΔK×n ,
which satisfy linear constraints that correspond to statements about scenario probabilities.
Consider, for instance, the previous example where the selection of action 1 affects the
scenario probabilities in one way, and the selection of both of actions 2 and 3 affects them
in some other way. This setting was shown to partition the set of feasible portfolios ZF into
8
four sets Z 1 , . . . , Z 4 defined by (4)-(7). Assume that there are three scenarios. A statement
that scenario s2 is more probable than scenario s3 no matter what actions are selected can
be modeled as
Pk,2 ≥ Pk,3 ∀k = 1, . . . , 4.
(11)
Dependence of the scenario probabilities on the DM’s actions can be modeled by imposing
linear constraints on these probabilities only on condition that a given subset of actions is
selected. For instance, stating that the selection of action 1 makes scenario s1 more probable
than scenario s2 or s3 can be modeled as
Pk,1 ≥ Pk,2 + Pk,3 for k = 2, 4.
(12)
Similarly, the statement that choosing both actions 2 and 3 makes (i) the probability of
scenario s1 greater than or equal to 50% and (ii) the probability of scenario s3 less than or
equal to 10% can be modeled with constraints
Pk,1 ≥ 0.5
Pk,3 ≤ 0.1
)
for k = 3, 4.
(13)
Using constraints (11)-(13), the set of feasible probability matrices for the four portfolio
sets Z k , k = 1, . . . , 4 and three scenarios becomes
P = {P ∈ Δ4×3 | Pk,2 ≥ Pk,3 ∀k = 1, . . . , 4,
Pk,1 ≥ Pk,2 + Pk,3 for k = 2, 4,
(14)
Pk,1 ≥ 0.5 for k = 3, 4,
Pk,3 ≤ 0.1 for k = 3, 4}.
3.4. Dominance structures
Given complete information about the scenario probabilities, the DM would select the
feasible portfolio z ∈ ZF corresponding to the highest expected value Ep(z) [zX] = zXT pT (z).
However, when the scenario probability matrix P varies over the feasible set P, different
expected values are associated with each portfolio z. In order to determine which portfolios
outperform others, we define dominance as follows.
Definition 1. Portfolio z dominates z 0 with regard to the set of feasible probability matrices
9
P denoted z P z 0 if and only if
for all P ∈ P
Ep(z) [zX] ≥ Ep(z) [z 0 X]
for some P ∈ P,
Ep(z) [zX] > Ep(z) [z 0 X]
(15)
(16)
where p(z) denotes the row of matrix P that contains the scenario probabilities associated
with the selection of portfolio z (see Equation (9)).
This definition states that portfolio z dominates portfolio z 0 if (i) z yields at least as high
expected value as z 0 for all feasible scenario probabilities, and (ii) portfolio z yields strictly
higher expected value than z 0 for at least some feasible scenario probabilities. Even though
the probabilities in computing the expected values of portfolios z and z 0 may be different,
the relation P is transitive, which is a desirable property for any partial ordering used for
deriving normative decision recommendations. All proofs are in the Appendix.
Lemma 1. The dominance relation P is transitive.
Dominance between two portfolios can be readily checked by minimizing and maximizing
the value difference between them subject to the requirement that the scenario probabilities
P belong to P. These are linear problems with optimal solutions at some extreme point of
P. Hence, to check dominance relations among many portfolios using the same probability
information, it is faster to solve the set of extreme points ext(P) first and to then compare
the value differences between the portfolios in these points.
Theorem 1. Let z, z 0 ∈ ZF , and let the set of feasible probability matrices be P ⊆ ΔK×n .
Then, z P z 0 if and only if
zXT p(z)T − z 0 XT p(z 0 )T } ≥ 0,
max zXT p(z)T − z 0 XT p(z 0 )T } > 0,
min
P ∈ext(P)
P ∈ext(P)
(17)
(18)
where ext(P) is the set of extreme points of P and p(z) denotes the row of matrix P that
contains the scenario probabilities associated with selecting portfolio z (see Equation (9)).
Figure 3 illustrates dominance relations among three portfolios z 1 , z 3 ∈ Z 1 and z 2 ∈ Z 2
for two scenarios s1 and s2 such that the set of feasible probability matrices is
P = {P ∈ Δ2×2 | P1,1 ≤ 0.6
P2,1 ≥ 0.5}.
10
1
0.8
z
Expected protfolio value
Expected protfolio value
Portfolio z 2 is dominated by portfolio z 1 , because its expected value (ranging from 0.1 to
0.5) is less than or equal to that of portfolio z 1 (ranging from 0.5 to 0.75) for all feasible
scenario probabilities p(z 1 ) = [P1,1 , P1,2 ] and p(z 2 ) = [P2,1 , P2,2 ]. Also, portfolio z 1 dominates
portfolio z 3 , because its expected value is 0.15 units greater than that of portfolio z 3 for each
feasible p(z 1 ) = p(z 3 ) = [P1,1 , P1,2 ].
1
0.6
0.4
z
3
0.2
0
0
0.2
0.4
P
0.6
0.8
1
1
0.8
0.6
0.4
z
2
0.2
0
0
0.2
0.4
P
0.6
0.8
1
2,1
1,1
Figure 3: Expected values of portfolios z 1 , z 3 ∈ Z 1 and z 2 ∈ Z 2 .
A rational DM would not choose a dominated portfolio. Therefore, it is reasonable to
propose the selection of portfolios which are not dominated by any other portfolio.
Definition 2. The set of non-dominated portfolios with regard to the set of feasible probability
matrices P is
(19)
ZN (P) = {z ∈ ZF | @z 0 ∈ ZF such that z 0 P z}.
A non-dominated portfolio is both (i) resilient in that it is not outperformed by any other
feasible portfolio and (ii) proactive in that it accounts for the effect that the actions may
have on scenario probabilities. In the example of Figure 3, there is only one non-dominated
portfolio so that ZN (P) = {z 1 }.
3.5. Additional information
The introduction of additional constraints on scenario probabilities can reduce the set of
non-dominated portfolios but cannot generate new non-dominated portfolios. More specifically, if the set of feasible probability matrices P˜ is a subset of P, then the set of non˜ is also a subset of ZN (P).
dominated portfolios ZN (P)
˜ may contain two portfolios
However, if P˜ is a subset of the ‘border’ of P, then ZN (P)
whose expected values coincide on this border, while one has strictly lower expected value
everywhere else in P and, thus, does not belong to ZN (P). Such a situation is illustrated in
Figure 3, where portfolio z 1 dominates z 2 . Assume that the additional information causes the
11
set of feasible probabilities for portfolios z 1 , z 3 ∈ Z 1 to become the single point [P1,1 , P1,2 ] =
(0.6, 0.4), and that for portfolio z 2 ∈ Z 2 to become the single point [P1,1 , P1,2 ] = (1, 0). Here,
the expected values of portfolios z 1 and z 2 are equal (=0.5) so that z 1 no longer dominates
z 2 . To rule out this possibility, we assume that P˜ includes some points from the relative
interior of P.
˜ ⊆ ZN (P).
Theorem 2. Let P˜ ⊆ P such that int(P) ∩ P˜ 6= ∅. Then, ZN (P)
When introducing additional constraints on scenario probabilities without any guidance,
these constraints may conflict each other so that the set of feasible probability matrices becomes empty (Salo and Hämäläinen, 1992). Based on experimental evidence, however, such
situations are unlikely if the DM first provides loose statements that do not limit the feasible
region too much, and tightens her earlier statements only if the initial recommendations are
not conclusive enough (Moskowitz et al., 1989).
3.6. Implications for decision support
All reasonable decision recommendations are contained in the set of non-dominated portfolios ZN (P), because any portfolio outside this set is outperformed by at least one nondominated portfolio. Furthermore, by Theorem 2, no portfolio outside ZN (P) can become
non-dominated if additional information about the scenario probabilities is elicited.
The set of non-dominated portfolios can be examined to derive action-specific recommendations by using the concept of core index, defined as follows (cf. Liesiö et al., 2007).
Definition 3. For a given set P of feasible probability matrices we define
Core index of action j :
Core actions :
Borderline actions :
Exterior actions :
CIj (P) = |{z ∈ ZN (P)|zj = 1}|/|ZN (P)|
XC (P) = {j ∈ {1, . . . , m} | CIj (P) = 1}
XB (P) = {j ∈ {1, . . . , m} | 0 < CIj (P) < 1}
XE (P) = {j ∈ {1, . . . , m} | CIj (P) = 0}.
All core actions are resilient choices, because they belong to all non-dominated portfolios
even if additional information about the scenario probabilities was given. Similarly, all
exterior actions can be rejected, because they do not belong to any non-dominated portfolios
even in light of additional information. This result is formalized in Corollary 1.
˜ and XE (P) ⊆
Corollary 1. Let P˜ ⊆ P such that int(P) ∩ P˜ 6= ∅. Then, XC (P) ⊆ XC (P)
˜
XE (P).
12
3.7. Computation of non-dominated portfolios
Solving the set of non-dominated portfolios ZN (P) can be carried out by first enumerating all feasible portfolios and then checking the dominance relations by using Theorem 1.
However, a considerably faster approach is to utilize the partition of the portfolios z ∈ ZF
into sets Z 1 , . . . , Z K . In particular, let ZNk (P) denote the portfolios that are non-dominated
among Z k , i.e.,
ZNk (P) = {z ∈ Z k | @z 0 ∈ Z k s.t. z 0 P z}.
The set ZNk (P) can be obtained by solving all Pareto optimal solutions to the r-objective
zero-one linear programming problem
v–max[zXT p1 , zXT p2 , . . . , zXT pr ],
z∈Z k
(20)
where the vectors p1 , . . . , pr are the k-th rows of the extreme point matrices {P 1 , . . . , P r } =
ext(P), i.e.,
{p1 , . . . , pr } = [Pk,1 , . . . , Pk,n ]T | P ∈ ext(P) .
(21)
This is a direct consequence of the following Lemma.
Lemma 2. Let the set of feasible probability matrices be P ⊆ ΔK×n , and let z, z 0 ∈ Z k .
Then, z P z 0 if and only if

 
zXT p1
z 0 XT p1



..
..


.
.

 
zXT pr
z 0 XT pr


,

where denotes that the inequality is strict on at least one element, and the vectors p1 , . . . , pr
are given by Equation (21).
The Pareto optimal solutions to Problem (20) can be found by using any solution algorithm for the general multiple objective zero-one linear programming (MOZOLP) problem (Villareal and Karwan, 1981; Kiziltan and Yucaoğlu, 1983; Liesiö et al., 2007, 2008).
Once the sets ZN1 (P), . . . , ZNK (P) have been obtained, ZN (P) can be determined by checking dominance relations (Theorem 1) between all pairs z, z 0 of portfolios that are included
in different sets z ∈ ZNk (P), z 0 ∈ ZN` (P), k 6= `. The following Lemma states that this
approach generates all non-dominated portfolios.
Lemma 3. Let the set of feasible probability matrices be P ⊆ ΔK×n . Then, ZN (P) ⊆
ZN1 (P) ∪ . . . ∪ ZNK (P).
13
The pairwise dominance checks are initialized by setting up an empty list of dominated
portfolios ZD = ∅. The dominance relation is checked between each pair of portfolios (z, z 0 )
such that z ∈ ZNk (P), z 0 ∈ ZN` (P), k = 1, . . . , K − 1, ` = k + 1 . . . , K. If z ∈ ZD or
z 0 ∈ ZD , then the check is omitted. Otherwise, the minimum and maximum value differences
zXT p(z)T − z 0 XT p(z 0 )T over the sets of feasible probability matrices P ∈ P are computed.
If the minimal value difference is non-negative and the maximal value difference is strictly
positive, then portfolio z 0 is added to the list of dominated portfolios, i.e., ZD ← ZD ∪ {z 0 }.
Conversely, if the minimal value difference z 0 XT p(z 0 )T − zXT p(z)T is non-negative and the
maximum value difference is strictly positive, then ZD ← ZD ∪ {z}. Once all pairwise
dominance relations are checked, the set ZD contains all dominated portfolios among those
S
S
in k ZNk (P). The set of non-dominated portfolios is, then, ZN (P) = k ZNk (P) \ ZD .
The computational effort for determining the set ZN (P) increases as a function of the
number of actions m, the number of scenarios n, the number of constraints on feasible portfolios and scenario probabilities, and the number K of sets into which the action-dependent
scenario probabilities partition the set of feasible portfolios ZF . Previous computational experiments suggest that, with no action-dependent scenario probability information, ZN (P)
can be computed within a reasonable time (i.e., a few minutes) when there are some 60
actions, 5 scenarios, and 10 constraints (Liesiö et al., 2007; Vilkkumaa et al., 2014). Because
all portfolios within the set Z k are characterized by the same probability information, this
result holds for the computation time of the non-dominated sets ZNk (P) as well.
The pairwise dominance checks between portfolios in the sets ZNk (P) may increase the
computation time significantly; nevertheless, based on our numerical experiments, the total
computation time remains reasonable with some 10-15 actions, 4 scenarios, 6-7 constraints,
and 6-8 portfolio sets. In these experiments, the actions’ impacts were randomly generated,
and there was a single feasibility constraint to limit the number of actions in the portfolio.
Constraints on scenario probabilities corresponded to ordinal statements and lower bounds
such that some of these constraints only applied if a certain subset consisting of one or two
actions was selected.
4. Illustrative example
4.1. Problem description
We illustrate our approach with a numerical example that builds on a case study
by Raynor and Leroux (2004). In this example, a team of managers at a high-tech company
selects a portfolio of R&D projects to help the company prepare for anticipated changes
in the operational environment. Towards this end, the team constructs four scenarios to
portray the alternative future environments. These scenarios are developed by focusing on
14
two principal drivers of uncertainty, namely market demand and regulation, which will be
realized at either a low or a high level each. The different combinations of these levels define
four business-technology scenarios, as shown in Figure 4.
Figure 4: Four possible business-technology scenarios. Source: Raynor and Leroux (2004).
The team of managers lists ten projects which, if all implemented, would guarantee
success in all scenarios. The estimated costs and impacts – represented by the net present
value (NPV) – of these ten projects in each of the four scenarios are shown in Table 1. The
budget is $76M. This corresponds to roughly 60% of the sum of the projects’ costs and, thus,
not all ten projects can be selected. Also, the company wishes to keep the balance between
(i) projects 1 through 4 which maintain current businesses and (ii) projects 5 through 10
which correspond to the development of entirely new technologies. In particular, the share
of both types in the project portfolio should be at least 30%. Moreover, project 5 is a
follow-up project for 8, meaning that it can only be selected if project 8 is selected. Finally,
the probability of high market demand can be affected if at least one of projects 2 and 9 is
selected.
The m actions in our model are these ten R&D projects. We denote the costs of these
P
projects by c = [c1 , . . . , c10 ]. The budget imposes feasibility constraint 10
i=1 ci zi ≤ $76M on
the selected portfolio. A second feasibility constraint z5 ≤ z8 ensures that the follow-up
project 5 can only be selected if project 8 is selected. The requirement that the share of both
project types (1-4 and 5-10) in the portfolio is at least 30% is modeled through the feasibility
P
P10
P10
P
z
and
z
≥
0.3
constraints 4i=1 zi ≥ 0.3 10
i
i
i=1
i=5
i=1 zi . Because selecting either one of
projects 2 or 9 affects the scenario probabilities, the set of feasible portfolios ZF is partitioned
into two sets Z 1 = {z ∈ ZF | z2 = z9 = 0} and Z 2 = {z ∈ ZF | z2 = 1 ∨ z9 = 1}. Therefore,
15
Project
1
2
3
4
5
6
7
8
9
10
Optimal portfolio value
s1
15,168
20,736
27,359
26,691
28,291
33,949
13,392
22,609
12,395
25,654
164,553
NPV
s2
27,956
47,706
22,335
28,006
59,417
32,592
31,069
11,262
40,957
31,879
241,562
($k)
s3
5,984
3,778
9,616
22,450
7,510
12,976
8,969
8,063
9,198
14,713
77,922
s4
15,840
12,506
4,652
37,404
14,998
20,551
26,302
14,978
7,975
24,900
131,088
Cost ($k)
cj
26,862
9,348
12,220
14,315
5,730
19,440
14,140
15,575
3,268
5,832
Table 1: Estimated NPVs of ten projects across the four scenarios, and the projects’ costs. Project
5 can only be selected if project 8 is selected. Bolded figures indicate the scenario-specific optimal
portfolios.
the feasible probability matrix P ∈ Δ2×4 .
The bolded numbers in Table 1 indicate the optimal portfolios in each scenario. The
values of these optimal portfolios imply that scenario 2 (in which the company’s new technology dominates the market) is the most desirable, followed by scenario 1 (in which the
company’s technology shares the market with alternative low-cost technologies) and scenario
4 (in which alternative low-cost technologies dominate) such that scenario 3 (in which both
the company’s technology and the alternative ones ‘tank’ in the market) is the least desirable.
We compute the set of non-dominated portfolios for three different sets of feasible probability matrices.
• In the first set P1 , there is no probability information except that the selection of either
project 2 or 9 affects the scenario probabilities:
P1 = Δ2×4 .
• In the second set P2 , the probability of high regulation is deemed higher than that of
low regulation no matter which portfolio is selected; moreover, the probability of high
market demand is at least 50% if at least one of projects 2 and 9 is selected:
P2 = {P ∈ P1 | Pk,1 + Pk,2 ≥ Pk,3 + Pk,4 for all k = 1, 2
P2,2 + P2,4 ≥ 0.5}.
16
• In the third set P3 , there is also a lower bound on the scenario probabilities resulting
from the statement that the probability of any scenario is at least 5% no matter which
portfolio is selected:
P3 = {P ∈ P2 | Pk,i ≥ 0.05 for all k = 1, 2, i = 1, . . . , 4}.
In set P1 , the probability of each scenario can vary over the full range between 0 and
1. In set P2 , the probabilities of scenarios s3 and s4 are bounded from above by 0.5 for
all portfolios. Moreover, the probability of scenario s1 is also bounded from above by 0.5
if either project 2 or project 9 is selected. Finally, in set P3 , we have p3 , p4 ∈ [0.05, 0.45]
and p1 , p2 ∈ [0.05, 0.85] for all portfolios. However, for those portfolios which contain either
project 2 or 9, the probability of scenario s1 has a narrower interval p1 ∈ [0.05, 0.45]. Thus,
for both sets P2 and P3 , the project-dependent probability information affects the range
of feasible probabilities for the second-most desirable scenario s1 in which the company’s
technology shares the market with alternative low-cost technologies.
4.2. Results
The non-dominated sets ZN (P1 ), ZN (P2 ) and ZN (P3 ) were computed in seven seconds
using Matlab on a standard laptop (2.60 GHz, 8 GB memory). These sets are illustrated in
Figure 5 through the projects’ core indices. The length of the bar shows the project’s core
index. The core projects with core index equal to one are marked with green bars, and the
borderline projects with core indices between zero and one with yellow bars. The core index
– and thus the length of the bar – for exterior projects is zero.
If no information about scenario probabilities is given, there are 9 non-dominated portfolios. There is one core (project 4) and one exterior project (project 1), but all other
projects are borderline. With the set of feasible probability matrices P2 , the number of
non-dominated portfolios drops to 4. Now, there are three core projects (projects 4, 5 and
8), and one exterior project (project 1). Imposing the lower bound of 5% on all scenario
probabilities (P = P3 ) reduces the set of non-dominated portfolios to just one portfolio consisting of projects {2, 3, 4, 5, 8, 9, 10}, so that also projects 6 and 7 become exterior. This is
because now the probability of scenario s4 – the only scenario in which it would be optimal
to select the costly projects 6 and 7 together with project 10 – is bounded from above by
0.45. In the other three scenarios, which have a higher combined probability than scenario
s4 , projects 4, 5 and 8 are optimally complemented with projects 2, 3, 9, and 10.
Interestingly, the decision recommendation {2, 3, 4, 5, 8, 9, 10} can be offered without precise point estimates about the scenario probabilities. Instead, it suffices to say that high regulation is more likely than low regulation, the selection of either project 2 or 9 makes high
17
|Z (P )|=4
Project number
|ZN(P1)|=9
N
|ZN(P3)|=1
2
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
10
10
10
0
0.2
0.4
0.6
Core index
0.8
1
0
0.2
0.4
0.6
Core index
0.8
1
0
0.2
0.4
0.6
0.8
1
Core index
Figure 5: Number of non-dominated portfolios and core indices for the projects corresponding to
three different sets of feasible probability matrices P1 , P2 and P3 .
market demand more likely than low market demand, and each of the four combinations of
high/low regulation and high/low market demand occurs with at least 5% probability. Even
if the 5% lower bound on the scenario probabilities is not imposed (so that the set of feasible
probability matrices is P2 ), it is possible to identify projects that should be selected (core
projects 4, 5 and 8) and those that should be rejected (exterior project 1).
The set of core projects 4, 5 and 8 for set P2 of feasible probability matrices is not the same
as the set of those projects that are included in all scenario-specific optimal portfolios, namely
4 and 10 (see the bolded figures in Table 1). This follows in part from the information about
scenario probabilities, but also from the interdependencies between the projects in these
portfolios. Due to the balance and budget constraints as well as the logical interdependency
between projects 5 and 8, it may not be meaningful to consider project recommendations
independently of other projects.
We conclude the example by examining how the decision recommendations depend on
the budget. Figure 6a shows the core indices of the ten projects as functions of the budget
with information set P2 . The darker the shade of the bar, the higher the core index. The
core indices in Figure 5 for set P2 correspond to the core indices in Figure 6a with budget
$76M.
The fact that all projects are exterior for budgets smaller than $13M indicates that
the budget needs to be at least this much (or, more precisely, c2 + c9 = $12.616M) to
satisfy the feasibility constraints. For budgets under $27M, the non-dominated portfolios
are combinations of projects 2, 4, 9 and 10. The interdependent projects 5 and 8 enter some
18
non-dominated portfolios when the budget reaches $47M. With a smaller budget it does not
pay off to implement the rather costly project 8, which is a precondition for implementing the
less expensive and more valuable project 5. For any budget over $81M, these two projects
are included in all non-dominated portfolios; as is project 2, the selection of which increases
the probability of high market demand. Project 1, on the other hand, is not included in
any non-dominated portfolio until the budget reaches $96M, because this project is very
expensive and does not yield exceptionally high NPV in any of the four scenarios.
(a) Scenario probabilities depend on selected projects.
(b) Scenario probabilities do not depend on selected projects.
Figure 6: Projects’ core indices as functions of the budget.
Figure 6b shows the projects’ core indices as functions of the budget when (i) the selected
projects do not affect the scenario probabilities, and (ii) regulation is more likely to be high
than low. Then, the feasible probability matrices are vectors P ∈ Δ1×4 , the feasible set of
which is Pˆ2 = P ∈ Δ1×4 |P1,1 + P1,2 ≥ P1,3 + P1,4 . Set Pˆ2 , therefore, corresponds to set P2
without the project-dependent probability information.
The differences between the non-dominated portfolios resulting from the projectdependent set P2 and the project-independent set Pˆ2 reflect the possibility of making proactive choices which help steer the future towards the desired scenarios. For instance, without
19
the project-dependent information, project 3 is included in some non-dominated portfolios
for all budgets in the range $22M-$63M (red oval in the left-hand side of Figure 6b). However, if the project-dependent information is taken into account, then project 3 should not
be selected for most budgets within this range (red oval in the left-hand side of Figure 6a).
This is because the selection of project 2 or project 9 increases the probability of high market
demand, thereby increasing the probability of scenario s4 in which project 3 would be a nonoptimal selection, and decreasing the probability of scenarios s1 and s3 in which the selection
of project 3 would be optimal (see Table 1); in fact, as noted before, selecting project 2 or 9
diminishes the range of feasible probabilities for scenario s1 from [0, 1] to [0, 0.5]. Similarly,
project 7 – which yields relatively high NPV in scenario s4 but relatively low NPV in scenario s1 – is not included in all non-dominated portfolios in the budget range $112M-$126M
for the project-independent probability information Pˆ2 . However, if the project-dependent
information is taken into account, this project should be selected for all budgets in the same
range (red ovals in the right-hand side of Figures 6a and 6b).
5. Discussion
The application of the model requires the specification of scenarios, the estimation of the
actions’ impacts in these scenarios, and the elicitation of statements about scenario probabilities. Typically, the scenarios are developed by (i) identifying a set of key uncertainties
and the possible outcomes of these uncertainties, (ii) forming all combinations of these outcomes, and (iii) constructing the scenarios as stories based on those combinations that are
strategically relevant, internally consistent, and ‘archetypical’ in that they describe generally different futures rather than variations of one theme (cf. Figure 4; Schoemaker, 1995;
Peterson et al., 2003; Raynor and Leroux, 2004).
Depending on the context, the actions’ impacts in these scenarios can be estimated
through expert assessments about the projects’ NPVs or their multi-attribute values, for
instance. More effort should be devoted to estimating the impacts of those actions that affect
scenario probabilities, because they have more influence on the selection decision than those
actions that do no affect these probabilities. In eliciting statements about the probabilities,
it is advisable to start with loose statements so that the feasible region does not become
empty, and to tighten these statements only if the initial recommendations are not conclusive
enough (Moskowitz et al., 1989).
Importantly, decision recommendations can be generated even if no information about
scenario probabilities is given. In this case, the set of non-dominated portfolios includes all
portfolios that would be recommended by decision rules based on the maximin and minimax
regret robustness measures (Kouvelis and Yu, 1997; Lempert et al., 2006; Ram et al., 2011).
20
In this setting, the maximin portfolios are those that yield the highest value in the worst-case
scenario, whereas the minimax regret portfolios yield the smallest maximal regret, where (i)
regret is defined as the difference between the portfolio value in the given scenario and that
of the best-performing portfolio in the same scenario, and (ii) the maximization is carried
out over all scenarios.
In some cases, such as when selecting a portfolio of large non-recurring investment
projects, it may be advisable to account for the DM’s risk preferences. In our framework,
the DM’s risk aversion can be captured by introducing constraints on risk measures. A risk
measure ρ maps each portfolio to a real-valued number such that portfolio z is less than or
equally risky as portfolio z 0 if ρ[z] ≥ ρ[z 0 ]. Here, the direction of the inequality is due to
the objective to maximize the portfolio value rather than to minimize loss. If, for instance,
the risk measure describes the value of a portfolio in the worst-case scenario, then a lower
value of this measure corresponds to higher risk. A lower bound γ on the risk measure can
be imposed to rule out portfolios that are too risky for the DM from the feasible set ZF so
that (1) becomes
ZF = {z ∈ {0, 1}m | Az T ≤ B, ρ[z] ≥ γ}.
(22)
Conditional Value-at-Risk (CVaR) measure, for instance, is the expected portfolio value
conditioned on the event that the value is in the worst α-quantile, i.e., CVaR[z] =
E[zX|zX ≤ F −1 (α)], where F is the cumulative distribution function of the portfolio
value (Rockafellar and Uryasev, 2000). With incomplete probability information, the WorstCase Conditional Value-at-Risk WCVaR[z] = minP ∈P CVaR[z] (Zhu and Fukushima, 2009)
can be used to measure portfolio risk. Liesiö and Salo (2012) show that in scenario-based
project portfolio models the requirement WCVaR[z] ≥ γ can be modeled through linear
constraints on the set of feasible portfolios. Such constraints can be directly introduced to
MOZOLP problem (22) to limit the WCVaR of feasible portfolios.
6. Conclusions
In this paper, we have developed a scenario-based portfolio methodology to support the
selection of action portfolios when the information about the scenario probabilities is possibly incomplete and may depend on which actions are selected. This model helps identify
portfolios that are resilient across the possible future scenarios in light of the available scenario probability information, and proactive in that they help steer the course towards the
desired scenario. In particular, action-specific decision recommendations can be provided
even with fairly loose constraints on scenario probabilities. Moreover, the model can accommodate different kinds of action interdependencies, including balance requirements and
logical constraints, which make traditional methods of cost-benefit analyses difficult to apply.
21
Our model is generic and can therefore be utilized in different contexts: for instance, to
develop strategic research agendas for research organizations, to identify the core issues in
conflict management and negotiation processes, or to help companies prepare for disruptive
shocks in the market environment. The focus of the model on both resilience and proactivity
resonates well with current approaches to addressing the risks of climate change, too: namely
(i) adaptation, i.e, building resilience towards changes in the climate conditions, and (ii)
mitigation, i.e., taking proactive measures to reduce net CO2 emissions (Hamin and Gurran,
2009; Moss et al., 2010).
There are several avenues for future research. First, the framework could be extended
to support multi-period portfolio selection processes in which the DM has the opportunity
to revisit the initial selection decision in a later period (cf. Huchzermeier and Loch, 2001;
Gustafsson and Salo, 2005). Second, uncertainty about the actions’ impacts in different
scenarios could be accommodated. Earlier work on this subject suggests that, to limit the
complexity of the model, such action-specific uncertainties should be characterized with the
help of intervals, instead of dividing the scenarios further into sub-scenarios, each of which
would correspond to specific realizations of these action-specific uncertainties (Liesiö et al.,
2008). Third, the model could be integrated with a game-theoretic framework to support
the selection of actions when both the actions’ impacts and the scenario probabilities may
be affected by the actions of other DMs. Such an integrated framework could provide
useful decision support in contexts such as corporate strategy development and conflict
management, in which many uncertainties often relate to the actions of others.
Appendix
Proof of Lemma 1: Assume z P z and z 0 P z 00 . Then for any P ∈ P : Ep(z) [zX] ≥
Ep(z0 ) [z 0 X] ≥ Ep(z00 ) [z 00 X]. Also, there exists P ∈ P such that Ep(z) [zX] > Ep(z0 ) [z 0 X] ≥
Ep(z00 ) [z 00 X]. These inequalities together imply that z P z 00 .
Proof of Theorem 1: We show that the two conditions of Definition 1 for portfolios z and
z 0 hold if and only if the minimization and maximization conditions of Theorem 1 hold.
Ep(z) [zX] ≥ Ep(z0 ) [z 0 X] ∀P ∈ P
⇔
zXT pT (z) ≥ zXT pT (z 0 ) ∀P ∈ P ⇔
min{zXT pT (z) − z 0 XT pT (z 0 )} ≥ 0 ⇔
P ∈P
min {zXT pT (z) − z 0 XT pT (z 0 )} ≥ 0,
P ∈ext(P)
where the last equivalence follows from that fact that the sets of the row vectors p(z) of
22
P ∈ P are polyhedra, and zXT pT (z) is linear in p(z) (e.g., Bertsimas and Tsitsiklis, 1997).
Furthermore,
∃P ∈ P s.t.
∃P ∈ P s.t.
Ep(z) [zX] > Ep(z0 ) [z 0 X]
zXT pT (z) > zXT pT (z 0 )
⇔
⇔
max{zXT pT (z) − z 0 XT pT (z 0 )} > 0 ⇔
P ∈P
max {zXT pT (z) − z 0 XT pT (z 0 )} > 0.
P ∈ext(P)
˜ z0 ∈
Proof of Theorem 2: Assume contrary to the claim that ∃z 0 ∈ ZN (P),
/ ZN (P). Then,
∃z ∈ ZF such that z P z 0 , which is equivalent to
zXT pT (z) − z 0 XT pT (z 0 ) ≥ 0 ∀P ∈ P
∃P ∗ ∈ P s.t. zXT pT (z) − z 0 XT pT (z 0 ) > 0.
∧
(.1)
(.2)
˜
Because P˜ ⊆ P, it holds that zXT pT (z) − z 0 XT pT (z 0 ) ∀P ∈ P.
˜ Let P ◦ = Pˆ + ε(Pˆ − P ∗ ). Let us denote by
By assumption, there exists Pˆ ∈ int(P) ∩ P.
∗
∗
◦
◦
. . . , Pk,n
], pˆ(z) = [Pˆk,1 , . . . , Pˆk,n ] and p◦ (z) = [Pk,1
, . . . , Pk,n
] the row vectors of
p∗ (z) = [Pk,1
matrices P ∗ , Pˆ and P ◦ , respectively, which result from selecting portfolio z ∈ Z k . Because
Pˆ ∈ int(P), ∃ε such that P ◦ ∈ P. By rearranging the terms we have
ε
1
P◦ +
P ∗ ≡ αP ◦ + βP ∗ ⇔
1+ε
1+ε
1 ◦
ε ∗
pˆ(z) =
p (z) +
p (z) ≡ αp◦ (z) + βp∗ (z) ∀z ∈ ZF .
1+ε
1+ε
Pˆ =
Note that α, β > 0. But then,
zXT pˆT (z) − z 0 XT pˆT (z 0 )T = zXT (αp◦ (z) + βp∗ (z))T − z 0 XT (αp◦ (z 0 ) + βp∗ (z 0 ))T
= α(zXT p◦ (z)T − zXT p◦ (z 0 )T ) + β(zXT p∗ (z)T − zXT p∗ (z 0 )T ) > 0.
|
{z
}
|
{z
}
≥0 by (.1)
>0 by (.2)
˜ implies that
The last inequality, together with the fact that zXT pT (z) − z 0 XT pT (z 0 ) ∀P ∈ P,
˜ which is a contradiction.
z P˜ z 0 . Thus, z 0 ∈
/ ZN (P),
˜ ⊆ ZN (P) of
Proof of Corollary 1: The result follows directly from the result ZN (P)
Theorem 2.
23
Proof of Lemma 2: By Theorem 1, z P z 0 if and only if
zXT p(z)T − z 0 XT p(z 0 )T } ≥ 0,
max zXT p(z)T − z 0 XT p(z 0 )T } > 0,
min
P ∈ext(P)
P ∈ext(P)
(.3)
(.4)
Denoting {p1 , . . . , pr } =
Because z, z 0 ∈ Z k , p(z) = p(z 0 ) = [Pk,1 , . . . , Pk,n ].
{[Pk,1 , . . . , Pk,n ]T | P ∈ ext(P)}, Equations (.3) and (.4) are equivalent to
zXT p ≥ z 0 XT p for all p ∈ {p1 , . . . , pr }, and
zXT p > z 0 XT p for some p ∈ {p1 , . . . , pr },
respectively, which proves the Lemma.
Proof of Lemma 3: Assume z ∈ ZN (P) ⊆ ZF . By Definition 2, @z 0 ∈ ZF such that
z 0 P z. Because Z 1 ∪ . . . ∪ Z K = ZF , there exists k such that z ∈ Z k . Now, @z 0 ∈ Z k ⊆ ZF
such that z 0 P z, whereby z ∈ ZNk (P).
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