Public Project Portfolio Optimization under a Participatory Paradigm

A new democracy paradigm is emerging through participatory budgeting exercises, which can be de�ned as a public space in which the government and the society agree on how to adapt the priorities of the citizenship to the public policy agenda. Although these priorities have been identi�ed and they are likely to be re�ected in a ranking of public policy actions, there is still a challenge of solving a portfolio problem of public projects that should implement the agreed agenda. is work proposes two procedures for optimizing the portfolio of public actions with the information stemming from the citizen participatory exercise. e selection of the method depends on the information about preferences collected from the participatory group. When the information is sufficient, the method behaves as an instrument of legitimate democracy. e proposal performs very well in solving two real-size examples.


Introduction
Even in the best scenarios, designing public policies is far from being an exact science, with quantitative determinations beyond all subjectivity.Without denying the objective content of the social interest, it is certain that the difficulty in apprehending it opens space to methods that seek to model the preferences of concrete individuals, able to express their preferences in a more or less consistent way.
Up to now, democracy in the distribution of public resources has been fundamentally expressed in (i) the action of groups empowered by the society to make budget decisions on its behalf (GESBD) (parliaments, communes, governing boards of public organizations), formed by members of the political class legitimized by the popular vote, but that respond to their personal and their party's interests instead of to the will of the electorate, (ii) the action of GESBDs formed by officials and experts appointed by the executive power that rather than interests of the electorate, re�ected only in a very indirect way, re�ect policies already designed by the executive.
(iii) the attempts of participatory budgeting carried out at local level where the population's priorities are directly heard by constituted authorities and are later re�ected in the distribution of resources once the compatibility with the opinions of the political class has been achieved.
Since its emergence in Porto Alegre, Brazil, participatory budgeting has spread to hundreds of Latin American cities and dozens of cities in other continents."Participatory budgeting" can be de�ned as a public space in which the government and the society agree how to adapt the priorities of the citizenship to the public policy agenda.e utility of these participatory exercises is that the government obtains information about the priorities of the participating social sectors and thus can perhaps identify programs that are of consensual bene�t.
However, the challenge that is still not approached by "participatory budgeting" is how to translate priorities and policies stemming from the exercise into a system of concrete social action projects, each with well-de�ned resources and falling within a frame of an approved general budget.
Although the priorities of the citizenship have been identi�ed and are likely to be re�ected in a ranking of public policy actions, it is still necessary to solve a portfolio problem of public projects that should implement the agreed agenda.
Portfolio selection is an optimization problem with exponential complexity.e set of possible portfolios is the power set of the projects applying for funding.e cardinality of the set of portfolios is 2  , where  is the number of projects.If synergetic projects and nonlinear and timing distributions effects are considered, the complexity of the resulting optimization model increases signi�cantly.So far, the participatory budgeting exercises have been unable to deal with this kind of problem.Two questions arise.How to obtain the information about the preferences of the participants in the exercise?How to use it later to explore the space of portfolios and �nd the best solutions compatible with the participants' satisfaction?
Basic structures of collective decision making and information provided by participatory budgeting exercises are discussed in Section 2. e way in which a public portfolio can be optimized is described in Section 3. Section 4 details the way in which a portfolio can be optimized when the group preference information comes from a project ranking.Section 5 presents a portfolio optimization method when the preference information is given as individual goals.ese methods are illustrated by real size examples in Sections 4 and 5. Finally, we draw some conclusions in Section 6.

Basic Structures for Collective Decision Making.
Let us consider a decision-maker group (DMG) composed of  actors with heterogeneous value systems.is collective is in charge of building a public project portfolio.According to Marakas [1], we can distinguish three basic structures for collective work oriented to decision making.
Figure 1(a) covers the case in which the whole group is responsible for the decision, there is a complete symmetry between the different DMs, and the �nal decision is made according to previously established rules that de�ne the way in which the group is "constituted." Figure 1(b) corresponds to situations in which symmetry is lost, there exists a DM in charge of the decision (hereinaer referred to as group supra-decision maker (GSDM)), who bases it on the opinions of the collective that processes a lower level of decision; the members of this collective only interact with the GSDM.e structure in Figure 1(c) shows a complete interaction amongst all participants, but the main responsibility still falls on the GSDM whose �nal judgment will be sustained by the best possible consensus of the members of the collective.All of the three structures assume a certain level of collaboration, even though it is not necessarily exempted from contradictions.ese three cases will be referred to as partner association, team, and committee, respectively.

Information Stemming from the Participatory Exercise and
Its Exploitation.Participatory budgeting exercises preferably take the form of Figure 1(c), with government representatives playing the role of GSDM.From this interaction, the GSDM can obtain the following preferential information from the actors representing the society: (A) opinions about certain priorities that the public agenda should contain, accepted by a signi�cant part of participants; (B) individual preferences of each participant about the public policy actions requiring budgetary resources ordered in a ranking of priorities; (C) if the public agenda success is measured through a set of indicators, the levels that each participant expects as a result of the budgeting exercise.

Given the social heterogeneity and con�icting interests, usually it is not possible to achieve consensuses in individual preferences (case B) or individual goals (C).
A portfolio is a subset of a set   of projects or social actions which are under consideration.To distribute resources means to �nd a feasible portfolio that meets speci�c requirements.In practice, the GSDM is the one who solves the portfolio problem by using the obtained information.But the ethics of democracy (not only the electoral factor, but democracy as a government by and for the people) require the GSDM to adhere as much as possible to the preferential information stemming from the participatory exercises.It is noteworthy that in (A) and (B) social preferences are not collected over project portfolios, but over concrete actions.Since the budget distribution is a problem of project, policy, or public action portfolios, the issue is how to translate the information from participatory exercises into preferences over portfolios.In (C), the aspirations collected about the social status are likely to be contradictory among different participants, but social status indicators will vary as a result of the portfolio that is decided to be supported.us, the information in mode (C) collects preferences over portfolios or over the result of portfolios.
Case (A) is predominant among real exercises of participatory budgeting.e GSDM seeks to adapt his/her agenda previously conceived according to the opinions received and on that basis to make the distribution of budgetary resources.e mind of GSDM processes the information, partly modi�es his/her own subjectivity, and decides the distribution of resources.e group representing the society is not the one solving the portfolio problem, but its opinions are taken into account.
In case (B), the public action rankings given by the participants are an expression of their preferences over projects, not over portfolios.Let us suppose that a method for integrating individual rankings is applied in collective ordering as the Borda score or a procedure based on exploitation of collective fuzzy preference relations (e.g., [2][3][4]).With the resulting group ordering, the GSDM has more information about social preferences regarding different actions that are to be budgeted.e GSDM shall use that information to �nd the best portfolio.
In the following section, we assume that the GSDM has at his/her disposal a ranking of priorities of public actions or projects stemming from the participatory exercise.

Finding the Best Portfolio via Multiobjective Optimization
As stated by Fernandez et al. [5] and Fernandez-Gonzalez et al. [6], the main difficulty for characterizing the "best public project portfolio� is �nding a mechanism to appropriately de�ne, evaluate, and compare social returns.�egardless of the varying de�nitions of the concept of social return, however, we can assert the tautological value of the following proposition.

Proposition 1. Given two social projects, A and B, with similar costs and budgets, A should be preferred to B if A has a better social return.
Ignoring, for a moment, the difficulties for de�ning the social return of a project portfolio, given two portfolios, C and D, with equivalent budgets, C should be preferred to  if and only if  has a better social return.us, the problem of searching for the best project-portfolio can be reduced to �nding a method for assessing social project returns, or at least a comparative way to analyze alternative portfolio proposals.If we do not have reliable information about speci�c social project returns, the social impact of a given portfolio may be represented by a set of "proxy" variables measuring different criteria of portfolio quality.Let us denote by { 1 , … ,   } such a set.Without loss of generality, we suppose that the GSDM's preferences are increasing with values of   .Assumption 1.Let  * and  * * be two feasible portfolios and  * = ( 1 , … ,   ) * and  * * = ( 1 , … ,   ) * * their respective image in the proxy objective space.en,  * is Paretodominated by  * * only if under the available information,  * * is preferred to  * from the GSDM's point of view.
Let  be the set of feasible portfolios.Feasibility is determined by budgetary constraints.Assuming that the problem is solved by and for the GSDM, the comparison of the social value of the two portfolios becomes meaningful.Let   and   be two feasible portfolios.Let us assume that   is an asymmetric preference relation de�ned by the GSDM on a subset of C × C. is is a subjective relation based on the information that the GSDM has about the portfolios.If       does not generate a loop in   , then we can accept that   has a greater value for the GSDM than   .e necessary condition for   to be the best portfolio with respect to   is that there is no   ∈  such that       .Furthermore, a sufficient condition for   to be the best portfolio with respect to   is       for all   ∈ .

Proposition 2. Under Assumption 1 the best portfolio for the GSDM is a nondominated solution to the problem
e proof is very simple.Suppose that   is the best portfolio being a dominated solution to problem (1).If   is dominated by   , then from Assumption 1,       .So,   does not hold the necessary condition to be the best portfolio.e choosing of a set { 1 , … ,   } depends on the available information about the portfolio social impact, or how it is perceived by the GSDM.At least we should distinguish the two cases: (i) when the preference information from the participants has been aggregated in a ranking of priorities (case (B) of Section 2.2); (ii) when the public agenda success is measured by a set of indicators, and the contribution of each project or public action to the improvement of these indicators is known (case (C) of Section 2.2).
In the next section, we propose a way to build the set { 1 , … ,   } and solve problem (1) in case (i).Case (ii) will be approached in Section 5.

Portfolio Selection When the Group
Preference Information Comes from a Project (Action) Ranking It is necessary to compare the quality of the possible portfolios in order to �nd the best one.is problem was �rstly approached by Fernandez et al. [5], under the assumption "the portfolio impact on a decision-maker's mind is determined by the number of supported projects and their particular rank." We shall accept that if project  is clearly ranked better than b, then  is admitted to have "more social impact" than .e GSDM should take this information from the ranking into account.e appropriateness of a portfolio is not only de�ned by the quality of the included projects, but also by the amount of contained projects.As in Fernandez et al. [5], the purpose will be to �nd the best portfolio by increasing the number of supported projects and controlling the possible disagreements with respect to the GSDM's preferences, which as it is assumed, are incorporated in the input ranking.Some discrepancies may be acceptable between the information provided by the ranking and the decisions concerning the approval (hence, supporting) of projects, whenever this fact increases the number of projects in the portfolio.However, this inclusion should be controlled because the admission of unnecessary discrepancies is equivalent to underestimating the ranking information.
Among different heuristics, we suggest one that is based on dividing the ranking in �ve categories; then, on this basis construct the speci�c preference relation.e categories are labeled (1) vanguard, (2) high-medium, (3) medium, (4) lowmedium, and (5) rearguard.If   is the set of projects, let us de�ne a "priority" function     → {1, 2, 3, 4, 5} whose image is the project label (for instance,   1   is ranked in the vanguard).
rough exercising such categorization, we may build up the preference relations according to the ranking, as explained below.
Strict preference (  ):        and |  |  1 or    and  is ranked better than b).
Weak preference (  ):    a and  are ranked equal, but  needs more funds than .
Strict outranking (aSb):      Note that  is an asymmetric relation expressing preference for one of the projects.Hence, a well-formed portfolio should be compatible with the information contained in .Let  be a project portfolio that obtains support, with aSb and   .In such a case, the inclusion of  into  would constitute a discrepancy with respect to the information contained in .us, each portfolio may be associated a discrepancy set in the following way: and according to Proposition 2, the best portfolio should be selected by the GSDM among its nondominated solutions.It is him/her who will �nally select the adequate compromise among the number of supported projects and the quality given by their rank.Compromise solutions will depend on the extent that the GSDM relies on the information provided by the rank ordering and facts such as pressures exerted on him/her to include high-ranked projects or to increase the number of approved projects, as well as his/her aversion to costly projects.

An Illustrative Example.
As illustration, let us consider the problem of distributing 2.5 billion dollars among 100 projects, all of them deserving individual �nancing.Information about these projects (costs in million dollars and their ranks according to some preference ordering) is shown in Table 1 (the data are �ctitious).Financing all projects would amount more than 5.5 billions, but the available funds are not enough to support them all.e traditional heuristics for solving this problem consider giving support by following the rank order until the available resources run out.If this method was used, the resources would reach up to project 22, implying a total expenditure of 2.468 billion dollars.
Let us suppose that the GSDM is not averse to making contradictory decisions with respect to the ranking information, thus being prepared to accept some discrepancies in order to increase the number of supported projects.Hence, he/she wants to evaluate nondominated solutions to problem (3).However, this problem exhibits exponential complexity, which complicates its solution through classical techniques of multiobjective optimization even for a medium-size instances [5].Since it is difficult to model GSDM preferences over its objective functions, it seems preferable to apply a method for building the Pareto optimal set, and from this result, the GSDM can select the best "compromise." For this reason we choose a multiobjective evolutionary algorithm (MOEA).One advantage of MOEAs is that they handle efficiently exponential complexity of the search space.According to Fernandez et al. [5], several works report successful experiments applying MOEAs to large 0-1 multiobjective knapsack problems.Unlike classical multiobjective optimization techniques, MOEAs are not sensitive to mathematical properties of the objective functions or the shape of the feasible region; moreover, they can obtain many nondominated solutions in only one run [7].Here, we apply the NSGA-II (nondominated sorting genetic Algorithm) developed by Deb et al. [8], which is the benchmark of evolutionary multiobjective optimization.e NSGA-II performance is fairly sensitive to the number of objective functions, but this is not a main drawback for solving problem (3).Its pseudocode is shown in Pseudocode 1 (adapted from [9]).
Binary encoding was used; a "1" in the individual mth allele means that the mth project belongs to this particular portfolio.One-point crossover and the standard mutation operator were implemented (cf.[10]).Binary tournament selection was performed as in [11].
e approach for handling constraints is based on the principle that any individual satisfying the constraints is always better than any individual that does not.If the feasible individuals in the population are classi�ed into  nondominated classes, then any unfeasible individual is relegated to class .is implies that in binary tournaments any feasible individual will have priority over any nonfeasible individual.
e parameters of the evolutionary search were: crossover probability = 1; mutation probability = 0.02; population size = 100 number of generations = 500.

Selection of Public Project Portfolios
When the Information about Preferences Is Given in terms of Goals When an individual decision maker gives his/her particular project ranking, he/she is expressing preferences about projects, not about portfolios.In this section we are interested in the case when each member of a participatory group is able to express his/her preferences through a set of aspiration levels which re�ect the desired state of the concerned social object.Since those levels result from the aggregated action of projects, such information re�ects preferences about portfolios.e best portfolio for a heterogeneous participatory group should be the most acceptable agreement.Let us recall aer Fernandez et al. [4] some ideas about collective agreement.When a heterogeneous group is making decisions, situations may arise that impede, even with the existence of the "constitution" rules, the determination of an acceptable agreement.ese situations arise due to strong contradictions amongst important coalitions of its members, for example, when a solid opposition of a numerically signi�cant minority exists and has to accept a preference that is supported by a weak majority of the group.To establish a collective agreement is only possible when unanimity or an appreciable level of consensus exists (cf.[4]).A de�nition of this last concept may be associated to the satisfying of the following two conditions.(a) ere is an important agreeing majority.
(b) ere is no appreciable disagreeing minority.
Fernandez et al. [4] alerted about the undesirable effects of the "majority dictatorship." is could have a negative in�uence on the stability of the group, making it less governable.Such "dictatorship" could be acceptable as an organizational principle with simple applications and that bases its rationality in the idea that more individuals should have weight more than that of lesser individuals (such a judgment ignores the importance of disagreement intensity)."Majority dictatorship" performs well in collectives that are disciplined and stable, mostly if the group is fundamentally comfortable accepting the majority principle as a basis for its decisions, and/or mechanisms for motivation and coercion exist in order to support group stability over the potential inconformity of a minority.As acceptable as the organizing principle might be, one should not make the mistake of assigning an ontological value that is generated by its rationality for every organization.Rejecting "majority dictatorship, " the best agreement should maximize the group satisfaction and minimize the strong opposition.We need to de�ne what a satisfactory/unsatisfactory portfolio is for each individual decision maker, followed by a way to �nd a good compromise between measures of satisfaction and unsatisfaction for the group.
In order to make a formal de�nition of satisfaction/dissatisfaction, let us introduce several new assumptions.
Assumption 2. Each individual decision maker in the participatory event is able to set a goal vector  * ∈ ℜ  which contains his/her aspiration levels for the relevant criteria of the concerned social object.Assumption 3.Each individual decision-maker in the participatory event is able to set a set of constraints which express budgetary limits and some particular preferences concerning classes of projects, social groups, or geographic regions.Let us denote by   the corresponding feasible region in the portfolio space.Assumption 4. Each member of the participatory group can set a fuzzy preference relation   (,  de�ned on the criterion space;   values on [0, ] represent the degree of credibility of the statement "x is at least as good as y." In order to be computable from the multicriteria description of x and y,   (,  can be built by using outranking methods [12], as is shown in Section 5.2.
Assumption 5.Each one of the group coalitions   is a sufficiently homogeneous group as to achieve a consensus on the fuzzy preference relation   parameters, the constraint setting, the projects which are considered very important ones, and the goal vector  *  .

93:
Non-supported proyect F 2: Some acceptable solutions in the decision space.
th decision maker (or the th coalition) may accept that "x is at least as good as y." ��������� �� Let  *  be the goal vector set by the th decisionmaker (or the th coalition).Suppose that   is a portfolio which is being considered by the group;   is its image in the criterion space.e th decision maker (or the th coalition) is said to be satis�ed by   if and only if the following conditions hold: (i)   (  ,  *  ) ≥ ; (ii)   holds the constraints imposed by the th decisionmaker; (iii) e projects which are considered very important by the th decision maker belong to   .
��������� �� e th decision-maker (or the th coalition) is said to be unsatis�ed by   if one of the following conditions hold: (i)   (  ,  *  ) < 0.5 and   ( *  ,   ) ≥ ; (ii)   does not hold the constraints imposed by the th decision maker; (iii) most of the projects which are considered very important by the th decision maker do not belong to   ; Let   be the union of the feasible regions   .Note that two integer functions  sat (  ) and  dis (  ) may be de�ned on   .For each   in   with image   in the criterion space, there is a number  sat obtained by counting the number of individual decision-makers which are satis�ed by   .On the other hand,  dis is the result of counting the unsatis�ed members.Note that  dis +  sat ≤ .e difference   ( dis +  sat ) is the number of decision makers which do not support a favorable consensus but are not against it.
e best agreement may be seen as the best compromise solution for the problem Maximize  sat , Minimize  dis .  ∈   .
(4) Group satisfaction and dissatisfaction are optimized by solving problem (4).Dominated solutions of (4) should not be good agreements.Only nondominated solutions of ( 4) satisfying a minimum level of satisfaction should be considered.At least the condition of "agreeing relative majority"  sat , >  dis is necessary.So, (4) can be transformed into Maximize  sat , Minimize  dis s.t.  ∈    sat , >  dis . (5) For solving (5), any non-linear multiobjective optimization method able to generate the Pareto frontier is acceptable.Once the set  of nondominated solutions has been found, the GSDM should determine the "best" compromise.Such a most acceptable solution corresponds to the current stage of the group members' preferences and beliefs.
Unfeasibility of ( 5) should be a consequence of very con�icting value systems within the group.is is a tough situation which must be treated with care.e reasons for strong con�ict should be determined.Further group discussions and the application of groupware techniques (e.g., [13,14]) would be necessary to close divergent beliefs, preferences, and constraint settings in order to achieve feasible solutions.(5).Solving problem (5) is not an easy task due to its size and the nonlinearity of its objective functions.We use evolutionary multiobjective optimization.Evolutionary algorithms are less sensitive to some mathematical properties of objective functions and constraints than the traditional optimization methods [9].Additionally, in problem (5) an evolutionary algorithm helps to handle its complexity with respect to the number of applicant projects because computer efficiency of  these methods is less sensitive to problem size in comparison to the traditional techniques.

An Evolutionary Algorithm to Solve Problem
Algorithm 1 is a direct adaptation of the NSGA-II [11].
In the binary tournament, two individuals are chosen randomly, the winner being the one with a smaller "rank." If both are in the same front, the tournament is won by the one who has a bigger crowding distance.

Example: Resources Assignment for Social Action Programs (Projects)
. is example is a �ctitious decision making situation in which the participatory group chooses among 100 different social policies (projects), each with a direct social return.is is measured by using a nine-component vector ( 1 ,  2 , … ,  9 ).  =   , the number of people belonging to the jth social category which receive the kth bene�t level from that policy or project.In this example  = 1, 2, 3 correspond to (extreme poverty, poverty, low-middle), and k = 1, 2, 3 to (high impact, middle impact, low impact).
Generate a size  random population , calculating the portfolio criteria.Determine the different fronts assigning each individual a "rank" that is, the front which it belongs to and its crowding distance.Execute the following as many times as generations Generate an offspring population and calculate their criteria Select the parents using binary tournament.eir crosses produce two descendants to whom the mutation operator is applied Combine the parents' population and offspring population: Evaluate the  value in GDM × .GDM is the group of  decision-makers Calculate the satisfaction and dissatisfaction level for each individual: If   (  ,  *  ) ≥   and   satis�es the -DM restrictions; and all projects which -DM considers very important belong to the portfolio, en e individual  is satis�ed If   (  ,  *  ) < 0.5 and   ( *  ,   ) ≥ , or   doesn't satisfy the -DM restrictions or a signi�cant part of projects that -DM considers very important is not in   , en e individual  is unsatis�ed Count  sat and  dis Determine the different fronts Select the new population so that the members of the �rst fronts belong to it, and if necessary, execute the Crowding-Sort Repeat the above-mentioned as another generation.A 1 e role of GSDM becomes rather that of a group-work facilitator.If the collective is formed as a representative sample of the social structure, the solutions obtained by our method re�ect the social preference, and the procedure may become a paradigm of the best forms of democracy.
Complete interaction and symmetry DM (b) A supra-decision maker interacting with the group DM (c) Complete interaction, no symmetry F 1: Basic structures for collective work.
T 1: Information about projects.
,  ∈   ×   such that ,  ∈ ,    .(2) (ii) a set of strict discrepancies    {,  ∈   ×  such that   ,  ∈ ,   }; (iii) a set of weak discrepancies    {,  ∈   ×   such that   ,  ∈ ,   }.Let   ,   , and   denote the respective cardinality of the above sets.Considering also the number of supported projects in C (denoted by   ), problem (1) can be transformed into D can be partitioned in (i) a set of absolute discrepancies    {,  ∈   ×   such that   ,  ∈ ,   }; 6.e GSDM can set a real value  (05     such that given (x,y) in the criterion space if   (,   , the PROCEDURE NSGA-II ( ′ , Number_of_Generations)