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A participatory budget model under uncertainty

16 Feb 2016-European Journal of Operational Research (North-Holland)-Vol. 249, Iss: 1, pp 351-358

TL;DR: This paper proposes a model for participatory budgeting under uncertainty based on stochastic programming, and suggests that this approach seems lacking, especially in times of crisis when public funding suffers high volatility and widespread cuts.

AbstractParticipatory budgets are becoming increasingly popular in many municipalities all over the world. The underlying idea is to allow citizens to participate in the allocation of a fraction of the municipal budget. There are many variants of such processes. However, in most cases they assume a fixed budget based upon a maximum amount of money to be spent. This approach seems lacking, especially in times of crisis when public funding suffers high volatility and widespread cuts. In this paper, we propose a model for participatory budgeting under uncertainty based on stochastic programming.

Topics: Participatory budgeting (67%)

Summary (1 min read)

1. Introduction

  • Most countries have a strict legal framework that regulates budgetary processes.
  • Furthermore, the elaboration of flexible budgets requires the use of multiple tools and methods such as Monte Carlo simulation, forecasting or game theory models (Verbeeten, 2006) Section 5 illustrates their methodology with a simple example.

3. The case of a single participant

  • Β would typically be stated by the technical staff supporting the process after listening to the problem owners concerning uncertainty aversion, with sensitivity analysis performed to assess its impact.
  • The selection of this parameter is critical, since it will affect the number of choices available.
  • In general, the lower β is, the bigger the number of feasible portfolios would be available but, also, the bigger chances of not meeting the specified targets.

4.1.1. Posting under uncertainty

  • The authors assume that projects are ordered according to their expected utility and a simple bookkeeping mechanism is available to avoid repeating portfolios already declined.
  • A participant may propose the portfolio ϝ where projects are gradually included when the proportion of samples satisfying the corresponding constraints is greater than β.

Stop;

  • (c) Approximate K(S, x) through the nondominated portfolio closest to the straight line joining x and B(S, x).
  • The authors just need to replace the corresponding steps in Algorithm 7 (and eliminate its first line) to obtain a much more affordable algorithm.

4.3. Arbitration under uncertainty

  • Method assumes an initial inefficient solution and suggests at each iteration, as new solution, a Pareto improvement with respect to the previous offer, see Raiffa et al. (2002) .
  • The process ends when no further Pareto improvements are possible.
  • Again, this assumes that uncertainty has been previously resolved after applying Algorithms 1-4.
  • As this may be expensive computationally, the authors could apply a similar approach to the BIM under uncertainty algorithm in Section 4.1.2.

5. An example

  • The final budget therefore includes the following five projects:.
  • Bike lane, Park, School, Theater and Trees.

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European Journal of Operational Research 249 (2016) 351–358
Contents lists available at ScienceDirect
European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Interfaces with Other Disciplines
A participatory budget model under uncertainty
J. Gomez
a,
, D. Rios Insua
b
,C.Alfaro
a
a
Rey Juan Carlos University, Department of Statistics and Operations Research, Fuenlabrada, 28943 Madrid, Spain
b
ICMAT-CSIC, 280 49 Madrid, Spain
article info
Article history:
Received 16 April 2014
Accepted 14 September 2015
Available online 25 September 2015
Keywords:
Participatory budgeting
Multicriteria decision making
Resource allocation
Decision making under uncertainty
Flexible budgeting
abstract
Participatory budgets are becoming increasingly popular in many municipalities all over the world. The un-
derlying idea is to allow citizens to participate in the allocation of a fraction of the municipal budget. There are
many variants of such processes. However, in most cases they assume a fixed budget based upon a maximum
amount of money to be spent. This approach seems lacking, especially in times of crisis when public funding
suffers high volatility and widespread cuts. In this paper, we propose a model for participatory budgeting
under uncertainty based on stochastic programming.
© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the
International Federation of Operational Research Societies (IFORS). All rights reserved.
1. Introduction
Over the last years there have been movements demanding in-
creased participation in public policy, especially at the local level
(Matheus & Ribeiro, 2009; Smith, 2009). For this reason, institu-
tions worldwide are promoting various participatory initiatives, see
Ríos Insua and French (2010) for reviews. A paradigm for these is par-
ticipatory budgeting (PB) which allows citizens to take part in the
allocation of a fraction of the available financial resources, typically,
in local governments and municipalities.
PBs have spread to over 1500 municipalities across the world since
its inception in Porto Alegre (Brazil) in 1989, see Sintomer, Herzberg,
Allegretti, and Rocke (2010). The dissemination of PBs started in Latin
America including countries such as Ecuador, Argentina or Uruguay.
In 2001, PBs expanded to Europe with Italy, France and Spain becom-
ing the main countries of initial adoption. Over the last years, PBs
have also been implemented in municipalities in Asia, Oceania and
Africa. More recently, PB processes have reached the USA where they
have been tested in large cities such as Chicago or New York.
There are many variants of PBs according to several factors such
as the number and duration of meetings or the roles assigned to offi-
cials (who typically promote the PB experience), technical staff (who
support the implementation of the PB by providing cost estimates, fa-
cilitate preference elicitation or suggest initial criteria for project as-
sessment) and citizens or participants (who provide input concerning
projects, preferences in various phases or criteria), see Alfaro, Gomez,
Corresponding author. Tel.: +34 914888414.
E-mail addresses: javier.gomez@urjc.es (J. Gomez), david.rios@icmat.es (D.R. Insua),
cesar.alfaro@urjc.es (C. Alfaro).
and Rios (2010) or Gomez, Ríos Insua, Lavin, and Alfaro (2013) for de-
tails. The amount of capital funds allocated through PBs varies widely
across experiences: there are places where the expenditure is lim-
ited to a small proportion of the municipal budget, whereas in other
locations, like Rubí (Spain) or Campinas (Brazil), citizens have been
allowed to decide how to spend the entire investment budget, see
Cabannes (2004) and Nebot (2004) for details. However, most of the
PB experiences incorporate quantities, such as costs or budget avail-
able, which are assumed to be fixed before the execution period be-
gins. They are, therefore, static budgets, see Kriens, van Lieshout, Roe-
men, and Verheyen (1983) or Horngren et al. (2010).
There is another type of budget called flexible (Horngren, Bhi-
mani, Datar, & Foster, 2002; Mak & Roush, 1994; Nam Lee & Soo Kim,
1994), with growing acceptance in the private sector. This is an im-
portant tool applied to perform budget uncertainty analysis, usually
through scenarios, especially in times of economic crisis. However,
the use of flexible budgets is unusual in the public sector as it en-
tails administrative and bureaucratic difficulties (Robinson & Ysander,
1981). Most countries have a strict legal framework that regulates
budgetary processes. For example, in Spain, the General Budgetary
Act requires approval of the budget before the fiscal year starts. In or-
der to ensure the adoption of flexible budget methods, it would be
necessary to introduce budget reforms by amending existing laws or
adopting new ones. This reform process is complex and could take a
long time, see Lienert and Jung (2004). Furthermore, the elaboration
of flexible budgets requires the use of multiple tools and methods
such as Monte Carlo simulation, forecasting or game theory models
(Verbeeten, 2006) and public administrations do not frequently have
experts in such fields.
http://dx.doi.org/10.1016/j.ejor.2015.09.024
0377-2217/© 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS).
All rights reserved.

352 J. Gomez et al. / European Journal of Operational Research 249 (2016) 351–358
We propose in this paper a model for PBs under uncertainty, com-
bining the recent interest in participatory and flexible budgeting. In
Section 2, we introduce the problem. Then, we briefly describe an ap-
proach that can be used to solve problems in which there is uncer-
tainty about the values of some of its parameters. Section 4 proposes a
scheme based on the joint chance constraints method, adapting typ-
ical participatory decision tasks (negotiation, voting, arbitration) to
the presence of stochastic elements. Section 5 illustrates our method-
ology with a simple example. We conclude with some remarks and
lines for future research.
2. Participatory budgeting under uncertainty
PBs (see Alfaro et al., 2010) provide citizens with the possibility of
jointly deciding how to spend an amount of public funds in neighbor-
hood investment projects. Methodologically, we assimilate PBs with
allocating limited resources among several projects subject to con-
straints, with the aim of somehow maximizing the satisfaction of
all participants. Some of the quantities involved in a PB, like project
costs, income, available budget,..., may be subject to considerable un-
certainty, which we shall denote with the symbol on top to describe
the corresponding random variable. Salo, Keisler, and Morton (2011)
provide various perspectives on resource allocation problems.
We thus incorporate uncertainty to the classical PB problem
(Alfaro et al., 2010; Ríos & Ríos Insua, 2008). Assume, therefore, that
agroupofn persons has to decide how to spend a budget
b. There is
asetX of q possible projects, X =
{a
1
,...,a
q
}.Projecta
i
has an esti-
mated cost
c
i
, and is evaluated with respect to m criteria with values
x
j
i
, j = 1,...,m. We assume that the criteria are initially proposed by
municipality technicians but may be subject to discussion with par-
ticipants. The random variables
b,
c
i
and
x
j
i
will be typically assessed
or estimated by the organization technical staff. We represent this
information as in Table 1, which is exemplified in Table 4.
A feasible budget for the PB problem is a subset of projects, de-
fined by the corresponding subset of indices F I =
{1, 2,...,q},
which satisfies all constraints, including the maximum budget one.
Formally, we represent this through
iF
c
i
b. (1)
This is a stochastic constraint, as both the left and right terms are
random variables. In addition, there may be other constraints that
further restrict the set of feasible budgets. We describe some of them
as an illustration:
1. Restrict the maximum investment on one type of projects: Due to lo-
gistic, political or economic reasons, we could consider assigning
a maximum amount c ofthebudgettobeinvestedinaparticular
subset F
1
I of projects. This could be represented through
iFF
1
c
i
c. (2)
2. Mutually exclusive projects: In some cases, due to their similarity,
the inclusion of some projects would entail the exclusion of oth-
ers. Analogously, there could be a maximum number k of projects
Table 1
Participatory budget under uncer-
tainty. Basic data.
Project Cost Performance
a
1
c
1
(
x
1
1
,...,
x
m
1
)

a
i
c
i
(
x
1
i
,...,
x
m
i
)

a
q
c
q
(
x
1
q
,...,
x
m
q
)
Table 2
Matrix of (random) utilities for the PB problem.
Participants
Project Cost 1 ... j ... n
a
1
c
1
u
1
1
...
u
j
1
...
u
n
1

a
i
c
i
u
1
i
...
u
j
i
...
u
n
i

a
q
c
q
u
1
q
...
u
j
q
...
u
n
q
of a certain type, say concerning cultural services, which we de-
note as JI, to be included in the final budget. Formally, we could
represent this constraint through
iF
y
i
k,with
y
i
= 1ifi J
y
i
= 0ifi / J
.
(3)
3. Dependent projects: Sometimes a project requires another one to
be in the final budget. As an example, suppose there is a project
concerning building a new geriatric center and another one to
build its parking. Clearly the second one makes sense only if the
geriatric center is built as well. We represent this type of con-
straints through
y
i
1
y
j
1
, y
i
1
, y
j
1
{0, 1}, for certain i
1
, j
1
I, (4)
where y
k
= 1(0) means that the kth project is (not) in the final
budget. In example (4), we can include project a
i
1
, only if project
a
j
1
has been included.
In what follows, to fix ideas, when modeling the PB problem we
shall include the (stochastic) budget constraint (1) and constraints of
the types (2)(4).
We assume that we may model each participant’s preferences
through a multiattribute utility function u
j
, j = 1,...,n, whose
expected value should be maximized, see e.g. French (1986).The
utility functions account for the preferences and risk attitudes of
participants. We shall further assume that such utility functions are
additive.
1
Thus, if w
jk
is the weight that the jth participant gives to
the kth criterion, his utility for a performance x =
(x
1
,...,x
m
) would
be
u
j
(x) =
m
k=1
w
jk
u
jk
(x
k
),
with w
jk
0,
m
k=1
w
jk
= 1, k = 1,...,m. If a participant disregards
one criteria, he/she just needs to give it weight zero. Once with
the utility functions, we associate with the PB problem a random
matrix where each entry
u
j
i
is the utility that the jth participant
would obtain if the ith project was in the final budget, where
u
j
i
=
m
k=1
w
jk
u
jk
(
x
k
i
). Thus, we propagate the uncertainty in Table 1
through the participants’ utility functions to obtain Table 2.
3. The case of a single participant
We first describe how to obtain the optimal budget for a single
participant, as it will be a basic ingredient for the multiple partici-
pant case. For the jth decision maker, we have to solve the following
problem which provides the maximum expected utility project port-
folio, where E stands for expected value of the corresponding random
variable:
max
FI
E(
u
j
(F)) =
iF
E(
u
j
i
)
s.t.
iF
c
i
b,
(5)
1
Additivity of utility functions require preferential independence conditions, rea-
sonably frequently verified in practice, see Von Winterfeldt and Edwards (1986).

J. Gomez et al. / European Journal of Operational Research 249 (2016) 351–358 353
and other possible constraints that, as we have mentioned, will be
of the types (2)(4). Note that in this formulation we are assuming
that the value (expected utility) of a portfolio is the sum of the values
of the included projects. For discussions in relation with subadditiv-
ity or superadditivity of portfolio values see references in Salo et al.
(2011).
Problem (5) is a stochastic programming problem, see Kall and
Wallace (1994) or Abdelaziz, Aouni, and Fayedh (2007) fordetails.We
may solve it, e.g., through the Chance-Constrained Programming ap-
proach, presented by Charnes, Cooper, and Symonds (1958). In it, the
stochastic problem is replaced by an equivalent deterministic prob-
lem whose solution is considered the stochastic solution. Two clas-
sic versions of chance-constrained problems are the individual chance
constraints (Charnes & Cooper, 1959; Wets, 1989)andthejoint chance
constraints (Miller & Warner, 1965), which we adopt here: we place a
lower bound
β on the probability that each stochastic constraint will
be jointly satisfied. Thus, our jth individual problem would be refor-
mulated as
max
FI
E(
u
j
(F)) =
iF
E(
u
j
i
)
s.t. Pr
iF
c
i
b,
iFF
1
c
i
c
β, β [0, 1]
iF
y
i
k, where
y
i
= 1ifi J
y
i
= 0ifi / J
y
i
1
y
j
1
, y
i
1
, y
j
1
{0, 1}.
(6)
β would typically be stated by the technical staff supporting the
process after listening to the problem owners concerning uncertainty
aversion, with sensitivity analysis performed to assess its impact. The
selection of this parameter is critical, since it will affect the number
of choices available. In general, the lower
β is, the bigger the number
of feasible portfolios would be available but, also, the bigger chances
of not meeting the specified targets.
4. Participatory scheme under uncertainty
We consider now the participatory scheme taking into account
the presence of uncertainty. Assuming that the participants provide
their utility functions and obtain their expected utilities, this leads
us to the classic PB problem (Ríos & Ríos Insua, 2008), except for the
stochastic constraints.
A possible PB solution scheme is summarized through
Algorithm 1, in which we shall have to discuss how to adapt
Algorithm 1: Finding a group portfolio.
Generate the set of possible project portfolios:
ϕ = {ϕ
1
, ϕ
2
,...,ϕ
S
};
Filter the feasible portfolios;
Estimate expected utilities for each participant for every
feasible portfolio;
Calculate the optimal solution for each participant;
if All participants prefer the same optimal portfolio then
The PB process ends;
else
Filter the Pareto portfolios;
Find a group agreement;
group decision tasks (negotiation, voting, arbitration) to the presence
of uncertainty.
Note that this refers to finding a group agreement and this may be
pursued in several ways, depending on how do we schedule the group
decision tasks. For example, among many other schedules, we could
find an agreement directly through arbitration; or, alternatively, we
could find it through negotiation, and, if no agreement is reached,
use a voting session; or, directly through voting. Gomez et al. (2013)
provide a framework to choose the most appropriate group decision
tasks schedule when designing classic PB processes.
We detail now various steps in Algorithm 1.Henceforth,weuseN
to refer to the sample size in the Monte Carlo approximations, a topic
well studied in the simulation literature, see e.g. Henderson and Nel-
son (2006), in relation with precision of MC estimates and ranking &
selection and multiple comparison methods. Through these, we are
able to choose appropriate sample sizes. Note that when these are
deemed too big, we may need to opt for the alternative computa-
tionally cheaper approaches discussed in Sections 4.14.3 which treat
uncertainty only when necessary. Thus, Algorithm 1 may be seen as
a brute force approach to finding a group portfolio that may turn out
to be computationally expensive.
Filter feasible portfolios:We split this step into two substeps:
1. Delete from the set
ϕ of possible portfolios those not satisfying
the deterministic constraints ((3) and (4) in our example).
2. Delete from
ϕ the portfolios not satisfying the stochastic con-
straints ((1) and (2) in our case), e.g. by applying Algorithm 2
2
.
Algorithm 2: Filter portfolios not satisfying stochastic con-
straints.
Generate c
j
i
c
i
, i = 1,...,q and b
j
b, j = 1,...,N;
g = totalPort = length);
for k = 1 to totalPort do
cont = 0;
for j = 1 to N do
if
iϕ
k
c
j
i
< b
j
and
iϕ
k
F
1
c
j
i
< c
then
cont = cont + 1;
if
cont
N
< β then
Delete ϕ
k
from ϕ;
g = g 1;
k = k + 1;
In it, if the proportion of samples of a portfolio that satisfies the
stochastic constraints is greater than
β, we consider that this port-
folio verifies such constraints.
Estimate expected utilities for each participant for every feasible port-
folio: Algorithm 3 provides a vector with Monte Carlo estimates of the
Algorithm 3: Expected utilities for ith participant.
for k = 1 to g do
util = 0;
for j = 1 to N do
for h = 1 to m do
Generate x
jh
i
x
j
i
;
util = util +
rϕ
k
m
l=1
w
li
u
r
li
x
jh
i
;
f (i, k) =
util
N
;
k = k + 1;
2
Henceforth, we shall use the expression ‘Generate x
j
x, j = 1,...,N’tomean
sample N observations
{x
j
}
N
j=1
from the distribution of
x.

354 J. Gomez et al. / European Journal of Operational Research 249 (2016) 351–358
expected utilities of the g feasible portfolios, for each of the n partici-
pants f
(i) = { f (i, 1), f (i, 2),...,f(i, g)}, i = 1,...,n. For example,
f (i, k) =
1
N
N
j=1
rϕ
k
m
l=1
w
il
u
r
il
(
x
i
)

is a Monte Carlo estimate of the expected utility that participant i
obtain with portfolio
ϕ
k
.
Filter the Pareto portfolios: Algorithm 4 identifies the dominated
set of portfolios,
δ = {δ
1
, δ
2
,...,δ
z
}, filtering them from ϕ, which
contains the feasible portfolios.
Algorithm 4: Obtain Pareto portfolios from ϕ.
δ =∅;
for i = 1 to g do
if ϕ
i
/ δ then
for j = 1 to g do
if (i = j) and
j
/ δ) then
if ( f [1, i] f [1, j])and...and ( f [n, i] f [n, j])
then
δ = δ ϕ
j
;
Delete from ϕ the portfolios in δ;
We discuss now how to adapt the tasks (negotiation, voting, arbi-
tration) that may be used in the last step of Algorithm 1, which refers
to finding a group agreement, when uncertainty is relevant.
4.1. Negotiation under uncertainty
If participants disagree on their preferred budget, they may try to
deal with the conflict through negotiation. There are several classes
of negotiation methods, as described by Kersten (2001). In this pa-
per, we focus on two of them: Posting, applied by Ríos and Ríos In-
sua (2008), and the Balanced Increment Method (BIM), see Ríos and
Ríos Insua (2010). We focus on incorporating uncertainty to them.
For a general discussion on the role of uncertainty in negotiations see
Raiffa, Richardson, and Metcalfe (2002), Neale and Fragale (2006) or
Moon, Yao, and Park (2011).
4.1.1. Posting under uncertainty
In this method, participants offer portfolios for discussion, and
eventual approval, to the other participants. The offer with highest
percentage of acceptance will be implemented, should this percent-
age be sufficiently high before a negotiation deadline is met.
Algorithm 5 describes how to support the ith participant in mak-
ing offers until the deadline, where we assume that uncertainty has
been resolved as in Algorithm 1, through filtering feasible portfolios
Algorithm 5: Posting.
j = 1, solution =∅;
repeat
if ϕ
i
j
has not been offered by another participant then
O = ϕ
i
j
ϕ
i
;
ith participant offers O, which is subject to a voting
process, with result
v
0
;
if
v
O
> T then
solution = O;
j = j + 1;
until
(solution =∅) or (negotiation deadline) ;
and computing expected utilities. Algorithm 5 may be applied to sup-
port the n participants in parallel.
Let
ϕ
i
be the set ϕ of nondominated feasible portfolios, obtained
from Algorithm 4, ordered according to the ith participant expected
utilities.
v
o
will be the number of votes that an offer O receives. T
will be the acceptance threshold. The process ends when a deadline
is reached or the number of participants who accept a specific offer
is greater than the required threshold.
A computationally less expensive approach handles uncertainty
during the negotiation itself, as in Algorithm 6 which checks the
Algorithm 6: Generation of a proposable portfolio.
=∅;
post
(i) = 0, i = 1,...,q;
d
j
= 0, e
j
= 0, j = 1,...,N;
Generate b
j
˜
b, j = 1,...,N;
repeat
if ( a
i
) satisfies constraints (3) and (4) then
Generate c
j
i
˜
c
i
, j = 1,...,N;
d
j
= d
j
+ c
j
i
;
if a
i
F
1
then
e
j
= e
j
+ c
j
i
;
p =
#{ j:d
j
b
j
e
j
c}
N
;
if p
β then
post(i) = 1;
= a
i
i = i + 1;
until (i > q);
projects that may be included in a proposed portfolio (condition
post
(i) = 1). We assume that projects are ordered according to their
expected utility and a simple bookkeeping mechanism is available to
avoid repeating portfolios already declined. We use d
j
and e
z
to re-
fer to type (1) and (2) constraints, respectively, where F
1
I are the
projects referred to in type (2) constraint. A participant may propose
the portfolio ϝ where projects are gradually included when the pro-
portion of samples satisfying the corresponding constraints is greater
than
β.
4.1.2. BIM under uncertainty
BIM is an iterative multilateral negotiation support method, based
on the discrete balanced increment solution, see Raiffa et al. (2002).
Starting from the disagreement point d, the method iteratively of-
fers (Kalai & Smorodinsky, 1975) solutions to participants. The pro-
cess ends when the parties accept the offered solution or there is no
agreement but the last offer is close enough to the Pareto set. Let u
t
j
be the expected utility level for the jth participant at the tth step;
S =
{x R
n
: x = (E(u
1
k
)),...,E(u
n
k
))) forsomefeasibleport-
folio
ϕ
k
} be the set of attainable values; d = (d
1
, ..., d
n
), be the dis-
agreement point, so that d
i
represents the utility level that the ith
participant would receive when no agreement is reached; P(S, d), the
set of Pareto solutions in S that improve upon d; K(S, d), the Kalai–
Smorodinsky solution of the arbitration problem (S, d); and B(S, d),
the bliss point associated with (S, d).
Algorithm 7 implements BIM, where we assume that
ϕ has been
obtained after applying Algorithms 14.
This may be a computationally expensive approach, as it assumes
that uncertainty has been resolved through Algorithms 14.Apos-
sible alternative replaces the computation of B(S, x) and K(S, x),
as follows, where, to simplify matters,
ξ denotes all constraints in
problem (6):

J. Gomez et al. / European Journal of Operational Research 249 (2016) 351–358 355
Algorithm 7: BIM.
Calculate P(S, d) = ϕ {x R
n
: x
i
d
i
};
Fix
α (0, 1);
Start with x
0
= d, t = 0;
Calculate B
(S, x
0
) and K(S, x
0
);
Offer K
(S, x
0
);
while K
(S, x
t
) not accepted by majority of participants do
if x
t
is close to K(S, x
t
) then
Stop;
else
x
t+1
= x
t
+ α(K(S, x
t
) x
t
);
t = t + 1;
Calculate B
(S, x
t
) and K(S, x
t
);
if K
(S, x
t
) = K(S, x
t1
) then
offer K(S, x
t
);
1. First, we compute B(S, x), solving the following stochastic pro-
gramming problem for each participant, whose optimal value is
B
j
(S, x)
max
FI
E(
u
j
(F))
s.t. ξ
E(
u
j
(F)) x
j
, j = 1,...,n.
(7)
2. Then, we calculate K(S, x). As its determination in discrete and
stochastic cases is expensive computationally, we propose this ap-
proach:
(a) Apply Algorithm 8 to generate a set
ϕ
of, at most, z random
portfolios
{ϕ
1
, ϕ
2
,...,ϕ
z
} satisfying E
u
j
(F)
x
j
, for j =
1,...,n.
Algorithm 8: Random portfolio generation.
ϕ
=∅;
while lengt h
) < z do
Generate a portfolio F ;
if (F is feasible) and (E
u
j
(F)
x
j
)then
ϕ
= ϕ
{F};
if ϕ
=∅then
Declare no solution through negotiation.
(b) Calculate the nondominated portfolios in ϕ
, applying
Algorithm 4 to
ϕ
.
(c) Approximate K(S, x) through the nondominated portfolio clos-
est to the straight line joining x and B(S, x).
We just need to replace the corresponding steps in Algorithm 7 (and
eliminate its first line) to obtain a much more affordable algorithm.
4.2. Voting under uncertainty
Achieving consensus in a negotiation is sometimes not possi-
ble since participants may have very different preferences. In other
circumstances, the PB process requires obtaining a quick solution.
Voting may then be a useful method to obtain it. Bartels (1986),
Macdonald and Rabinowitz (1993) or Nurmi (2002) discuss issues
in relation with voting and uncertainty. Voting can be performed
through different rules, such as simple majority, approval voting or
Borda count, see Brams and Fishburn (2002) or Nurmi (2010) for ref-
erences. We shall use approval voting (Brams & Fishburn, 1983).
Assume first that uncertainty has been resolved as explained
through Algorithms 14. In order to solve the PB problem with ap-
proval voting, (1) each participant votes for his acceptable portfolios
(those that exceed his expected utility threshold); (2) votes are ag-
gregated; (3) project portfolios are ordered according to the number
of votes; and (4) the feasible portfolio with highest number of votes
is offered as solution.
Alternatively, we could deal with uncertainty during voting, lead-
ing to the following steps, where each participant votes based on ap-
proval voting, taking into account the constraints:
1. The kth participant orders projects a
j
based on expected utilities.
Assume, with no loss of generality, that
E(
u
k
(a
1
)) ··· E(
u
k
(a
j
)) E(
u
k
(a
j+1
)) ··· E(
u
k
(a
q
)).
Then, he votes according to Algorithm 6,wherepost(i) = 1(0)
means now that the kth participant votes (does not vote) for the
ith alternative.
2. Votes are aggregated in
vot es(i), and alternatives ordered accord-
ing to the number of votes. Suppose we label them as follows
vot es(1) vot es(2) ··· vot es(q),
where vot es(1) refers to the most voted project and vot es(q) to the
project which received the smallest number of votes.
3. Based on the order of
vot es(i), we apply Algorithm 6,wherepost(i)
means now whether project a
i
is in the final budget (post(i)=1) or
not (post(i)=0).
4.3. Arbitration under uncertainty
Arbitration (Efremov, Insua, & Lotov, 2009; Raiffa, 1953; Thom-
son, 1994) is a dispute resolution mechanism involving a third actor
who makes a final decision for a group, based on justice and fairness
concepts, once the opinions and reasoning of different participants
have been presented. Rosenthal (1978), Babcock and Taylor (1996) or
Bollen, Euwema, and Müller (2010) discuss issues in relation with ar-
bitration under uncertainty.
We propose an approach based on balanced concessions as in
Algorithm 9,seeRíos and Ríos Insua (2010) for further details. This
Algorithm 9: Arbitration algorithm with uncertainty resolved.
Calculate optimal solution for each participant:
D
0
i
= D
i
(S, d), i = 1, ..., n;
Calculate bliss point b
0
= B(S, d) = (D
1
0
, ..., D
n
0
);
Calculate x
0
= B
1
(S, b
0
) = d and
ˆ
K
0
= K(S, x
0
);
Offer alternative associated with
ˆ
K
0
;
while offer is not accepted unanimously by participants do
if x
t
or b
t
is close to
ˆ
K
t
then
Stop;
else
Calculate C
t+1
= α(b
t
ˆ
K
t
);
t = t + 1;
Calculate b
t
= b
t1
C
t
; x
t
= B
1
(S, b
t
) and
ˆ
K
t
= K(S, x
t
);
if
ˆ
K
t
=
ˆ
K
t1
then
Offer alternative associated with
ˆ
K
t
;
method assumes an initial inefficient solution and suggests at each
iteration, as new solution, a Pareto improvement with respect to the
previous offer, see Raiffa et al. (2002). The process ends when no fur-
ther Pareto improvements are possible. An equitable way to conduct
this would be to increment at each step the participants’ utilities in
such a way that it implies a balanced concession, proportional to the

Citations
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Journal ArticleDOI
TL;DR: A fuzzy technique is proposed for order preference based on the similarity to an ideal solution for the personalized ranking of projects in a participatory budget (PB) based on an empirical example from a Poznan PB project ( Poland).
Abstract: In this study, a fuzzy technique is proposed for order preference based on the similarity to an ideal solution for the personalized ranking of projects in a participatory budget (PB). A PB is a group decision-making process where citizens distribute public resources among a set of city investment proposals. The dynamic growth in the popularity of PB during the last 10 years has been due to a significant increase in the number of projects submitted and the demonstrable weakness of the traditional majority vote. The rationality of decision-makers is restricted by the large number of possible options from which voters can choose only a few within a limited amount of time, and thus there is no opportunity to review all of the projects. Appropriate decision support tools can assist with the selection of the best outcome and help to address the growth of PB processes. The ranking of PB projects is a specific problem because multi-criteria comparisons are based on non-quantitative criteria, i.e., nominal and fuzzy criteria. The “Technique for Order Preference by Similarity to Ideal Solution” (TOPSIS) method aims to minimize the distance to the ideal alternative while maximizing the distance to the worst. In a fuzzy extension of TOPSIS, the ratings of alternatives and the weights of the criteria are fuzzy numbers or linguistic variables. The major modification required to the TOPSIS method for PB is that the perfect objective solution does not exists among the maximum and minimum values for the criteria. Thus, the subjective choice is the ideal solution for the decision maker and the negative ideal solution is the most dissimilar solution. This study describes the application of fuzzy TOPSIS with a modification for PB based on an empirical example from a Poznan PB project (Poland).

79 citations


Cites background from "A participatory budget model under ..."

  • ...Theoretical studies have focused on communication, deliberation, and decision making [5], modeling under uncertainty [6], designing general frameworks [7], and experimental solutions [8], [9], but most of these existing solutions only consider support for administrative tasks related to PBs, rather than the actual decisionmaking process....

    [...]



Journal ArticleDOI
Abstract: Purpose The purpose of this paper is to present a model to support governmental local managers in public budget optimization, based on an integration of methods. It was constructed to fill the gap related to weights definition in problematic, commonly performed subjective assessments. This model supports the decision making in budget distribution identifying the importance of sectors in local governments, captured by historical data. Design/methodology/approach The model was developed following three steps: the first step included the exploitation of the characteristics of local sectors represented by city departments and the data collection procedure using time series (TS). In the second one, the weights regarding the importance of each city department were calculated by the UTASTAR method and based on historical data from the first step. Finally, an objective function was formulated using linear programming and constraints based on law specifications, and as a result, an optimized projection for public budget distribution was performed. Findings The results demonstrated that the model can be more efficient to weights definition, considering the behavior of preferences by historical data and supporting local public resources optimization, also to comply with the legislation, being able to predict or project future values available on the budget. Research limitations/implications The theoretical and practical implications are related with a novelty in recognizing the weights for criteria by a historical behavior of preferences. It can be bringing important directions for budget distribution. The main limitation detected in this study was the difficulty to formulate an assessment involving an integrated opinion from local managers and the population. Practical implications First of all, with the correct allocation of resources, the government has a greater advantage to capture investments from the negotiation with development entities and banks. Second, an efficient local government management can promote compliance with legislation and more transparent public policies. Social implications The correct distribution of resources affects the life quality for citizens, since the government acts as a provider of essential services for the population like education, safety, health, particularly for citizens who depend exclusively on the services offered by the local government. Moreover, it can also affect the environment as resources for garbage collection, disposal services and sanitation and, finally, affect the city development such as infrastructure, taxes, etc. Originality/value It might be considered an original contribution mainly by the development of a procedure to capture values for weights by TS and meeting the manager’s requirements, based on analytical, statistical and mathematical tools integrated.

12 citations


Posted Content
01 Jan 2017
Abstract: This article is an attempt to present and analyse the changes in the scale and scope of the significance of participatory budgets introduced in the years 2014-2016 in most Polish cities with district rights (MNP), as well as to determine the importance of activities in the field of marketing communication for the effective implementation of the objectives related to its functioning. The analysis carried out in the article concerning PB development in MNP clearly indicates the rise of interest of this form of citizens’ participation in deciding about MNP expenses. Simultaneously, differences in scale and range of implemented PB among examined groups might be visible. As an example, the city of Wroclaw shows that an adequate marketing communication of the city as well as local project leaders with inhabitants is a fundamental factor influencing forming an active participation among inhabitants.

7 citations


Cites background from "A participatory budget model under ..."

  • ...…(radio, television, press, the Internet, telephone networks) and through medium such as billboards, posters, audiotapes and video, CD-ROMs, etc. (Kotler & Keller, 2012); direct marketing - the use of post, phone, fax, e-mail or the Internet to communicate directly or encouraging specific…...

    [...]

  • ...…- the use of post, phone, fax, e-mail or the Internet to communicate directly or encouraging specific recipients to a response and a dialogue (Kotler & Keller, 2012); events and experiences marketing - activities and programs organized by the municipality, which are aimed at daily or…...

    [...]

  • ...…municipality, which are aimed at daily or occasional interactions with recipients, including sports, cultural, entertainment, charity events, etc. (Kotler & Keller, 2012) and finally interactive marketing - activities and programs on the Internet, aimed at drawing the recipients into interaction…...

    [...]

  • ...…etc. (Kotler & Keller, 2012) and finally interactive marketing - activities and programs on the Internet, aimed at drawing the recipients into interaction and direct or indirect increase of awareness, improving the image, increasing the interest in the offer of municipality (Kotler & Keller, 2012)....

    [...]


Journal ArticleDOI
TL;DR: A negotiation support system (NSS) with a theoretical modeling that considers the aspects of human personality and negotiator’s behavior to assist the decision-making of public managers and stakeholders in democratic bargaining processes and support social-efficient outcomes is introduced.
Abstract: Purpose This paper aims to introduce a negotiation support system (NSS) with a theoretical modeling that considers the aspects of human personality and negotiator’s behavior to assist the decision-making of public managers and stakeholders in democratic bargaining processes and support social-efficient outcomes. Design/methodology/approach A game theoretical modeling of public participatory negotiations characterized by complete and perfect information is explored with the inclusion of personality aspects and negotiation styles. The importance of the negotiation knowledge disclosure in the sequential bargains of participative budgeting is highlighted by an experiment with 162 state-owned companies’ managers and graduate students to present the contribution of the system’s applicability. Findings A considerable number of Pareto-efficient deliberation agreements are obtained with few interactions when the negotiation strategies and the personality aspects of opponents and stakeholders are freely available (a symmetry in the public negotiation knowledge). In addition to the set of Pareto-efficient agreements, those with the best social outcome (i.e. that maximize the group satisfaction despite individual losses) are observed when the informational tool for personality and negotiation style inference is enabled. Originality/value Many scholars argue for Pareto-efficient allocation instead of equal divisions of resources within participative democracies and public governance. This work provides a new system with an empirical application and theoretical modeling which may support those arguments based on the nonverbal negotiation aspects.

6 citations


References
More filters

Journal ArticleDOI
TL;DR: The paper presents a method of attack which splits the problem into two non-linear or linear programming parts, i determining optimal probability distributions, ii approximating the optimal distributions as closely as possible by decision rules of prescribed form.
Abstract: A new conceptual and analytical vehicle for problems of temporal planning under uncertainty, involving determination of optimal sequential stochastic decision rules is defined and illustrated by means of a typical industrial example. The paper presents a method of attack which splits the problem into two non-linear or linear programming parts, i determining optimal probability distributions, ii approximating the optimal distributions as closely as possible by decision rules of prescribed form.

2,335 citations


"A participatory budget model under ..." refers background in this paper

  • ...Two clasic versions of chance-constrained problems are the individual chance onstraints (Charnes & Cooper, 1959; Wets, 1989) and the joint chance onstraints (Miller & Warner, 1965), which we adopt here: we place a ower bound β on the probability that each stochastic constraint will e jointly…...

    [...]


Book
01 Jan 1972
Abstract: 1. The Accountant's Role in the Organization. 2. An Introduction to Cost Terms and Purposes. 3. Cost-Volume Profit Analysis. 4. Job Costing. 5. Activity-Based Costing and Activity-Based Management. 6. Master Budget and Responsibility Accounting. 7. Flexible Budgets, Variances, and Management Control: I 8. Flexible Budgets, Variances, and Management Control: II. 9. Inventory Costing and Capacity Analysis. 10. Determining How Costs Behave. 11. Decision Making and Relevant Information. 12. Pricing Decisions and Cost Management. 13. Strategy, Balanced Scorecard, and Strategic Profitability Analysis. 14. Cost Allocation, Customer-Profitability Analysis, and Sales-Variance Analysis. 15. Allocation of Support Department Costs, Common Costs and Revenues. 16. Cost Allocation: Joint Products and Byproducts. 17. Process Costing. 18. Spoilage Rework, and Scrap. 19. Quality, Time, and the Theory of Constraints. 20. Inventory Management, Just-in-Time, and Backflush Costing. 21. Capital Budgeting and Cost Analysis. 22. Management Control Systems, Transfer Pricing, and Multinational Considerations. 23. Performance Measurement, Compensation, and Multinational Considerations.

1,855 citations


Journal ArticleDOI
Abstract: A two-person bargaining problem is considered. It is shown that under four axioms that describe the behavior of players there is a unique solution to such a problem. The axioms and the solution presented are different from those suggested by Nash. Also, families of solutions which satisfy a more limited set of axioms and which are continuous are discussed. WE CONSIDER a two-person bargaining problem mathematically formulated as follows. To every two-person game we associate a pair (a, S), where a is a point in the plane and S is a subset of the plane. The pair (a, S) has the following intuitive interpretation: a = (a1, a2) where ai is the level of utility that player i receives if the two players do not cooperate with each other. Every point x = (x1, x2) e S represents levels of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they do cooperate. We are interested in finding an outcome in S which will be agreeable to both players. This problem was considered by Nash [3] and his classical result was that under certain axioms there is a unique solution. However, one of his axioms of independence of irrelevant alternatives came under criticism (see [2, p. 128]). In this paper we suggest an alternative axiom which leads to another unique solution. Also, it was called to our attention by the referee that experiments conducted by H. W. Crott [1] led to the solution implied by our axioms rather than to Nash's solution. We also consider the class of continuous solutions which are required to satisfy only the axioms of Nash which are usually accepted. We give examples of families of such solutions.

1,609 citations


"A participatory budget model under ..." refers methods in this paper

  • ...BIM under uncertainty BIM is an iterative multilateral negotiation support method, based n the discrete balanced increment solution, see Raiffa et al. (2002). tarting from the disagreement point d, the method iteratively ofers (Kalai & Smorodinsky, 1975) solutions to participants....

    [...]


Book
01 Jan 1986

836 citations


"A participatory budget model under ..." refers methods in this paper

  • ...We assume that we may model each participant’s preferences hrough a multiattribute utility function uj, j = 1, . . . , n, whose xpected value should be maximized, see e.g. French (1986)....

    [...]


MonographDOI
31 Mar 2007
Abstract: Preface Part I. Fundamentals 1. Decision Perspectives On four approaches to decision making 2. Decision Analysis On how individuals should and could decide 3. Behavioral Decision Theory On the psychology of decisions on how real people do decide 4. Game Theory On how rational beings should decide separately in interactive situations 5. Negotiation Analysis On how you should and could collaborate with others Part II. Two-Party Distributive (Win-Lose) Negotiations 6. Elmtree House On setting the stage for adversarial bargaining 7. Distributive Negotiations: The Basic Problem On the essence of noncooperative, win-lose negotiations 8. Introducing Complexities: Uncertainty On deciding to settle out of court and other problems of choice under uncertainty 9. Introducing Complexities: Time On entrapments and downward escalation on real and virtual strikes 10. Auctions and Bids On comparing different auction and competitive bidding procedures Part III. Two-Party Integrative (Win-Win) Negotiations 11. Template Design On brainstorming alone and together on deciding what must be decided 12. Template Evaluation On deciding what you need and want 13. Template Analysis (I) On finding a joint compromise for a special simple case 14. Template Analysis (II) On finding a joint compromise for the general case 15. Behavioral Realities On learning how people do negotiate in the laboratory and the real world 16. Noncooperative Others On how to tackle noncooperative adversaries Part IV. External Help 17. Mostly Facilitation and Mediation On helping with people problems 18. Arbitration: Conventional and Nonconventional On how a neutral joint analyst might help 19. What Is Fair? On principles for deciding joint outcomes 20. Parallel Negotiations On negotiating without Negotiating Part V. Many Parties 21. Group Decisions On organizing and managing groups 22. Consensus On how to achieve a shared agreement for all 23. Coalitions On the dynamics of splitting and joining subgroups 24. Voting On anomalies of collective action based on voting schemes 25. Pluralistic Parties On dealing with parties fractured by internal conflict 26. Multiparty Interventions On the role of external helpers in multiparty negotiations 27. Social Dilemmas On the conflict between self-interest and group interest References Note on Sources Index

641 citations


"A participatory budget model under ..." refers background or methods in this paper

  • ...For a general discussion on the role of uncertainty in negotiations see Raiffa, Richardson, and Metcalfe (2002), Neale and Fragale (2006) or Moon, Yao, and Park (2011)....

    [...]

  • ...BIM under uncertainty BIM is an iterative multilateral negotiation support method, based n the discrete balanced increment solution, see Raiffa et al. (2002). tarting from the disagreement point d, the method iteratively ofers (Kalai & Smorodinsky, 1975) solutions to participants....

    [...]

  • ...…Calculate bt = bt−1 − Ct ; xt = B−1(S, bt) and K̂t = K(S, xt); if K̂t = K̂t−1 then Offer alternative associated with K̂t ; ethod assumes an initial inefficient solution and suggests at each teration, as new solution, a Pareto improvement with respect to the revious offer, see Raiffa et al. (2002)....

    [...]


Frequently Asked Questions (1)
Q1. What have the authors contributed in "A participatory budget model under uncertainty" ?

In this paper, the authors propose a model for participatory budgeting under uncertainty based on stochastic programming.