Smooth Optimal Decision Strategies
for Static Team Optimization Problems
and Their Approximations
Giorgio Gnecco1,2 and Marcello Sanguineti2
1
Department of Computer and Information Science (DISI), University of Genova
Via Dodecaneso, 35, 16146 Genova, Italy
2
Department of Communications, Computer, and System Sciences (DIST)
University of Genova
Via Opera Pia 13, 16145 Genova, Italy
[email protected], [email protected]
Abstract. Sufficient conditions for the existence and uniqueness of
smooth optimal decision strategies for static team optimization problems
with statistical information structure are derived. Approximation methods and algorithms to derive suboptimal solutions based on the obtained
results are investigated. The application to network team optimization
problems is discussed.
Keywords: Team utility function, value of a team, statistical information structure, approximation schemes, suboptimal solutions, network
optimization.
1
Introduction
Decision makers (DMs) cooperating to achieve a common goal, expressed via
a team utility function, model a variety of problems in engineering, economic
systems, management science and operations research, in which centralization is
not feasible and so distributed optimization processes have to be performed. Each
DM has at disposal various possibilities of decisions generated via strategies, on
the basis of the available information that it has about a random variable, called
state of the world. In the model that we adopt, the information is expressed via
a probability density function, so we have a statistical information structure [13,
Chapter 3].
In general, one centralized DM that, relying on the whole available information, maximizes the common goal, provides a better performance than a set of
decentralized DMs, each of them having partial information. However, centralization is not always feasible. For example, each DM may have access only to
local information that cannot be instantaneously exchanged. Alternatively, the
cost of making the whole information available to one single DM may be too
high with respect to having several DMs with different information. This is often the case, e.g., in communication and computer networks extending in large
J. van Leeuwen et al. (Eds.): SOFSEM 2010, LNCS 5901, pp. 440–451, 2010.
c Springer-Verlag Berlin Heidelberg 2010
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441
geographical areas, production plants, energy distribution systems, and traffic
systems in large metropolitan areas divided into sectors.
In the team optimization problems that we address in this paper, the information of each DM depends on the state of the world but is independent of the
decisions of the other DMs. These are called static teams, in contrast to dynamic
teams, for which each DM’s information can be affected by the decisions of the
other members. However, many dynamic team optimization problems can be
reformulated in terms of equivalent static ones [28].
Static teams were first investigated by Marschak and Radner [22,23,24], who
derived closed-form solutions for some cases of interest. Then, dynamic teams
were studied [5]. Unfortunately, closed-form solutions to team optimization problems can be derived only under quite strong assumptions on the team utility
function and the way in which each DM’s information is influenced by the state
of the world (and, in the case of dynamic teams, by the decisions previously taken
by the other DMs). In particular, most available results hold under the so-called
LQG hypotheses (i.e., linear information structure, concave quadratic team utility, and Gaussian random variables) and with partially nested information, i.e,
when each DM can reconstruct all the information available to the DMs that
affect its own information [7,12]. However, as remarked in [9], these assumptions
are often too simplified or unrealistic. For more general problems, closed-form
solutions are usually not available, so one has to search for suboptimal solutions.
In this paper, we derive sufficient conditions for the existence and uniqueness
of smooth optimal decision strategies, for static team optimization problems with
statistical information structure. Then, we show that a sufficiently high degree of
smoothness of the optimal decision strategies is a useful property when searching
for suboptimal solutions.
The paper is organized as follows. Section 2 introduces definitions and assumptions and formulates the family of static team optimization problems under
consideration. Section 3 investigates existence and uniqueness of smooth optimal strategies for such problems. Section 4 examines some consequences of the
obtained results in developing approximation methods and algorithms to derive
suboptimal solutions. Section 5 discusses the application of our results to static
network team optimization problems.
2
Problem Formulation
The context in which we shall formalize the optimization problem and derive
our results is the following.
– Static team of n decision makers (DMs), i = 1, . . . , n.
– x ∈ X ⊆ Rd0 : vector-valued random variable, called state of the world,
describing a stochastic environment. The vector x models the uncertainties
in the external world, which are not controlled by the DMs.
– yi ∈ Yi ⊆ Rdi : vector-valued random variable, which represents the information that the DM i has about x.
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– si : Yi → Ai ⊆ R: measurable strategy of the i-th DM.
– ai = si (yi ): decision that the DM i chooses on the basis
nof the information yi .
n
n
– u : X × Πi=1
Yi × Πi=1
Ai ⊆ RN → R, where N = i=0 di + n: real-valued
team utility function.
– The information that the n DMs have on the state of the world x is modelled
by an n-tuple of random variables y1 , . . . , yn , i.e., by a statistical information
structure [6] represented by a joint probability density ρ(x, y1 , . . . , yn ) on the
n
set X × Πi=1
Yi .
We shall address the following family of static team optimization problems.
Problem STO (Static Team Optimization with Statistical Information). Given the statistical information structure ρ(x, y1 , . . . , yn ) and the team
utility function u(x, y1 , . . . , yn , a1 , . . . , an ), find
sup v(s1 , . . . , sn ) ,
s1 ,...sn
where
n
v(s1 , . . . , sn ) = Ex,y1 ,...,yn {u(x, {yi }n
i=1 , {si (yi )}i=1 )} .
The quantity sups1 ,...sn v(s1 , . . . , sn ) is called the value of the team.
Throughout the paper, we make the following three assumptions. For Ω ⊆ Rd ,
by C(Ω) we denote the space of continuous functions on Ω; for a positive integer
m > 0, by C m (Ω) we denote the spaces of functions on Ω, which are continuous
together with their partial derivatives up to the order m.
A1. The sets X, Y1 , . . . , Yn are compact, and A1 , . . . , An are bounded closed
intervals. For an integer m ≥ 2, the team utility u is of class C m on an open set
n
n
Yi × Πi=1
Ai , and ρ a (strictly) positive probability density
containing X × Πi=1
n
on X × Πi=1 Yi , which can be extended to a function of class C m on an open set
n
containing X × Πi=1
Yi .
A concave function f defined on a convex set Ω has concavity at least τ > 0
if for all u, v ∈ Ω and every supergradient1 pu of f at u one has f (v) − f (u) ≤
pu · (v − u) − τ v − u2 . If f is of class C 2 (Ω), then a necessary condition for its
concavity at least τ is supu∈Ω λmax (∇2 f (u)) ≤ −τ , where λmax (∇2 f (u)) is the
maximum eigenvalue of the Hessian ∇2 f (u).
n
Yi ×
A2. There exists τ > 0 such that the team utility function u : X × Πi=1
n
Πi=1 Ai is separately concave in each of the decision variables, with concavity at
least τ (i.e., if all the arguments of u are fixed except the decision variable ai ,
then the resulting function of ai has concavity at least τ ).
Assumption A2 is motivated by tractability reasons and encountered in practice. For example, in economic problems it can be motivated by the “law of
diminishing returns”, i.e., the fact that the marginal productivity of an input
usually diminishes as the amount of output increases [23, p. 99 and p. 110].
1
For Ω ⊆ Rd convex and f : Ω → R concave, pu ∈ Rd is a supergradient of f at u ∈ Ω
if for every v ∈ Ω it satisfies f (v) − f (u) ≤ pu · (v − u) .
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443
A3. For every n-tuple {s1 , . . . , sn } of strategies, the strategies defined as
n
ŝ1 (y1 ) = argmax Ex,y2 ,...,yn |y1 {u(x, {yi }n
i=1 , a1 , {si (yi )}i=2 )} ∀y1 ∈ Y1 ,
a1 ∈A1
...
n−1
ŝn (yn ) = argmax Ex,y1 ,...,yn−1 |yn {u(x, {yi }n
i=1 , {si (yi )}i=1 , an )} ∀yn ∈ Yn
an ∈An
do not lie on the boundaries of A1 , . . . , An , respectively.
The interiority condition in Assumption A3 can be imposed a-priori, by
strongly penalizing the team utility function on the boundary. Simple examples of problems for which Assumptions A1, A2 and A3 hold simultaneously can
be constructed by starting from a problem in which there is no interaction among
n
the DMs (i.e., u(x, y1 , . . . , yn , s1 (y1 ), . . . , sn (yn )) = i=1 ui (x, yi , . . . , yn , s1 (yi )),
so that the assumptions are easy to impose), then adding to the team utility
function a sufficiently small smooth interaction term.
3
Existence and Uniqueness of Smooth Optimal
Strategies
The next theorem (which takes the hint from [13, Theorem 11, p. 162], and
extends it to a higher degree of smoothness) gives conditions guaranteeing that
Problem STO has a solution made of an n-tuple of strategies that are Lipschitz
continuous together with their partial derivatives up to a certain order. For
limitations of space, we only sketch the proof for n = 2; details can be found
in [8, Chapter 5].
Theorem 1. Let Assumptions A1, A2 and A3 hold. Then Problem STO admits
an n-tuple (so1 , . . . , son ) of C m−2 optimal strategies with partial derivatives that
are Lipschitz up to the order m − 2.
Sketch of proof. Let n = 2. Consider a sequence {sj1 , sj2 } of pairs of strategies,
indexed by j ∈ N+ , such that limj→∞ v(sj1 , sj2 ) = sups1 ,s2 v(s1 , s2 ) (such a sequence exists by the definition of supremum). From the sequence {sj1 , sj2 }, we
generate another sequence {ŝj1 , ŝj2 } defined by
ŝj1 (y1 ) = argmax M1j (y1 , a1 ) ∀y1 ∈ Y1 ,
a1 ∈A1
ŝj2 (y2 ) = argmax M2j (y2 , a2 ) ∀y2 ∈ Y2 ,
a2 ∈A2
where for every (y1 , a1 ) ∈ Y1 × A1 and (y2 , a2 ) ∈ Y2 × A2 , we let
M1j (y1 , a1 ) = Ex,y2 |y1 {u(x, y1 , y2 , a1 , sj2 (y2 ))} ,
M2j (y2 , a2 ) = Ex,y1 |y2 {u(x, y1 , y2 , ŝj1 (y1 ), a2 )} .
Since the probability density ρ(x, y1 , y2 ) is of class C m and strictly positive on an
open set containing X ×Y1 ×Y2 , we obtain that the conditional density ρ(x, y2 |y1 )
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G. Gnecco and M. Sanguineti
is of class C m on the compact set X × Y1 × Y2 and the team utility function u
is of class C m on the compact set X × Y1 × Y2 × A1 × A2 . So M1j , as an integral
dependent on parameters, is of class C m on the compact set Y1 × A1 , with upper
bounds on the sizes of its partial derivatives up to the order m independent
of y1 , a1 , and j. In particular, it is easy to show that M1j has concavity at least τ
in a1 . By such continuity and concavity properties of M1j with respect to a1 ,
for all y1 ∈ Y1 the set argmaxa1 ∈A1 M1j (y1 , a1 ) consists of exactly one element.
An analogous conclusion holds for argmaxa2 ∈A2 M2j (y2 , a2 ). So ŝj1 and ŝj2 are
well-defined.
Let y 1 , y 1 ∈ Y1 . By the definition of ŝj1 , exploiting the concavity τ of M1j
with respect to a1 and taking the supergradient 0 of M1j with respect to the
second variable at (y 1 , ŝj1 (y 1 )) and (y 1 , ŝj1 (y 1 )), respectively, we get
and
M1j (y 1 , ŝj1 (y 1 )) − M1j (y 1 , ŝj1 (y 1 )) ≤ −τ |ŝj1 (y 1 ) − ŝj1 (y 1 )|2
(1)
M1j (y 1 , ŝj1 (y 1 )) − M1j (y 1 , ŝj1 (y 1 )) ≤ −τ |ŝj1 (y 1 ) − ŝj1 (y 1 )|2 .
(2)
By (1) and (2) we obtain
|M1j (y 1 , ŝj1 (y 1 )) − M1j (y 1 , ŝj1 (y 1 ))| + |M1j (y 1 , ŝj1 (y 1 )) − M1j (y 1 , ŝj1 (y 1 ))|
≥ 2τ |ŝj1 (y 1 ) − ŝj1 (y 1 )|2 .
(3)
Let L1 > 0 (which can be chosen independently of j) be an upper bound on the
Lipschitz constant of M1j . Then by (3) we obtain 2L1 y 1 − y 1 ≥ 2τ |ŝj1 (y 1 ) −
ŝj1 (y 1 )|2 , i.e.,
L1 j j |ŝ1 (y 1 ) − ŝ1 (y 1 )| ≤
y 1 − y 1 ,
(4)
τ
which proves the Hölder continuity of ŝj1 , hence its continuity and measurability.
Continuity and measurability of ŝj2 can be proved in the same way. Then it makes
sense to evaluate v(ŝj1 , ŝj2 ), and by construction we have v(ŝj1 , ŝj2 ) ≥ v(sj1 , sj2 )).
Let us focus on the strategies of the first DM. The next step consists in
showing that there exists a subsequence of {ŝj1 } that converges uniformly to
a strategy so1 ∈ C m−2 (Y1 ) with Lipschitz (m − 2)-order partial derivatives. By
∂M j Assumption A3, for every y1 ∈ Y1 ŝj1 (y1 ) is interior, so ∂a11 = 0.
j
a1 =ŝ1 (y1 )
∂ŝj
1
Then, by the Implicit Function Theorem, for every k = 1, . . . , d1 we get ∂y1,k
=
−1
∂ 2 M1j
∂ 2 M1j
∂ 2 M1j
−
, where
≤ −τ < 0 by the concavity at least τ
j2
j2
∂ŝj ∂y
∂ŝ1
1
1,k
∂ŝ1
of M1j in a1 and its smoothness.
As M1j is of class C m , by taking higher-order partial derivatives we conclude
that ŝj1 (y1 ) is locally of class C m−1 . As this holds for every y1 ∈ Y1 , it is of
class C m−1 on all Y1 . Since M1j has upper bounds on the sizes of its partial
Smooth Optimal Decision Strategies
445
derivatives up to the order m that are independent of y1 , a1 , and j, then for every (i1 , . . . , id1 ) such
that i1 + . . . + id1 = m − 1, there exists a finite upper bound
m−1 j
∂
ŝ1
on i1
, which is independent of y1 and j. So, one can easily show
id
∂y1,1 ,...,∂y1,d11 that for every
1 ) such that i1 + . . . + id1 = m − 2, the functions of the
(i1 , . . . , id
∂ m−2 ŝj1
id
i1
∂y1,1 ,...,∂y1,d1
1
sequence
are equibounded and have the same upper bound on
their Lipschitz constants, so they are uniformly equicontinuous on Y1 . Hence, by
Ascoli-Arzelà’s Theorem [1, Theorem 1.30, p. 10], such sequence admits a subsequence that converges uniformly to a function defined on Y1 , which is also
Lipschitz, with the same upper bound on its Lipschitz constant as above.
By integrating m − 2 times, we conclude that also the integrals of these
subsequences converge uniformly to the integrals of the limit functions. Therefore, there exists a subsequence of {ŝj1 } that converges uniformly to a strategy so1 ∈ C m−2 (Y1 ) with Lipschitz (m − 2)-order partial derivatives. Similarly,
one proves that there exists a subsequence of {ŝj2 } that converges uniformly to
so2 ∈ C m−2 (Y1 ) with Lipschitz (m − 2)-order partial derivatives.
By the continuity of the functional v(s1 , s2 ) on C(Y1 ) × C(Y2 ) with the respective sup-norms, finally we obtain v(so1 , so2 ) = limj→∞ v(ŝj1 , ŝj2 ) = sups1 ,s2
v(s1 , s2 ) .
The next theorem show that, under additional conditions, the optimal n-tuple
of smooth strategies is unique. We denote by C(Yi , Ai ) the set of continuous
functions from Yi to Ai with the sup-norm. Without loss of generality, we restrict
the spaces of admissible strategies to C(Yi , Ai ), as one can show that under the
assumptions of Theorem 1 any optimal strategy coincides almost everywhere
with a continuous function. To simplify the statement, in Theorem 2 we consider
the case of n = 2 DMs, but it can be extended to n ≥ 2 DMs.
Theorem 2. Let the assumptions of Theorem 1 hold with m ≥ 3 and n = 2,
∂2
β1,2
and let also τ < 1, where β1,2 = max(a1 ,a2 )∈A1 ×A2 ∂a1 ∂a2 u(x, y1 , y2 , a1 , a2 ).
Then (so1 , so2 ) given in Theorem 1 is the unique optimal pair of strategies in
C(Y1 , A1 ) × C(Y2 , A2 ).
Sketch of proof. Inspection of the first part of the proof of Theorem 1 shows
that there exists a (possibly nonlinear) operator T : C(Y1 , A1 ) × C(Y2 , A2 ) →
C(Y1 , A1 ) × C(Y2 , A2 ) such that
T1 (s1 , s2 )(y1 ) = argmax Ex,y2 |y1 {u(x, y1 , y2 , a1 , s2 (y2 ))} ∀y1 ∈ Y1 ,
a1 ∈A1
T2 (s1 , s2 )(y2 ) = argmax Ex,y1 |y2 {u(x, y1 , y2 , T1 (s1 , s2 )(y1 ), a2 )} ∀y2 ∈ Y2 .
a2 ∈A2
Let (so1 , so2 ) ∈ C(Y1 , A1 ) × C(Y2 , A2 ) be an optimal pair of strategies. Then
it is easy to see that (so1 , so2 ) = T (so1 , so2 ) is a necessary conditions for its
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G. Gnecco and M. Sanguineti
optimality. By Assumption A3 and the compactness of Y1 and Y2 , for any
(s1 , s2 ) ∈ C(Y1 , A1 ) × C(Y2 , A2 ) the strategies T1 (s1 , s2 ) and T2 (s1 , s2 ) belong
respectively to the interiors of C(Y1 , A1 ) and C(Y2 , A2 ). So, Problem STO is reduced to an unconstrained infinite-dimensional game theory problem, for which
one can apply the techniques developed in [17] to study the stability of Nash
equilibria. This can be done since every pair of optimal strategies for Problem
STO constitutes a Nash equilibrium for a two-player game, for which the individual utilities J 1 and J 2 are the same and are equal to v(s1 , s2 ). By using the norm
Eyi {(si (yi ))2 } on C(Y1 , A1 ) and C(Y2 , A2 ) (instead of the usual sup-norms),
computing the Frechét derivatives of the integral functional v up to the second
order and applying [17, Theorem 1, formula (1)], one can show that, for m ≥ 3,
T is a contraction operator with contraction constant bounded from above by
2
β1,2
τ2
< 1. So, T has at most a unique fixed point (so1 , so2 ) ∈ C(Y1 , A1 )×C(Y2 , A2 ),
which by Theorem 1 coincides with (so1 , so2 ).
4
Approximation Methods and Algorithms
In this section, we discuss how the existence and uniqueness of an optimal ntuple of strategies with a sufficiently high degree of smoothness can be exploited
when searching for suboptimal solutions to Problem STO.
4.1
Estimates of the Accuracy of Suboptimal Solutions by
Nonlinear Approximation Schemes
In [10, Propositions 4.2 and 4.3] we have shown that, for a degree of smoothness m in Assumption A1 that is linear in maxi {di } (i.e., the maximum dimension of the information vectors yi ), the smooth optimal strategies so1 , . . . , son
(whose existence is guaranteed by Theorem 1), can be approximated by suitable nonlinear approximation schemes modelling one-hidden-layer neural networks [11] with Gaussian and trigonometric computational units, with upper
bounds on the approximation errors of order k −1/2 , where k is the number
of computational units used in such schemes. This is an instance of the socalled blessing of smoothness [21]. We are currently investigating the extension
of such results to other nonlinear approximation schemes with sigmoidal and
spline computational units. The numerical results in [2,3,4] show that often these
approximation schemes (which belong to the wider family of variable-basis approximation schemes [15,16]) are able to find accurate suboptimal solutions to
team optimization problems with high-dimensional states, using a small number of parameters to be optimized. Variable-basis approximation schemes have
been successfully exploited also in other optimization tasks (see the references
in [30,31]).
4.2
Application of Quasi-Monte Carlo Methods
Another consequence of a sufficiently high degree of smoothness of the optimal
strategies is that it allows the application of quasi-Monte Carlo methods [20]
Smooth Optimal Decision Strategies
447
and related ones (such as Korobov’s method; see [29] and [14, Chapter 6])
for the approximate computation of the multidimensional integrals v(so1 , . . . , son )
and v(s̃1 , . . . , s̃n ), where s̃1 , . . . , s̃n are smooth approximations of so1 , . . . , son . For
example, upper bounds on the error in the approximate evaluation of a multidimensional integral by quasi-Monte Carlo methods can be obtained via KoksmaHlawka’s inequality [20, p. 20], which requires that the integrand has a finite
variation in the sense of Hardy and Krause [20, p. 19]. Considering, e.g., the
case of an integrand f defined on a r-dimensional unit-cube [0, 1]r , the most
common formula [20, p. 19, formula (2.5)] used to prove that f has a finite
variation in the sense of Hardy and Krause requires that f ∈ C r ([0, 1]r ) (i.e.,
its degree of smoothness has to be at least equal to the number of variables).
With the obvious changes
n in notation, Theorem 1 provides such a degree of
smoothness, for m ≥ i=0 di + 2.
4.3
Algorithms for Suboptimal Solutions
Finally, we investigate some implications of our results in the development of
algorithms to find suboptimal solutions to Problem STO. For simplicity of exposition, we consider the case of n = 2 agents.
Recall that under the assumptions of Theorem 2, the operator T defined in
the proof of such theorem is a contraction operator. Then, given any initial pair
of smooth suboptimal strategies (s̃01 , s̃02 ) and the unique (and a-priori unknown)
optimal one (so1 , so2 ), for every positive integer M one has
o
M
(y
)|,
max
|s
(y
)
−
s̃
(y
)|
max max |so1 (y1 ) − s̃M
1
2
1
2 2
2
≤
2
β1,2
τ2
y1 ∈Y1
M
y2 ∈Y2
max
max
y1 ∈Y1
|so1 (y1 )
−
s̃01 (y1 )|,
max
y2 ∈Y2
|so2 (y2 )
−
s̃02 (y2 )|
,
(5)
where
M
M 0 0
(s̃M
(s̃1 , s̃2 )
1 , s̃2 ) = T
(6)
2
β1,2
τ2
and
< 1. So, for the algorithm (6), the upper bound (5) shows that the rate
of convergence to the optimal pair of strategies is exponential in M .
In practice, however, the operator T itself has to be replaced by a finitedimensional approximating operator. Consider, e.g., an approximation scheme
in which one searches for suboptimal strategies of the form
s̃1 =
h
j=1
cj,1 φj,1 and s̃2 =
h
cj,2 φj,2 ,
j=1
where the positive integer h and the basis functions {φj,1 }hj=1 and {φj,2 }hj=1 are
fixed, and {cj,1 }hj=1 , {cj,2 }hj=1 are real coefficients to be optimized. Let
ũ(x, y1 , y2 , {cj,1 }, {cj,2 }) = u(x, y1 , y2 , s̃1 (y1 ), s̃2 (y2 )) .
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Then, one can replace (6) by
0
M
M
{cj,1 }, {c0j,2 } ,
({cM
j,1 }, {cj,2 }) = T̃h
(7)
where one chooses the approximating operator T̃h such that
T̃h,1 ({cj,1 }, {cj,2 }) = argmax Ex,y1 ,y2 {ũ(x, y1 , y2 , {ĉj,1 }, {cj,2 })} ,
(8)
{ĉj,1 }
T̃h,2 ({cj,1 }, {cj,2 }) = argmax Ex,y1 ,y2 {ũ(x, y1 , y2 , Th,1 ({cj,1 }, {cj,2 }), {ĉj,2 })} (9)
{ĉj,2 }
(for simplicity, we are assuming that there exist unique maxima). Exploiting
Assumption A2, one can show that finding the argmax in (8) and (9) for fixed
M
{cM
j,1 }, {cj,2 } requires one to solve two stochastic finite-dimensional concave optimization problems, to which the information-based-complexity results [27] and
the efficient algorithms described in [19, Chapter 14] may be applied.
Subjects of future research include studying the properties of the abovedefined operator T̃h and of other approximating operators. In particular, it is
of interest finding conditions under which
– the operator T̃h is a contraction operator (like T );
– the minimum positive integer h and the minimum number of elementary
operations of the algorithms described in [19, Chapter 14], required to find
a suboptimal solution to Problem STO with an error at most ε > 0, grow
“slowly” with respect to 1/ε.
5
Network Team Optimization
For static network team optimization problems [10], our smoothness results take
on a simplified form. For these problems, the team utility function u can be
written as the sum of a finite number of individual utility functions ui , each
one associated with a single DM (e.g., a router) or with a shared resource in
the network (e.g., a communication link). In addition, each ui depends only on
a subset of the DMs. This situation can be described by a multigraph, where the
DMs are the nodes and there is an edge between two DMs if and only if both
appear in a same individual utility function.
Figure 1 gives an idea of a network team optimization problem modeling
a store-and-forward packet-switching telecommunication network (see [2,4]).
Suppose that the DMs are n routers acting as members of a same team (i.e., they
aim to maximize a common objective, decomposable into the sum of several individual objectives related, e.g., to the congestion of the links). Each router has
at its disposal some private information (e.g., the total lengths of its incoming
packet queues). Assume also that the traffic flows can be described by continuous variables. Then, on the basis of its private information, each router decides
how to split the incoming traffic flows into its output links. The network team
optimization problem consists in finding optimal (or nearly optimal) n-tuples of
Smooth Optimal Decision Strategies
449
DM4
DM1
DM3
DM2
DM5
Fig. 1. An example of store-and-forward packet-switching telecommunication network
(left). An example of graph model with buffers at the nodes (right).
strategies according to some given optimality criterion (for simplicity we ignore
any dynamics in the problem, and we model it as a static one).
Compared with a general instance of Problem STO, the particular structure
of a static network team optimization allows various simplifications:
– For any n-tuple of strategies, the integral v(s1 , . . . , sn ) can be decomposed
into the sum
n of a finite number of integrals, each usually dependent on
smoothness
less than i=0 di real variables. So, the minimum degree m of n
required to apply [20, p. 19, formula (2.5)] is usually less than i=0 di + 2
(compare with the general case in Section 4).
– Since the strategy of each DM is influenced only by those of its neighbors in
the network, Assumption A3 may be easier to impose.
– One can show that an extension of Theorem 2 to n > 2 DMs can be formulated in terms of interaction terms βi,j , where (i, j) are pairs of different
DMs in the team. For a static network team optimization problem, usually
most of the βi,j are equal to 0 (since the interaction of each DM is limited
to its neighbors in the graph), so such extension takes a simplified form.
As to specific applications to static network team optimization problems, our
smoothness results may be applied, e.g., to stochastic versions of the congestion,
routing, and bandwidth allocation problems considered in [18, Lectures 3 and 4],
which are stated in terms of smooth and concave individual utility functions.
Acknowledgement. The authors were partially supported by a grant “Progetti
di Ricerca di Ateneo 2008” of the University of Genova, project “Solution of
Functional Optimization Problems by Nonlinear Approximators and Learning
from Data”.
450
G. Gnecco and M. Sanguineti
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