Smooth Optimal Decision Strategies for Static Team

Smooth Optimal Decision Strategies

for Static Team Optimization Problemsand Their Approximations

Giorgio Gnecco1,2 and Marcello Sanguineti2

1 Department of Computer and Information Science (DISI), University of GenovaVia Dodecaneso, 35, 16146 Genova, Italy

2 Department of Communications, Computer, and System Sciences (DIST)University of Genova

Via Opera Pia 13, 16145 Genova, [email protected], [email protected]

Abstract. Sufficient conditions for the existence and uniqueness ofsmooth optimal decision strategies for static team optimization problemswith statistical information structure are derived. Approximation meth-ods and algorithms to derive suboptimal solutions based on the obtainedresults are investigated. The application to network team optimizationproblems is discussed.

Keywords: Team utility function, value of a team, statistical informa-tion structure, approximation schemes, suboptimal solutions, networkoptimization.

1 Introduction

Decision makers (DMs) cooperating to achieve a common goal, expressed viaa team utility function, model a variety of problems in engineering, economicsystems, management science and operations research, in which centralization isnot feasible and so distributed optimization processes have to be performed. EachDM has at disposal various possibilities of decisions generated via strategies, onthe basis of the available information that it has about a random variable, calledstate of the world. In the model that we adopt, the information is expressed viaa probability density function, so we have a statistical information structure [13,Chapter 3].

In general, one centralized DM that, relying on the whole available informa-tion, maximizes the common goal, provides a better performance than a set ofdecentralized DMs, each of them having partial information. However, central-ization is not always feasible. For example, each DM may have access only tolocal information that cannot be instantaneously exchanged. Alternatively, thecost of making the whole information available to one single DM may be toohigh with respect to having several DMs with different information. This is of-ten 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

Smooth Optimal Decision Strategies 441

geographical areas, production plants, energy distribution systems, and trafficsystems in large metropolitan areas divided into sectors.

In the team optimization problems that we address in this paper, the infor-mation of each DM depends on the state of the world but is independent of thedecisions of the other DMs. These are called static teams, in contrast to dynamicteams, for which each DM’s information can be affected by the decisions of theother members. However, many dynamic team optimization problems can bereformulated in terms of equivalent static ones [28].

Static teams were first investigated by Marschak and Radner [22,23,24], whoderived closed-form solutions for some cases of interest. Then, dynamic teamswere studied [5]. Unfortunately, closed-form solutions to team optimization prob-lems can be derived only under quite strong assumptions on the team utilityfunction and the way in which each DM’s information is influenced by the stateof the world (and, in the case of dynamic teams, by the decisions previously takenby the other DMs). In particular, most available results hold under the so-calledLQG hypotheses (i.e., linear information structure, concave quadratic team util-ity, and Gaussian random variables) and with partially nested information, i.e,when each DM can reconstruct all the information available to the DMs thataffect its own information [7,12]. However, as remarked in [9], these assumptionsare often too simplified or unrealistic. For more general problems, closed-formsolutions are usually not available, so one has to search for suboptimal solutions.

In this paper, we derive sufficient conditions for the existence and uniquenessof smooth optimal decision strategies, for static team optimization problems withstatistical information structure. Then, we show that a sufficiently high degree ofsmoothness of the optimal decision strategies is a useful property when searchingfor suboptimal solutions.

The paper is organized as follows. Section 2 introduces definitions and as-sumptions and formulates the family of static team optimization problems underconsideration. Section 3 investigates existence and uniqueness of smooth opti-mal strategies for such problems. Section 4 examines some consequences of theobtained results in developing approximation methods and algorithms to derivesuboptimal solutions. Section 5 discusses the application of our results to staticnetwork team optimization problems.

2 Problem Formulation

The context in which we shall formalize the optimization problem and deriveour results is the following.

– Static team of n decision makers (DMs), i = 1, . . . , n.– x ∈ X ⊆ R

d0 : vector-valued random variable, called state of the world,describing a stochastic environment. The vector x models the uncertaintiesin the external world, which are not controlled by the DMs.

– yi ∈ Yi ⊆ Rdi : vector-valued random variable, which represents the informa-

tion that the DM i has about x.

442 G. Gnecco and M. Sanguineti

– si : Yi → Ai ⊆ R: measurable strategy of the i-th DM.– ai = si(yi): decision that the DM i chooses on the basis of the information yi.– u : X × Πn

i=1Yi × Πni=1Ai ⊆ R

N → R, where N =∑n

i=0 di + n: real-valuedteam utility function.

– The information that the n DMs have on the state of the world x is modelledby an n-tuple of random variables y1, . . . , yn, i.e., by a statistical informationstructure [6] represented by a joint probability density ρ(x, y1, . . . , yn) on theset X × Πn

i=1Yi.

We shall address the following family of static team optimization problems.

Problem STO (Static Team Optimization with Statistical Informa-tion). Given the statistical information structure ρ(x, y1, . . . , yn) and the teamutility function u(x, y1, . . . , yn, a1, . . . , an), find

sups1,...sn

v(s1, . . . , sn) ,

wherev(s1, . . . , sn) = Ex,y1,...,yn {u(x, {yi}n

i=1, {si(yi)}ni=1)} .

The quantity sups1,...snv(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 integerm > 0, by Cm(Ω) we denote the spaces of functions on Ω, which are continuoustogether with their partial derivatives up to the order m.

A1. The sets X, Y1, . . . , Yn are compact, and A1, . . . , An are bounded closedintervals. For an integer m ≥ 2, the team utility u is of class Cm on an open setcontaining X ×Πn

i=1Yi ×Πni=1Ai, and ρ a (strictly) positive probability density

on X ×Πni=1Yi, which can be extended to a function of class Cm on an open set

containing X × Πni=1Yi.

A concave function f defined on a convex set Ω has concavity at least τ > 0if for all u, v ∈ Ω and every supergradient1 pu of f at u one has f(v) − f(u) ≤pu · (v − u)− τ‖v − u‖2. If f is of class C2(Ω), then a necessary condition for itsconcavity at least τ is supu∈Ω λmax(∇2f(u)) ≤ −τ , where λmax(∇2f(u)) is themaximum eigenvalue of the Hessian ∇2f(u).

A2. There exists τ > 0 such that the team utility function u : X × Πni=1Yi ×

Πni=1Ai 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 prac-tice. For example, in economic problems it can be motivated by the “law ofdiminishing returns”, i.e., the fact that the marginal productivity of an inputusually diminishes as the amount of output increases [23, p. 99 and p. 110].1 For Ω ⊆ R

d 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) .


A3. For every n-tuple {s1, . . . , sn} of strategies, the strategies defined as

s1(y1) = argmaxa1∈A1

Ex,y2,...,yn |y1{u(x, {yi}ni=1, a1, {si(yi)}n

i=2)} ∀y1 ∈ Y1 ,

. . .

sn(yn) = argmaxan∈An

Ex,y1,...,yn−1 |yn{u(x, {yi}ni=1, {si(yi)}n−1

i=1 , an)} ∀yn ∈ Yn

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 exam-ples of problems for which Assumptions A1, A2 and A3 hold simultaneously canbe constructed by starting from a problem in which there is no interaction amongthe DMs (i.e., u(x, y1, . . . , yn, s1(y1), . . . , sn(yn)) =

∑ni=1 ui(x, yi, . . . , yn, s1(yi)),

so that the assumptions are easy to impose), then adding to the team utilityfunction a sufficiently small smooth interaction term.

3 Existence and Uniqueness of Smooth OptimalStrategies

The next theorem (which takes the hint from [13, Theorem 11, p. 162], andextends it to a higher degree of smoothness) gives conditions guaranteeing thatProblem STO has a solution made of an n-tuple of strategies that are Lipschitzcontinuous together with their partial derivatives up to a certain order. Forlimitations of space, we only sketch the proof for n = 2; details can be foundin [8, Chapter 5].

Theorem 1. Let Assumptions A1, A2 and A3 hold. Then Problem STO admitsan n-tuple (so

1, . . . , son) of Cm−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, s

j2} of pairs of strategies,

indexed by j ∈ N+, such that limj→∞ v(sj1, s

j2) = sups1,s2

v(s1, s2) (such a se-quence exists by the definition of supremum). From the sequence {sj

1, sj2}, we

generate another sequence {sj1, s

j2} defined by

sj1(y1) = argmax

a1∈A1

M j1 (y1, a1)∀y1 ∈ Y1 ,

sj2(y2) = argmax

a2∈A2

M j2 (y2, a2)∀y2 ∈ Y2 ,

where for every (y1, a1) ∈ Y1 × A1 and (y2, a2) ∈ Y2 × A2, we let

M j1 (y1, a1) = Ex,y2 |y1{u(x, y1, y2, a1, s

j2(y2))} ,

M j2 (y2, a2) = Ex,y1 |y2{u(x, y1, y2, s

j1(y1), a2)} .

Since the probability density ρ(x, y1, y2) is of class Cm and strictly positive on anopen set containing X×Y1×Y2, we obtain that the conditional density ρ(x, y2|y1)


is of class Cm on the compact set X × Y1 × Y2 and the team utility function uis of class Cm on the compact set X × Y1 × Y2 ×A1 ×A2. So M j

1 , as an integraldependent on parameters, is of class Cm on the compact set Y1×A1, with upperbounds on the sizes of its partial derivatives up to the order m independentof y1, a1, and j. In particular, it is easy to show that M j

1 has concavity at least τ

in a1. By such continuity and concavity properties of M j1 with respect to a1,

for all y1 ∈ Y1 the set argmaxa1∈A1M j

1 (y1, a1) consists of exactly one element.An analogous conclusion holds for argmaxa2∈A2

M j2 (y2, a2). So sj

1 and sj2 are

well-defined.Let y′

1, y′′1 ∈ Y1. By the definition of sj

1, exploiting the concavity τ of M j1

with respect to a1 and taking the supergradient 0 of M j1 with respect to the

second variable at (y′1, s

j1(y

′1)) and (y′′

1, sj1(y

′′1)), respectively, we get

M j1 (y′

1, sj1(y

′′1)) − M j

1 (y′1, s

j1(y

′1)) ≤ −τ |sj

1(y′′1) − sj

1(y′1)|2 (1)

andM j

1 (y′′1, s

j1(y

′1)) − M j

1 (y′′1, s

j1(y

′′1)) ≤ −τ |sj

1(y′1) − sj

1(y′′1)|2 . (2)

By (1) and (2) we obtain

|M j1 (y′

1, sj1(y

′′1)) − M j

1 (y′1, s

j1(y

′1))| + |M j

1 (y′′1, s

j1(y

′1)) − M j

1 (y′′1, s

j1(y

′′1))|

≥ 2τ |sj1(y

′′1) − sj

1(y′1)|2 . (3)

Let L1 > 0 (which can be chosen independently of j) be an upper bound on theLipschitz constant of M j

1 . Then by (3) we obtain 2L1‖y′′1 − y′

1‖ ≥ 2τ |sj1(y

′′1)−

sj1(y

′1)|2 , i.e.,

|sj1(y

′′1) − sj

1(y′1)| ≤

√L1

τ

√‖y′′

1 − y′1‖ , (4)

which proves the Holder continuity of sj1, hence its continuity and measurability.

Continuity and measurability of sj2 can be proved in the same way. Then it makes

sense to evaluate v(sj1, s

j2), and by construction we have v(sj

1, sj2) ≥ v(sj

1, sj2)).

Let us focus on the strategies of the first DM. The next step consists inshowing that there exists a subsequence of {sj

1} that converges uniformly toa strategy so

1 ∈ Cm−2(Y1) with Lipschitz (m − 2)-order partial derivatives. By

Assumption A3, for every y1 ∈ Y1 sj1(y1) is interior, so ∂Mj

1∂a1

∣∣∣a1=sj

1(y1)= 0 .

Then, by the Implicit Function Theorem, for every k = 1, . . . , d1 we get ∂sj1

∂y1,k=

−(

∂2Mj1

∂sj12

)−1∂2Mj

1

∂sj1∂y1,k

, where(

∂2Mj1

∂sj12

)≤ −τ < 0 by the concavity at least τ

of M j1 in a1 and its smoothness.

As M j1 is of class Cm, by taking higher-order partial derivatives we conclude

that sj1(y1) is locally of class Cm−1. As this holds for every y1 ∈ Y1, it is of

class Cm−1 on all Y1. Since M j1 has upper bounds on the sizes of its partial


derivatives up to the order m that are independent of y1, a1, and j, then for ev-ery (i1, . . . , id1) such that i1 + . . .+ id1 = m−1, there exists a finite upper bound

on

∣∣∣∣∣ ∂m−1sj1

∂yi11,1,...,∂y

id11,d1

∣∣∣∣∣, which is independent of y1 and j. So, one can easily show

that for every (i1, . . . , id1) such that i1 + . . . + id1 = m − 2, the functions of the

sequence

{∂m−2sj

1

∂yi11,1,...,∂y

id11,d1

}are equibounded and have the same upper bound on

their Lipschitz constants, so they are uniformly equicontinuous on Y1. Hence, byAscoli-Arzela’s Theorem [1, Theorem 1.30, p. 10], such sequence admits a sub-sequence that converges uniformly to a function defined on Y1, which is alsoLipschitz, with the same upper bound on its Lipschitz constant as above.

By integrating m − 2 times, we conclude that also the integrals of thesesubsequences converge uniformly to the integrals of the limit functions. There-fore, there exists a subsequence of {sj

1} that converges uniformly to a strat-egy so

1 ∈ Cm−2(Y1) with Lipschitz (m − 2)-order partial derivatives. Similarly,one proves that there exists a subsequence of {sj

2} that converges uniformly toso2 ∈ Cm−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 re-

spective sup-norms, finally we obtain v(so1, s

o2) = limj→∞ v(sj

1, sj2) = sups1,s2

v(s1, s2) . �

The next theorem show that, under additional conditions, the optimal n-tupleof smooth strategies is unique. We denote by C(Yi, Ai) the set of continuousfunctions from Yi to Ai with the sup-norm. Without loss of generality, we restrictthe spaces of admissible strategies to C(Yi, Ai), as one can show that under theassumptions of Theorem 1 any optimal strategy coincides almost everywherewith a continuous function. To simplify the statement, in Theorem 2 we considerthe 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,

and let alsoβ1,2

τ < 1, where β1,2 = max(a1,a2)∈A1×A2

∣∣∣∣ ∂2

∂a1∂a2u(x, y1, y2, a1, a2)

∣∣∣∣.Then (so

1, 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 showsthat 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) = argmaxa1∈A1

Ex,y2 |y1{u(x, y1, y2, a1, s2(y2))} ∀y1 ∈ Y1,

T2(s1, s2)(y2) = argmaxa2∈A2

Ex,y1 |y2{u(x, y1, y2, T1(s1, s2)(y1), a2)} ∀y2 ∈ Y2.

Let (so′1 , so′

2 ) ∈ C(Y1, A1) × C(Y2, A2) be an optimal pair of strategies. Thenit is easy to see that (so′

1 , so′2 ) = T (so′

1 , so′2 ) is a necessary conditions for its


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) belongrespectively to the interiors of C(Y1, A1) and C(Y2, A2). So, Problem STO is re-duced to an unconstrained infinite-dimensional game theory problem, for whichone can apply the techniques developed in [17] to study the stability of Nashequilibria. This can be done since every pair of optimal strategies for ProblemSTO constitutes a Nash equilibrium for a two-player game, for which the individ-ual utilities J1 and J2 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 Frechet derivatives of the integral functional v up to the secondorder 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 byβ21,2τ2 < 1. So, T has at most a unique fixed point (so′

1 , so′2 ) ∈ C(Y1, A1)×C(Y2, A2),

which by Theorem 1 coincides with (so1, s

o2). �

4 Approximation Methods and Algorithms

In this section, we discuss how the existence and uniqueness of an optimal n-tuple of strategies with a sufficiently high degree of smoothness can be exploitedwhen searching for suboptimal solutions to Problem STO.

4.1 Estimates of the Accuracy of Suboptimal Solutions byNonlinear Approximation Schemes

In [10, Propositions 4.2 and 4.3] we have shown that, for a degree of smooth-ness m in Assumption A1 that is linear in maxi{di} (i.e., the maximum di-mension of the information vectors yi), the smooth optimal strategies so

1, . . . , son

(whose existence is guaranteed by Theorem 1), can be approximated by suit-able nonlinear approximation schemes modelling one-hidden-layer neural net-works [11] with Gaussian and trigonometric computational units, with upperbounds on the approximation errors of order k−1/2, where k is the numberof computational units used in such schemes. This is an instance of the so-called blessing of smoothness [21]. We are currently investigating the extensionof such results to other nonlinear approximation schemes with sigmoidal andspline computational units. The numerical results in [2,3,4] show that often theseapproximation schemes (which belong to the wider family of variable-basis ap-proximation schemes [15,16]) are able to find accurate suboptimal solutions toteam optimization problems with high-dimensional states, using a small num-ber of parameters to be optimized. Variable-basis approximation schemes havebeen successfully exploited also in other optimization tasks (see the referencesin [30,31]).

4.2 Application of Quasi-Monte Carlo Methods

Another consequence of a sufficiently high degree of smoothness of the optimalstrategies is that it allows the application of quasi-Monte Carlo methods [20]


and related ones (such as Korobov’s method; see [29] and [14, Chapter 6])for the approximate computation of the multidimensional integrals v(so

1, . . . , son)

and v(s1, . . . , sn), where s1, . . . , sn are smooth approximations of so1, . . . , s

on. For

example, upper bounds on the error in the approximate evaluation of a multidi-mensional integral by quasi-Monte Carlo methods can be obtained via Koksma-Hlawka’s inequality [20, p. 20], which requires that the integrand has a finitevariation in the sense of Hardy and Krause [20, p. 19]. Considering, e.g., thecase of an integrand f defined on a r-dimensional unit-cube [0, 1]r, the mostcommon formula [20, p. 19, formula (2.5)] used to prove that f has a finitevariation in the sense of Hardy and Krause requires that f ∈ Cr([0, 1]r) (i.e.,its degree of smoothness has to be at least equal to the number of variables).With the obvious changes in notation, Theorem 1 provides such a degree ofsmoothness, for m ≥

∑ni=0 di + 2.

4.3 Algorithms for Suboptimal Solutions

Finally, we investigate some implications of our results in the development ofalgorithms to find suboptimal solutions to Problem STO. For simplicity of ex-position, we consider the case of n = 2 agents.

Recall that under the assumptions of Theorem 2, the operator T defined inthe proof of such theorem is a contraction operator. Then, given any initial pairof smooth suboptimal strategies (s0

1, s02) and the unique (and a-priori unknown)

optimal one (so1, s

o2), for every positive integer M one has

max{

maxy1∈Y1

|so1(y1) − sM

1 (y1)|, maxy2∈Y2

|so2(y2) − sM

2 (y2)|}

≤(

β21,2

τ2

)M

max{

maxy1∈Y1

|so1(y1) − s0

1(y1)|, maxy2∈Y2

|so2(y2) − s0

2(y2)|}

, (5)

where(sM

1 , sM2 ) = T M (s0

1, s02) (6)

andβ21,2τ2 < 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 finite-

dimensional approximating operator. Consider, e.g., an approximation schemein which one searches for suboptimal strategies of the form

s1 =h∑

j=1

cj,1φj,1 and s2 =h∑

j=1

cj,2φj,2 ,

where the positive integer h and the basis functions {φj,1}hj=1 and {φj,2}h

j=1 arefixed, and {cj,1}h

j=1, {cj,2}hj=1 are real coefficients to be optimized. Let

u(x, y1, y2, {cj,1}, {cj,2}) = u(x, y1, y2, s1(y1), s2(y2)) .


Then, one can replace (6) by

({cMj,1}, {cM

j,2}) = T Mh

({c0

j,1}, {c0j,2}), (7)

where one chooses the approximating operator Th such that

Th,1({cj,1}, {cj,2}) = argmax{cj,1}

Ex,y1,y2{u(x, y1, y2, {cj,1}, {cj,2})} , (8)

Th,2({cj,1}, {cj,2}) = argmax{cj,2}

Ex,y1,y2{u(x, y1, y2, Th,1({cj,1}, {cj,2}), {cj,2})} (9)

(for simplicity, we are assuming that there exist unique maxima). ExploitingAssumption A2, one can show that finding the argmax in (8) and (9) for fixed{cM

j,1}, {cMj,2} requires one to solve two stochastic finite-dimensional concave op-

timization problems, to which the information-based-complexity results [27] andthe efficient algorithms described in [19, Chapter 14] may be applied.

Subjects of future research include studying the properties of the above-defined operator Th and of other approximating operators. In particular, it isof interest finding conditions under which

– the operator Th 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 finda 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 takeon a simplified form. For these problems, the team utility function u can bewritten as the sum of a finite number of individual utility functions ui, eachone associated with a single DM (e.g., a router) or with a shared resource inthe network (e.g., a communication link). In addition, each ui depends only ona subset of the DMs. This situation can be described by a multigraph, where theDMs are the nodes and there is an edge between two DMs if and only if bothappear in a same individual utility function.

Figure 1 gives an idea of a network team optimization problem modelinga 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., theyaim to maximize a common objective, decomposable into the sum of several in-dividual objectives related, e.g., to the congestion of the links). Each router hasat its disposal some private information (e.g., the total lengths of its incomingpacket queues). Assume also that the traffic flows can be described by continu-ous variables. Then, on the basis of its private information, each router decideshow to split the incoming traffic flows into its output links. The network teamoptimization problem consists in finding optimal (or nearly optimal) n-tuples of


DM1

DM2

DM3

DM5

DM4

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 ignoreany dynamics in the problem, and we model it as a static one).

Compared with a general instance of Problem STO, the particular structureof a static network team optimization allows various simplifications:

– For any n-tuple of strategies, the integral v(s1, . . . , sn) can be decomposedinto the sum of a finite number of integrals, each usually dependent onless than

∑ni=0 di real variables. So, the minimum degree m of smoothness

required to apply [20, p. 19, formula (2.5)] is usually less than∑n

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 inthe network, Assumption A3 may be easier to impose.

– One can show that an extension of Theorem 2 to n > 2 DMs can be for-mulated in terms of interaction terms βi,j , where (i, j) are pairs of differentDMs in the team. For a static network team optimization problem, usuallymost of the βi,j are equal to 0 (since the interaction of each DM is limitedto its neighbors in the graph), so such extension takes a simplified form.

As to specific applications to static network team optimization problems, oursmoothness 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 “Progettidi Ricerca di Ateneo 2008” of the University of Genova, project “Solution ofFunctional Optimization Problems by Nonlinear Approximators and Learningfrom Data”.


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