Generalized Proportional Allocation Mechanism Design for Unicast Service on the Internet

0.26 MB·

by  Abhinav Sinha

Generalized Proportional Allocation Mechanism Design for
Unicast Service on the Internet
4
1 Abhinav Sinha and Achilleas Anastasopoulos
0
EECS Department, University of Michigan, Ann Arbor, MI 48109
2
{absi,anastas}@umich.edu
b
e
F
Abstract. In this report we construct two mechanisms that fully implement social welfare
3
maximising allocation in Nash equilibria for the case of a single infinitely divisible good subject

]
to multiple inequality constraints. The first mechanism achieves weak budget balance, while
T
G the second is an extension of the first, and achieves strong budget balance. One important
. applicationofthismechanismisunicastservice ontheInternet whereanetworkoperatorwishes
s
c to allocate rates among strategic users in such a way that maximise overall user satisfaction
[
while respecting capacity constraints on every link in the network. The emphasis of this work is

3
on full implementation, which means that all Nash equilibria of the induced game result in the
v
0 optimal allocations of the centralized allocation problem.
6
7
1
.
1
May 4, 2013
0
4
1
:
v
i 1. Centralised Problem
X
r
a
Consider a set , of of Internet agents (an agent is considered a pair of
N = f1;2;:::;Ng N
source and destination users) that communicate over pre-specified routes on the Internet. Each
R R
agent , communicates at an information rate (where is the set of non-negative
i 2 N xi 2 + +
R
real numbers). Agent’s valuation for an overall rate allocation , can be written
N
x = (xi)i2N 2 +
as
v~i(x) = vi(xi) 8 i 2 N
R R
where , for all , which indicates that agent ’s satisfaction only depends on its
vi : + ! i 2 N i
own information rate allocation . Due to capacity constraints on the utilized links, allocation
xi
to agents is constrained by a number of inequality constraints. It is assumed that although some
agentsmayshareinformationcontent(e.g., watchthesamevideostream), aseparatedatastream
is transmitted for each agent. This transmission technique is referred to as unicast service.
1
2 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS
Each agent has a fixed pre-determined route. The route for agent is the set of links that
Li i
agent uses for his communication, and is the set of all available links. We also
i L = [i2NLi
define the sets of agents utilizing link as . Finally, for any agent
l
l 2 L N = fi 2 N j l 2 Lig i
we denote and for any link we denote .
l l
Li = jLij l N = jN j
Thenetworkadministratorisinterestedinmaximizingthesocialwelfareunderthelinkcapacity
constraints. This centralized problem (CP) is formally defined below.
(CP) X
max vi(xi)
x
i2N
(C ) s.t.
1 xi (cid:21) 0 8 i 2 N
(C ) and X
l l
2 (cid:11)jxj (cid:20)
8 l 2 L
j2Nl
Specifically, constraints (C ) are the inequality constraints on allocation, which as mentioned
2
above, can be interpreted as capacity constraint for every link , in the network. In this
l 2 L
interpretation would be representative of the QoS requirement of agent combined with the
l
(cid:11)j j
specific architecture on link . As an example, for all links , where
l 1 l
represents the packet error prlobability for link fo(cid:11)rja=paRckj(e1t(cid:0)(cid:15)eljn)coded with chlan2neLl jcoding ra(cid:15)tej
l
.
Rj
1.1. Assumptions. Our analysis would be done under the following assumptions.
(A1) For all agents, , where the sets are arbitrary subsets of , the set of strictly
vi((cid:1)) 2 Vi Vi R VR0
increasing, strictly concave, twice differentiable functions with continuous
+ !
second derivative.
(A2) is finite . This also implies that is finite and bounded and
0 0
vi(0) 8 i 2 N vi(x) 8 i 8 x
since ’s are concave.
vi
(A3) There are at least two agents on each link i.e. .
l
N (cid:21) 2 8 l 2 L
(A4) The optimal solution of (CP) has at least two non-zero components at each link i.e. if
and then we assume
l l l ?
S(x) := fi 2 N j xi > 0g S (x) := S(x)\N jS (x )j (cid:21) 2 8
. (where is the solution of (CP))
?
l 2 L x
In addition, the coefficients in (C ) are all strictly positive, i.e. , . Also,
l l
2 (cid:11)j > 0 8 j 2 N 8 l 2 L
for well-posedness of the problem we take .
l

> 0 8 l 2 L
Assumption (A1) is made in order for the centralized problem to have a unique solution and
for this solution to be characterized by the KKT conditions. (A2) is a mild technical assumption
that is required in the proof of Lemma 3.7. Assumption (A3) is made in order to avoid situations
where there is a link constraint involving only one agent. Such case requires special handling
in the design of the mechanism (since in such a case there is no contention at the link), and
destructs from the basic idea that we want to communicate. Finally (A4) is related to (A3) and
MECHANISM DESIGN FOR UNICAST SERVICE 3
is made in order to simplify the exposition of the proposed mechanism, without having to define
corner cases that are of minor importance. We will present our proposed mechanism in a concise
way using assumption (A4) and generalisations will be discussed separately in Section 5.
1.2. Necessary and Sufficient Optimality conditions. We can write the following four
KKT conditions, which are generally necessary, but in our case (due to all constraints being affine
and strict concavity of ) they will also be sufficient. For that we first define the Lagrangian
vi
0 1
X X X X
L(x;(cid:21);(cid:23)) = vi(xi)(cid:0) (cid:21)l (cid:11)ljxj (cid:0)
lA+ (cid:23)ixi
i2N l2L j2Nl i2N
We will write KKT conditions without explicitly referring to ’s and just using the fact that
(cid:23)i
and . With the assumptions above, it’s easy to see that the KKT
? ? ?
(cid:23)i (cid:21) 0 (cid:23)ixi = 0 8 i 2 N
conditions below will give rise to a unique as the optimiser for (CP).
?
x
KKT conditions:
a) Primal Feasibility:
and X
? l ? l
xi (cid:21) 0 8 i 2 N (cid:11)jxj (cid:20)
8 l 2 L
j2Nl
b) Dual Feasibility:
?
(cid:21)l (cid:21) 0 8 l 2 L
c) Complimentary Slackness:
0 1
X
(cid:21)?l  (cid:11)ljx?j (cid:0)
lA = 0 8 l 2 L
j2Nl
d) Stationarity:
X s.t.
0 ? ? l ?
vi(xi) = (cid:21)l(cid:11)i 8 i 2 N xi > 0
l2Li
X s.t.
0 ? ? l ?
vi(xi) (cid:20) (cid:21)l(cid:11)i 8 i 2 N xi = 0
l2Li
2. Different Formulations of the Centralised Problem
The designer’s task is to ensure the above optimum allocation is made. This clearly requires
the knowledge of even when constraints (C ) and (C ) are completely known. The premise
vi 1 2
of the problem, however, is that we are dealing with agents who are and for each of
strategi
whom, their own valuation function is their private information. One way forward for the
vi((cid:1))
designer could be to simply ask each agent to report their private information and announce the
solution of (CP), with reported functions in place of , for allocation. Apart from the fact that
vi
asking to report a function creates a practical communication problem, the main problem with
this is that the agents could report untruthfully and end up getting a strictly better allocation
(e.g., by reporting a which has higher derivative than original at every point). In mechanism
vi
4 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS
1
design terminology, the allocation function arising out of (CP) isn’t even partially implementable
Restricting ourselves to a certain class of utility functions (quasi-linear utilities), provides the
additional flexibility of penalising agents for reporting untruthfully by imposing taxes/subsidies.
In this way, another related problem is created which is implementable, and which is equivalent
to (CP) as far as allocation is concerned. This leads us to the following additional assumption
about users’ utilities
(A5) All agents have quasi-linear utilities, i.e. the overall utility functions can be expressed as
ui(x;t) = vi(xi)(cid:0)ti 8 i 2 N
R
where in addition to allocation we have introduced taxes .
N
t = (ti)i2N 2
Note that under assumption (A5), agent pays tax if and receives a subsidy if .
i ti > 0 ti < 0
Taxes affect utilities linearly and overall utility itself is valuation after adjustment for taxes (total
monetary representation of one’s state of happiness).
Because we talk about social welfare as our main objective, the centralised problem (CP) isn’t
complete until we fix who owns the good that is being allocated. Then one will have to further
check whether including their welfare in the objective function changes the optimum allocation.
As it turns out, under the assumption of utilities and cost of providing the good
quasi-linear
being zero for the owner, optimum doesn’t change even if we involve the seller’s welfare. In this
regard, there are two interesting ways of reformulating (CP), as elaborated below.
2.1. First Reformulation of CP: Weak budget balance. We now introduce agent as
0
the owner of the good (called the seller). The seller doesn’t have any costs for producing and
providing the good, i.e. his valuation is the zero function. This could be interpreted as the good
being already produced and ready to be provided, so those costs don’t come into consideration
for the seller as well as the designer. His utility is linear (since valuation is zero) and his revenue
is the total tax paid by the agents, P .
i2N ti
We define centralised problem (CP ) as
1
(CP ) X X
1 max ui(x;t)+ tj
x;t
i2N j2N
s.t. (C ) and (C )
1 2
where now, instead of just taking agent’s valuations into account, we maximise the sum of their
overall utilities, with the addition of seller’s utility (which is only his revenue) - each agent pays
a tax , all of which goes to the seller, who has no valuation and therefore has utility equal to
ti
1
Thiscanbededucedfromtherevelationprinciple. Indeediftherewasamechanismthatevenpartiallyimplements
the allocation function arising out of (CP), then there would exist also a truthful implementation. However, as
shown with the above example, such an implementation will always fail.
MECHANISM DESIGN FOR UNICAST SERVICE 5
sum of taxes. Anticipating that a rational seller will only sell if his revenue is non-negative we
can add a weak budget balance (WBB) constraint, which states
(WBB) X
tj (cid:21) 0:
j2N
2.2. Second Reformulation of CP: Strong budget balance. In this case, in contrast to
(CP ), there is no separate seller. We can alternatively say that the agents are themselves the
1
owners of the good and are only looking to distribute the good (which they collectively own) in
a way such that sum of utilities is maximised. Therefore strong budget balance (SBB) constraint
is needed. This means that for the system , no money has been introduced from the outside
N
and the agents wish that no excess money remains undistributed after the allocation either.
The new centralised problem (CP ) resulting from the above interpretation can be stated as
2
(CP ) X
2 max ui(x;t)
x;t
i2N
s.t. (C ) and (C )
1 2
(SBB) and X
ti = 0:
i2N
. The two problems defined above will be shown to be equivalent to (CP) where since the
original problem (CP) did not involve taxes, we will talk of equivalence only in terms of optimum
allocation, . Notethatduetodifferentconditionsontaxesinthetwo, two differentmechanisms
?
x
will be needed to implement them.
It is straightforward to see that (CP ) and (CP) are completely equivalent - due to constraint
2
, the objective for (CP ) is independent of and is exactly the same as objective for (CP),
SBB 2 t
with same remaining constraints. Now for (CP ) and (CP ), since the constraints on are the
1 2 x
same in (CP ) and (CP ) and the dependent part of the objective in (CP ) in independent
1 2 x(cid:0) 2
of and is the same as the objective of (CP ), we can see that (CP ) and (CP ) are equivalent.
t 1 1 2
The two equivalences above automatically give the third one i.e. (CP) and (CP ).
1
The above equivalences mean that not only will be the same, but also that the necessary
?
x
and sufficient conditions describing it will be the same i.e. KKT conditions, for and , will be
? ?
x (cid:21)
exactly the same for all three problems (additionally we will show and constraints
WBB SBB
to be satisfied in respective formulations). This fact will be used in Sections 3, 4 where the
KKT conditions from Section 1 will be treated as if they have been written for (CP ), (CP ),
1 2
respectively.
In Section 3, we will present a mechanism that fully implements (CP ) in Nash Equilibria
1
(NE), while in Section 4 we will modify our mechanism to fully implement (CP ) in NE.
2
6 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS
3. A Mechanism with Weak Budget Balance
In this section we refer to (CP ) as the centralised problem. So we have all the agents in
1 N
plus the seller and social welfare is in terms of everyone’s utility (including seller’s).
We will define a mechanism, in a way that doesn’t require knowledge of , whose game-form
vi
will have NE in pure strategies such that the allocation which corresponds to the equilibria of
the game-form is same across all equilibria and is equal to the unique optimiser of (CP ), . In
?
1 x
addition, the mechanism will be such that everyone involved (including the seller) will be weakly
better-off at equilibrium than not participating at all.
3.1. Information assumptions. Assume that is a private information of agent and
2 vi((cid:1)) i
nobodyelseknowsit . Let bethesetofcommoninformationbetweenallagents, containingthe
I
information about full rationality of each agent. Finally, let be the knowledge of the designer,
Id
containing the information about constraints (C ), (C ), the fact that and that
1 2 Vi (cid:26) V0; 8i 2 N
the seller has valuation.
0
3.2. Mechanism. Formally, wehaveasetofenvironments . Wehaveseenfrom
V = (cid:2)i2NVi
KKT, how each element of can be mapped to an allocation which maximises social welfare
?
V x
for that set of utilities. The allocation achieves the maximum of (CP ), and correspondingly
?
x 1
R
any tax satisfying (WBB) would do.
N
t 2
In our mechanism, the designer would define an action (message) space for each agent
Si
. We denote the set of action profiles for all agents. In addition the designer
i 2 N S = (cid:2)i2NSi
R R
defines and announces the that maps every vector of messages
N N

ontra
t h : S ! + (cid:2)
received from the agents into an allocation vector and a tax vector. The designer would then ask
every agent to choose a message from the set based on which allocations (and taxes)
i 2 N Si
would be made. The seller is not asked to take any action, so as far as strategic decision making
is concerned, we don’t need to consider him any further. It is implicit in our mechanism in this
section that when the tax is imposed, the seller gets revenue (or utility) of P .
t i2N ti
Specifically, the designer would ask each agent to report , . This
l
si = (yi;pi) pi = (pi)l2Li
includes their demand for the good and the “price”, for each constraint they are involved in,
R R
that they believe everyone should pay. This means . For received messages
Li
Si = + (cid:2) +
the contract will
s = (s1;:::;sN) = (y;P) = (y1;:::;yN;p1:::;pN) hi(s) = (hx;i(s);ht;i(s))
be defined for each as follows.
i 2 N
2
ThisassumptioniscrucialbecauseitraisesthequestionofthevalidityofNEasasolutionconceptoftheresulting
game, since that would require that all agents have complete information about everyone’s utilities. We believe
thisiaaseriousprobleminthisentirelineofresearchandthataBayesianformulationwouldbemoreappropriate.
However,in this work we accept the justification–weakin our opinion–givenby Reichelstein and Reiter in [1] and
Groves and Ledyard in [2].
MECHANISM DESIGN FOR UNICAST SERVICE 7
Ifthereceived demandvectoris theallocationis .
y = (y1;:::;yN) = 0 x = (x1;:::;xN) = 0
Otherwise it is evaluated by first generating a scaling factor through
r
l
r = minr ;
l2L
where
8
>>>>< Pj2N
ll(cid:11)ljyj; if jSl(y)j (cid:21) 2
(1) rl = >>>>: Pj2N
ll(cid:11)ljyj (cid:0)fl(yi); iiff Sl(ly) = fig
+1 jS (y)j = 0
with
l

l
f (yi) = l :
(cid:11)iyi(yi +1)
Using these previously defined quantities, the allocation and taxes are
(2)
hx;i(s) = xi = ryi
(3) X
l
ht;i(s) = ti = ti
l2Li
0 1
(cid:16) (cid:17) X
tli = xi(cid:11)lip(cid:22)l(cid:0)i +(pli (cid:0)p(cid:22)l(cid:0)i)2 +(cid:17)p(cid:22)l(cid:0)i pli (cid:0)p(cid:22)l(cid:0)i 
l (cid:0) (cid:11)ljxjA;
j2Nl
where is a small enough positive constant (described in proof of Lemma 3.7) for any link
(cid:17) l 2 Li
we define by
l
p(cid:22)(cid:0)i
(4) 1 X 1 X
l l l
p(cid:22)(cid:0)i := l pj = l pj:
jN nfigj j2Nlnfig N (cid:0)1 j2Nlnfig
The quantity is calculated by averaging the quoted prices for link over all agents other than
l
p(cid:22)(cid:0)i l
who use that link (note that due to assumption (A3) every link has at least 2 agents). The
i
interpretation of prices in this mechanism is closely related to agent ’s willingness to pay
l
pj j
for consuming resource on link . Since we have the problem of information elicitation for each
l
agent’s type ( ), quoting of prices and demand is used as a way of eliciting by comparing
0
vi vi(xi)
it appropriately with prices.
The quantity creates allocation by dilating/shrinking a given demand vector on to
hx;i(s) y
one of the hyperplanes defined by the constraints in (C ), specifically, that hyperplane for which
2
thecorresponding allocationistheclosesttoorigin(theallocationcouldalsobeattheintersection
of multiple hyperplanes). Another way to describe this is to say that dilates/shrinks to
hx;i y
the boundary of the feasible region defined by (C ) constraints. Since all the ’s are positive,
l
2 (cid:11)j
this means that all other constraints in (C ) are satisfied for the allocation automatically (shown
2
later). Additionally, the separate definition for when is to ensure (as it will be
l l
r jS (y)j (cid:20) 1
8 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS
shown later) that there are no equilibria with . This is required since we are only
l
jS (y)j (cid:20) 1
3
dealing with achieving solutions to (CP) which satisfy assumption (A4).
The mechanism gives rise to a one-shot game , played by all the agents in , where action
G N
sets are and utilities are given by
(Si)i2N
u^i(s) = vi(xi)(cid:0)ti = vi(hx;i(s))(cid:0)ht;i(s) 8 i 2 N
We will say that maximising social welfare for (CP ) has been , if all
1 fully implemented in NE
outcomes (all possible NE) of this game produce allocation and all agents in plus the seller
?
x N
are better-off participating in the mechanism than opting out (getting allocation and taxes).
0
The second property is known as .
individual rationality
3.3. Results.
Theorem 3.1 (Full Implementation). For game , there is a unique allocation, , corre-
G x
sponding to all NE. Moreover, , the maximiser of (CP). In addition, individual rationality
?
x = x
is satisfied for all agents and for the seller.
Thetheorem willbeprovedbya sequence ofresults, inwhichwewill characterise all candidate
NE of by necessary conditions until we are left with only one family of NE candidates. We will
G
then show that has NE in pure strategies, and that all of them result in allocation .
?
G x = x
Finally, individual rationality will be checked.
Lemma 3.2 (Primal Feasibility). For any action profile of game , constraints
s = (y;P) G
(C ) and (C ) are satisfied at the corresponding allocation.
1 2
Proof. Constraint (C ) is clearly always satisfied. For , constraint (C ) is also clearly
1 y = 0 2
satisfied. We will now show (C ) is satisfied for any . In that case (since there
2 y 6= 0 r < +1
exists at least one link with and thus ). Now, for any link , we have
q q
q jS (y)j (cid:21) 1 r < +1 l
the following two cases. If then the allocation to agents on that link is clearly zero
l
jS (y)j = 0
(since and ), so (C ) for those links is satisfied. If we have
l
xi = ryi yi = 0 2 jS (y)j (cid:21) 1
X X X
l X
l l l l l l
(cid:11)jxj = r (cid:11)jyj (cid:20) r (cid:11)jyj (cid:20) P l (cid:11)jyj =
j2Nl j2Nl j2Nl j2Nl(cid:11)jyj j2Nl
where the first inequality holds because is the minimum of all ’s. The second inequality will be
l
r r
(cid:3)
equality if and will be strict only if (see second sub-case in eq. (1)).
l l
jS (y)j (cid:21) 2 jS (y)j = 1
Feasibility of allocation for action profiles is a direct consequence of using projections of
demand on to the feasible region. Now we will prove that all agents using a link quote the
y
same price for it, this is brought about by the 2nd tax term P . This is a way of
l l 2
l2Li(pi (cid:0)p(cid:22)(cid:0)i)
penalizing users with higher taxes just for quoting a different price than average, at each link.
3
Note that for the given allocation function, is equivalent to .
l l
jS (y)j=0;1 jS (x)j=0;1
MECHANISM DESIGN FOR UNICAST SERVICE 9
Lemma 3.3. At any NE of , price for any link, is same forall agents, i.e.,
l l
s = (y;P) G pi = p
, (where we denote the common price at any link by ).
l l
8 i 2 N 8 l 2 L l p
Proof. Suppose there exist a link for whichprices atequilibrium are notall equal.
q
q (pi)i2Nq
Clearly then there is an agent q for whom q q (this can be seen from eq. (4)). We
j 2 N pj > p(cid:22)(cid:0)j
will show that this agent can deviate by only reducing price for link and be strictly better off,
q
thereby contradicting the equilibrium condition.
Takethedeviationbyagent , where 0q q q andhisdemandremainsthesame. Then
j pj = p(cid:22)(cid:0)j < pj
the difference between utilities for agent after and before deviation would arise only because
j
of change in taxes (since allocations haven’t changed for any agent) and moreover the difference
would only arise from tax terms corresponding to link (refer to (3)). We’ll have
q
X
0q q 2 q 0q q q q
(cid:1)u^j = (cid:0)(pj (cid:0)p(cid:22)(cid:0)j) (cid:0)(cid:17)p(cid:22)(cid:0)j(pj (cid:0)p(cid:22)(cid:0)j)(
(cid:0) (cid:11)kxk)
k2Nq
X
q q q q q q q 2
+(cid:17)p(cid:22)(cid:0)j(pj (cid:0)p(cid:22)(cid:0)j)(
(cid:0) (cid:11)kxk)+(pj (cid:0)p(cid:22)(cid:0)j)
k2Nq
h X i
q q q q q q q
= (cid:0)0(cid:0)0+(pj (cid:0)p(cid:22)(cid:0)j) (cid:17)p(cid:22)(cid:0)j (
(cid:0) (cid:11)kxk)+(pj (cid:0)p(cid:22)(cid:0)j) > 0
| {z } k2Nq | {z }
>0 | {z } >0
(cid:21)0 by Lemma 3:2
which shows that the above deviation is a profitable one.
Hence at equilibrium, for any link, the price quoted for that link by any user using that link
(cid:3)
is the same, we denote the common price vector by .
l
p = (p )l2L
Now that we established Lemma 3.3, we can talk in terms of the common price vector at
equilibrium rather than different price vectors for all agents, in fact we can identify each NE
candidate profile with , with .
l
s = (y;P) s = (y;p) p = (p )l2L
We will later see how will take the place of dual variables when we compare equilibrium
p (cid:21)
conditions with KKT conditions, hence we identify the following condition as dual feasibility.
Lemma 3.4 (Dual Feasibility). .
l
p (cid:21) 0 8 l 2 L
Proof. This is also by design, since any agent is asked to select a price vector in R . (cid:3)
Li
i +
Following is the property that solidifies the notion of prices as dual variables, since here we
claim that inactive constraints do not contribute to payment at equilibrium. This notion is very
similar to the centralised problem, where if we know certain constraints to be inactive at the
optimum then the same problem without these constraints would be equivalent to the original.
The 3rd term in the tax function facilitates this by charging extra taxes for inactive constraints
when the agent is quoting higher prices than the average of remaining ones, thereby driving prices
down.
10 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS
Lemma 3.5 (Complimentary Slackness). At any NE of game with corresponding
s = (y;p) G
allocation satisfies
x
0 1
X
pl (cid:11)lkxk (cid:0)
lA = 0 8 l 2 L
k2Nl
Proof. Suppose there is a link for which at NE P q q and q . Again, we
q k2Nq (cid:11)kxk <
p > 0
will show that deviation makes any agent better off. Take the deviation
q 0q q
j 2 N pj = p (cid:0)(cid:15) > 0
and no deviation in the demand. We can write the difference after and before deviation as
X
0q q 2 q 0q q q q
(cid:1)u^j = (cid:0)(pj (cid:0)p ) (cid:0)(cid:17)p (pj (cid:0)p )(
(cid:0) (cid:11)kxk)
k2Nq
X
q q q q q q q 2
+(cid:17)p (p (cid:0)p )(
(cid:0) (cid:11)kxk)+(p (cid:0)p )
k2Nq
X (cid:16) X (cid:17)
2 q q q q q q
) (cid:1)u^j = (cid:0)((cid:0)(cid:15)) (cid:0)(cid:17)p ((cid:0)(cid:15))(
(cid:0) (cid:11)kxk)+0+0 = (cid:15) (cid:0)(cid:15)+(cid:17)p (
(cid:0) (cid:11)kxk)
k2Nq k2Nq
So if we take such that
(cid:15)
(cid:26) (cid:27)
X
q q q q
min (cid:17)p (
(cid:0) (cid:11)kxk); p > (cid:15) > 0
|{z} k2Nq |{z}
>0 | {z } >0
>0
(cid:3)
then . Taking such an is possible because LHS above is positive.
(cid:1)u^j > 0 (cid:15)
Lemma 3.6 (Stationarity). At any NE of game , and corresponding allocation,
s = (y;p) G
, we have
x
X if
0 l l
vi(xi) = p (cid:11)i 8 i 2 N xi > 0
l2Li
X if
0 l l
vi(xi) (cid:20) p (cid:11)i 8 i 2 N xi = 0
l2Li
Proof. At any NE , agent ’s utility as a
0 0 0
s i u^i(si;s(cid:0)i) = vi(hx;i(si;s(cid:0)i)) (cid:0) ht;i(si;s(cid:0)i)
function of his message , with fixed, should have a global maximum at
0 0 0
si = (yi;pi) s(cid:0)i si =
. This would mean that if this function was differentiable w.r.t. at , the partial
0
(yi;p) yi si
derivatives w.r.t. at should be . However, since our allocation dilates/shrinks demand
0
yi si 0
vector on to the feasible region, it could be the case that increasing and decreasing gives
0 0
y yi
allocations lying on different hyperplanes, meaning that the transformation from to is
0 0
y x
different on both sides of and therefore may not be differentiable w.r.t at . The
0
yi u^i yi yi
important thing here however is to notice that right and left derivatives exist, it’s just that they
may not be equal. Hence we can take derivatives on both sides of as (noting that derivative
yi
of the other two terms in utility will be zero due Lemma 3.3)
0 1 0 1
(5) uy^ii0(cid:12)(cid:12)(cid:12)(cid:12)yi0#yi = vi0(xi)(cid:0)lX2Lipl(cid:11)liA xyi0i0(cid:12)(cid:12)(cid:12)(cid:12)yi0#yi; uy^ii0(cid:12)(cid:12)(cid:12)(cid:12)yi0"yi = vi0(xi)(cid:0)lX2Lipl(cid:11)liA xyi0i0(cid:12)(cid:12)(cid:12)(cid:12)yi0"yi: