A Mechanism for Multi-unit Multi-item Commodity Allocation in
Economic Networks
Pankaj Mishra
1,2
, Ahmed Moustafa
1
and Fenghui Ren
2
1
Department of Computer Science, Nagoya Institute of Technology, Gokiso, Naogya, Japan
2
School of Computing and Information Technology, University of Wollongong, Wollongong, Australia
Keywords:
Economic Networks, Multi-Unit Homogeneous Resource Allocation, Diffusion Mechanism, Procurement
Auction Theory.
Abstract:
In this work, we introduce a novel resource allocation mechanism that aims to maximise the social welfare
of the market in procurement auctions. Specifically, we consider a market setting with multiple units of
homogeneous resources. In such settings, buyers submit their resource requests to a limited number of known
providers. This limited number of providers might in turn lead to a provider monopoly in the market and a
scarcity of the resources. To address this problem, we propose a novel information diffusion-based resource
allocation mechanism for resource allocation in procurement auctions. The proposed mechanism focuses on
procuring multiple units of homogeneous resources. In this regard, the proposed mechanism incentivises the
providers to truthfully diffuse the procurement information to their neighbours. This information diffusion
aids the buyers to procure the required amounts of commodities/resources at the minimum possible prices.
In addition, the proposed mechanism gives fair chances to the distant providers to fairly participate in the
procurement auction. Further, we prove that the proposed mechanism minimises the procurement costs, with
no deficits, compared to the Vickrey-Clarke-Groves mechanism. Finally, based on the experiments, we show
that the proposed mechanism has comparatively lesser procurement costs.
1 INTRODUCTION
The procurement of multiple units of different types
of resources has become a challenging problem. Nor-
mally, the procurement of resources in competitive
markets takes place through procurement auctions
(reverse auctions) (Krishna, 2009). In those procure-
ment auctions, sellers with the lowest offered price are
the winners. Further, the procurement cost is com-
puted based on an adopted pricing policy, such as
first-price auction, second-price auction, etc.
Generally, designing an optimal resource procure-
ment mechanism depends on finding an optimal win-
ner determination policy and an optimal pricing pol-
icy. An optimal winner determination policy (WDP)
is usually implemented in auction paradigms for dif-
ferent real-world market settings (Samimi et al., 2016;
Weber et al., 1998; Prasad et al., 2016; Zaman and
Grosu, 2013; Wu et al., 2018). However, the conven-
tional auction mechanisms (Myerson, 1981; Mishra
et al., 2020a; Mishra et al., 2020b), mainly focus on
maximising the revenue of the owners of the auction.
In classical auctions, the owner (buyer is the owner
in the procurement) of the auction is only aware of
a limited number of bidding participants (sellers are
the participants). This might lead to a monopoly in
the market (drop-in competition) and also a shortage
of resources. Also, the unbalanced supply or demand
in the market might affect the stability of the market
(Tobin, 1969). Therefore, there is a need for a pro-
curement mechanism that is capable of controlling the
number of participants as per the resource demands.
In this regard, an information diffusion-based mech-
anism become an appropriate choice. Specifically,
information diffusion would invite distant sellers to
take part in auctions and satisfy the market demands.
In the literature, those diffusion aided resource allo-
cation mechanisms are used in several e-commerce
platforms to advertise their products to remote buyers
(Fieldman and Chaube, 2020; Sepehrian et al., ; Mon-
eypenny and Flinn, 2009). Lately, (Zhao et al., 2018;
Li et al., 2017) introduced a set of diffusion-based
mechanisms in social networks to sell different re-
sources. In specific, (Zhao et al., 2018; Li et al., 2017)
presented diffusion-based mechanisms to reduce the
total procurement cost. In this regard, the existing
Mishra, P., Moustafa, A. and Ren, F.
A Mechanism for Multi-unit Multi-item Commodity Allocation in Economic Networks.
DOI: 10.5220/0010844400003116
In Proceedings of the 14th International Conference on Agents and Artificial Intelligence (ICAART 2022) - Volume 1, pages 265-273
ISBN: 978-989-758-547-0; ISSN: 2184-433X
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
265

work focused on selling a single unit of resources
(Li et al., 2018; Li et al., 2017) as well as multi-
item single-unit resource allocation (Kawasaki et al.,
2019; Takanashi et al., 2019). As a result, the ex-
isting diffusion-based mechanisms are being adopted
for crowd-sourcing aided bulk data collection (Shen
et al., 2019; Zhang et al., 2019).
Briefly, the existing diffusion-based auction
mechanisms have the potential to address the chal-
lenges in designing procurement auctions. However,
those existing mechanisms cannot be directly adopted
for the problem we aim to solve in this research,
i.e., designing procurement auctions for multi-unit
multi-item resource allocation. Because, the existing
diffusion-based mechanisms are mainly seller-centric
(seller is the owner), but the procurement auctions are
buyer-centric. Besides, the existing mechanisms are
designed for single-unit or unit-demand resource al-
locations, with no budget constraints. Therefore, to
the best of our knowledge, no known diffusion-based
mechanisms are designed for a multi-unit multi-item
procurement settings. Owing to this, in this research,
we focus on designing a diffusion-based collaborative
mechanism for a multi-unit multi-item procurement
mechanism. Such that, it not only keeps a check on
competition in the market but also encourages coop-
erative behaviour amongst independent sellers. Such
procurement mechanisms could be used in different
procurement problems such as the procurement of
vehicles (Remli and Rekik, 2013), crops, milk, etc,
(Vykhaneswari and Devi, ; Nuthalapati et al., 2020).
Also the proposed mechanism is incentive compat-
ible, individually rational and an optimal payment
mechanism for multi-unit multi-item resource alloca-
tion in economic networks. To summarise, the contri-
butions of this research are as follows: (1) We propose
a novel diffusion mechanism for multi-unit multi-item
resource allocation in economic networks; (2) we in-
troduce a practical information propagation mecha-
nism to disclose the private information of the partic-
ipating sellers; and (3) Then, we introduce a contest
function based iterative auction-based group determi-
nation strategy.
The rest of this paper is organised as follows: we
first discuss the model and the different key defini-
tions in modelling the proposed mechanism in Section
2. Section 3 presents the proposed (DMMP) mech-
anism. In Section 4, the properties of the proposed
DMMP mechanism is presented. Section 5 discusses
the experimental results. Finally, the paper is con-
cluded in Section 6.
2 THE MODEL
We consider an economic network with a single buyer
denoted as b having multi-unit resource request of k
types of non-substitute-able heterogeneous resources
in set K, denoted as Q
b
= {q
b,1
,q
b,2
,... ,q
b,k
}, k ∈ K,
where Q
b
is termed as the resource package or simply
the package. This procurement request is submitted
directly or through diffusion to the set N of n inde-
pendent sellers denoted as N = {s
1
,. .. ,s
n
}. In this
context, an economic network is represented as a di-
rected acyclic graph G ≡ (V,E), where V = N ∪{b} =
{s
1
,. .. ,s
n
} ∪ b representing the set of all the nodes
(including the buyer and all the sellers reachable to
buyer b), whereas E represents the set of edges be-
tween these nodes representing the neighbourhood re-
lationship. For any node i, j ∈ V (i 6= j), if there is a
directed edge from i to j, then it means j is the di-
rect successor of i, and i is the direct predecessor of
j and the edge is represented as e
i j
= 1, else e
i j
= 0,
e
i j
∈ E. Further, for all the nodes i ∈ V , its set of
immediate children is termed as neighbours and de-
noted as Ng
i
⊆ V , s.t., for j ∈ Ng
i
there exists an edge
e
i j
∈ E between node i and j. Further, a set of direct
successor for node i ∈ G is termed as neighbours de-
noted as Ng
i
, whereas direct predecessor is termed as
parent node denoted as P
i
. In this regard, if there is a
path between two nodes i, j ∈ V (i 6= j), the distance
between i, j is denoted as dist(i, j), and if no path ex-
ists, then dist(i, j) = ∞. Also, we represent set of all
the predecessors as Ng
all
i
, s.t. Ng
all
i
= { j ∈ V : 0 <
dist(i, j) < ∞}. Similarly, set of all the predecessors
P
all
i
= { j ∈ V : 0 < dist( j, i) < ∞}. Also let d
i
> 0,
represent the depth for node ∀i ∈ N, representing the
shortest path from buyer b to seller i. In our setting,
initially, buyer b has no prior information about all the
sellers in the market.
In such a market setting for multi-unit multi-item
procurement auctions, it would be ideal for the buyer
to be reachable to maximum possible sellers, so as to
procure the resources at the minimum possible price.
However, initially, buyer b can only submit its pro-
curement request to its neighbour nodes j ∈ Ng
b
.
Also, submitting its procurement request to distant
potential sellers would incur extra cost on the buyer’s
budget. Therefore, to avoid this extra cost, we pro-
pose a diffusion-based mechanism, that encourages
each seller to invite their respective neighbours to par-
ticipate in the procurement. In specific, firstly, buyer b
submit its procurement request to set of neighbours in
Ng
b
. Then, all the seller j ∈ Ng
b
would diffuse the in-
formation to their respective neighbour set Ng
j
. The
subgraph formed with root node j is termed as local
economic network of the node j ∈ V denoted as G
j
≡
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
266

Figure 1: Local economic networks.
Figure 2: Global economic networks with three local net-
works.
(V
j
,E
j
) ≡ (Ng
j
,E
j
). In this regard, independent local
economic networks are connected to form a complete
economic network denoted as G = G
1
∪· ·· ∪G
k
. This
whole economic network G resembles a rooted tree,
wherein the buyer b is the root node and a set N of
potential sellers are leaf nodes.
For instance, Figure 1 depicts an example of eco-
nomic network. In this regard, upon receiving a re-
source request from the buyer b, b’s immediate neigh-
bours along with their local economic networks are
combined to form an economic network as depicted
in Figure 2.
Further, it should be noted that the resource pack-
age request Q
b
from buyer b denotes the minimum
resource demand, such that buyer b aims to procure at
least Q
b
package of resources from a single or a group
of sellers in an economic network through a diffusion
mechanism. In this context, the objective of the buyer
is to procure at-least Q
b
at minimum possible price
from least number of sellers. Because, with the in-
crease in number of sellers, would lead to increase in
transaction overhead, also referred as chaining cost in
communication domain (Kayal and Liebeherr, 2019).
Each seller i ∈ N has two private values, i.e., per-
unit valuation of the resource (bid-density) v
i
≥ 0 and
the maximum quantity of resource the seller is avail-
able for selling denoted as A
i
= {a
i,1
,a
i,2
,. .. ,a
i,k
}.
Also, ∀ j ∈ N, let set of all the sellers except seller
j denoted as N
− j
, s.t., N
− j
≡ N \ j. For example,
in Figure 2, for the seller s
1
, Ng
s
1
≡ {s
4
,s
5
}, and
Ng
all
s
1
= {s
4
,s
5
,s
8
,s
9
,s
11
,s
13
,s
14
}.
Further, in any multi-item market setting, combi-
natorial auctions that allow bidders to bid on com-
binations (bundles or packages) of items make busi-
ness sense when there are bundles of items that have a
combined valuation to bidders higher than the sum of
their individual valuations. Such items are said to be
complementary. In addition to that, we assume that
each seller has incentive in selling their maximum
possible resources to avoid wastage of their remain-
ing resources. Therefore, it is practical to assume that
sellers give discounted price over bulk procurement.
To simplify the valuation, we consider two types of
valuation, namely, valuation for bulk purchases and
single item purchases. Therefore, ∀i ∈ N, its truth-
ful type is represented as θ
i
= (bv
i
,sv
i
,A
i
,Ng
i
), where
bv
i
, sv
i
, A
i
and Ng
i
are the per unit valuation for bulk
purchase (i.e, all the offered resources), the per unit
valuation for single item purchase, offered quantity
of resource and set of neighbours, respectively. Fur-
ther, the type profile of all the sellers is denoted as
θ
θ
θ = (θ
1
,. .. ,θ
n
). Let θ
θ
θ
−i
be the type profile for sell-
ers except i, s.t θ
θ
θ = (θ
θ
θ
−i
,θ
i
). Also, let Θ
i
be the type
space for seller i, s.t., Θ = (Θ
i
,. .. ,Θ
n
) = (Θ
−i
,Θ)
be the the type profile space for all sellers. Also,
we consider a strategic setting, wherein sellers might
not report their true type to maximise their utility,
this reported type of the seller i ∈ N is denoted as
θ
0
i
≡ (bv
0
i
,sv
0
i
,A
0
i
,Ng
0
i
), where bv
0
i
, sv
0
i
, A
0
i
and Ng
0
i
are
the reported per unit valuation for bulk purchase (i.e,
all the offered resources), the per unit valuation for
single item purchase, offered quantity of resource and
set of neighbours, respectively. Also, let
/
0 be the
default reported type, when i ∈ N had not received
the information I
P
i
≡ (Q
i
,b) from its parent seller or
seller i ∈ N do not want to participate in the procure-
ment, where Q
i
denotes the minimum resource pack-
age requested by seller i, whereas b resembles the pro-
curement information from buyer b. In this context,
we assume that, if a seller i ∈ N is not invited, then
the mechanism will not observe any action from that
seller, called feasible type profile denoted as F(θ
θ
θ
0
),
s.t. F(θ
θ
θ
0
) ⊆ θ
θ
θ
0
. Further, the diffusion mechanism for
feasible type profile is defined as follows:
Definition 1. A diffusion mechanism M in the eco-
nomic network is denoted by an allocation policy
π = (π
1
,π
2
,. .. ,π
n
) and a payment policy pay =
(pay
1
, pay
2
,. .. , pay
n
), where π
i
: θ
θ
θ → {0,1}, π : θ
θ
θ →
R and pay
i
denotes the payment received by the seller
i ∈ N from buyer b.
Given the type profile θ
θ
θ
0
= (θ
0
1
,. .. ,θ
0
(n)
) ∈
F(θ
0
), the payment policy p
p
pa
a
ay
y
y(θ
θ
θ
0
) =
(pay
1
(θ
θ
θ
0
),. .. , pay
(n)
(θ
θ
θ
0
)) represents the amount
of money each seller would be given at the end
of the resource procurement. For seller i ∈ N,
if pay
i
(θ
θ
θ
0
) ≥ 0, then it receives pay
i
(θ
θ
θ
0
) from
the buyer and if pay
i
(θ
θ
θ
0
) ≤ 0, then it will pay to
the buyer. In this regard, the allocation policy
π(θ
θ
θ
0
) = (π
1
(θ
θ
θ
0
),. .. ,π
n
(θ
θ
θ
0
)) represents the resource
allocation. We have, π
i
= 1 if seller i is among the
winning seller, else π
i
= 0.
In this context, for the diffusion mechanism M =
A Mechanism for Multi-unit Multi-item Commodity Allocation in Economic Networks
267

(π, p
p
pa
a
ay
y
y), we assume that there is no cost for a seller to
spread the procurement information to its neighbours.
Thus, for seller i ∈ N of type profile θ
0
i
, given a feasi-
ble type profile θ
θ
θ
0
∈ F(θ) of all sellers, then the utility
of seller i’s is defined as the payment received mi-
nus the expected payment based on its true valuation.
In this context, we say that a diffusion mechanism is
individually rational if the utility of every seller in-
volved is non-negative as long as it performs its ac-
tions truthfully, i.e., reports the valuation truthfully
no matter how many neighbours it invites to join the
mechanism and how many resources it is willing to
sell, as defined in Theorem 1. It should be noted that
the definition does not rely on diffusion and disclosed
quantity of resources as we do not want to force the
seller to invite others and sell all its resources to guar-
antee him a non-negative gain. Further, if all the sell-
ers are willing to report their valuations truthfully for
the reported quantities of resource, we say the mecha-
nism satisfies the property of incentive compatibility.
However, in this mechanism, sellers also need to in-
vite their neighbours. Thus, we want to incentivise
sellers not only to report their truthful bids but also to
invite all their neighbours, as defined in Theorem 2.
In the next section, we present a novel diffusion
mechanism for multi-unit homogeneous resource al-
location wherein a seller collaborates within their lo-
cal economic network. Then, finally, the resource is
served by a group of sellers having the minimum val-
uation for the minimum requested resource. Also, all
the related sellers who contributed to inviting the win-
ner are given incentives for diffusion.
3 DIFFUSION BASED
MULTI-UNIT MULTI-ITEM
PROCUREMENT
In this section, we introduce a novel procurement
mechanism for decentralised multi-unit multi-item re-
source allocation. This proposed mechanism, i.e.,
Diffusion based Multi-Unit Multi-Item Procurement
(DMMP) is based on information diffusion tech-
nique. In specific, DMMP mechanism adopts a
reverse-auction paradigm (Krishna, 2009), wherein
each buyer/seller submits a multi-unit multi-item of
resource request to their neighbours sequentially. In
this context, a seller i ∈ N and its respective neigh-
bours Ng
i
is called local economic network.
Firstly, buyer b submits its minimum package of
resource request Q
b
to all the sellers in set Ng
b
. Then,
seller ∀ j ∈ Ng
b
are encouraged through incentives to
diffuse this information within their respective local
economic network. Finally, package Q
b
is allocated
to a single or a group of sellers and its corresponding
payment is computed. Besides, all the intermediate
sellers between the buyer and the winner are given
incentives for information diffusion. In specific, in
this novel DMMP mechanism, two independent pro-
curements are carried out, namely local procurement
and global procurement. The local procurement takes
place within the local economic networks, whereas
global procurement takes place between sellers and
the buyer. In this regard, once the seller agrees to be-
come the part of the global network by reporting its
type θ
0
, then the DMMP mechanism computes diffu-
sion information for that seller. In addition, mecha-
nism computes bids for local and global procurement
sequentially on behalf of the sellers. In this context,
it should be noted that the set of neighbours remains
the same throughout in local and global procurement.
Also, we assume that, any seller can participate in the
global procurement , only if participates in local pro-
curement.
Briefly, whole mechanism can virtually
1
be di-
vided into three major stages, that is, (1) information
propagation, (2) group determination, and (3) winner
determination, discussed in the following subsections.
3.1 Information Propagation
In this subsection, we present the information dif-
fusion stage in the DMMP mechanism. In specific,
upon receiving Q
b
≡ {q
b,1
,q
b,2
,. .. ,q
b,k
} from b, all
the sellers i ∈ Ng
b
diffuse the information message
I
i
= (Req
i
,b) to their neighbours Ng
i
, wherein, Req
i
=
{req
i,1
,req
i,2
,. .. ,req
i,k
} is minimum local package
request for Ng
i
, i.e., minimum quantity required by
its neighbours to participate in the local procurement,
computed by the mechanism using Equation 1 ∀k ∈ K
and b represent the buyer b’s information.
req
i,k
=





q
i,k
, if i = b.
req
P
i
,k
− a
0
i,k
, if req
p
i
,k
> a
0
i,k
.
0, if q
p
i
≤ q
0
i
.
(1)
In this way, mechanism would model the diffusion
information I
i
for all the sellers sequentially based on
the reported type θ
0
i
∀i ∈ N. Note that, from Equation
1, diffusion information reveals only the difference
of the resources request for each of the sellers. This
difference-mechanism is designed to preserve the pri-
vacy of the sellers. Also it promotes participation of
the sellers and maintain the competition in the over-
allprocurement process. In real-world setting, assum-
1
stages are interdependent and run simultaneously
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
268

ing that seller would reveal the true resource require-
ments from its ancestors is unrealistic. This is unre-
alistic because any greedy seller would be interested
in selling its resource first and then inviting other sell-
ers. Therefore, this difference-mechanism is in ac-
cordance with the rational behaviour of the sellers.
Further, to avoid information diffusion for infinitely
larger economic networks, diffusion phase will con-
tinue until enough resources are available in the over-
all economic network. In this context we set the max-
imum available resources is reached at-least A
max
or
T
max
time is reached. The value of A
max
is computed
as A
max
= Q
b
× δ, where, δ represents the maximum
resource factor. This value is decided such that, there
should be enough resource available in the global eco-
nomic network to maintain the competition. Also,
too much availability of the resource would lead to
wastage of resources as well as bidder drop prob-
lem (Baranwal and Vidyarthi, 2015). Therefore δ
should be chosen such that it maintains the trade-off
between supply and demand in the global market. In
this regard, A
max
is computed such as, for k = 3, then
for Q
b
≡ (q
1,b
,q
2,b
,q
3,b
) ≡ (5,7,8), then for δ = 2
A
max
≡ Q
b
× δ ≡ (5,7,8) × 2 ≡ (10,14, 16). In ad-
dition to that, in order to restrict the infinitely infor-
mation diffusion, we set the maximum time T
max
, this
is there to stop the algorithm if A
max
is never reached.
This maximum time step is fixed at the beginning of
the mechanism. So once algorithm has A
max
or T
max
is
reached, the mechanism moves to the next stage, i.e,
group determination discussed in the next subsection.
Algorithm 1 gives the full pseudocode for information
propagation stage.
This algorithm takes buyer b’s information i.e., lo-
cation and request Q
b
along with the whole network
information G ≡ (V, E), δ and T
max
as input. Besides,
it has temporary (arbitrary) neighbour list which is a
queue used for traversing the neighbour list. After
that, the algorithm en-queues the neighbours of buyer
list. Then the algorithm runs until maximum T
max
or
A
max
volume of resources available, and dequeue the
top element from the arbitrary neighbours’ list and
diffuses the information to its neighbours. Finally, it
enqueues, sequentially, all its neighbours to the neigh-
bour list and arbitrary list and update the value of the
level of each seller and computes the maximum depth
d
max
. In the next subsection, we will present the sec-
ond stage of the DMMP mechanism i.e., Group For-
mation.
3.2 Group Determination
In this subsection, we present the second stage of the
novel DMMP mechanism, i.e., the group determina-
Algorithm 1: Information Propagation.
Input: G ≡ (V,E), Q
b
, δ, T
max
Result: d
max
1 A
max
← Q
b
× δ ; sn ← b ;
2 A
total
, h
max
← 0 ;
3 req
sn,k
← q
b,k
; ∀k ∈ K ;
4 Req
sn
= {req
sn,k
: ∀k ∈ K} ;
5 I
s
n ≡ (Req
sn
;
/
0) ;
6 neighbour list[b][][∅];
neighbour list arbitrary[b][] ← R
b
;
7 while t = T
max
or A
total
≥ A
max
do
8 for neighbour list arbitrary[sn] 6= ∅ do
9 node ←
DEQU EUE(neighbour list arbitrary[sn])
node ← []diffuseI
sn
≡ (Q
sn
;b) ;
10 seller list; seller list arbitrary ← node
;
11 end
12 sn ← DEQU EUE(seller list arbitrary) /*
changing the seed node */ ;
13 compute Req
sn
using Equation 1 */ ;
14 A
total
+ = Q
sn
;
15 if Ng
sn
6=
/
0 then
16 if sn /∈ neighbour list then
17 d
max
+ = 1
18 end
19 neighbour list[sn];
neighbour list arbitrary[sn] ← Ng
sn
;
20 t + + ;
21 end
22 end
tion stage. After receiving the information messages
I
i
from ancestor node P
i
, on behalf of seller i ∈ N,
the proposed mechanism starts to fill the requested
package Req
i
. In fact, the proposed mechanism does
this based on an iterative auction, sequentially for
all the local networks from bottom to top. In spe-
cific, all the seller at level j ∈ N submit their bids
bid
1
i
≡ (bv
+
i
,sv
+
i
,Q
i
) to their parent node P
j
, where
bv
+
i
, sv
+
i
and Q
i
are the updated valuations and the
combined available bids. Then, finally, after form-
ing groups all the seller’s bids can now be submit-
ted directly to buyer b. In this context, it should be
noted that, in any economic market, any greedy seller
would prefer to sell its resource first then would try
to sell others resource. Therefore, we assume that
seller would consume the resource from its neigh-
bours, only when its available resource is less than
the resource request from its respective parent. In
specific, if a
0
i,r
< req
p
i
,r
, then the mechanism would
aid the seller i to collaborate with its neighbour(s)
and submit the bid the combined resource in the local
procurement. Similarly, if a
0
i,k
< q
b,k
, then only the
mechanism would reveal the combined resource in its
submitted bid bid
2
i
. Then, at the end of local procure-
A Mechanism for Multi-unit Multi-item Commodity Allocation in Economic Networks
269

ment i.e., after receiving reported type bid
1
i
,∀ j ∈ Ng
i
,
set of local winners L
i
is elected by performing iter-
ative auction. In specific, winning seller in each it-
eration is decided based on customised contest suc-
cess function (CSF) (Skaperdas, 1996). A CSF de-
termines each sellers probability of winning the local
auction in terms of other sellers bidding package. In
specific, based on CSF, a local winner in each itera-
tion is determined from the local network with seed
node i ∈ N, lw ∈ Ng
i
is elected as lw ≡ min(cs f
j
),
∀ j ∈ Ng
i
, whereas cs f
j
is computed using Equation
2.
cs f
i
=
∑
k∈K
(q
i,k
)
σ
∗ v
0
i
∑
N
j6=i
(q
j,k
)
σ
xv
0
j
(2)
where, 0 < σ ≤ 1 represents the noise parameter
in the contest, interpreted as the marginal increase in
probability with the increase in valuation (Shen et al.,
2019). Towards this end, seller i would update its total
available resources Q
i
using Equation 3 ∀k ∈ K
q
i,k
= a
0
i,k
+
∑
l∈L
i
q
j,k
(3)
where, L
i
is set of local winner from all the lo-
cal iterative auctions. In this regard, mechanism con-
ducts a iterative auction based matching for all the
seed node belonging to same height h sequentially
with continuous iteration. In each round of auction for
seed node i ∈ N, mechanism elicits a single winner.
The iteration continues until minimum package Q
i
is
filled or maximum waiting time W
d
has been reached,
where d denotes the depth of the seed node. In addi-
tion, in each iteration XOR bids (Leyton-Brown et al.,
2000) are submitted, i.e. each local seller can win
only once. This will reduce the complexity of the iter-
ative auction by reducing the number of combinations
in each iteration.
Further, at the end of each iteration for local auc-
tion for seed node i ∈ N, mechanism computes the
per unit valuation for each of the winning sellers in
L
i
using Equation 4 at which each of the local winner
j ∈ L
i
would get its payment, if seller i is the winner
in the procurement.
v
l
=





min(v
0
j
)∀ j ∈ Ng
0
i
\ l, if Ng
i
6= {null}.
max(v
0
i
,v
0
j
), if |Ng
i
| = 1.
0, otherwise.
(4)
where, v
0
j
= bv
0
j
, if all the resources in the offered
package Q
j
is allocated, else v
0
j
= sv
0
j
. Intuitively the
above equation depicts that, if there are more than one
neighbours, i.e., |Ng
i
| > 1, then the valuation of the
local winner is computed based on the VCG mecha-
nism (Krishna, 2009). However, if there is only one
local neighbour, i.e., |Ng
k
| = 1, then the valuation is
the maximum valuation among seller i and j ∈ Ng
i
;
otherwise its set to be zero. Similarly, the valuation
for the seed seller is updated based on the valuation
of the local winner using Equation 5.
v
i
=
(
v
0
i
, if Ng
i
= {null}.
v
0
i
×(q
0
i,k
−
∑
∀l∈L
i
q
0
l,k
)+v
l∈L
×
∑
∀l∈L
i
q
0
l,k
q
i,k
, otherwise.
(5)
Then, after end of the iterative auction at depth
d, mechanism moves upward at depth d − 1. In spe-
cific, at depth d, ∀i ∈ N, where d
i
= d, seller i sub-
mit their bids to parent P
i
∈ N. In this way, lo-
cal groups are formed within the local economic net-
works in a decentralised manner. In this regard, max-
imum time-step w
d
for which iterative auction will
continue at depth d is computed as w
d
= ((d
max
−d)+
1) ∗ (T
max
/d). Finally, after completion of auctions in
local networks, a new global economic network with
updated valuation and available resources is formed,
represented as G
0
⊆ G.
In the next subsection, we will present the third
and the final stage of the DMMP mechanism i.e., win-
ner determination and payment distribution.
3.3 Winner Determination
In this section, we present the third stage i.e., win-
ner determination, wherein, payment of the winning
seller(s) are computed. In addition to that, rewards
for all the sellers who has contributed by informa-
tion diffusion are computed. Specifically, the mech-
anism determines a single winner w which can fill
all the requested resources in the package Q
b
, s.t.
w ≡ min(cs f
i
), ∀i ∈ G
0
, whereas cs f
i
is computed us-
ing Equation 2. Also, all the sellers in the path path
bw
from buyer b to the winning seller w are rewarded
for diffusing the information to winner w. This path
path
bw
is called as winning path and represented as
path
bw
≡ {b,...,w}. Let, v
∗
D
= min
i∈D
v
0
i
be the min-
imum reported valuation in the subset D ⊆ N and the
corresponding seller is represented as w
∗
D
, and then
v
0
w
= v∗
∗
N
whereas w = w
∗
N
. Similarly, v
∗
D\i
denotes
the minimum valuation and w
∗
D\i
when seller i ∈ D
does not participate. Also to simplify the notations,
let v
∗
N
∗
\i
= v
∗
−i
.
Definition 2. feasible global neighbours (N∗) is a set
of all sellers for buyer b having req
b
≤ q
0
k
, s.t k ∈ N
and N∗ ⊆ N
Definition 3. A critical seller set (C), is a set of all
the sellers in winning path path
bw
including the local
winner of the winning seller, i.e., C ≡ {c
1
,. .. ,c
h
,L
c
h
},
where i ∈ path
bw
, w = w
∗
N
. This critical seller C set
is an ordered set, s.t., d
c
1
⊃ d
c
2
⊃,. .. ,⊃ d
c
h
⊃ d
L
c
h
,
ICAART 2022 - 14th International Conference on Agents and Artificial Intelligence
270

where d
c
i
denotes the depth of the node c
i
from buyer
b.
In this regard, the allocation policy for diffusing
mechanism DMMP is computed using Equation 6.
π
i
(θ) =
(
1 if i ∈ C, v
0
i
= v
∗
−(i+1)
.
0, if i /∈ C.
(6)
Intuitively, in DMMP mechanism, the first criti-
cal seller who has the least per unit valuation when
critical seller i + 1 is removed from the economic net-
work is the winner. Then the bid density bd
i
∀i ∈ C
is computed using Equation 7., bid density represents
the value at which payment for a seller will be calcu-
lated
bd
i
=









v
∗
−i
+ ε
i
if i = w.
ε
i
, if i ∈ C
−w
.
v
0
L
i
, if i = L
w
.
0, Otherwise.
(7)
where, ε
i
represents the reward factor for seller
i ∈ C for diffusing the information to its neighbours,
which is computed using Equation 8
ε
i
=
v
∗
−(α
i+1
,w
∗
α
i
)
− v
∗
−α
i
|C − 1|
(8)
where, α
i
= 1, if i ∈ C and i ∈ N
∗
, else α
i
= 1 if
i ∈ C and i /∈ N
∗
and P
i
∈ N
∗
, represent the closest
ancestor node in N
∗
if i /∈ N
∗
According to above bid density calculation policy,
winning seller’s bid density is sum of vcg payment
v
∗
−w
, i.e., winner is paid the second lowest per unit
valuation. In addition, winning seller is rewarded γ
w
for diffusing the information to its neighbour. On the
other hand, all the other critical sellers are rewarded
γ
i
for information diffusion. Intuitively, reward is the
decrease in payment for buyer b for seller i ∈ C dif-
fusion action. In particular, it is change in payment
for buyer b when seller i + 1 along with the seller w
∗
−i
(when seller i do not participate) do not participate
and when seller i ∈ C do not participate in the pro-
curement. Finally, bid density of local winner is same
as computed during its local procurement based on
VCG mechanism.
To the end, based on the valuation computed using
Equation 7 and 8, payment pay
i
is computed for all
the critical sellers in set C.
pay
i
=
(
bd
i
× (q
0
i
− q
0
L
i
), if L
i
6= null and i ∈ C.
bd
i
× q
0
i
, otherwise.
(9)
Further, the total payment given by the buyer b to
all the sellers in winning path path
bw
is computed as
pay =
∑
i∈C
pay
i
In this way, winners and their respective payment
are computed, also rewards for all the sellers in win-
ning path is computed for information diffusion. In
the next section we would discuss the properties of
the proposed DMMP mechanism.
4 PROPERTIES OF DMMP
In this section, we prove that DMMP mechanism is in-
dividual rational (IR) (Theorem 1) and incentive com-
patible (IC) (Theorem 2). The proofs for both the
Theorems are in the appendix.
Theorem 1. A diffusion mechanism M = (π, p
p
pa
a
ay
y
y) is
IR, if u
i
(θ
i
,(θ
i
,θ
θ
θ
0
−i
)) ≥ u
i
(θ
i
,(θ
0
i
,θ
θ
θ
00
−i
)), ∀i ∈ N, all
θ
0
i
∈ Θ
i
, where (θ
0
i
,θ
00
−i
) ∈ F(θ
0
i
,θ
0
−i
). In this theorem
we will prove that DMMP is individually rational.
Theorem 2. A diffusion mechanism M = (π, p
p
pa
a
ay
y
y) is
IC, if u
i
(θ
i
,(θ
i
,θ
θ
θ
0
−i
)) ≥ u
i
(θ
i
,(θ
0
i
,θ
θ
θ
00
−i
)), ∀i ∈ N, all
θ
0
i
∈ Θ
i
, where (θ
0
i
,θ
00
−i
) ∈ F(θ
0
i
,θ
0
−i
). In this theorem
we will prove that DMMP is incentive compatible.
5 EXPERIMENTAL RESULTS
In this section, we present the experimental results
performed to evaluate the performance of the novel
DMMP mechanism based on two conventional mech-
anisms. In this regard, we compare the performance
of the following three mechanisms: (1) Simple Pro-
curement: This is the classical procurement, wherein
the buyer procures only from its neighbours through
iterative first price reverse auctions; (2) Fixed Re-
ward: In this mechanism, sellers are given a fixed
reward ω for the diffusion of information, whereas
payment is based on a second price auction: and (3)
DMMP: In this mechanism, the buyer is not aware of
all the sellers in the economic network until its neigh-
bour directly or indirectly diffuses the procurement
information; whereas, payment and rewards are com-
puted using DMMP.
In our experimental setting, we consider the pro-
curement of five different types of resources i.e., k = 5
and set the maximum neighbours for each node as 4,
i.e. 0 < |Ng
i
| ≤ 4, ∀i ∈ V , and for fixed reward, ω
is set to root of the per-unit valuation of the corre-
sponding node. In this regard, we randomly generate
economic networks and set available resource pack-
age for each seller. Besides, we set that |Ng
b
| = 4,
s.t., buyer has at least four sellers directly reachable
which can collectively fill the requested package Q
b
,
i.e., Q
b
⊆ {A
i
: ∀i ∈ Ng
b
}. Further, values for Q
b
A Mechanism for Multi-unit Multi-item Commodity Allocation in Economic Networks
271

Figure 3: Impact of value of δ on the total procurement cost.
and A
i
∀i ∈ N, for all the five types of resources are
sampled from a random generator which takes val-
ues [200,1000] units. Further, both the per unit val-
uations, i.e., sv
i
and bv
i
∀i ∈ N is also drawn from
a random generator which takes values [20,50] s.t.
sv
i
> bv
i
. In this setting, for the economic network,
we run all mechanism for 20 times. Then we evalu-
ate the results to show the merits of adopting DMMP
mechanism for multi-unit multi-item resource pro-
curement based on the cost of the procurement for
the buyer. Besides, in our experimental setting, we
intend to analyse the impact of change in the value
of δ on the cost of procurement. Therefore, we de-
signed six different experimental settings concerning
the value of δ, such as, δ = (1,6). Finally, all the
mechanism are implemented in Python 3 and the ex-
periments are performed on Intel Xeon 3.6GHz 6 core
processor with 32 GB RAM.
From Figure 3, it can be observed the procurement
cost is minimum for DMMP mechanism as compared
to the other two mechanisms. An interesting observa-
tion here is, initially, procurement cost decreases with
the increase in value of δ. However, later the procure-
ment cost increases with the increase in value of δ.
For instance, at δ = 3, procurement cost is the least,
but at δ = 5 procurement cost rises. This is possibly
because of the increase in the number of nodes in the
winning path, which leads to an increase in the to-
tal rewards distributed. Overall, the experimental re-
sults highlight that the novel DMMP mechanism out-
performs the other two mechanisms and demonstrate
its efficiency for procurement of multi-unit multi-item
resources through economic networks.
6 CONCLUSION
In this research, we present a novel information
diffusion-based resource allocation mechanism in
economic networks. In specific, we consider the eco-
nomic networks where independent buyers submit
their multi-unit multi-item resource requests to mul-
tiple independent sellers. In this regard, the proposed
mechanism aids the buyers to procure the required re-
sources from a group of distant sellers with the min-
imum possible prices. Besides, the proposed mecha-
nism encourages the independent sellers to share their
available resources amongst each other. Also, the pro-
posed DMMP mechanism guarantees that every seller
receives an incentive to reveal their truthful type and
invite all their neighbouring sellers to participate in
the procurement. In this context, rewards are given
to all the sellers in the winning path for diffusing in-
formation. Most importantly, those rewards do not
increase the buyer’s payment. In fact, the buyer’s pay-
ment is even improved as compared to the VCG mech-
anism. As for future work, we plan to focus on mul-
tiple buyers with multi-unit heterogeneous resource
combinatorial auctions in economic networks.
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