Assigning Business Processes: A Game-Theoretic Approach
Evangelos D. Spyrou and Dimitris Mitrakos
School of Electrical and Computer Engineering, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece
Keywords:
Business process, market, assignment game, characteristic function, coalition formation.
Abstract:
Business processes are essential for the successful growth of an organisation. Business models aim to organise
such processes and invoke the necessary processes for particular tasks. Such a mechanism is responsible for the
appropriate response and time management of the business strategy of a corporation. To this end, we formulate
a scenario where business processes need to be invoked according to the expenditures of their operation, in
order to complete a transaction in a market model. We model such a mechanism using game theory and we
produce a characteristic function that we maximise, in order to reduce the cost of the business processes.
We employ the well-known assignment game to form business process coalitions and minimise the business
operation cost in the market.
1 INTRODUCTION
A current trend in the business oriented research is
the emergence of business intelligence, since it re-
flects on real problems that businesses deal and their
respective solutions (Chen et al., 2012). In particu-
lar, bid data has shown specific trends that businesses
follow in their market domains (Minelli et al., 2012).
Bid take place in a real-time fashion; hence, real-time
strategic interactions constitute a major issue in busi-
ness intelligence. Such decisions may introduce enor-
mous consequences to the development of a business;
thus, uncertainties must be taken into serious con-
siderations by managers. A risk-minimum estimate
needs to be identified, in order to proceed with busi-
ness process deployment. This belongs to a class of
problems named strategic business planning (Sonteya
and Seymour, 2012).
Business analysts face unique issues when at-
tempting to address a specific strategic problem. De-
cision time is usually limited. The potential of a
wrong decision increases the cost and overall conse-
quences of a business plan. In most of the cases, real-
time strategic planning is associated with cost. Hence,
a business analyst needs to have expert knowledge, in
order to act in a timely fashion and address the busi-
ness problem by keeping the cost at reasonable levels.
Business planning can be thought of a complex sys-
tem (Snowden and Boone, 2007); hence, often, de-
bates in meeting rooms are at hand, in order to come
up with the ideal strategic plan. Business processes
can assist to such a problem by trying to automate re-
sponses to real world business problems (Scheer et al.,
2004).
Game theory utilises models of conflict and coop-
eration (Von Neumann et al., 2007), between business
processes in this paper. Aumann (Aumann and Dreze,
1974) provides the difference between cooperative
and non-cooperative games. We deal with the class of
cooperative games in this paper. Agreements between
players can take place before the start of a cooperative
game. The cooperation in such games is given by a
set of players a set of strategies, and a set of payoffs
that represent the outcome of the strategies played, in
the form of a utility. A coalition is characterised by
the achievement of the coordination of the members’
strategies (Saad et al., 2009), (Curiel, 1988). As we
read in (Weber, 1994), should we consider the busi-
ness process assignment as a market game, a business
process not participating in the coalition does not af-
fect the trading within the coalition. Thus, the cor-
responding strategy profiles and their respective utili-
167

ties define the characteristic function of our game for-
mulation. Furthermore, when considering a Cournot
game in which business processes will select quanti-
ties, the outcome of the coalition depends on the be-
haviour assumption of the business processes outside
the coalition.
We aim to construct a cooperative model and de-
fine a characteristic function in order to maximise
the efficiency of a business process network. There
has been other research works that dealt with busi-
ness issues in a game theoretic and cooperative man-
ner (Binmore and Vulkan, 1999), (Yahyaoui, 2012),
(Katsanakis and Kossyva, 2012), (Li et al., 2002),
(Yasir et al., 2010). We encapsulate the characteris-
tic function to indicate the formed coalitions between
business processes. Games that involve forming of
coalitions are distinguished between the ones which
include transferable utilities and the ones that with
non-transferable utilities. In the former, utility is di-
vided within a coalition and in the latter, it is difficult
to show what the utility can define when a coalition is
formed.
In this paper, we address the assignment of busi-
ness processes to operational business processes that
need to be executed with time constraints. We
produce a cooperative coalition formation game-
theoretic model and we solve it to provide the optimal
business process assignment. Specifically we show
the following contributions:
• We build a game theoretic model of the business
process assignment problem
• We construct a characteristic function based on
the time/cost for an execution of a business pro-
cess
• We show that it may be more efficient for a busi-
ness process to form a coalition with another pro-
cess to fulfill a business process if the time spent
is less than the direct business process assignment
• We show the distribution of the time of comple-
tion with our model
This paper is structured as follows: Section 2 pro-
vides the method of our game-theoretic formulation,
section 3 gives results on a specific scenario and sec-
tion 4 provides the conclusions of our approach.
2 COOPERATIVE BUSINESS
PROCESS ASSIGNMENT
We consider a simple market model where business
processes may be assigned to intermediate (relay)
business processes within a time frame, in order to
save cost. Initially, we produce a cooperative model
for the relay business process selection problem and
we attempt to distribute the saved completion time
between different business processes. This will be
accomplished once we manage the completion time
appropriately. To this end, we propose a sellers - buy-
ers approach based on (Shapley and Shubik, 1971) to
solve the completion time distribution issue. We de-
fine the coalition formation game as an ordered pair
< N, φ >, where N = 1, 2, 3...N is the set of business
processes and φ is the value of the characteristic func-
tion, which is given over 2
n
possible coalitions of N.
Also, note that φ(
/
0) = 0. If we have a set of all busi-
ness processes in a coalition, then we can claim that
we have formed a grand coalition. On the other hand,
if our set does not include the entire set of the business
processes, the resulting subset is called coalition. The
price that a coalition C is worth, is obtained by the
value of the characteristic function φ(C). This value
constitutes the maximum common payoff of the busi-
ness processes in C upon cooperation. We denote the
T value that a source business process j requires to
complete by allocating it to another business process
further up the process stack k by T
jk
. Also, denote the
time demand of completion time required for a buyer
business process be allocated to a seller business pro-
cess by T
ji
. Furthermore, denote the completion time
value of the i
th
seller to her own offer to the comple-
tion time saving of the cooperative coalition by c
i
and
the the value of the j
th
buyer to the cooperation of the
i
th
seller by T
i j
. We denote the value required by the
relay business process to reach the target next busi-
ness process as T
f rwd
. At this point, we assume that
c
i
= T
f rwd
, since a seller business process has T
f rwd
amount of completion time, favouring its correspond-
ing source business process. Note that the sellers pay-
off will not be maximised unless its time of comple-
tion is compensated. On the contrary, the cooperation
between a buyer and a seller and the completion time
that will be saved constitutes the requirement of each
buyer to form a coalition with a seller business pro-
cess. Therefore we calculate T
i j
as
T
i j
= T
jk
− T
ji
(1)
if T
ji
> c
i
, then a value exists that both the sellers
and buyers select. When we refer to time of business
process completion saving, we mean the selection of
the best completion time required to finish the process
and move to the next business process of the business
plan network.
2.1 The Characteristic Function
We define the characteristic function φ(C) as the max-
imum completion time saving that the business pro-
Seventh International Symposium on Business Modeling and Software Design
168

cesses accomplish by cooperating between them, in
order to form a coalition. Let B and S denote the buy-
ers and the sellers respectively. If there is only one
business process present in the coalition, the business
process has no cooperation process; hence, its prefer-
ence is the direct transmission to the business process
with the less completion time T . Thus, we have
φ(C) = 0 if |C| = 0 or 1 (2)
since, there is no improvement we have
φ(C) = 0 if (C ∩ B =
/
0) or (C ∩ S =
/
0) (3)
In order to further explain equation (3), for a connec-
tion efficiency improvement to take place, a coalition
that consists of buyers and sellers business processes
must be established. Hence, we will separate business
processes into buyer and seller pairs respectively. Be-
fore we move into business process assignment, we
provide the simplest form of coalition, which is given
below:
σ
i j
= max[0, T
i j
− c
i
] if i ∈ B and j ∈ N (4)
Equation (4) states that cooperation between a buyer
and a seller business process will be instantiated only
if the direct connection requires a larger completion
time than the cooperative connection. In the case
that the direct connection is better than the cooper-
ative connection, σ
i j
= 0. Our aim is to calculate the
function φ for reasonably large coalitions; hence, we
are trying to identify the best buyers assignments to
the respective sellers business processes, which max-
imize the time efficiency and minimise the cost. This
is represented as
φ(C) = max[σ
i
1
, j
1
+ σ
i
2
, j
2
+ ... + σ
i
n
, j
n
] (5)
where n = min[|C ∩ B|, |C ∩ S|]. We maximize (5)
through all the arrangements of the players i in C ∩
B and j in C ∩ S. As we can see, we can formulate
the assignment game as a linear programming (LP)
problem. Let mn be a set of binary decision variables
that satisfy
x
i j
=
(
1, if i relay process assigned to process j
0, otherwise
(6)
where i = 1, 2, 3...m and j = 1, 2, 3...n. Each bi-
nary variable indicates whether a business process i
acting as a relay process will be allocated to a busi-
ness process j that is wishes to execute.
We denote as ξ the total time saving of the coop-
erative coalition formation and we formulate the busi-
ness process relay selection problem as an LP.
Maximize ξ =
∑
i∈B
∑
j∈S
σ
i j
x
i, j
(7)
s.t
∑
i∈B
x
i j
≤ 1 for i = 1, 2, ...m
∑
i∈S
x
i j
≤ 1 for i = 1, 2, ...n
The first constraint states that each relay business pro-
cess may be assigned to at most one business process.
The second constraint specifies that every business
process has to be connected to at least one relay busi-
ness process. Solving this LP problem will give us
the maximum completion time saved when a coalition
is formed of B relay business processes and S source
business processes. Hence, we have
ξ
max
= φ(B ∪ S) (8)
Thereafter, we transform the LP problem to its equiv-
alent matrix formulation.
maxc
T
x
s.t. A ·x ≤ b
x ≥ 0
(9)
Note that the constraint x ≤ 1 has been folded into the
constraint A ·x ≤ b
2.2 Business Process Selection Core
We proceed to the core of the coalition, which should
not be empty or consisting of one business process
only. According to Shapley and Shubik (Shapley and
Shubik, 1971), the core of the relay selection game
is the set of solutions of the dual LP problem of the
assignment problem. In this paper, we introduce the
Lagrangian dual. We take the nonnegative Lagrangian
multipliers (y, λ) to the constraints Ax ≤ b and x ≥ 0
as follows
L (x, y, λ) = c
T
x +y
T
(b −Ax) + λ
T
x (10)
which serves as an upper bound of the characteristic
function (9), whenever x is feasible or not. Therefore,
max
x
L (x, y, λ) bounds the optimum of (9). In order
to obtain the upper bound, we have to solve the fol-
lowing
min
y,λ
max
x
L (x, y, λ) =
min
y,λ
max
x
c
T
x +y
T
(b −Ax) + λ
T
x
(11)
We have the third equality since c − A
T
y +λ 6= 0 and
we may select an appropriate x such that L (x, y, λ)
goes to infinity. Therefore, we have a finite bound
when c −A
T
y + λ = 0. By taking the strong duality
of the linear program, the optimum of (9) coincides
with (Boyd and Vandenberghe, 2004). We can make
the formulation more simple by assuming that b > 0,
Assigning Business Processes: A Game-Theoretic Approach
169

since T is always positive. We denote P as a convex
set on x and f
i
(x) = (i = 1, ...m) as a set of convex
functions. Moreover, we define the general min-max
problem as
min
x∈P
max
i∈[k]
f
k
(x) (12)
where [k] = {1, ...m} is a set of indexes. For detailed
report on the solving method of this problem, we refer
the reader to (Spyrou and Mitrakos, 2017).
We have to mention that the dual problem consists
of m +n variables
y = [q
1
, ...q
m
, r
1
, ...., r
n
]
Moreover, we can see that the constraints of the prob-
lem are
q
i
+ r
j
≥ σ
i j
∀i ∈ B and ∀ j ∈ S (13)
Essentially, the solution of min-max problem is equiv-
alent to the solution of φ(B ∪ S). The remark above
dictates the incentive of relay and source business
processes to cooperate. Specifically, q
i
and r
j
com-
prise the T values that a relay business process i and a
source business process j receive, in order to perform
a cooperative transmission. Furthermore, the vector
y = [q
1
, ...q
m
, r
1
, ...., r
n
] provides the distribution of
the T enhancement and the equivalence of the dual
problem with the solution of φ(B ∪ S) constitutes an
imputation of the coalition formation relay selection
game. Additionally, from (5) and (13)
∑
i∈C∩B
q
i
+
∑
C∈S
r
i
≥ φ(C), ∀C ⊂ S (14)
Thus, we defined the core of the relay selection game
using (12) and (14),since we encapsulate the imputa-
tion efficiency and the fact that an improvement move
on the coalition cannot be made.
2.2.1 Completion Time Distribution
The enhancement of completion time T is shared be-
tween the business processes. On the other hand, the
T completion time enhancement cannot be transferred
to the relay business processes, unless the source busi-
ness processes receive the value first; however, ev-
ery source business process my be a relay business
process; thus, transferring the completion time T en-
hancement. Thereafter, we construct the T func-
tion, distinguished between the cooperative and self-
ish business processes and prevents any undesired be-
havior. The T function of the relay business process i
at a cooperative scenario is given by
T
i
[n] = T
i
[n −1] + c
i
+ q
i
(15)
Note that c
i
is the T completion time received by
the relay business process i and q
i
is the amount of
T available for the relay business process to proceed
with a cooperation. On the other hand, the T com-
pensation of the relay business process needs to be
provided by the source business process. Thus, the T
function of source business process j is given by
T
j
[n] = T
j
[n −1] − c
j
− q
j
(16)
Note that, even after transferring c
i
+ q
i
amount of
completion time T to the relay business process i, the
source business process j will have φ
j
of T enhance-
ment.
3 RESULTS
We consider a simple market and we take two sce-
narios on board; the first scenario is a single source
process - relay business process and the second is a
multiple source - relay business processes scenario.
3.1 Single Source and Relay Business
Process
As we can see in figure 1, the source business pro-
cess executes by reaching the end-business process
directly, since its incentive is not to cooperate with the
relay business process. This is the case due to the fact
that the completion time of going directly to the end-
business process is less than the completion time after
forming the coalition with the relay business process.
On the other hand, in figure 2, we see that the
source process forms a coalition with the relay busi-
ness process, since it is in its benefit to cooperate,
since it saves completion time. This is only the sim-
plest scenario. We investigate thoroughly a more
complicated scenario with a network of business pro-
cesses. In both figures the solid lines represent the
preferences of the source business process, while the
dashed lines the discarded choices.
Figure 1: Source Business Process Direct communication
with End-Business Process.
Seventh International Symposium on Business Modeling and Software Design
170

Figure 2: Source Business Process Direct forming Coalition
with Relay Business Process.
3.2 Multiple Source and Relay Business
Processes
The objective is to maximise the time completion of
the business processes that will be formed after the
cooperation and the coalitions formed. In our sce-
nario, business process 1 is the business process that
the business plan leads to, business processes 2 − 4
are the relay business processes and business pro-
cesses 6 − 8 are the source business processes. The
source processes do not form coalitions with all the
relay business processes. In particular, source busi-
ness processes 6 and 8 form 3 coalitions respectively
and source business process 7 establishes 4 coalitions
respectively. These coalitions include a connection to
the end business process, in order to show the differ-
ence in the T value and the necessity of cooperation
between the business processes. The respective com-
pletion time values reside in table 1. Finally the com-
pletion time values between the relay business pro-
cesses 2, 3, 4 and the end-business process are 1, 1.5, 3
respectively.
Table 1: Network connections and T values.
Source Process Seller Connections T
6 1,2,3 23,20,16
7 1,2,3,4 16,12.5,11,12
8 1,3,4 21,15.10,14
Thereafter, we provide the reader with the val-
uations of the business processes (sellers and buy-
ers). Notably, the sellers’ valuations are calculated
by simply obtaining the completion time values re-
quired to transmit operate a business process to busi-
ness process 1. On the other hand, the valuations of
the buyers are estimated by the result of equation (1),
which gives us the difference of completion time be-
tween the one-hop connection of each buyer with the
end business process and the completion time of each
buyer with a seller. Note that when a connection does
not exist between a seller and a buyer, we set the re-
sult of (1) as 0. We provide the valuations in the table
2, which is counted in days of completion time.
Table 2: Business Processes and Completion Time
Values.
Source Seller Val Buyer Val
(i) (c
i
) T
i6
T
i7
T
i8
2 1.5 3 3.5 0
3 2 7 5 6.5
4 2 0 4 7
The result of the assignment of the most appropri-
ate seller to a buyer is the enhancement of completion
time value per pair. The outcome of the game is a
profit matrix that shows the resulting completion time
enhancement from coalitions between source and re-
lay business processes. We provide this information
in table 3. Furthermore we highlight the optimal
assignment between relay and source business pro-
cesses in bold numbers, which indicate the maximi-
sation of the completion time enhancement for each
pair that form a coalition.
We are able to identify the core solution of the
problem by solving the dual LP problem described
in the previous section. However, we are not in the
position to claim that the core solution is unique. One
of the core solutions obtained, which is close to the
Shapley value (Shapley, 1988)
y = [2.1708 0.95 2.55 0.55 3.30 2.44] (17)
Furthermore, we can derive from Table 1 that the
completion time required for a direct communication
of the buyer business process with the destination -
business process 1 - is 45. On the other hand, the suc-
cessful cooperation between the sellers and the buyers
is 17.5, which gives us an completion time enhance-
ment of 68.1%. In order to accomplish that we need
to connect business process 6 with business process
2, business process 8 with business process 3 and fi-
nally, business process 7 with business process 4. We
Table 3: Completion Time Enhancement.
Buyers
6 7 8
Sellers 2 1.5 2 0
3 5 3 4.5
4 0 2 5
Assigning Business Processes: A Game-Theoretic Approach
171

can see the final configuration of the business process
selection process in figure 3.
Figure 3: Final Transmission Configuration.
4 CONCLUSIONS
In this paper we attempted to approach the collabo-
ration between business processes in order to accom-
plish tasks of a business plan. To that end, business
processes either connect directly with the business
process they require to finish the task or they form
coalitions by finding a relay business process to con-
nect to, depending on the business process completion
time.
Subsequently, the business processes establish a
cooperative network in a game theoretic manner. Our
model is based on combinatorial optimisation, which
target the maximisation of the completion time en-
hancement when a relay and a source business pro-
cess cooperate. We derived the characteristic function
used in our game, the coalition core and the credit that
each business process has for playing the relay selec-
tion game. We evaluated a simple and a more com-
plicated scenario, which indicated the fact that using
cooperative business process cooperation the process
network exhibits a better completion time. This is due
to the fact that each source business process gets as-
signed to the relay business process that has the best
completion time enhancement.
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