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Incentive Alignment of Business Processes


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Many definitions of business processes refer to business goals, value creation, profits, etc. Nevertheless, the focus of formal methods research on business processes lies on the correctness of the execution semantics of models w.r.t. properties like deadlock freedom, liveness, or completion guarantees. However, the question of whether participants are interested in working towards completion-or in participating in the process at all-has not been addressed as of yet. In this work, we investigate whether inter-organizational business processes give participants incentives for achieving the business goals: in short, whether incentives are aligned within the process. In particular, fair behavior should pay off and efficient completion of tasks should be rewarded. We propose a game-theoretic approach that relies on algorithms for solving stochastic games from the machine learning community. We describe a method for checking incentive alignment of process models with utility annotations for tasks, which can be used for a priori analysis of inter-organizational business processes. Last but not least, we show that the soundness property is a special case of incentive alignment.
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Incentive Alignment of Business Processes
Tobias Heindel[0000000333718564] and Ingo Weber[0000000248335921]
Chair of Software and Business Engineering, Technische Universitaet Berlin, Germany
Many definitions of business processes refer to business goals,
value creation, profits, etc. Nevertheless, the focus of formal methods
research on business processes lies on the correctness of the execution
semantics of models w.r.t. properties like deadlock freedom, liveness, or
completion guarantees. However, the question of whether participants
are interested in working towards completion – or in participating in the
process at all – has not been addressed as of yet.
In this work, we investigate whether inter-organizational business pro-
cesses give participants incentives for achieving the business goals: in
short, whether incentives are aligned within the process. In particular,
fair behavior should pay off and efficient completion of tasks should be
rewarded. We propose a game-theoretic approach that relies on algorithms
for solving stochastic games from the machine learning community. We
describe a method for checking incentive alignment of process models
with utility annotations for tasks, which can be used for a priori analysis
of inter-organizational business processes. Last but not least, we show
that the soundness property is a special case of incentive alignment.
inter-organizational business processes
incentive align-
ment ·collaboration ·choreography
1 Introduction
Many definitions of what a business process is refer to business goals [
] or
value creation [
], but whether process participants are actually incentivized to
contribute to a process has not been addressed as yet. For intra-organizational
processes, this question is less relevant; motivation to contribute is often based
on loyalty, bonuses if the organization performs well, or simply that tasks in a
process are part of one’s job. Instead, economic modeling of intra-organizational
processes often focuses on cost, e.g. in activity-based costing [
], which can be
assessed using model checking tools [9] or simulation [5].
For inter-organizational business processes, such indirect motivation cannot
be assumed. A prime example of misaligned incentives was the $2.5B write-off
in Cisco’s supply chain in April 2001 [
]: success of the overall supply chain
was grossly misaligned with the incentives of individual participants. (This hap-
pened despite the availability of several game theoretic approaches for analyzing
incentive structures for the case of supply chains [
].) Furthermore, modeling
incentives accurately is actually possible in cross-organizational processes, e.g.,
2 Tobias Heindel and Ingo Weber
based on contracts and agreed-upon prices. With the advent of blockchain tech-
nology [
], it is possible to execute cross-organizational business processes or
choreographies as smart contracts [
]. The blockchain serves as a neutral,
participant-independent computational infrastructure, and as such enables col-
laboration across organizations even in situations characterized by a lack of trust
between participants [
]. However, as there is no central role for oversight, it is
important that incentives are properly designed in such situations, e.g., to avoid
unintended –possibly devastating– results, like those encountered by Cisco. In
fact, a main goal of the Ethereum blockchain is, according to its founder Vitalik
Buterin, to create “a better world by aligning incentives”1.
In this paper, we present a framework for incentive alignment of inter-
organizational business processes based on game theory. We consider bpmn
models with suitable annotation concerning the utility
of activities, very much
in the spirit of activity-based costing (abc) [
, Chapter 5]. In short, fair behavior
should pay off and participants should be rewarded for efficient completion of
process instances. In more detail, we shall consider bpmn models as stochastic
games [
] and formalize incentive alignment as “good” equilibria of the resulting
game. Which equilibria are the desirable ones depends on the business goals
w.r.t. which we want align incentives. In the present paper, we focus on proper
completion and liveness of activities. Interestingly, the soundness property [
will be rediscovered as the special case of incentive alignment within a single
organization that rewards completion of every activity.
The overall contribution of the paper is a framework for incentive alignment
of business process models, particularly in inter-organizational settings. Our
approach is based on game theory and inspired by advances on the solution of
stochastic games from the machine learning community, which has developed
algorithms for the practical computation of Nash [
] and correlated equilib-
ria [
]. The framework focuses on checking incentive alignment as an a priori
analysis of business processes specified as bpmn models with activity-based utility
annotations. Specifically, we:
describe a principled method for translating bpmn-models with activity-based
costs to stochastic games [24]
propose a notion of incentive alignment that we prove to be a conservative
extension of Van der Aalst’s soundness property [2],
3. illustrate the approach with a simplified order-to-cash (o2c) process.
We pick up the idea of incentive alignment for supply chains [
] and set out to
apply it in the realm of inter-organizational business processes. From a technical
point of view, we are interested in extending the model checking tools for cost
analysis [
] for bpmn process models to proper collaborations, which we model as
stochastic games [
]. This is analogous to how the model checker prism has been
extended from Markov decision processes to games [
]. We keep the connection
with established concepts from the business process management community by
1, accessed 8-3-2020
2We shall use utility functions in the sense of von Neumann and Morgenstern [19].
Incentive Alignment of Business Processes 3
showing that incentive alignment is a conservative extension of the soundness
property (see Theorem 1). Our approach hinges on algorithms [
] for solving
the underlying stochastic games of bpmn process models, which are sufficient for
checking incentive alignment.
The remainder of the paper is structured as follows. We introduce concepts
and notations in Section 2. On this basis, we formulate two versions of incentive
alignment in Section 3. Finally, we draw conclusions in Section 4. The proof of
the main theorem can be found in the extended version [8].
2 Game theoretic concepts and the Petri net tool chest
We now introduce the prerequisite concepts for stochastic games [
] and ele-
mentary net systems [
]. The main benefit of using a game theoretic approach
is a short list of candidate definitions of equilibrium, which make precise the
idea of a “good strategy” for rational actors that compete as players of a game.
We shall require the following two properties of an equilibrium: (1) no player
can benefit from unilateral deviation from the “agreed” strategy and (2) players
have the possibility to base their moves on information from a single (trusted)
mediator. The specific instance that we shall use are correlated equilibria [
as studied by Solan and Vieille [
We take ample space to review the latter
two concepts, followed by a short summary of the background on Petri nets.
We use the following basic concepts and notation. The cardinality and the
powerset of a set
are denoted by
, respectively. The set of real
numbers is denoted by
and [0
is the unit interval. A probability
distribution over a finite or countably infinite set
is a function
whose values are non-negative and sum up to 1, in symbols
The set of all probability distributions over a set Mis denoted by (M).
2.1 Stochastic games, strategies, equilibria
We proceed by reviewing core concepts and central results for stochastic games [
introducing notation alongside; we shall use examples to illustrate the most im-
portant concepts. The presentation is intended to be self-contained such that no
additional references should be necessary. However, the interested reader might
want to consult standard references or additional material, e.g., textbooks [
handbook articles [11], and surveys [26]. We start with the central notion.
Definition 1 (Stochastic game).
Astochastic game
is a quintuple
hN, S, A, q, uithat consists of
a finite set of players N={1,...,|N|} (ranged over by i, j, in, etc.);
a finite set of states S(ranged over by s, s0, sn, etc.);
Nash equilibria are a special case, which however have drawbacks that motivate
Aumann’s work on the more general correlated equilibria [3,10].
4 Tobias Heindel and Ingo Weber
check stock
pick up deliver
payment success
no success
pick up return
write off
payment sold
damage report
OK damaged
Fig. 1. A a simplified order-to-cash process
a finite, non-empty set of action profiles
i=1 Ai
(ranged over by
a, an
etc.), which is the Cartesian product of a player-indexed family
, each of which contains the actions of the respective player (ranged
over by ai, ai
n, etc.);
a non-empty set of available actions
, for each state
player i;
probability distributions
· | s, a
), for each state
and every
action profile
, which map each state
s0|s, a
), the transition
probability from state sto state s0under the action profile a; and
the payoff vectors
s, a
) =
s, a
, . . . , u|N|
s, a
, for each state
and every action profile a=ha1, . . . , a|N|i ∈ A.
Note that players always have some action(s) available, possibly just a dedicated
idle action, see e.g. [13].
The bpmn model of Fig. 1can be understood as a stochastic game played by
a shipper, a customer, and a supplier. Abstracting from data, precise timings,
and similar semantic aspects, a state of the game is a state of an instance of
the process, which is represented as a token marking of the bpmn model. The
actions of each player are the activities and events in the respective pool, e.g., the
Incentive Alignment of Business Processes 5
ship task, which Supplier performs after receiving an order from the Customer
and payment of the postage fee to Shipper. Action profiles are combinations
of actions that can (or must) be executed concurrently. For example, sending
the order and receiving the order after the start of the collaboration may be
performed synchronously (e.g., via telephone). The available actions of a player in
a given state are the tasks or events in the respective pool that can be executed
or happen next – plus the idle action. The transition probabilities for available
actions in this bpmn process are all 1, such that if players choose to execute
certain tasks next, they will be able to do so if the chosen activities are actually
available actions. As a consequence, all other transition probabilities are 0.
One important piece of information that we have to add to a bpmn model
via annotations is the utility of tasks and events. In analogy to the abc method,
which attributes a cost to every task, we shall assume that each task has a certain
utility for every role – and be it just zero. Utility annotations are the basis for the
subsequent analysis of incentive alignment, vastly generalizing cost minimization.
Note that, in general, it is non-trivial to chose utility functions, especially in
competitive situations. However, the o
cprocess comes with natural candidates
for utilities, e.g., postage fees can be looked up from one’s favorite carrier, the
cost for gas, maintenance, and personnel for shipping is fairly predictable, and
finally there is the profit for selling a good.
A single instance of the o
cprocess exhibits the phenomenon that Customer
has no incentive to pay. However, we want to stress that – very much for the same
reason – Shipper would not have any good reason to perform delivery, once the
postage fee is paid. Thus, besides the single instance scenario, we shall consider
an unbounded number of repetitions of the process, but only one active process
instance at each point in time.
In the repeating variant, the rational reason
for the shipper to deliver (and return damaged goods) is expected revenue from
future process instances.
One distinguishing feature of the o
ccollaboration is that participants do not
have to make any joint decisions. Let us illustrate the point with another example.
Alice and Bob are co-founders of a company, which is running so smoothly that
it suffices when, any day of the week, only one of them is going to work.
Alice suggests that their secretary Mrs. Medina could help them out by rolling
a 10-sided die each morning and notifying them about who is going to go to work
that day, dependent on whether the outcome is smaller or larger than six. This
elaborate process (as shown in Fig. 2), lets Bob work 60% and Alice 40% of the
days, respectively. Alice’s reasoning behind it is the observation that Alice is 50%
more efficient than Bob when it comes to generating revenue, as indicated by the
amount of $ signs in the process.
In game theoretic terminology, Mrs. Medina is taking the role of a common
source of randomness that is independent of the state of the game and does not
need to observe the actions of the players. The specific formal notion that we
shall use is that of an autonomous correlation device [25, Definition 2.1].
We leave the very interesting situation of interleaved execution of several process
instances for future work.
6 Tobias Heindel and Ingo Weber
Mrs. Medina
Fig. 2. The To work or not to work? collaboration
Definition 2 (Autonomous correlation device).
An autonomous correla-
tion device is a family of pairs
n}iN, dninN
(that is indexed over
natural numbers nN) each of which consists of
a family of finite sets of signals
, (additionally) indexed over players; and
a function
that maps lists of signal vectors
hx1, . . . , xn1i ∈ Qn1
k=1 Mk
to probability distributions
dnhx1, . . . , xn1i ∈
)over the Cartesian
product Mn=Q|N|
i=1 Mi
nof all signal sets Mi
We shall refer to operators of autonomous correlation devices as mediators, which
guide the actions of players during the game.
Each correlation device for a game induces an extended game, which proceeds
in stages. In general, given a game and an autonomous correlation device, the
th stage begins with the mediator drawing a signal vector
xn∈ Mn
i=1 Mi
according to the device distribution
dnhx1, . . . , xn1i
– e.g., Mrs. Medina rolling
the die – and sending the components to the respective players – the sending of
messages to Bob and Alice (in one order or the other). Then, each player
an available action
. This choice can be based on the respective component
of the signal vector
xn∈ Mn
, information about previous states
of the
, and moves
of (other) players from the history.
After all players
made their choice, we obtain an action profile an=ha1
n, . . . , a|N|
While playing the extended game described above, each player makes obser-
vations about the state and the actions of players; the role of the mediator is
special insofar as it does not need and is also not expected to observe the run of
In the present paper, we only consider games of perfect information, which is suitable
for business processes in a single organization or which are monitored on a blockchain.
Incentive Alignment of Business Processes 7
the game. The “local” observations of each player are the basis of their strategies.
Definition 3 (Observation, strategy, strategy profile).
An observation at
stage nby player iis a tuple h=hs1, xi
1, a1, . . . , sn1, xi
n1, an1, sn, xi
one state sk, signal xi
k, and action profile ak, for each number k < n,
the current state sn, also denoted by sh, and
the current signal xi
The set of all observations is denoted by
). The union
) =
of observations at any stage is the set of observations of player
. A strategy is
a map
)from observations to probability distributions over
actions that are available at the current state of histories, i.e.,
), for all histories
). A strategy profile is a player-indexed
family of strategies {σi}iN.
Thus, each of the players observes the history of other players, including the
possibility of punishing other players for not heeding the advice of the mediator.
This is possible since signals might give (indirect) information concerning the
behavior of players in the past, as remarked by Solan and Vieille [
, p. 370]:
by revealing information about proposed actions of previous rounds, players can
check for themselves whether some player has ignored some signal of the mediator.
The data of a game, a correlation device, and a strategy profile induce
probabilities for finite plays of the game, which in turn determine the expected
utility of playing the strategy. Formally, an autonomous correlation device and a
strategy profile with strategies for every player yield a probabilistic trajectory of
a sequence of “global” states, signal vectors of all players, and complete action
profiles, dubbed history. The formal details are as follows.
Definition 4 (History and its probability).
Ahistory at stage
is a tuple
h=hs1, x1, a1, . . . , sn1, xn1, an1, sn, xnithat consists of
one state
, signal vector
, and action profile
, for each number
k < n
the current state sn, often denoted by sh, and
the current signal vector xn.
The set of all histories at state
is denoted by
). The union
) =
)of histories at arbitrary stages is the set of finite histories. The
probability of a finite history
hs1, x1, a1, . . . , sn1, xn1, an1, sn, xni
in the
context of a correlation device
, an initial state
, and a strategy profile
defined as follows, by recursion over the length of histories.
n= 1:PD,s,σ (hs1, x1i) = (0if s6=s1
n > 1:PD,s,σ (h~, an1, sn, xni) = ph~i(an1)
| {z }
q(sn|sn1, an1)pdn1(xn)
| {z }
8 Tobias Heindel and Ingo Weber
Again, note that the autonomous correlation device does not “inspect” the states
of a history, in the sense that the distributions over signal vectors
are not
parameterized over states from the history, but only over previously drawn signal
vectors – whence the name.
Definition 5 (Mean expected payoff).
The mean expected payoff of player
for stage
D, s, σ
) =
PD,s,σ (h)
k=1 ui
sk, ak
hs1, x1, a1,, sn+1, xn+1 i.
At this point, we can address the question of what a good strategy profile
is and fill in all the details of the idea that an equilibrium is a strategy profile
that does not give players any good reason to deviate unilaterally. We shall tip
our hats to game theory and use the notation (
πi, σi
)for the strategy profile
which is obtained by “overwriting” the single strategy
of player
with a
(which might, but does not have to be different); thus, the expression
πi, σi
)’ denotes the unique strategy subject to equations (
πi, σi
(πi, σi)j=σj(for i6=j).
Definition 6 (Autonomous correlated ε-equilibrium).
Given a positive
ε >
0, an autonomous correlated
-equilibrium is a pair
hD, σi
, which
consists of an autonomous correlation device
and a strategy profile
for which
there exists a natural number
such that for any alternative strategy
any player i, the following inequality holds, for all nn0and all states sS.
n(D, s, σ)¯γi
nD, s, (σi, σ∗−i)ε(1)
Thus, a strategy is an autonomous correlated
-equilibrium if the benefits that
one might reap in the long run by unilateral deviation from the strategy are
negligible as
can be arbitrarily small. In fact, other players will have ways to
punish deviation from the equilibrium [25, § 3.2].
2.2 Petri nets and their operational semantics
We shall use the definitions concerning Petri nets that have become established
in the area of business processes management [2].
Definition 7 (Petri net, marking, and marked Petri net).
APetri net is
a triple N= (P, T , F)that consists of
a finite set of places P;
a finite set of transitions
that is disjoint from places, i.e.,
; and
a finite set of arcs F(P×T)(T×P)(a.k.a. the flow relation).
An input place (resp. output place) of a transition
is a place
p, t
(resp. (
t, p
). The pre-set
(resp. post-set
) of a transition
is the set of all input places (resp. output places), i.e.,
t={pP|pis an input place of t}t={pP|pis an output place of t}.
Incentive Alignment of Business Processes 9
Amarking of a Petri net
is a multiset of places
, i.e., a function
that assigns to each place
a non-negative integer
0. A marked Petri
net is a tuple
= (
P, T , F, m0
)whose first three components (
P, T , F
)are a
Petri net and whose last component
is the initial marking, which is a marking
of the latter Petri net.
One essential feature of Petri nets is the ability to execute several transitions
concurrently – possibly several occurrences of one and the same transition.
However, we shall only encounter situations in which a set of transitions fires. To
avoid proliferation of terminology, we shall use the general term step. We fix a
Petri net N= (P, T, F )for the remainder of the section.
Definition 8 (Step, step transition, reachable marking).
Astep in the
is a set of transitions
. The transition relation of a step
a marking
to another marking
, in symbols
, if the following two
conditions are satisfied, for every place pP.
1. m(p)≥ |{tt|pt}|
2. m0(p) = m(p)− |{tt|pt}| +|{tt|pt}|
We write
holds for some step
and denote the reflexive
transitive closure of the relation
. A marking
is reachable in a marked
Petri net N= (P, T , F, m0)if m0[im0holds, in the net (P, T, F).
For a transition
, we write
instead of
m[{t}i m0
. Thus the empty
step is always fireable, i.e., for each marking
, we have an “idle” step
Recall that a marked Petri net
= (
P, T , F, m0
)is safe if all reachable
have at most one token in any place, i.e., if they satisfy
for all
. Thus, a marking
corresponds to a set
0; for convenience, we shall identity a safe marking
with its set
of places
. The main focus will be on Petri nets that are safe and extended
free choice, i.e., if the pre-sets of two transitions have a place in common, the
pre-sets coincide. Also, recall that the conflict relation, denoted by #, relates two
transitions if their pre-sets intersect, i.e.,
, for
t, t0T
; for
extended free choice nets, the conflict relation is an equivalence relation. We call
a marked Petri net an elementary net system [
] if all pre-sets and post-sets of
transitions are non-empty and every place is input or output to some transition.
The latter encompass the following class of Petri nets that is highly relevant to
formal methods research of business processes.
Definition 9 (Workflow net (WF-net)).
A Petri net
= (
P, T , F
)is a
Workflow net or WF-net, for short, if
there are unique places
i, o P
such that
is not an output place of any
transition and ois not an input place of any transition and
if we add a new transition
and the two arcs (
o, t
t, i
), the resulting
directed graph (PT∪ {t}, F ∪ {(o, t),(t, i)})is strongly connected.
Finally, let us recall the soundness property [1]. A Workflow net is
10 Tobias Heindel and Ingo Weber
sound if and only if the following three requirements are satisfied: (1) op-
tion to complete: for each case it is always still possible to reach the state
which just marks place end, (2) proper completion : if place end is marked
all other places are empty for a given case, and (3) no dead transitions:
it should be possible to execute an arbitrary activity by following the
appropriate route
where end is place
,each case means every marking reachable from the initial
,state means marking, marked means marked by a reachable mark-
ing, activity means transition, and following the appropriate route means after
executing the appropriate firing sequence.
3 Incentive alignment
Soundness of business processes in the sense of Van der Aalst [
] implies termina-
tion if transitions are governed by a strongly fair scheduler [
]; indeed, such a
scheduler fits the intra-organizational setting. However, as discussed for the o
process model, unfair scheduling practices could arise in the inter-organizational
setting if undesired behavior yields higher profits. We consider incentive alignment
to rule out scenarios that lure actors into counterproductive behavior. We even
can check whether all activities in a given bpmn model with utility annotations
are relevant and profitable.
As bpmn models have established Petri net semantics [
], it suffices to consider
the latter for the game theoretic aspects of incentive alignment. As a preparatory
step, we extend Petri nets with utility functions as pioneered by von Neumann
and Morgenstern [
]. Then we describe two ways to associate a stochastic game
to a Petri net with transition-based utilities: the first game retains the state space
and the principal design choice concerns transition probabilities; the second game
is the restarting version of the first game. Finally, we define incentive alignment
in formally based on stochastic games and show that the soundness property for
Workflows nets [
] can be “rediscovered” as a special case of incentive alignment;
in other words, the original meaning of soundness is conserved, and thus we
extend soundness conservatively in our framework for incentive alignment.
3.1 Petri nets with utility and role annotations
We assume that costs (respectively profits) are incurred (resp. gained) per task
and that, in particular, utility functions do not depend on the state. Note that the
game theoretic results do not require this assumption; however, this assumption
does not only avoid clutter, but also retains the spirit of the abc method [
and is in line with the work of Herbert and Sharp [9].
Definition 10 (Petri net with transition payoffs and roles).
For a set of
roles R, a Petri net with transition payoffs and roles is a triple (N, u, ρ)where
N= (P, T , F, m0)is a marked Petri net with initial marking m0,
Incentive Alignment of Business Processes 11
Fig. 3. Extending Petri nets with role and utility annotations
u:R → TRis a utility function, and
ρ:T * R
is a partial function, assigning at most one role to each transition.
The utility
)of a step
is the sum of the utilities of its elements, i.e.,
ui(t) = Pttui(t), for each role i∈ R.
As a consequence of the definition, the idle step has zero utility. We have included
the possibility that some of the transitions are not controlled by any of the roles
(of a bpmn model) by using a partial function from transitions to roles; we take
a leaf out of the game theorist’s book and attribute the missing role to nature.
Fig. 3displays a Petri net on the left. The names of the places
p1, . . . , p4
be convenient later. In the same figure on the right, we have added annotations
that carry information concerning roles, costs, and profits in the form of lists of
role-utility pairs next to transitions. E.g., the transition
is assigned to role
and firing
results in utility
, i.e., one unit of cost. The first role in
each list denotes responsibility for the transition and we have omitted entries
with zero utility. We also have colored transitions with the same color as the
role assigned to it. If we play the token game for Petri nets as usual, each firing
sequence gives cumulative utilities for each one of the roles; each transition gives
an immediate reward. These rewards will influence the choice between actions
that are performed by roles as made precise in the next subsection.
There are natural translations from bpmn models with payoff annotations for
activities to Petri nets with payoffs and roles (relative to any of the established
Petri net semantics for models in bpmn [
]). If pools are used, we take one role
per pool and each task is assigned to its enclosing pool; for pairs of sending and
receiving tasks or events, the sender is responsible for the transition to be taken.
The only subtle point concerns the role of nature. When should we blame nature
for the data on which choices are based? The answer depends on the application
at hand. For instance, let us consider the o
cmodel of Fig. 1: whether or not
the goods will be damaged during shipment is only partially within the control
of the shipper; thus, we shall blame nature for any damage or praise her if
everything went well against all odds. In a first approximation, we simply let
nature determine whether goods will arrive unscathed.
3.2 Single process instances and the base game with fair conflicts
We now describe how each Petri net with transition payoffs and roles gives rise
to a stochastic game, based on two design choices: each role can execute only
12 Tobias Heindel and Ingo Weber
one (enabled) transition at a time and conflicts are resolved in a probabilistically
fair manner. For example, for the net on the right in Fig. 3, we take four
p0, p1, p2, p3
, one for each reachable marking. The Petri net does not
prescribe what should happen if roles
both try to fire transitions
simultaneously if the game is in state
. The simplest probabilistically
fair solution consists of flipping a coin; depending on the outcome, the game
continues in state
or in state
. For the general case, let us fix a safe, extended
free-choice net (
N, u, ρ
)with payoffs and roles whose initial marking is
the marked net Nis an elementary net system (e.g., a WF-net).
Definition 11 (The base game with fair conflicts).
X ℘T
be the
partitioning of the set of transitions into equivalence classes of the conflict
relation on the set of transitions, i.e.,
t} | tT}
; its
members are called conflict sets. Given a safe marking
and a step
-enabled sub-step is a step
that is enabled at the marking
, is
contained in the step
, and contains one transition of each conflict set that has a
non-empty intersection with the step, i.e., such that all three of
|{X∈ X | tX6
hold. We write
if the step
is a maximal
m-enabled sub-step of the step t.
The base game with fair conflicts
hN, S, A, q, ui
of the net (
N, u, ρ
)is defined
as follows.
The set of players N:=R ∪ {⊥} is the set of roles and nature,/∈ R.
The state space
is the set of reachable markings, i.e.,
The action set of an individual player
Ai:={} ∪ {{t} | tT, ρ
) =
which consists of the empty set and possibly singletons of transitions, where
) =
)is not defined. We identify an action profile
i=1 Ai
with the union of its components aSiNai.
In a given state
, the available actions of player
are the enabled transitions,
i.e., Ai(m) = {{t} ∈ Ai|m[ti}.
q(m0|m, t) = Pt0vmts.t. m[t0im0QX∈X s.t. tX6=
ui(m, t) = Pttui(t)if i∈ R and u(m, t)=0, for all tT, and mP.
Let us summarize the stochastic game of a given Petri net with transition payoffs
and roles. The stochastic game has the same state space as the Petri net, i.e., the
set of reachable markings. The available actions for each player at a given marking
are the enabled transitions that are assigned to the player, plus the “idle” step.
Each step comes with a state-independent payoff, which sums up the utilities
of each single transition, for each player
. In particular, if all players chose to
idle, the corresponding action profile is the empty step
, which gives 0payoff.
The transition probabilities implement the idea that all transitions of an action
profile get a fair chance to fire, even if the step contains conflicting transitions.
Let us highlight the following two points for a fixed marking and step: (1) given
a maximal enabled sub-step, we roll a fair “die” for each conflict set where the
“die” has one “side” for each transition in the conflict set that also belongs to the
sub-step (unless the “die” has zero sides); (2) there might be several choices of
maximal enabled sub-steps that lead to the same marking. In the definition of
Incentive Alignment of Business Processes 13
transition probabilities, the second point is captured by summation over maximal
enabled sub-steps of the step and the first point corresponds to a product of
probabilities for each outcome of “rolling” one of the “dice”.
We want to emphasize that if additional information about transition proba-
bilities are known, it should be incorporated. In a similar vein, one can adapt the
approach of Herbert and Sharp [
], which extends the bpmn language with prob-
ability annotations for choices. However, as we are mainly interested in a priori
analysis, our approach might be preferable since it avoids arbitrary parameter
guessing. The most important design choice that we have made concerns the role
of nature, which we consider as absolutely neutral; it is not even concerned with
progress of the system as it does not benefit from transitions being fired.
Now, let us consider once more the o
cprocess. If the process reaches the
state in which customer’s next step is payment, there is no incentive for paying.
Instead, customer can choose to idle, ad infinitum. In fact, this strategy yields
maximum payoff for the customer. The bpmn-model does not give any means for
punishing customer’s payment inertia. However, even earlier there is no incentive
for shipper to pick up the goods. Incentives in the single instance scenario can
be fixed, e.g., by adding escrow. However, in the present paper, we shall give yet
a different perspective: we repeat the process indefinitely.
3.3 Restarting the game for multiple process instances
The single instance game from Definition 11 has one major drawback. It allows
to analyze only a single instance of a business process. We shall now consider a
variation of the stochastic game, which addresses the case of multiple instances in
the simplest form. The idea is the same as the one for looping versions of Workflow
nets that have been considered in the literature, e.g., to relate soundness with
liveness [
, Lemma 5.1]: we simply restart the game in the initial state whenever
we reach a final marking.
Definition 12 (Restart game).
A safe marking
is final if it does not
intersect with any pre-set, i.e., if
, for all transitions
; we write
if the marking
is final, and
m6 ↓
if not. Let
hN, S, A, q, ui
be the base
game with fair conflicts of the net (
N, u, ρ
). The restart game of the net (
N, u, ρ
is the game hN, ˚
S, ˚
A,˚q, uiwith
S=S\ {m00 P|m00 ↓};
˚q(m0|m, t) = (q(m0|m , t)if m06=m0
q(m0|m, t) + Pm00 q(m00 |m, t)if m0=m0
for all
m, m0˚
; and the available actions restricted to
, i.e.,
) =
Ai(s), for s˚
For WF-nets, the variation amounts to identifying the final place with the initial
place. The passage to the restart game is illustrated in Fig. 4. The restart game of
our example is drastically different from the base game. Player
will be better off
“cooperating” and never choosing the action
, but instead idly reaping benefits
14 Tobias Heindel and Ingo Weber
Fig. 4. Restarting process example
by letting players
do the work. As a consequence, the transition
probably never occur since the responsible role has no interest in executing it.
Thus, if we assume that the process may restart, the net from Fig. 3is an example
where incentives are aligned w.r.t. completion but not with full liveness.
3.4 Incentive alignment w.r.t. proper completion and full liveness
We now formalize the idea that participants want to expect benefits from taking
part in a collaboration if agents behave rationally – the standard assumption of
game theory. The proposed definition of incentive alignment is in principle of
qualitative nature, but it hinges on quantitative information, namely the expected
utility for each of the business partners of an interorganizational process.
Let us consider a Petri net with payoffs (
N, u, ρ
), e.g., the Petri net semantics
of a bpmn model. Incentive alignment amounts to existence of equilibrium
strategies in the associated restart game
hN, ˚
S, ˚
A,˚q, ui
(as per Definition 12) that
eventually will lead to positive utility for every participating player. The full
details are as follows.
Definition 13 (Incentive alignment w.r.t. completion and full liveness).
Given an autonomous correlation device
, a correlated strategy profile
eventually positive if there exists a natural number
such that, for all larger
natural numbers
n > ¯n
, the expected payoff of every player is positive, i.e., for
all iN,¯γi
n(D, m0, σ)>0. Incentives in the net (N, u, ρ)are aligned with
proper completion if, for every positive real
ε >
0, there exist an autonomous
correlation device
and an eventually positive correlated
-equilibrium strat-
egy profile
of the restart game
hN, ˚
S, ˚
A,˚q, ui
such that, for every natural
, there exists a history
)at stage
n > ¯n
with current
state sh=m0that has non-zero probability, i.e., PD,m0 (h)>0;
full liveness if, for every positive real
ε >
0, there exist an autonomous correla-
tion device
and an eventually positive correlated
-equilibrium strategy pro-
of the restart game
hN, ˚
S, ˚
A,˚q, ui
such that, for every transition
for every reachable marking
, and for every natural number
, there
exists a history
hm0, x1, a1, . . . , sn1, xn1, an1, sn, xni ∈ Hn
stage n > ¯nwith tan1and PD,m0(h)>0.
Both variations of incentive alignment ensure that all participants can expect
to gain profits on average, eventually; moreover, something “good” will always
Incentive Alignment of Business Processes 15
be possible in the future where something “good” is either restart of the game
(upon completion) or additional occurrences of every transition.
There are several interesting consequences. First, incentive alignment w.r.t.
full liveness implies incentive alignment w.r.t. proper completion, for the case
of safe, conflict-free elementary net systems where the initial marking is only
reachable via the empty transition sequence; this applies in particular to Workflow
nets. Next, note that incentive alignment w.r.t. full liveness implies the soundness
property for safe, free-choice Workflow nets. The main insight is that correlated
equilibria cover a very special case of strongly fair schedulers, not only for the case
of a single player. However, we can even obtain a characterization of soundness
in terms of incentive alignment w.r.t. full liveness.
Theorem 1 (Characterization of the soundness property).
be a
Workflow net that is safe and extended free-choice; let (
N, ρ :T→ {Σ},1
)be the
net with transition payoffs and roles where
is a unique role,
ρ:T→ {Σ}
the unique total role assignment function, and
is the constant utility-1 function.
The soundness property holds for the Workflow net
if, and only if, we have
incentive alignment w.r.t. full liveness in (N, ρ :T→ {Σ},1).
The full proof can be found in the extended version [
, Appendix A]. However,
let us outline the main proof ideas. The first observations is that, w.l.o.g.,
schedulers that witness soundness of a WF-net can be assumed to be stochastic;
in fact, truly random scheduling is strongly fair (with probability 1). Somewhat
more detailed, if a WF-net is sound, the scheduler is the only player and scheduling
the next best random transition at every point in time yields maximum payoff for
the single player. Now, the random choice of a transition at each point in time is
the simplest example of an equilibrium strategy (profile); moreover, no matter
what the current reachable state of the net, any transition will occur again with
non-zero probability, by soundness of the net.
Conversely, incentive alignment w.r.t. strong liveness entails that the unique
player – which we might want to think of as the scheduler – will follow a strategy
that will eventually fire a transition of the “next instance” of the “process”. In
particular, we always will have an occurrence of an initial transition by which we
mean a transition that consumes the unique token from the initial marking. After
firing an initial transition (of which there will be one by the structure of the net)
we are in a state that does not allow us to fire another initial transition. However,
strong liveness entails that it has to occur with non-zero probability again if we
follow a witnessing equilibrium strategy (profile). Thus, with probability 1, the
“current instance” of the “process” will complete such that we will again be able
to fire an initial transition.
Finally, the reader may wonder why we consider the restarting game. First,
let us emphasize that the restart games are merely a means to an end to reason
about incentive alignment of bpmn models with suitable utility annotations
by use of their execution semantics, i.e., Petri nets with transition payoffs and
roles. If these Petri nets do not have any cycles, one could formalize the idea
of incentive alignment using finite extensive form games for which correlated
equilibria have been studied as well [
]. However, this alternative approach is
16 Tobias Heindel and Ingo Weber
only natural for bpmn models without cycles. In the present paper we have opted
for a general approach, which does not impose the rather strong restriction on
nets to be acyclic. Notably, while we work with restart games, we derive them
from arbitrary free-choice safe elementary net systems – i.e., without assuming
that the input nets are restarting. The restart game is used to check whether
incentives are aligned in the original Petri net with transition payoffs and roles.
4 Conclusions and future work
We have described a game theoretic perspective on incentive alignment of inter-
organizational business processes. It applies to bpmn collaboration models that
have annotations for activity-based utilities for all roles. The main theoretical
result is that incentive alignment is a conservative extension of the soundness
property, which means that we have described a uniform framework that applies
the same principles to intra- and inter-organizational business processes. We have
illustrated incentive alignment for the example of the order-to-cash process and
an additional example that is tailored to illustrate the game theoretic element of
The natural next step is the implementation of a tool chain that takes a
bpmn collaboration model with annotations, transforms it into a Petri net with
transition payoffs and roles, which in turn is analyzed concerning incentive
alignment, e.g., using algorithms for solving stochastic games [
]. A very
challenging venue for future theoretical work is the extension to the analysis of
interleaved execution of several instances of a process.
Acknowledgments We would like to thank the anonymous referees for their
detailed comments and suggestions on a previous version of this paper.
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