In the previous sections we gave an exemplary overview of how agent-based modeling contributed to research on two fundamental issues (“tensions”) of managerial science. In this section, by employing an example related to the “differentiation versus integration” tension, we seek to illustrate how agent-based simulation could contribute to research in the domain of management accounting.
4.2 An agent-based simulation model
In this section, we describe an agent-based model which reflects major features of the Bushman et al.’s (
1995) principal-agent model as there are interdependencies between business units, linear additive incentive schemes based on the business unit or more aggregate performance measures and linear additive errors related to performance measures. However, given the characteristic properties of agents in ABM (see Sect.
2.1), there are also major differences compared to Bushman et al.’s formal model like, for example, the “solution strategy” of the unit managers (i.e. stepwise improvement rather than maximization) due to limited information about the solution space.
The central issue of our illustrative subject of investigation is the intersection of interdependencies between business units on the one hand and performance measures used in the incentive scheme for business unit managers on the other. Hence, the simulation model has to allow for the representation of different structures of interdependencies. For this, the concept of NK fitness landscapes (Kauffman
1993; Kauffman and Levin
1987, see Sect.
2.2) provides an appropriate simulation approach (Davis et al.
2007) as it allows interactions between attributes to be mapped in a highly flexible and controllable way.
In our model, artificial organizations, consisting of business units and a central office, including an accounting department, search for solutions providing superior levels of organizational performance for an
N-dimensional decision problem. In particular, at each time step
t (
t = 1, …,
T) in the observation period, our artificial organizations face an
N-dimensional decision problem
d
!t
,
d
2t
, …,
d
Nt
. Corresponding to the formal platform of the NK model, the
N single decisions are binary decisions, i.e.
\(d_{it} \in \left\{ {0,1} \right\} , { }(i = 1, \ldots ,N)\). With that, over all configurations of the
N single decisions, the search space at each time step consists of 2
N
different binary vectors
\({\mathbf{d}}_{t} \equiv (d_{1t} , \ldots ,d_{Nt} )\). Each of the two states
d
it
∊ {0; 1} makes a certain contribution
C
it
(with 0 ≤
C
it
≤ 1) to the overall performance
V
t
of the organization. However, in accordance with the NK framework, the contribution
C
it
to overall performance may not only depend on the single choice
d
it
; moreover,
C
it
may also be affected by
K other decisions,
K ∊ {0, 1, …,
N − 1}. Hence, parameter
K reflects the level of interactions, i.e. the number of other choices
d
jt
,
j ≠
i which also affect the performance contribution of decision
d
it
. For simplicity’s sake, it is assumed that the level of interactions
K is the same for all decisions
i and stable over observation time
T. More formally, contribution
C
it
is a function
c
i
of choice
d
it
and of
K other decisions:
$$C_{it} = c_{i} (d_{it} ,d_{it}^{1} , \ldots ,d_{it}^{K} )$$
(4)
In line with the NK model, for each possible vector (
d
i
,
d
i
1
, …,
d
i
K
) the contribution function
c
i
randomly draws a value from a uniform distribution over the unit interval, i.e.
U [0, 1]. The contribution function
c
i
is stable over time. Given Eq.
4, whenever one of the choices
d
it
,
d
ti
1
, …,
d
it
K
is altered, another (randomly chosen) contribution
C
it
becomes effective. The overall performance
V(
d
t
) of a configuration
d
t
of choices is represented as the normalized sum
10 of contributions
C
it
, which results in
$$V_{t} = V({\mathbf{d}}_{t} ) = \frac{1}{N}\sum\limits_{i = 1}^{N} {c_{i} \left( {d_{it} ,d_{it}^{1} , \ldots ,d_{it}^{K} } \right)} .$$
(5)
Vice versa, depending on the interaction structure, altering
d
it
might not only affect
C
it
, but also the contribution of
C
jt,j≠i
to “other” decisions
j ≠
i. Hence, altering
d
it
could provide further positive or negative contributions (i.e. spillover effects) to overall performance
V
t
. In the most simple case, no interactions between the single choices
d
it
exist, i.e.
K = 0, and the performance landscape has a single peak. In contrast, a situation with
K =
N − 1 for all
i reflects the maximum level of interactions, and the performance landscape would be maximally rugged (e.g. Altenberg
1997; Rivkin and Siggelkow
2007).
So far, the model describes interactions between decisions but not between business units. For this, we assume that decisions i are delegated to business units. In particular, our organizations have M business units indexed by r = 1, …, M. Let each business unit r have primary control over a subset with N
r
decisions of the N decisions with the units’ subsets being disjoint so that \(\sum _{r = 1}^{M} N^{r} = N\). The overall organizational N-dimensional decision problem \({\mathbf{d}}_{t} \equiv (d_{1t} , \ldots ,d_{Nt} )\) can then also be expressed by the combination of “partial” decision problems as \({\mathbf{d}}_{t} = \left[ {{\mathbf{d}}_{t}^{1} \ldots {\mathbf{d}}_{t}^{r} \ldots {\mathbf{d}}_{t}^{M} } \right]\) with each unit’s decisions related only to its own partial decision problem \({\mathbf{d}}_{{\mathbf{t}}}^{r} \equiv (d_{1t}^{r} , \ldots ,d_{{N^{r} t}}^{r} )\).
According to the basic behavioral assumptions of agent-based models as sketched in Sect.
2, our decision-making units do not have the cognitive capabilities to survey the whole solution space, i.e. the entire performance landscape, at once. Rather they are limited to exploring the performance landscape stepwise. As familiar in ABM, this is reflected in our model by a form of
local search combined with a
hill-
climbing algorithm (e.g. Levinthal
1997; Chang and Harrington
2006; Levinthal and Posen
2007). In every period, each of the business units makes a choice out of three options: keeping the status quo, i.e. the choice
\({\mathbf{d}}_{t - 1}^{r*}\) made by the department
r in the last period, or opting for one of two adjacent alternatives discovered randomly. These alternatives are “neighbors” of
\({\mathbf{d}}_{t - 1}^{r*}\), i.e. the status quo of the partial configuration, where “neighborhood” is specified in terms of the
Hamming distance, i.e. the number of dimensions in which two vectors differ. In particular, each department randomly discovers one alternative
\({\mathbf{d}}_{t}^{r1}\) which differs with respect to
one of the single decisions that unit
r is responsible for and which has a
Hamming
distance
h equal to 1 with
\(h({\mathbf{d}}_{t - 1}^{r*} ,{\mathbf{d}}_{t}^{r1} ) = \sum\nolimits_{i = 1}^{{N^{r} }} {\left| {d_{i,t - 1}^{r*} - d_{i,t}^{r1} } \right| = 1} \,\). A second option
\({\mathbf{d}}_{t}^{r2}\) is discovered in which
two bits are altered compared to the (partial) status quo configuration, i.e. with
\(h({\mathbf{d}}_{t - 1}^{r*} ,{\mathbf{d}}_{t}^{r2} ) = \sum\nolimits_{i = 1}^{{N^{r} }} {\left| {d_{i,t - 1}^{r*} - d_{i,t}^{r2} } \right| = 2}\). From the three options,
\({\mathbf{d}}_{t - 1}^{r*}\),
\({\mathbf{d}}_{t}^{r1}\) and
\({\mathbf{d}}_{t}^{r2}\), each unit’s manager seeks to identify the best configuration while assuming that the other units
q do not change their prior sub-configuration
\({\mathbf{d}}_{t - 1}^{q*} , { }q \ne r, q = 1, \ldots ,M\).
Which configuration is most preferable from a unit head’s perspective is determined by her/his preferences. We assume that managers are interested in increasing their compensation compared to the status quo salary according to the incentive scheme given. The compensation in each period
t is based on a
linear
additive function with possibly two components: First, the rewards depend on the normalized sum
\(B_{t}^{rOWN} \,\) of those contributions
C
it
resulting from the subset
\({\mathbf{d}}_{t}^{r}\) of decisions delegated to the unit and this reflects unit
r’s “own” performance contribution—corresponding to a unit’s performances in Bushman et al. (
1995):
$$B_{t}^{rOWN} ({\mathbf{d}}_{t}^{r} ) = \frac{1}{N} \cdot \sum\limits_{i = 1 + p}^{{N^{r} }} {C_{it} } {\text{ with }}p = \sum\limits_{s = 1}^{r - 1} {N^{s} } {\text{ for }}r > 1{\text{ and }}p = 0{\text{ for }}r = 1.$$
(6)
Second, to harmonize the interests of head of unit
r with the organization’s performance, the performances achieved through the subset of decisions
\({\mathbf{d}}_{t}^{q}\) assigned to other units
q ≠
r could be part of the value base of compensation of unit
r. Hence, similar to Bushman et al. (
1995), the compensation might also depend on the firm’s performance as given by
$$V_{t} = V_{t} ({\mathbf{d}}_{t} ) = \sum\limits_{\begin{subarray}{l} r = 1 \\ \end{subarray} }^{M} {B_{t}^{rOWN} } \,$$
(7)
which leads to the overall basis for compensating business unit
r’s head
$$B_{t}^{r} ({\mathbf{d}}_{t} ) = \alpha \cdot B_{t}^{rOWN} ({\mathbf{d}}_{t}^{r} ) + \beta \cdot V_{t} ({\mathbf{d}}_{t} )$$
(8)
Thus, similar to Bushman et al.’s (
1995) model, it depends on the values of
α and
β to which extent unit
r’s performance and/or firm performance—as an aggregate performance measure in terms of Bushman et al.—is rewarded.
Our baseline model allows for one mode of coordination, which reflects most purely the form of coordination applied in Bushman et al.’s (
1995) model: In a fairly decentralized mode, in each period each unit’s head chooses one of the three optional partial vectors
\({\mathbf{d}}_{t - 1}^{r*}\),
\({\mathbf{d}}_{t}^{r1}\) and
\({\mathbf{d}}_{t}^{r2}\) related to the decisions the unit head is in charge of, and the overall configuration
d
t
results as a combination of these decentralized decisions—without any intervention from central office or any consultation with the other unit (for this and further modes of coordination, see Siggelkow and Rivkin (
2005) and Dosi et al. (
2003)). Hence, similar to Bushman et al. (
1995), only the incentive system is at work for coordination, and the role of central office is confined to ex post evaluation of the performance of choices made by the unit managers.
Ex post evaluation of the unit managers’ choices is supported by an accounting department which provides the central office and the units with performance information about the choices made, i.e. about the status quo configurations. In particular, at the end of period
t − 1 and before making the decision in period
t, the unit managers receive the compensation for period
t − 1 according to the incentive scheme, and in the course of being rewarded they are informed about the
\({\mathbf{d}}_{t - 1}^{r*}\) s’ performances as measured by the accounting department. However, the accounting department is not able to measure
\({\mathbf{d}}_{t - 1}^{r*}\) s’ performances perfectly. Instead, the accounting department makes measurement errors
\(y({\mathbf{d}}_{t - 1}^{r*} )\), which we assume to be independent and normally distributed random variables—each with mean zero, variance σ
r
and stable within the observation period
T. Hence, the measured value base for unit
r’s “own” performance contribution to their status quo option—in deviation from Eq.
6—is given by
$$\tilde{B}_{t}^{rOWN} ({\mathbf{d}}_{t - 1}^{r*} ) = B_{t}^{rOWN} ({\mathbf{d}}_{t - 1}^{r*} ) + y({\mathbf{d}}_{t - 1}^{r*} ) \,$$
(9)
which principally corresponds to Bushman et al.’s (
1995) model (see Eq.
1). Accordingly, firm performance and the overall value base for compensation resulting from the status quo configuration, as measured by the accounting department are afflicted by errors. Thus,
\(V_{t} ({\mathbf{d}}_{t - 1}^{r*} )\) and
\(B_{t}^{r} ({\mathbf{d}}_{t - 1}^{r*} )\) are modified to
\(\tilde{V}_{t} ({\mathbf{d}}_{t - 1}^{r*} )\) and
\(\tilde{B}_{t}^{r} ({\mathbf{d}}_{t - 1}^{r*} )\), respectively, corresponding to Eqs.
7,
8 and
9.
Hence, we assume that the unit managers in a period t, when choosing one out of options \({\mathbf{d}}_{t - 1}^{r*}\), \({\mathbf{d}}_{t}^{r1}\) and \({\mathbf{d}}_{t}^{r2}\), remember the compensation they received in the last period t − 1 and, from that, infer the value base \(\tilde{B}_{t}^{r} ({\mathbf{d}}_{t - 1}^{r*} )\) for the compensation of \({\mathbf{d}}_{t - 1}^{r*}\) as imperfectly measured by the accounting department. Additionally, our managers have some memorial capacities: Whenever, in the course of the search process, an option \({\mathbf{d}}_{t}^{r1}\) or \({\mathbf{d}}_{t}^{r2}\) is discovered which had already been chosen in periods ≤t−2, each unit manager remembers the compensation received for that period and, thus, the imperfectly measured value base for the compensation based on that configuration.
Let us summarize the information structure captured in the agent-based model: In the search process, the decision-making agents, i.e. the unit heads, stepwise discover the space of configurations
d and the performances related to those configurations. Further, the heads of unit
r stepwise learn about the measurement errors which the accounting numbers are afflicted with in relation to each configuration that has been implemented.
11 Moreover, the unit heads remember these (potentially imperfectly) measured performance numbers which are assumed to be stable in time. However, the unit heads have limited knowledge of each other’s actions. In particular, they assume that the other units stay with the status quo of their partial decisions.
4.4 Comparing the research approaches
In short, the major result of Bushman et al.’s (
1995) paper is that with increasing (decreasing) cross-unit interactions—all other things being equal—it becomes more (less) useful to base compensation on aggregate performance measures. The authors derive this finding from a formal analysis and provide empirical evidence. The agent-based model presented in this paper confirms the main result of the Bushman et al. (
1995) paper. Hence, we have three methodological approaches at hand leading, in principle, to the same result. Of course, on the one hand this is good news for the agent-based simulation approach; however, on the other hand the question arises as to whether the agent-based model yields additional insights which are captured neither in the formal analysis nor in the empirical study by Bushman et al. (
1995)—or in other words:
What is the marginal contribution of the agent-
based model in our illustrative investigation? This section aims to answer this question by comparing the three research approaches in order to highlight their specific features for our example. In the following, we discuss major aspects on rather a condensed level; additionally Table
4 provides a comparative overview of the components as captured in the three approaches.
Table 4
Overview of the major components of the three models
(1) Organizational levels | Two: firm and unit heads | Four: “corporate CEO”, “group CEO”, “division CEO” and “plant manager” | Two: central office and unit heads |
(2) Interdependencies among business units | Level of cross-unit interactions as number of units whose performance might be affected by a business unit Intensity of cross-unit interactions as marginal product of unit managers’ efforts on (other) units’ outcome Positive spillover effects only | Product diversification (related and unrelated) and geographical diversification Intra-firm sales; segment disclosures; entropy measures (Not controlled for) | Level of cross-unit interactions based on single decisions delegated to the units and affecting the performance contributions of other decisions (NK model) Intensity of cross-unit interactions as random variables Positive and negative spillover effects |
(3) Performance measures used in the incentive scheme | Continuously scalable ratio of unit performance versus firm performance | Annual compensation plans: four levels of aggregation (corporate, group, division and plant performance) and their averaged weights for each of four managerial levels Long-term compensation: e.g. stock options | Model: continuously scalable ratio of unit performance versus firm performance Results presented: unit performance versus firm performance as two distinct incentive schemes |
(4) Coordination mechanisms beyond incentive system | Though not explicitly described, “decentralized” in terms of the agent-based simulation model presented in this paper | (Not controlled for) | “Decentralized”: i.e. no coordination between units beyond the incentive system “Central”: proposals by units to central office which combines units’ proposals and makes final decision |
(5) Error/noise related to performance measures in ex post-evaluation | Normally distributed, mean zero, any covariance Decision makers informed about properties of errors/noise | (Not controlled for) | Normally distributed, mean zero, covariance zero Decision makers learn stepwise about measurement errors |
Bushman et al.’s (
1995) formal analysis, based on a principal-agent model, solves an optimization problem and yields the ratio of weights of aggregate-to-unit performance measures in the
optimal contract in an
explicit form. In particular, the weight ratio of performance measures is given as a function of both interactions among business units and (in-)accuracy (noise) related to performance measures as explanatory variables (Bushman et al.
1995, p. 107). From this, the form of how results of the agent-based simulations are given differs substantially: the results are generated by (extensive) numerical experiments and, in this sense, we simulated further parameter settings in order to gain deeper insights into the relation between the relative weights of aggregate-to-unit performance measures, level of intra-firm interactions, accounting error and overall firm performance.
21 However, as Chang and Harrington (
2006) put it, simulation results “ultimately are a collection of examples, perhaps many examples… but still noticeably finite” (p. 1277). Nevertheless, results of simulation experiments and extensive sensitivity analyses can be used to derive regression models (Leombruni and Richiardi
2005; Epstein
2006a).
With respect to the (in-)accuracy of performance measures, first of all, it should be mentioned that this aspect is not captured in the empirical study of Bushman et al. (
1995) due to limitations of data availability (p. 110). On the other side, the principal-agent model of Bushman et al. allows for a detailed analysis of the effects of the measures’ (in-)accuracy on the optimal contract: In particular, the authors derive that in the optimal contract the relative weights of aggregate to unit-related performance measures equals the ratio of signal-to-noise-ratios of the aggregate to unit-related measures which, in a way, reproduces the general finding of Banker and Datar (
1989) for the optimal weighting of multiple performance measures in linear contracts. In a further case discrimination, Bushman et al. (
1995) investigate the situation where the covariance between the performance measures related to the business units is zero and this case corresponds to our agent-based model where the measurement errors related to the measures of units’ performance are independent (see also Lambert
2001, p. 22). For this case, Bushman et al. find that, as mentioned before, the relative use of aggregate performance measures in the optimal contract increases if the cross-unit impact of business units increases and the noise in the aggregate performance measures decreases which corresponds to results of our agent-based model. Moreover, Bushman et al.’s principal-agent model allows for further insights for cases where the correlation between units’ performance measures is not zero. However, this case is not captured in our agent-based model. Notwithstanding this difference another aspect merits a comment: In the principal-agent model the unit managers are assumed to be well informed about the signal-to-noise-ratios related to the performance measures in advance (i.e. before contracting). In contrast, unit managers in our agent-based model stepwise learn about the measurement errors. These differences in the knowledge about errors related to performance measures in a way also reflect the different assumptions about the decision makers’ (bounded) rationality in “traditional” economic modeling versus ABM.
To figure out potential further insights yielded by the agent-based approach, we find it useful to furthermore address those aspects which are additionally reflected or controlled for in the model presented here.
The principal-agent model maps a rather reduced form of organizational structure, meaning, in particular, that no coordination mechanisms other than incentives are used, which corresponds to the Decentralized mode in our simulation model. In contrast, the empirical analysis presumably includes companies which apply various mechanisms of coordination—in addition to the incentive systems under investigation—but does not control for these coordination mechanisms. Hence, the fact that the major theoretical insight (i.e. the more cross-unit interactions there are, the more weight there is on aggregate performance measures) is supported by the empirical part may be regarded as an indication that this finding is robust against richer coordinative forms. However, the question arises as to under which circumstances and to what extent this is the case. Using an agent-based simulation model allows an explicit analysis of richer organizations with, for example, more sophisticated modes of intra-firm coordination (which might be hardly tractable in formal models). We tried to illustrate this by introducing the Central mode of coordination. In particular, analyzing the Central mode indicates that the general finding of Bushman et al.’s paper holds but that, first, the relevance of the incentive system seems to be reduced compared to the Decentralized mode and, second, the search processes differ substantially too.
The agent-based model allows further insights resulting from the underlying behavioral assumptions compared to those of the principal-agent model in Bushman et al. (
1995), who assume a utility-maximizing individual who is able to survey the whole solution space and to identify the individually optimal solution instantaneously. However, when the researcher is interested in how alternative incentive schemes, accounting errors and intra-firm interdependencies affect solutions discovered by less gifted decision makers, an agent-based model, as presented in this paper, could provide some further insights. For example, our model allows an investigation of procedural aspects like the
speed of performance enhancements, the
frequency of discovering the global maximum, or how many “wrong” decisions are made during the search process. Findings of this procedural nature are more or less precluded in the principal-agent model and would be particularly difficult to obtain in an empirical analysis (Davis et al.
2007). In particular, when the speed and costs of adaptations are of interest, e.g. due to turbulent environments, these findings may be particularly useful.
In this sense, ABM might also bear the potential to investigate to what extent findings provided by analytical models hold if some of the underlying assumptions are relaxed (e.g. Axtell
2007; Davis et al.
2007; Leitner and Behrens
2013). Our exemplary study might be regarded as an attempt in this direction, since it indicates that a major finding of the related principal-agent model is robust against relaxing assumptions about the cognitive capabilities of the decision-making agent and even holds for rather myopic agents which, however, are equipped with some foresight as to the outcomes of options and memory of accounting numbers.
Hence, against the background of our modeling effort and the extensions as discussed in Sect.
4.3.4, the potential contributions of ABM for research in management accounting could be summarized in five items: (1) Agent-based models allow the investigation of management accounting issues in rich organizational contexts including, for example, various coordination mechanisms, heterogeneous agents, and various forms of intra-organizational complexity. (2) ABM could help to study the effects of different errors in accounting numbers in interaction with each other and with respect to organizational performance. (3) When procedural aspects of management accounting are of interest—be it due to turbulence in the environment, the learning capabilities of the agents or since the development of the management accounting system itself is to be investigated—agent-based models allow us to study the relevant processes into detail. (4) The “micro–macro interaction” as incorporated in agent-based models enables researchers in management accounting to derive consequences for the system’s overall performance which result from, for example, the use of accounting techniques on the micro level. (5) ABM might allow us to investigate to what extent findings of, for example, principle-agent models hold if some of the underlying assumptions are relaxed. However, there are various shortcomings of ABM which we subsequently address in a broader perspective.