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Parkinson's Law Quantified: Three Investigations on Bureaucratic Inefficiency

We formulate three famous, descriptive essays of C.N. Parkinson on bureaucratic inefficiency in a quantifiable and dynamical socio-physical framework. In the first model we show how the use of recent opinion formation models for small groups can be used to understand Parkinson's observation that decision making bodies such as cabinets or boards become highly inefficient once their size exceeds a critical 'Coefficient of Inefficiency', typically around 20. A second observation of Parkinson - which is sometimes referred to as Parkinson's Law - is that the growth of bureaucratic or administrative bodies usually goes hand in hand with a drastic decrease of its overall efficiency. In our second model we view a bureaucratic body as a system of a flow of workers, which enter, become promoted to various internal levels within the system over time, and leave the system after having served for a certain time. Promotion usually is associated with an increase of subordinates. Within the proposed model it becomes possible to work out the phase diagram under which conditions bureaucratic growth can be confined. In our last model we assign individual efficiency curves to workers throughout their life in administration, and compute the optimum time to send them to old age pension, in order to ensure a maximum of efficiency within the body - in Parkinson's words we compute the 'Pension Point'.


I. INTRODUCTION
C.N. Parkinson has studied socio-dynamical systems with special focus on bureaucratic and administrative bodies. In his famous book [1] he comprises 10 essays which give detailed descriptions of microscopic mechanisms leading to macroscopic phenomena like exponential growth of administrations, impact on the size of executive councils on efficient decisionmaking, and the relationship between the age of retirement and a company's vital promotion scheme. These essays are not quantitative, however, the description of these microscopic mechanisms is so clear and brilliant, that they can be easily cast into dynamical models.
We pick three of these essays which are especially well suited for this purpose.
In the first essay (chapter 4 in [1]), Parkinson conjectured on historical evidence that there is a characteristic group size beyond which the ability of this group for efficient decisionmaking, i.e. the finding of consensus, is considerably diminished. The empirical findings based on government size, found in the 1950s [1], are still valid today to a large extend [15].
By proposing an opinion formation model for small groups we provide an understanding for this hypothesis through the existence of finite size effects which lead to this characteristic size ('Coefficient of Inefficiency') of decision-making bodies.
In the second essay (chapter 1 in [1]), we formalize the so-called 'Parkinson's Law'. We propose dynamical equations for his model of bureaucratic growth and solve them explicitly.
As the main result we derive the phase diagram of this model for the macroscopic evolution of an administrative body. One phase corresponds to an exponentially growing body, the other to a shrinking one.
The third essay (chapter 10 in [1]), studies the interrelation between the age of retirement and promotion schemes with the motif force to ensure the presence of career opportunities and prevent the lack thereof, a phenomenon also known as 'Prince Charles Syndrome'. There we show the existence of a unique maximum for the efficiency of a bureaucratic body at the optimum life-time of a worker in the system. Given a promotion scheme and the hierarchy in the system, the optimal age of retirement can be computed. This paper is organized as follows: in Section II we review work on the relationship between the size of a decision-making body and its efficiency [2]. In Section III we propose and solve the dynamical equations for a model of an administrative body which evolves according to Parkinson's Law. The question when an official within such a body should retire to ensure a maximum of overall efficiency of the administrative body is left for the final Section IV. We interpret our results and conclude in Section V.

II. THE COEFFICIENT OF INEFFICIENCY
Parkinson discovered what he called the life cycle of a cabinet. He studied the membership of Britain's highest executive council, the cabinet, spanning 700 years, from 1257 to 1955.
He found that an initially small council steadily increased in size, until growing beyond a size of 20, shortly after which it was superseded by a new council with the same fate. The British cabinet went five times through this life cycle within the aforementioned timespan.
He also lists the cabinet sizes of his time revealing that cabinets beyond a size of 21 are only found in communistic countries. He concludes that it is critical for cabinets to have memberships below this characteristic size (21) for efficient decision-making within this council (the 'Coefficient of Inefficiency'). His explanation for this phenomenon was that as group size increases the influence of individual members decreases (not only because there are simply more of them, but also because the group is more likely to dissociate into separate subgroups). The less influence a member bears, the more easily new members are admitted to the council which in turn decreases their influence further. In the following we ask if similar observations can be made with today's data on cabinets. We then propose a model for opinion formation in small groups whose dependence on the group-size nicely resembles the transition conjectured by Parkinson.

Governmental Efficiency and Cabinet Size
We compiled a database of cabinet sizes N, for 197 countries and self-governing territories with data drawn from the CIA [3]. The results reported in [2] showed that the majority of cabinet sizes falls in the range between 13 and 20 [16]. To compare cabinet size with its efficiency we made use of indicators provided by the United Nations Development Programme [4] and the indicator Political Stability (PS) provided by the World Bank [5]. We processed the data by computing the average of the indicator values for each N and tested against the hypothesis that there exists no correlation between them, see  Fig. 1(a), and between cabinet size and a series of governance indicators from [5] revealing that larger cabinets coincide with a more unstable political climate, Fig. 1(b). A linear fit to the data reveals that the value of N separating above-average ranking countries from below-average ranking ones is found around 20 for each indicator. For more details see [2].

Opinion Formation in Small Groups
Decision-making in small groups can be cast into a dynamical opinion formation model [6]. Each member of the committee is represented as a node in a network and holds a binary opinion, say 0/1. Each node has a connectivity k which is the number of undirected links to other nodes, representing a social influence (interactions, such as discussions) two agents exert upon each other [7,8]. Each node shares k undirected links with other nodes. Thus for N > k + 1 the graph is not fully connected and nodes appear which are not directly linked.
We assume the total network to be connected, i.e. there exist no disjoint subnetworks.
According to Parkinson groups such as cabinets are typically highly clustered. Therefore a sensitive choice for the fixed network architecture of our model is a small-world network, which has already been shown to be of paramount importance in modelling social interactions [9]. Each node is connected with its k nearest neighbors, then with probability e each link is randomly rewired. As a dynamical rule governing the interactions between connected agents we chose a majority rule [10,11,12,13] with a predefined threshold h ∈ (0.5, 1], see [14].
When the update is carried out random sequentially, i.e. within one iteration each node is updated one after the other in a random order, this yields exactly the opinion formation model studied in [15]. Here a node adopts the state of the majority of its k neighbors only if this majority lies above h, otherwise the node keeps its previous internal state.
To anticipate the evolution of a system specified by the above characteristics, consider For N < 10 we obtain always consensus here, except the N = 8 case. We then find two regimes approximated by a linear fit and separated by a shaded area, where we conjecture the 'Coefficient of Inefficiency' to lie. Decreasing the connectivity of the group's network, parameter setting (6, 0.6, 0.1), red left triangles, shifts the position of the shaded transition area and increases the tendency toward dissensus. Adjusting the threshold such that we recover the pure majority rule, (8, 0.5 + ǫ, 0.1), green right triangles, has no impact on D (N ) due to our choice of k, except that we do not find the 'Charles I' scenario in this case. Lowering e and therefore increasing the spatial correlations in the network hardens the finding of consensus too, as can be seen from the settings (8, 0.6, 0.05) (magenta diamonds).
with the same opinion (consensus) or whether there are at least two factions of opposite opinions (dissensus).
The question whether efficient decision-making in the group is possible or not translates into the question how likely the group evolves into final state of consensus or dissensus. We define dissensus D (N) as the order parameter of the model: let S i be the initial population of nodes with internal state 0 and S f the final population in this state. We then have  We now present a model to understand Parkinson's intuition for administrative growth.
We derive the growth rate λ explicitly, depending on the model parameters r and p. Further we introduce a drop-out rate γ, the probability for an official to quit his job, as well as the age of retirement τ R , which gives the typical administrative life-span. Consider γ and τ R to be fixed and measurable. The question to determine τ R is left to the next section.
Let us denote the number of promoted staff that has served for τ years at time t, n + τ (t) and the number of subordinate staff n − τ (t), analogously. Let the age of retirement be τ R , the time between entry and end of service. We then have for the number of total promoted/subordinate staff N ± (t) = τ R τ =0 n ± τ (t). Further, let p τ be the probability to get promoted after τ years of service, r the average number of subordinates one gets appointed if promoted, and γ the drop-out rate. We then arrive at the update equations Assume that no official enters at a promoted level, n + 0 (t) = 0. By introducing p as the average probability to get promoted, p = , Eq. 4 takes the form Combining this with Eq. 3 in the definition of N − (t) we obtain the update equation for the total number of subordinate staff, where we define q − = n ± τ R (t) N − (t) and assume that the growth of the system is stationary. Using Eq. 2 in the definition for N + (t) = τ R τ =0 n + τ (t) we get the update equation for the total number of promoted staff as, where q + = n ± τ R (t) To obtain q − let us rewrite Eq. 6 as N − (t + 1) = aN − (t). From Eq. 3 we find that Here we can plug in Eq. 4, iterate N − (t − τ R − 1) forward for τ R + 1 times and divide by an overall N − (t) to obtain a self-consistent equation For q + we start with Eq. 2 and iterate it τ R times backward. Making use of Eqs. 3 and 4 and using n + 0 (t) = 0 we get Applying Eq. 3 for τ times we obtain a sum which can be written as an incomplete geometric series leading to Combining Eq. 4 and Eq. 6 we get the helpful relation n − Plug this in Eq. 11 and divide both sides by N + (t) to finally where µ is the ratio of promoted to subordinate staff µ = N + (t) N − (t) . Let λ be the growth rate of the entire staff [N + (t) + N − (t)] ∝ e λt . Parkinson's Law then takes the form of the eigenvalue problem of a triangular matrix We find the eigenvalues immediately in the diagonal. This system allows one meaningful solution for λ and µ, namely λ = a and µ = b a−c . For the other solution, λ = c, µ diverges. We are finally in the position to answer the century, if not millennium old question how a system of bureaucrats can be organized such that it does not grow exponentially. The phase diagram in Fig. 4 shows the two phases, one of exponential growth, one of shrinking, for administrative bodies evolving according to Parkinson's Law. The line separating these two phases (given by λ = 1) shows how the parameters r (number of subordinates) and p (average promotion probability) should be adjusted to keep the size of the bureaucracy constant (drop-out rate was set to γ = 0.01). I: Stages of efficiency of an official. He enters at stage 1. After the respective number of years in service (number in brackets next to Parkinson's Coding) he advances to the next stage. If he does not reach a promoted position by reaching stage 6, he will enter the career described by the right column. We also show the values of the efficiency for official i, ǫ i , we assign to each stage.

IV. AGE OF RETIREMENT
To fully understand Parkinsonian growth of administrations we now determine the age of retirement τ R . Parkinson suggests that the optimal value for τ R has to be found on the basis of an administrative body's promotion scheme. The age of retirement thus does not depend on the actual age of the official whose retirement we are considering, but on his possible successors which should be given the possibility to advance in their careers. If this is not ensured a mechanism nowadays called 'Prince Charles Syndrome' sets in, i.e. the official gets frustrated by a lack of promotion opportunities, and individual efficiency decreases.
We assume the hierarchical structure of the administrative body to be built according to the motif force that 'Officials try to maximize subordinates, not rivals', i.e. each official seeks to achieve a number r of subordinates. One can picture this structure as a pyramid consisting of horizontal layers of officials holding an equally high office. On the hierarchical level above there are then 1/r times less, on the level below r times more officials. Let us label the levels in the pyramid by an index l, then there are r l officials at level l. If we fix the number of levels to L we have N = L l=0 r l and l ∈ {0, . . . , L}. We assign to each official i an efficiency ǫ i which depends on the years he has served and his current position. We use the career stages given as in Tab. I, and assign an ǫ i to each stage, increasing linearly.
These career stages are used as a tribute as in [1], but one could equally well work with We numerically study this system in the following way. It is initialized with randomly distributed ages (and therefore efficiencies) and r = 2. One time step in the simulation corresponds to one year. At each iteration each official ages by one year and his efficiency is updated according to Tab. I. If he reaches the age τ R he retires and his position becomes vacant. One of his two subordinates is chosen randomly to replace him, leaving a vacant position one level below, triggering a cascade of promotions. When the lowest hierarchical level L is reached the vacant position is taken by a new official who enters the system at stage 1. This process is iterated until the average efficiency in the system -ǫ i N , the mean of ǫ i over all officials i = 1, . . . , N -approaches a stationary value. Simulation results are shown in Fig. 5. We vary τ R and compute the average efficiency ǫ i N for systems of sizes L = 3, . . . , 7 and ∆L =1 (green slash-dotted lines), 2 (red slashed lines), 3 (blue solid lines). For increasing L the system approaches a stationary state where the efficiency only depends on ∆L. Accordingly we find three bundles according to ∆L = 1, 2, 3 with curves for different values of L discernible. Curves of same color and style correspond to increasing L from bottom to top. For each bundle we find a unique maximum indicating the optimal choice for τ R . For values of τ R below this maximum officials retire too early to reach higher efficiencies, above this point the Prince Charles Syndrome becomes a danger. We further see that the flatter the hierarchy, i.e. higher ∆L, the lower the optimal age of retirement. Recall that we defined the ratio between promoted and subordinate staff as (14) For large system sizes the ratio µ indeed approaches a stationary value. So ∆L determines µ and the best choice for τ R , which in turn can be used to compute the growth rate λ from the previous model.

V. CONCLUSIONS
We quantified three famous, descriptive essays of C.N. Parkinson on bureaucratic inefficiency in a dynamical socio-physical framework. In the first model we showed how the use of recent opinion formation models for small groups could be used to understand Parkinson's observation -which we showed is still valid in modern data -that decision making bodies such as cabinets or boards become highly inefficient once their size exceeds a critical 'Coefficient of Inefficiency'. We showed how this characteristic size arises due to finite-size effects in the process of forming of coalitions in small systems. Parkinson's Law states that the growth of bureaucratic or administrative bodies usually goes hand in hand with a drastic decrease of its overall efficiency. In our second model we pictured a bureaucratic body as a system of a flow of workers through an administrative body. Officials enter, become promoted to various internal levels within the system over time, and leave the system after having served for a certain time, they retire. Within the proposed model we showed how to compute the phase diagram under which conditions bureaucratic growth can be confined.