Proxyeconomics, An agent based model of Campbell's law in competitive societal systems

by   Oliver Braganza, et al.

In many areas of society we rely on competition to better achieve societal goals. Ideally, competition motivates effort and efficiently allocates resources. However, due to imperfect information, competition generally depends on quantitative proxy measures in order to assess performance. This leads to an increasing use of such quantitative proxies in modern societies. Examples include: in science, the publication count of an author, in healthcare, the number of patients treated or in business, the profit achieved. Importantly, some practices may optimize proxy performance but not the actual societal goal. In such cases, individual decisions and cultural practices may shift away from the societal goal and toward the proxy. Such processes have been described by a law attributed to Charles Goodhart or Donald T. Campbell, most pithily phrased as: 'When a measure becomes a target it ceases to be a good measure.' While the original mentions of this law address policy determination or education respectively, we propose it applies to any competitive societal system: Any proxy measure in a competitive societal system becomes a target for the competing individuals (or groups). Here, we construct an agent based model to explore the basic components and dynamics of such a process. The model combines an effort incentivization mechanism from economic multitasking theory and contest theory with a slower process of cultural evolution.



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1 Competitive Societal Systems

Competitive systems in society generally endeavor to further an abstract goal. For instance the scientific system is intended to generate true and relevant research. In order to incentivize effort toward this goal and achieve an efficient allocation of resources, individual agents in the system are subject to a competition for resources such as funding and positions. Importantly, competition must by necessity rely on some set of proxy measures, in order to rank individuals with respect to their putative performance. The societal system performs the dual task of collecting the information to create the proxy measure(s) and creating the institutions that implement competition. Additionally, it may provide the institutional framework to constrain the space of possible practices by measures independent from the proxy, for instance through professional norms or sanctions. While modern societies are increasingly relying on competitive systems and the proxy measures they require, there are prominent voices throughout multiple fields arguing that this is leading to corruption of the measures and decreased performance of the systems (Fig.1; Campbell, 1979; Wilsdon et al., 2015; Bénabou and Tirole, 2016; Baker, 2016; Fochler et al., 2016; Smaldino and McElreath, 2016; Stiglitz et al., 2010; Gigerenzer and Gray, 2013). Indeed, it is conceivable that a highly proxy-oriented system leads to substantial effort and resource misallocation and detrimental effects on the actual societal goal. Importantly, this would remain hidden as long as the proxy remained the central evaluative tool of system performance. We suggest the term proxyeconomy for a highly proxy-oriented, or in Campbell’s terms ’corrupted’, system. Extensive evidence, such as an extremely high fraction of non-reproducible science, or a dissociation between measures of economic growth and societal welfare among industrialized nations, suggest that real systems may indeed be proxy-oriented to an undesirable degree (Open Science Collaboration, 2015; Wilkinson and Pickett, 2009; Stiglitz et al., 2010).

Figure 1: Campbell’s law in competitive societal systems, Illustrative examples of proxy measures (red) used to attain a more abstract societal goals (blue) in various competitive societal systems. Experts in each of these fields have described processes akin to Campbell’s law, suggesting a general applicability of the law to competitive societal systems.

However, despite the similarity in the arguments in these diverse competitive systems, and the profound implications for societal welfare, we lack a coherent theory of the underlying processes. We believe this has two main reasons, one empirical, one theoretical due to restrictive disciplinary boundaries. A general empirical problem is posed by an inherent complexity/ambiguity gradient from the abstract societal goal to the quantitative proxy measures. Specifically, the aspects of the societal goal that are most difficult to clearly define or quantify are precisely those aspects in which proxy and goal may diverge. Accordingly, much of the evidence suggesting a corruption of practices tends to be qualitative, voiced in interviews, editorials and surveys (Fochler et al., 2016; Baker, 2016; Hicks et al., 2015; Wilsdon et al., 2015). Additional goal aspects may be costly to assess or become apparent only after long time frames. Such aspects can be quantified as alternative proxies and will remain unaffected by corruptive pressures as long as they play a minor role in competition. For instance scientific reproducibility and aspects of financial risk taking play a minor role in competition and may present a quantitative measure of corruption (Open Science Collaboration, 2015; Bénabou and Tirole, 2016). Another potential quantitative measure of divergence are specific readouts of corruption, such as retraction counts (Brembs et al., 2013) or implausibly small sample sizes (Smaldino and McElreath, 2016). However, due to the complexity of societal systems any individual study providing evidence in any of these domains remains difficult to interpret in isolation. Accordingly, a coherent theoretical foundation may help by linking evidence and predictions across studies. The second challenge to a comprehensive theory of proxy-based competition is the transdisciplinary nature of the problem. We identify three fundamental questions such a theory must address, or building blocks it must specify, traditionally treated in separate disciplines. Importantly, all three questions arise directly from the core problem of proxy based competition in societal systems (Fig. 1):

1. What is the informational relation between proxy measures and the societal goal, in the space of all possible practices/actions?
2. How do individual agents make decisions, given potentially conflicting information and value dimensions?
3. How does selection/ cultural evolution affect the system?

We separate these questions for narrative convenience and conceptual clarity, though they are complexly interrelated. While the first and third questions address primarily the system level (complex systems and control theory; information economics; sociology; cultural evolution theory) the second question concerns psychological mechanisms (multitasking theory; contest theory; behavioral/neuro- economics; behavioral ethics). In the following, we describe an agent based computational model, constructed to integrate confluent insights across these disciplines and formalize feedback loops between levels. The overall goal is to determine the principal components necessary to capture the processes described above and explore possible formal specifications. In the process, we outline the tenets of a unified theory of what we will here call ’Proxyeconomics’ or ’A theory of proxy based competition’, which draws on all above mentioned disciplines and applies to any proxy-based competitive system maintained by society to serve an abstract goal.

2 The Model

We developed the model in order to capture the following general notions, deemed to represent convergent insights by the respective disciplines.

1. There exists an informational difference between proxy measures and societal goal, particular in highly complex societal systems. The proxy can be viewed as the ’objective function’ of a complex optimization problem. (Flake, 1998; Stiglitz et al., 2010; Manheim and Garrabrant, 2018; Amodei et al., 2016).
2. Individuals are motivated by not only their personal egoistic interests but also by an intrinsic moral motivation (Hochman et al., 2016; Pittarello et al., 2015; Chugh and Kern, 2016; Holmstrom, 1989; Bénabou and Tirole, 2016). In competition, a desire to professionally survive suggests some form of (proxy based) reference dependence, well captured by a sigmoidal utility function (Kahneman and Tversky, 1979; Mishra, 2014; Delgado et al., 2008; Nieken and Sliwka, 2008; Gonzales et al., 2017).
3. Variability in culturally/professionally transmitted practices and selection through competition introduces cultural evolution at the system level (Smaldino and McElreath, 2016).
Importantly, there will be interactions including feedback loops within and between individual and system level. The information affecting individual decisions will depend on the information at the system level. The competition experienced by individuals relates to the competitive selection at the system level. Finally, and most importantly, proxy performances affect outcomes primarily as a relative measure, i.e. by their relation to other agents’ proxy performances. This implies feedback loops for i) psychologically driven proxy performance, ii) selection driven proxy performance and iii) between the two processes. An agent based modeling approach is ideally suited to capture these notions including the feedback loops within and between levels. The model draws from information/control theory and economic principal agent theory by viewing a societal system as an information processing device or principal, respectively (Manheim and Garrabrant, 2018; Holmstrom and Milgrom, 1991, see below). A neuroeconomically plausible (utility and reference based, Wilson and Kirman, 2016) effort incentivization mechanism is modeled using concepts from prospect theory and multitasking theory (Kahneman and Tversky, 1979; Holmstrom and Milgrom, 1991; Mishra, 2014; Bénabou and Tirole, 2016). Simultaneously, the processes described by Campbell’s law are modeled as a form of cultural evolution, on a slower but overlapping timescale (McElreath and Boyd, 2007; Smaldino and McElreath, 2016). The combination of these approaches allows to investigate the tension between positive and negative effects of competition, namely effort incentivization and screening versus Campbell’s law. It furthermore provides a way to probe the power of ’intrinsic motivations’ to bound a long term evolution towards the worst possible practices, as suggested by Smaldino and McElreath (2016). The main guiding principle in model construction was to use the simplest plausible and intuitive specifications, available within the current literature of the respective disciplines. Furthermore, the discipline specific literature we cite, often purports to apply only to specific societal systems, such as companies or science. Here we draw attention to the general nature of many of the relevant processes, thereby suggesting a more widespread applicability. The next five sections will sequentially introduce our model specifications in the three building blocks mentioned above, preceded by a short section on competition and model time course.

2.1 Competition and Model Time-course

A central variable, which is likely to impact individual decisions as well as system level selection, is competition. For a given population, we define competition simply as the fraction of ’losing’ agents , i.e. agents in danger of (professionally) dying due to insufficient proxy-performance at a given time point. For instance, in intense competition () an agent has to be in the top 10% of proxy-performers to be a ’winner’. This then affects both the psychological decision mechanism (agents desire to win) and system level selection as described below.
The model proceeds in time steps, each of which is subdivided into a decision phase and an evolution phase. In the decision phase all agents are drawn in random order to adjust their efforts to maximize utility given their preferences, observed competitive rank (of proxy performance) and talent. In the evolution phase, losing proxy-performers are subject to probabilistic death, and are instantaneously replaced by offspring from winning proxy-performers, such that the population size stays constant. Together, this models a competitive system in which agents make their decisions based on their individually observed proxy-performances of all other agents, but are unsure about the exact time frame of competition, i.e. to which degree other agents might still change their proxy performance before a selection event might occur. The continuous repeated comparisons, occurring in consecutive time steps resemble what has been called ’multi-contest tournament’ (Fu et al., 2015).

2.2 The Practice Space - Informational Relation of Proxy and Goal

We conceptualize the informational relation between proxy and goal with respect to the value-creation associated with specific cultural practices. A cultural practice is defined as a specific pattern of actions, that is predominantly learned and transmitted within a given cultural entity (laboratory, company, …), but may additionally be subject to individual agency. Four fundamental groups of practices can be intuitively differentiated based on the degree to which they are beneficial or detrimental to the proxy and goal value (Fig 2A). For instance, in any specific societal context, we may consider if any practices exist that exclusively contribute to the proxy measure (or goal). The existence of such practices motivates the dimensional reduction of the practice space to two dimensions representing the proxy and the goal (similar to Holmstrom and Milgrom, 1991; Bénabou and Tirole, 2016). The degree to which the proxy captures true and incorruptible information () about the societal goal can then be represented as the angle between the main axes (goal angle; , ), where information is given by Eq. 1.


Intuitively, information measures the effect of practices exclusively oriented towards the proxy (what we will call fully corrupted) on the societal goal. A good proxy () captures sufficient information about the societal goal, such that even a fully corrupted system produces positive outcomes for the actual societal goal (Fig.2B). In contrast, when full proxy orientation produces negative externalities which outweigh the beneficial effects of competition with respect to the societal goal, this is captured by and (Fig.2B). Externalities beyond the dimensions described above, i.e. that are independent of both proxy and goal, are not considered in the current model.

Figure 2: The practice space, Individual practices may contribute to different degrees to the proxy measure and the societal goal, motivating the mapping of all practices to a two-dimensional practice space. Sections of the practice space which are beneficial/ detrimental to the proxy/goal are color coded (see colored labels in A). The degree to which the proxy captures true and incorruptible information about the societal goal can then be represented as the angle between the main axes, denoted goal angle () with information . Exclusively proxy oriented practices are located on the horizontal axis (proxy measure). Accordingly, when such (corrupted) practices are neither beneficial nor detrimental to the societal goal, then and (A). Presumably, in most cases the proxy captures sufficient incorruptible information about the goal such that even even fully proxy oriented practices contribute to the societal goal ( (B). However, full proxy orientation may also lead to negative outcomes for the actual societal goal ( (C). When the goal component orthogonal to the proxy is labeled ’goal (oc)’.

We can most easily think of the proxy as a simple scalar metric, such as the journal impact factor or publication count. In fact, proxies can be better thought of as arbitrarily complex objective functions, designed to best capture the actual societal goal. In analogy to the economic revealed preference approach, we can also think of the proxy as the set of attributes that factually determine selection within the competitive system (the revealed preference of the system). In both these analogies, the societal system is conceptualized as an information processing device, collecting complex input information and converting it to actionable output metrics (competitive rankings). The societal goal is defined as the arbitrary set of properties society wants to optimize. Though this is difficult to define in any specific context, we can justify it as a theoretical entity based on general information theoretic considerations ranging from oversimplification and measurement error to a range of optimization induced errors (Manheim and Garrabrant, 2018; Amodei et al., 2016). For instance, in practice, proxy measures are heavily determined by the ease and cost of measurement and communication. Note that, in the present model, we define the societal goal as only those goal-aspects, which are privately known to individual agents. This definition of the societal goal may be most productive, since aspects of the goal unknown at the individual and the institutional level may be particularly difficult to address. Individuals are assumed to have superior knowledge as to the precise profiles of their actions and their implications. Indeed, there are numerous reports of individuals in competitive systems who state that competition is impeding their ability to act according to the societal goal, implying private knowledge of implications for this goal which is not captured by the proxy (for instance see Baker, 1992; Bénabou and Tirole, 2016, for examples from science and banking, respectively).

2.3 Effort choice - Multitasking

To intuitively capture the notion of proxy-orientation of a given agent’s practices, we describe all practices by their ’practice angle’ within the practice space (; Fig. 3

A-C). The agents effort level is assumed to be the length of a vector along this practice angle and the true contributions to proxy and goal value are the projections onto the main axes. This intuitively captures a positive effort to output relation, where only the type of output depends on practice orientation. To allow both, an intrinsic (goal) and a competitive (proxy) incentive to elicit effort, we adapt an economic principal-agent multitasking model (Fig.

3A). Note that in contrast to traditional multitasking models, practice angles are primarily established through a process of cultural evolution (i.e at the system level). The practice angle of an agent reflects the average outcome of all the different actions and choices she makes in an individual time step, i.e. it may include strategies such as cheating or sabotaging a fraction of the time, which would translate to a lower average .

Figure 3: Agent decision model, Agents derive utility both from contributing to the societal goal and performing well according to the proxy measure (blue, red respectively). An individual agent (index i) produces goal value () and proxy value () as determined by the practice angle () and effort (), simply as the projection of the effort vector onto the main axes (eqs. 2 and 3; A-C), where the goal angle () is the angle between the main axes (see Fig. 2). The utility derived in the proxy dimension (denoted prospect) depends not on the absolute proxy value but on the relative ranking of proxy values and the survival threshold according to eqs. 5 and 6. Panels D, E show illustrative prospect functions (dark red) for competition and respectively. The survival threshold (vertical spotted line in D, E) is the salient reference proxy value separating losers from winners, where competition () denotes the fraction of losers. Accordingly, for or , the survival threshold is the or the the percentile of the population distribution of proxy values respectively (grey histograms in D, E respectively).

Accordingly, the effort of the agent is modeled as the length of the vector with orientation and the resulting proxy and goal values are the projections onto the main axes capturing the practice dependent trade-off (Eq 2,3).


The utility derived from the goal performance is simply the goal value multiplied by a constant (), determining the relative psychological weight of the moral incentive (Eq 4).


2.4 Effort Choice - Competition as Incentive

The utility derived from the proxy value, denoted ’prospect’ (), is determined through competition (Fig. 3D, E). It depends on the relation of the own proxy value and the proxy value required for professional survival (the survival threshold, ). The survival threshold is the proxy value which separates winners and losers for a given level of competition. For instance indicates that the survival threshold is the proxy value at the percentile of the distribution of all proxy values, i.e. an agent has to be in the top 10% proxy performers to be a winner. The survival threshold is assumed to be the salient reference point with respect to which agents evaluate the utility of their own proxy value. Accordingly, they will derive negative/positive utility if they are below/above the survival threshold. In the first version of this manuscript/model we have considered a prospect function based on a Gaussian uncertainty distribution around the survival threshold based on Nieken and Sliwka (2008); McDermott et al. (2008); Mishra (2014). Though the main results were similar (see manuscript history on arXiv), this function was not scale invariant with respect to proxy value. Therefore we here choose a similarly sigmoid, but scale invariant prospect function, namely that of cumulative prospect theory (Eq 5; Tversky and Kahneman, 1992) .


Additionally, we assume agents are loss averse, i.e negative prospects are multiplied by 2.25 (Eq 6; Tversky and Kahneman, 1992).


Each agent , when she is drawn, chooses her effort level to maximize her individual utility given the observed current proxy performances of all other agents (Eq 7),


where effort cost is given by:


Here is the talent of the agent, i.e. a constant determining the relative cost of effort independent of . Agents are given variable talent according to such that the effort to proxy-performance relation depends on both individual practice orientation and talent. This allows to investigate the simultaneous effects of useful and wasteful signaling bringing together the concepts of screening and Campbell’s Law. In order for individual agents to compute the complex maximization problem described above, we assume they consider a limited range of effort adjustments, namely [-10, -5, -1, -0.5, -0.1, 0, 0.1, 0.5, 1, 5, 10]. The rationale behind this is that the agent intuitively judges what might happen if she changes her effort just a little or a lot but lacks the computational capacity for perfect maximization. Nevertheless, if the system is stable enough, she will iteratively approach the optimal effort. Note, that alternative effort choice lists (e.g. -10 to 10 in 0.1 increments) did not change model outcomes but dramatically increased the computational burden of agents (and the model). Furthermore, the specific range (-10 to 10) simply covers the magnitude of plausible effort changes in the system arising from the arbitrary mean talent choice of 10.

All competitive societal systems mentioned above involve highly trained professionals who are likely to derive substantial disutility from the prospect of not being able to perform their profession. Accordingly, we believe the degree of loss aversion is conservative. Furthermore, this framing suggests that agents may be willing to remain in the contest despite substantial disutility. In other words, there may be a severely muted participation constraint. However, note that an indirect participation constraint arises naturally in our model due to diminishing returns in the loss frame combined with selection.

2.5 Selection and Evolution

When all agents have been randomly drawn to adjust their efforts, the ranking of proxy-performances is reassessed. Each potential loser (

) is then subject to professional death with probability = selection pressure (

). Upon each death a position opens up, which allows a randomly chosen ’winning’ proxy-performer () to professionally reproduce, passing her practice angle on to her offspring. This specification keeps the population size constant, modeling a societal system with fixed size and resource consumption. Increased competition is realized through an increased throughput of new agents, as is common in many highly competitive societal systems.
During practice inheritance, mutates stochastically, such that . We reason that the magnitude of the mutation rate models the complexity of the societal system. Large mutation rates reflect potentially large and frequent effects of practice changes on proxy and goal values, since in a very complex system there are an infinite number of possible practices, and the effect of minor practice changes may be highly nonlinear. In this context it is important to remember that the practice space represents a dimensional reduction to the orthogonal components of proxy and goal. Accordingly, mutations can be thought of as comprising arbitrary changes in additional independent dimensions.

2.6 Implementation/ Model development/ Parameter choice

The model was implemented in the agent based modeling framework ’Mesa0.8’ in Python3.6 and run on a standard Windows7, 64bit operating system. Code will be made available upon publication (or request)

The main parameter of interest was competition, i.e. do different levels of competition lead to different dynamics/equilibria of proxy and goal performances. Accordingly, the model and analysis code was set up to allow an exploration of the impact of all remaining parameters on the interaction between competition and proxy/goal performance in the short and long run. Table 1 shows an overview of the explored parameter space with the parameters shown here highlighted in boldface. A number of additional parameters were varied to probe robustness, but lead to no change, and will be reported at appropriate locations throughout the manuscript. The model was initialized with population size N (typically 100), where every agent received a practice angle

drawn randomly from a uniform distribution between proxy and goal axes

and initial effort 0. In exploratory analysis we found no effect of population size N or the initial effort on final equilibria or long term dynamics. Model runs were typically repeated 10 times for each level of competition to obtain measures of the mean and spread of system behavior. In each model run, all agents in a population compete against each other via their proxy performances as described above.

Parameter Values Description Main effect (if parameter is increased)
global parameters
population size () 100, 50, 25, 10 number of agents in the system decreased variability over models
goal angle () 45, 90, 135 angle () defining true (incorruptible) proxy information determines the detrimental effect of proxy orientation, the optimal level of competition decreases
competition () [0.1, 0.2, …, 0.9] competitive pressure, i.e. the fraction of potential losers per round complex effects on effort, evolution and utility (see main text)
individual agent parameters
mean talent () 10, 5, 20 constant defining the cost of effort independent of increased effort

talent standard deviation (

1, 0 standard deviation of talent within the population increased effort spread
goal scale () 1, 2, 0 scaling factor of psychological goal valuation (relative to experienced prospect value) increased effort, increased equilibrium practice angle
loss aversion () 2.25, 1 factor by which negative prospects are multiplied increases effort proportional to competition
prospect exponent 0.88, 0.5 determines curvature (degree of diminution of marginal returns) of prospect function decreases oscillatory dynamics, increases effort
system level evolutionary parameters
selection pressure () 0.01, 0.001, 0.1 probability of death for each losing agent in each round increased speed of convergence to equilibria
practice mutation rate () 2, 0, 1, 5 standard deviation () of practice angle mutations during inheritance increased variability over time and speed of convergence to equilibria
Table 1: Parameter space, Overview of the investigated parameter space. Parameters shown in the results section are in boldface. The remaining parameters were tested to ensure robustness.

3 Results

We will now introduce the model with an exemplary set of plausible parameters and a relatively short time course (50 time steps). Specifically, we consider a system in which the proxy contains some incorruptible and some corruptible information about the societal goal (). Goal scale () is set to one implying comparable weighting of proxy and goal incentive. Selection and mutation are mild (), implying a probability of death for losers at each time step and small changes of practice orientation during inheritance. At initialization each agent receives a random practice angle between full corruption and full goal orientation

and normally distributed talent


3.1 Iterative effort choice algorithm

We begin by describing the emergent behavior resulting from iterative effort choice. In brief, at each time step, agents in random order adjust their effort given the observed but uncertain proxy performances of competing agents and their personal parameters (including prospect function, practice angle and talent). This simple contest specification produced a range of behaviors observed in experimental economics, including potent effort incentivization, a discouragement effect and effort bifurcation (Dechenaux et al., 2014). Importantly, we did not consider or anticipate any of these effects during model design, but rather strove to create the simplest effort choice algorithm applicable in a step wise agent based model with plausible preferences, information acquisition and computation of individual agents. Fig. 4A-C shows four exemplary model runs ranging from very low (; Fig. 4A-C, leftmost column) to very intense competition (; Fig. 4A-C, rightmost column). Fig. 4

A shows how each individual agent’s effort develops over time in response to the other agents. Each column within a subpanel shows an individual agent, where effort is color-coded. Within the first few (¡10) time steps each agent approaches a relatively stable individual equilibrium effort. This first rapid equilibration leads to a mean population effort, which is greater for greater levels of competition, and is succeeded by slower processes. In higher competition individual agents are forced to increase effort to cross the higher survival threshold, which in turn affects the population distribution of proxy values and the resulting survival threshold. This positive feedback loop leads to higher mean effort in higher competition but also promotes a counterbalancing effect, termed ’discouragement’. Discouragement results from the flattening of the prospect function when the observed survival threshold is far beyond the own proxy performance. In such cases reducing effort leads to savings in effort cost which outweigh the only slightly reduced utility from output. Discouragement becomes more frequent in more competitive systems, leading to an increase in variance of effort at the population level. At the individual level, agents similarly show more variable behavior in higher competition. Notice, how some agents progressively increase effort in order to compete, but eventually become discouraged (Fig.

4A, black arrows). The resulting decrease in the survival threshold may in turn provide other agents with a prospect of winning, such that they can gain utility by increasing effort (Fig. 4A, grey arrows). Thus, our iterative effort choice algorithm reproduces a remarkable range of empirically observed but incompletely understood phenomena during contest situations, given a neuroeconomically plausible individual decision model.
Additional to these psychologically driven effects, competition determines selection by defining the proportion of losers. Agents with insufficient proxy-performance (losers) are subject to probabilistic professional death with , indicated by white rectangles (Fig. 4A, two examples labeled with white arrows). Upon a death, the free slot is immediately filled up by the offspring of a winning agent. Given, that competition determines the proportion of losers and losers have uniform probability of death, selection events increase proportionally with competition.

Figure 4: Short term dynamics

Short term dynamics (50 time steps), with . Data are collected from 10 model runs per competition level (). A-C show four levels of competition (major columns) while D shows . A and B show individual agents while C and D show population means from repeated model runs as data points. A) Agent Dynamics, Each subpanel shows effort over time for each of the 100 agents (random run for given ). Columns within subpanels represent individual agents or positions. White squares indicate death/birth events (two example highlighted with white arrows). Black arrows indicate examples of agents becoming discouraged by competition. Grey arrows indicate examples of agents increasing effort to gain the prospect of winning. B) Agent Values, Realized proxy and goal values of each agent at the 50th time step. Values are the result of the current chosen effort given the individual agents practice angle (compare Fig 3B). Each data point represents an agent (agents accumulated from 10 runs per panel). Talent is color coded. The black and grey arrows indicate highly proxy or goal oriented agents, respectively. Note that the agents indicated by the grey arrow are outperforming those indicated by the black arrow concerning their goal performance, but not proxy performance. C) Model Dynamics, Model level dynamics of proxy and goal value (top) and the mean practice angle (bottom). D) Model Values, Final (step 50) mean proxy and goal values (left) and utility (middle) as a function of competition. Data on mean proxy and goal values can be transformed to illustrate mean effort level and practice angle (right) as in Fig.3A. The shaded area around lines represents the standard deviation over model runs.

Figure 5: continued

3.2 Realized value in the practice space

The relative ability of individual agents to compete depends on two not directly observable parameters, namely talent and practice orientation. Talent allows greater effort independent of practice orientation. Allowing competition to screen for talent for a given set of practices thus implies screening for higher goal performance. The proxy here can be seen as valuable signal of the goal. At the same time, however, high proxy performance may signal higher proxy orientation of the practice implying wasteful or even detrimental signaling. To visualize both processes simultaneously, we plot the realized practice-effort pairs of individual agents back into the practice space with color-coded talent (Fig.4B). Here each data point represents the endpoint of the effort vector (as in Fig. 3A-C) of an individual agent at time step 50. The resulting goal and proxy performances correspond to the projections onto the main axes, as shown in Fig. 3B). Across all levels of competition, but most prominently for high competition, we observe a screening effect as higher talent corresponds to higher effort and proxy performance. We furthermore observe a dominant effect of the competitive incentive on realized effort levels, as outcomes tend to organize in vertical lines, i.e. they cluster around a specific proxy value. Further analysis showed, that the vertical line corresponded to proxy-values just above the emergent survival threshold for a given model run. Agents just below this threshold, where the prospect function is steepest, tend to either increase effort attempting to enter the win-domain or decrease effort further, to save effort cost. This leads to effort bifurcation, particularly when judging effort by proxy performance. Another interesting emergent phenomenon is a competition dependent reversal of relative effort expenditure of agents with high practice angles. In low competition (Fig. 4B, c = 0.1, leftmost panel) agents with high practice angles are forced to put in extra effort, but are still able to compete, leading to particularly high relative contributions to the societal goal. By contrast, when competition is intense (Fig. 4B, c = 0.9, rightmost panel) agents with more goal oriented practices can no longer compete on the proxy scale and are preferentially discouraged, even if they have high talent. Notice, that the emergent practice-effort realizations cover several qualitatively distinct domains across the practice space, where observable proxy performance only partially predicts unobservable goal performance. Some agents are peak proxy performers, while only moderately contributing to the actual societal goal (Fig. 4B, black arrow). Simultaneously, some highly talented agents are ’losing’ in competition, while actually outperforming many ’winners’ regarding the actual societal (Fig. 4B, grey arrow). More generally, the model predicts that for any observed proxy performance, there is a mix of agents with highly varying degrees of proxy orientation. Thus, our model allows to simultaneously explore the beneficial as well as detrimental signaling effects of proxy measures, predicting competition dependent mixes between varying degrees of proxy orientation for any observed proxy performance.

3.3 Short term system dynamics

Preferential discouragement of goal oriented agents might translate to a divergence between proxy and goal performance at the system level. To visualize variability at the system level the following data are displayed as mean and standard deviation over model runs where each run is represented by the population mean of its agents. Indeed, mean proxy and goal performance of populations of agents show distinct competition dependent dynamics (Fig. 4C, top). Note the initial steep rise of value (within the first 10 steps), as effort first adjusts to an initialization independent level (Models initialized with effort levels of 0, 1, 10 or 20 all converged to the same values during this initial phase). Following this initial phase, value tends to progressively increase due to the positive feedback loop between individual effort and survival threshold. However, at more intense competition there is also an increasing divergence between proxy and goal value (Fig. 4C, top right). One driver of this effect is the preferential discouragement of agents with high goal orientation, implying an overall effort redistribution toward the proxy (a psychological mechanism of Campbell’s Law). A second potential driver is a selective removal of goal oriented agents due to inferior proxy performance (an evolutionary mechanism of Campell’s Law). The degree to which this happens can be assessed by viewing the dynamics of the population mean practice angle, which is independent of effort ((Fig. 4C, bottom). With a goal angle of , the initial uniform distribution of practices between proxy and goal leads to an initial population mean practice of . However, as selection removes agents with high practice angles, replacing them with offspring from agents with lower angles, the population mean evolves toward the proxy. Thus, our model reproduces a central finding by Smaldino and McElreath 2016. As expected, the speed of evolution towards the proxy is proportional to the intensity of competition since the quantity (and practice dependence) of selection events drives population change of practices.

3.4 System outcomes over competition

To obtain a more complete view of proxy and goal performances we next plot the mean final values of this short period (step 50) as a function of competition (Fig. 4D, left, data points represent model runs). Note that for both proxy and goal value, there is an optimal level of competition. Very low competition implies low effort incentivization, leading to very low value creation in both proxy and goal domains. As competition increases there is increasing effort incentivization but simultaneously proxy and goal values begin to diverge, due to the dual process of selective discouragement and evolution described above. Another potential negative effect of competition is lower mean utility of the participating agents (Fig. 4D, middle panel). Though each agent individually maximizes utility, the necessity of being relatively better than other agents pushes the system to relatively high effort levels implying high disutility from effort cost. The effect is mediated by disutility from negative prospects, increasing total effort expenditure, as agents continuously attempt to break out of the loss frame (Kräkel, 2008). However, ultimately the proportion of agents with negative prospects remains fixed by competition, implying that disutility due to being in the loss frame will remain constant at the system level. Notice, that we have not modeled positive utility from a flat wage component, since it does not affect the utility maxima of individual agents and thus system behavior. An added flat-wage utility (10 or 20 utils) increased utility by a constant independent of competition, linearly shifting the utility distribution (Fig. 4D, middle) upward. Also notice however, that as our agents model trained professionals, we assume they may accept substantial disutility rather than switch to a different profession, which may involve additional costly training. Finally, in Fig. 4D the right panel shows an overview of the population mean of practice orientations analogous to Fig. 3B and 4B. Competition is color coded, such that the effect on both mean effort (vector length) and practice orientation (vector angle) are directly visualized. Thus, the model at the system level similarly captures the trade-off between positive and negative effects of competition, and reproduces the notion of an optimal level of competition resulting from both effort and information dynamics (Bénabou and Tirole, 2016; Manheim and Garrabrant, 2018).

Figure 6: Long term dynamics

Long term dynamics, (2000 time steps), with . For panel descriptions please refer to Fig. 4. In (A) only a subset of agents is depicted.

Figure 7: continued

3.5 Long term dynamics

So far, we have explored the model dynamics in the short term (50 time steps), where psychological mechanisms dominate. To test how prolonged selection and evolution will impact the distribution of practices and the resulting outcomes, we next ran the same set of models for 2000 time steps (Fig. 6). In a previous similar model, practices were found to inevitable evolve to full corruption (Smaldino and McElreath, 2016). Our model allows to test if a moral incentive component (and the corresponding knowledge) has the power to bound this negative evolution. Indeed, the range of existing practices in our model evolves toward the proxy for all levels of competition (Fig. 6A, D). However, even in the most intense competition, an equilibrium is reached, beyond which the average practice no longer becomes more corrupted (Fig. 6D). This result is robust for 20000 time steps, but not when goal valuation is removed by setting (not shown). Thus, our model suggests, that an intrinsic motivation to work towards the societal goal can bound the evolution toward fully proxy oriented practices.

4 Discussion

We have presented an agent based computational model, sketching the central components of a socio-economic theory denoted ’Proxyeconomics’. The theory is motivated by the central insight, that any societal competition to achieve an abstract goal must rely on proxy measures which invariably capture only partial information, resulting in a susceptibility to corruption. Outlining the theory required integrating building blocks across several disciplines and attempting to capture the theoretical and empirical findings therein in minimal plausible formalizations:
1. There exists an informational difference between proxy measures and societal goal, particular in highly complex societal systems (Flake, 1998; Stiglitz et al., 2010; Manheim and Garrabrant, 2018; Amodei et al., 2016).
2. Individuals are motivated by not only their personal egoistic interests but also by an intrinsic moral motivation (Hochman et al., 2016; Pittarello et al., 2015; Chugh and Kern, 2016; Holmstrom, 1989; Bénabou and Tirole, 2016). In competition, a desire to ’win’ suggests some form of survival threshold separating winners from losers implying a) loss aversion and b) diminishing marginal utility with increasing distance to this reference point (Mishra, 2014; Delgado et al., 2008; Nieken and Sliwka, 2008; Gonzales et al., 2017).
3. Variability in culturally/professionally transmitted practices and selection through competition introduces cultural evolution, a process beyond individual agency (Smaldino and McElreath, 2016).

The main contribution, is to synthesize these diverse literatures into a coherent theory of proxy based competition. Beside this general conceptual contribution, our model yielded two specific results: First, the agent based implementation of a contest model reproduced a remarkable range of empirically observed behaviors from experimental contest theory, which were not considered during model design. For instance, competition led to discouragement of low performers producing a bifurcaton of efforts (Dechenaux et al., 2014; Sheremeta, 2016). Second, the complete model suggests that intrinsic moral incentives may bound the long term evolution to bad practices, even in the absence of additional societal constraints. Below, we will first discuss these two specific results and then proceed to discuss further literature informing the three central points introduced above, towards a more general theory of proxy based competition.

4.1 Agent based contest model

The relation of our model to the economic contest literature deserves particular mention, since to the best of our knowledge we are the first to implement a neuroeconomically plausible iterative effort choice heuristic in an agent based model of contests.

Bradler et al. (2016) demonstrate the isolated effect of psychological competition on effort, providing a compelling reason to clearly separate the psychological effects of the contest from actual outcomes, such as selection. Our simple specification of a contest heuristic was driven by convergent neuroeconomic and psychological evidence on i) a neural representation of utility, ii) multiple sources of input to this utility and iii) reference dependence of this utility (Wilson and Kirman, 2016). It further provided agents with plausible input information and computational capacity. Intriguingly, the emergent behavior reproduced many empirical findings, the origins of which are still incompletely understood (Dechenaux et al., 2014). One result is a natural discouragement of agents performing far below the survival threshold. While traditionally, a ’discouragement effect’ is thought to result from agent heterogeneity (Dechenaux et al., 2014), our model predicts it even in cases without agent heterogeneity, simply due to chance and uncertainty. For the same reason, our model predicts a bifurcation of effort, particularly in the presence of heterogeneity, but even without. Additionally, the sigmoidal prospect function leads to optimal effort incentivization at intermediate levels of competition, consistent with experimental and field observations (Harbring and Irlenbusch, 2008; Orrison et al., 2004; Bradler et al., 2016). A particularly interesting set of studies in our context are investigations of contests with the possibility of sabotage (Harbring and Irlenbusch, 2008; Falk et al., 2008). A mixed strategy containing productive effort as well as sabotage is analogous to a low practice angle in our model. Indeed, increased competition (a smaller fraction of winners or increased prize spread) lead to a redistribution of effort towards sabotage. Note, that these studies permit agency over the proportion of sabotage-effort, while our model has so far allowed only selective disincentivization of agents with high practice angles, albeit with similar results. In general, the investigation of contest incentives in explicit multitask settings is a crucial avenue for future research (Sheremeta, 2016).

Finally, our iterative effort choice algorithm makes predictions about the temporal evolution of effort expenditure, including it’s variability in time and the path of convergence to potential equilibria. In this, it resembles previous experimental contest models based on reinforcement learning

(van den Bos et al., 2013) or belief-updating (Fu et al., 2015). A specific prediction, consistent with casual experimental observation, is variability of individual effort over time and over the population, which is proportional to the level of competition (see Harbring and Irlenbusch, 2008; Dutcher et al., 2015).

4.2 The evolution towards bad practices

A previous agent based evolutionary model of cultural practices has reported a robust evolution toward the worst possible practices Smaldino and McElreath (2016). The central unifying element with the present model is the existence of a practice variable, which alters the relation between true (goal) performance and selected (proxy) performance, and that this variable can be imperfectly inherited ( in the present study). Given the complexity of cultural systems, these assumptions are nearly self evident, and the resulting detrimental evolution becomes a powerful prediction. The authors have therefore probed the ability of a specific mechanism, namely probabilistic sanctioning, to counteract detrimental evolution, but found it to be ineffective. Here we have modeled a psychological mechanism, namely the influence of an intrinsic psychological incentive on effort expenditure, and observe that it is able to bound the evolution to fully corrupted practices. Specifically, our model led to an equilibrium corruption level, which was substantial, but still markedly different from full corruption. Indeed, this result better matches the results of a meta-analysis of psychological sample sizes by Smaldino and McElreath (2016), in which sample-sizes are consistently too low, but substantially above minimum.

4.3 Beyond the model

Next, we will discuss the more general background of our model, including specific shortcomings and potential extensions, pointing toward a comprehensive theory of proxy based competition in societal systems.

4.4 Informational perspective - Relation to artificial intelligence/ machine learning

The general approach of the current model was to view competitive societal systems as information processing devices, which collect information to create the proxy and can ultimately be characterized by their realized competitive decisions. This approach allows to apply insights from a recent literature on artificial intelligence and machine learning

(Amodei et al., 2016; Manheim and Garrabrant, 2018). The proxy measure is analogous to a realized objective function and the competitive pressure corresponds to optimization pressure. We suggest that all the principal types of challenges that occur during machine learning optimization also apply in competitive societal systems, including reward hacking, negative side effects and scalable oversight (Amodei et al., 2016). For instance, the process of Campbell’s law is closely related to reward hacking. Negative side effects include reduced value in dimensions which do not directly impact either proxy or goal. For instance, Fochler et al. (2016) suggest young scientists progressively constrict their valuation repertoire to competitivity, at the cost of originally intrinsically valued ’sociability’. (such as sociability Fochler). Scalable oversight describes the process of balancing the cost of creating the proxy and monitoring corruption with the benefits from better proxy information. One main reason why proxy measures will imperfectly reflect the societal goal is because they need to be cost efficient. For instance, the information contained in an impact factor could be arguably arbitrarily increased by adding more reviewers or perhaps even having labs reproduce the findings. Evidently, societal systems must balance the cost of improving the proxy with the potential detrimental effects of an imperfect proxy. Note that both factors are likely to interact with competition. More frequent competitive evaluations will in themselves be more costly to perform while simultaneously incurring a higher unobserved cost due to corruption pressures. Introducing probabilitstic and/or delayed assessment and correction mechanisms for corrupted practices may lead to new dynamic equilibira (but see Smaldino and McElreath, 2016).
Summarily, the analogy to machine learning suggests two central conclusions: 1. There is an optimal optimization-pressure which crucially depends on the information captured in the proxy. 2. Problems are likely to arise through both continuous system change and the optimization pressure itself, necessitating a continuous higher level assessment process. In other words, we should continuously expect, attempt to measure and mitigate corruption in any competitive societal system. Importantly, the term corruption should be understood not primarily as driven by ill intent, but rather as an inevitable system level force arising through actions based on imperfect information.

4.5 Psychological/ Economic perspective

Our theory draws on a large experimental literature addressing the individual level decision mechanism. However, it must be stated clearly, that the actual mechanisms are far from understood. For instance, it is unknown what the relative motivational power of moral and competitive incentives is, particularly for real professionals. Compared to laboratory settings both the moral incentive (e.g. treating a patient well) and the competitive incentive (actual professional survival), may be substantially more powerful. Experimental investigations, such as Hochman et al. (2016); Pittarello et al. (2015); Mazar et al. (2008) provide essential insight about the potential psychological mechanisms during such incentive conflicts. Similarly, it is unknown what the actual effect of competition on motivation is, particularly in multitask settings. Behavioral economic, including neuroeconomic, studies are providing valuable insights on the neural mechanisms underlying effort-overinvestment in contests (Delgado et al., 2008). More generally, the fields of behavioral economics, psychology and neuroeconomics are beginning to converge on a set of answers as to actual, empirically validated, decision mechanisms. Specifically, they suggest a computationally bounded mechanism with multiple (moral/ egoistic), potentially reference dependent, valuation inputs converging into a single utility computation (Wilson and Kirman, 2016). While we have attempted to capture these confluent insights into our simple decision model, future research, and more complex empirically validated decision models will unquestionably yield a superior basis for generative agent based models.
Finally, there is substantial empirical evidence for interindividual differences in moral/egoistic drive. For instance, there is direct experimental evidence for an increased propensity to sabotage in males than females (Dato and Nieken, 2014), which may partially explain observed productivity differences (Larivière et al., 2013). Variable incentive strengths could be easily modeled as variations in goal scale (or prospect function variability). Accordingly, this kind of model could inform on the outcomes of empirically observed differences between genders or in ’machiavellanism’ scales (Niederle and Vesterlund, 2011; Tijdink et al., 2016).

4.6 Sociological/ Cultural evolution perspective

The sociological/ cultural perspective explicitly acknowledges the complexity of cultural practices and the degree to which these are determined beyond the individual level. We have adapted the model by Smaldino and McElreath (2016), in order to capture the slow evolution of a large body of cultural information within professional practices. Additional, factors which may prove highly relevant to long and short term outcomes is the network structure between agents, and the precise formulation of information transmission between them. In our model, all agents observe the noisy proxy performances of all other agents, implying full network connectivity. Though this may seem unrealistic, the idea that professionals are generally aware of their approximate competitive standing seems justified. To verify robustness against the full connectivity assumption, we probed if restricting the sampled proxy performances of other agents to 9 or 22 neighboring agents altered outcomes, but this had no effects besides increasing noise. Finally, we have not shown models including agency over the practice angle. In first models including agency over the practice angle during utility maximization [], implemented simultaneously with agency over effort, practices converged to the same range as observed by selection, but much faster. Notably, the hill shaped relation between competition and output remained almost identical. Future models could include varying degrees of agency over the practice angle. Additionally, practice angles could be influenced by social forces such as the formation of social norms (Epstein, 2001). While, social-norm transmission is implied in the hereditary transmission of practices modeled here, they may also be determined more directly through neighboring agents. This is likely to introduce additional nonlinear effects, as locally normative practices emerge or collapse.

4.7 Political implications

Finally, we note, that our model may have a number of political implications. Political conservatives will frequently argue, that a current system (and the resulting proxy measures) should be maintained. They may support this view by arguing that i) the current proxy measure has good information and works well and ii) the cost of improving on the current measure would create considerable unwarranted costs. On the other hand, progressives tend to argue that i) the current proxy measure is leading to substantial corruption and ii) the benefits of improving the measure would outweigh the resultant costs. The present theory mediates between these views in several ways. Firstly, it elucidates how both sides are likely to simultaneously be correct on the first count, and suggests that both sides will find substantial evidence supporting their respective view. Secondly, it suggests that the questions of how corrupted the proxy really is and whether to improve on the proxy measure is cost-effective can be approached empirically. Finally, it is important to note that the institutional mechanisms to create the proxy may be primarily determined by current proxy winners. Accordingly, we might generally expect some inertia in societal systems where current proxy winners would decrease their competitive prospects (or the valuation of their life legacy) by questioning the proxy or altering the mechanisms of proxy creation.

4.8 Conclusion

We have outlined a transdisciplinary theory of proxyeconomics, which applies whenever a societal system employs proxy measures to mediate competition in order to promote an abstract goal. Our agent based computational model synthesizes several major insights across disciplines into a formal framework, suggesting a central role of competition on effort expenditure, individual utility, selection and cultural evolution. Accordingly, there may be an optimal level of competition, depending on the complexity of the system and the preferences of the participating agents. Furthermore, we have demonstrated, that an individual level decision mechanism, which includes an intrinsic goal oriented motivation component, can bound the evolution to proxy oriented practices. More generally, the model provides a conceptual and predictive framework to empirically assess to which degree actual societal systems may be wastefully oriented towards proxy measures. Importantly, it includes a mechanistic account of how a system can remained locked in to a relatively proxy oriented state, even if all individual agents are conscious of the the true goal. This may help to explain diverse phenomena ranging from consistent statistical underpowering in science to global inaction to counter the threats posed by global warming.

5 Acknowledgements

I thank Heinz Beck for gracious support and Christina Selenz, Laura Ewell, Klaus G. Troitzsch and Gerben Ter Riet for many helpful comments and advice on the various versions of this manuscript. The project was funded through the VW-Foundation program ’Originalitaetsverdacht’.

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