in order to aquire a public service that requires an administrative procedure. The study is based in an agent-based model.

Transcription

1 Bribery Dynamics: An agent-based approach with evolutionary learning Working Paper Omar A. Guerrero Graduate Student Department of Computational Social Science George Mason University Abstract This paper studies the dynamics of bribery that occurs between citizens and bureaucrats when the formers go to the laters in order to aquire a public service that requires an administrative procedure. The study is based in an agent-based model. Citizens adapt their bribes levels through social learning. Bureaucrats develop their decisions through both individual and social learning mechanisms. The act of bribing can be caught and punish with certain probability, provoking the agents to inccur in different types of costs. Unlike most agent-based models of corruption, the penalization for bureacrats consists on a loss of utlility rathaer than jail time, or suspension from the game. This is intended to be a representative condition of the incentives structure of many countries in which this type of bribery is common. Bribing also generates a social costs among the population of citizens. This work is focused in two objectives: replicate the notion of bribery stable levels as a social convention through evolutionary social learning, and to compare two scenarios for hypothesis testing. The experiment testes the idea that societies with disparities in the social cost of corruption across the population, bribery remains higher than in those with flat social cost distribution. Furthermore, the former will be better off than the later. The model accomplishes both objectives, and provides useful insights about the direction and implementation of anti-bribery public policies. 1

2 1 Introduction The problem of corruption has caught the attention of social scientists due to its strong implications with social welfare and economic development. The bribery dynamics between citizens and bureaucrats (hereafter abbreviated as citizenbureaucrat ) is of special interest for countries with developing economies since, in many cases, although it implies a social cost, it seems to be a natural mechanism for economic and social agreements. From the perspective of economic development, this problem creates barriers to individuals who want to get involved in the economic life of the country. This paper studies the dynamics of corruption at such level, trying to replicate the conditions under which it occurs in developing regiongs like Latin America. Once the emergence of bribery is replicated, I test the hypothesis that, in societies with disparities in the impact of the social cost of corruption, bribery remains higher than in those with flat social cost distribution. 1.1 Literature Review From economic studies, we can identify three main broad classes of methodologies by which corruption has been analyzed: (I) neoclassical models, (II) experimental models and (III) agent-based models (ABM). The first class uses traditional economic theory models which involve centralized mechanisms of social interaction and utility maximizing agents. This methodology usually employs econometric tools, with empirical data across countries, in order to validate the models. The first attempt to explain corruption with this type of model was developed by Rose-Ackerman (1975), finding that the incentive structure in the public sector is the key factor to reduce bribery against its counterparts in the private sector. Another pioneering theoretical model can be found in Shleifer and Vishny (1993), where the strength of the government to prevent bureaucrats to impose bribes to private agents is an explanatory variable of the costs of economic development. Models with econometric validation can be found in Gupta et al. (1998) who show the strong relation between corruption and income distribution across different countries. Tanzi (1998) provides evidence that the cost of corruption in terms of economic growth is the main argument why fighting against corruption has to be a necessary part of the state reform for developing countries. An important relation between bureaucrats income and the level of corruption is shown in van Rijckeghem and Weder (1997) and Montinola and Jackman (2002). Class II makes extensive use of experimental economics. This approach consists in the design of controlled experiments in laboratories with real subjects. Researchers in this area usually try to test economic models from Class (I), or draw new conclusions that are difficult to support with aggregate statistical data but can be tested in controlled experiments. Representative studies conducted in this field are found in Abbink et al. (2002), who makes an experimental design based on a game theoretic framework and bribing reciprocity. Bilotkach (2006) presents an experimental model of corruption, considering some conditions of developing countries, and compared the results with the ones obtained by a game-theoretic model, which characterizes equilibriums according to macroscopic population dynamics conditions. Class III consists on the use of computer simulations using agent-based models. This type of implementations are usually oriented towards the study of a large population of agents who interact and learn inside a social network. Investigators will draw conclusions about the patterns and dynamics of the system, as a result of individual behavior of agents and their interactions. The first agent-based model that uses a game-theoretic framework to study the dynamics of corruption with 2

3 heterogeneous agents can be found in Hammond (2000). Situngkir (2004, 2005); Situngkir and Khanafiah (2006) conducted the first attempt to theorize with the structural factors of corruption through agent-based models. Traditional economic models are more concerned with establishing that self-interested agents will produce efficient outcomes as long as there is an effective enforcement of law. The experimental models, in general, have shown that, when testing with real subjects, cooperative behaviour emerges from the interaction of agents. Furthermore, they have shown that there are more efficient outcomes than predicted by theory 1. In the case of agent-based models, researchers usually try to study such cooperative dynamics in large populations and in games with longer rounds, which are impossible to replicate with real subjects. This work is the first to present an agent-based model of corruption using game-theoretic-experimental framework and focusing on the bribery dynamics at the citizen-bureaucrat level. In general, the literature in economics of corruption has been concerned with this problem, at the firm level-government (Rose-Ackerman, 1975; Gupta et al., 1998; Abbink et al., 2002; Shleifer and Vishny, 1993). In this case the assumption of high computational capabilities for calculating the best response strategies may be reasonable due to the nature of the agents and the game: commonly a one shot-game where the firm can have enough resources and information to plan the best strategy 2. Even some experimental studies that work with individual subjects, like in Abbink et al. (2002), are focused in the firm s case while the experimental design is more suitable for the study at the citizen-bureaucrat level. The reason for such persistence may rely on the significant amounts of money involved in government contracts, the direct impact of a bargaining decision on a public policy, and the availability of data related to the topic. Another reason for this focus is that, in developed economies, bribery between citizen and bureaucrat is less frequent due to the relatively high wages across sectors, and the presence of a modern State Proposal One of the difficulties with bribery at the citizen-bureaucrat level is the lack of data. The closest sources of information are surveys about frequency and types of bribery GAUSSC (2006), and personal experience. Although such lack of data, some attempts to theorize with the basic principles of this problem can be found in Situngkir and Khanafiah (2006) and Rose-Ackerman (1975). In the case of some developing economies, bribery at this level is a common practice (Doren, 2008). As discussed in the previous section, there are serious doubts about the degree to which the conclusions and implications extracted from corruption models can be extended to the context of developing countries. I identify four reasons that motivate my work on this topic. 1. Limited understanding of the real dynamics at the citizen-bureaucrat level. 2. Improper representation of the bribery negotiation game at this level (simultaneous prisoner s dilemma). 3. Lack of learning mechanisms that are able to represent the computational capabilities of agents. 4. Poor contextualization of the structural characteristics of developing countries. 1 See Hoffman et al. (1998) for further discussions. 2 Even knowing that the assumption of having complete information is still far from reality. 3 The modern State "is an association which, acting through laws as promulgated by a government endowed to this end with coercive power, maintains within a community territorially demarcated the universal external conditions of social order". (MacIver, 1964) 3

4 The aim of this work is to study the dynamics of bribery at the citizen-bureaucrat level through an agent-based model that makes use of the sequential repeated games framework, evolutionary and heuristic learning mechanisms for the agents, and a contextualization of developing countries. The agent-based simulation replicates the process by which a developing country develops a stable level of bribery that seems to naturally emerge from the system. The previous idea suggests that at some point in the evolution of the system, there is a natural bribery level which becomes a dominant strategy. In the context of cooperative games, Meidinger et al. (1998) have shown some evidence that trust and reciprocity in games results in stability, but in the long run, the usage of such mechanisms is reduced. Joyce et al. (1995) have shown the presence of reciprocity in experiments where trust needs to be developed, along with punishment and award mechanisms. Furthermore, Montinola and Jackman (2002), compare the results from the Investment Game 4 with a simulation where agents learn by reinforcement learning, providing evidence of the reduction of reciprocity in the long run. However, as stated by Abbink et al. (2002),... reciprocity is only one aspect of corruption, therefore these findings cannot be transferred directly to corruption scenarios. The first piece of evidence of a stabilized natural level of bribery is found in Chakrabarti (2001), who provides a general equilibrium model where bribing becomes a dominant strategy for agents. Still, critiques to general equilibrium models in terms of their feasibility in computational terms and top-down construction are well known. Axtell (2005) argues that the existence of a convergence mechanism and the pre-determinacy of the system s result based on the agent s preferences and endowments can yield some enlightenment to have reasonable doubts of how realistic a general equilibrium model can be when having a heterogeneous population of agents. This work is organized in as follows. Section 2 presents some structural characteristics of certain types developing countries that should be considered when designing a corruption model. The main characteristics are: (i) a differentiated justice system for public administrators from the one used by common citizens, (ii) the monopoly power of bureaucratic unions, (iii) the capacity of some social sectors to hedge from the social cost of corruption and (iv) the presence of very long administrative procedures. In section 3 the model is explained in detail. In section 4 I simulate the dynamics by which stable bribery emerges, following the pattern-oriented methodology (POT). Then I conduct a the calibration and verification of the model. Section 5 presents the results of an experiment where I test the hypothesis that, in societies with disparities in the impact of social cost of corruption, bribery remains higher than in those with a flat distribution. Furthermore, I show that the formers are better off than the laters. Next, I provide robustness test for the results, in order to generate valid conclusions. Finally, section 6 shows the conclusions and implications for anti-bribery policy-making. 2 Contextualizing the Model Although several advances have been made in terms of explaining the mechanisms under which corruption exists and how to prevent or punish it, the extent to which these results can be replicated and applied to developing countries is still a debatable issue. One of the reasons is the profound structural differences, historically produced by cultural and traditional factors. Seriously challenging ideas and discussions in the topic have came from the Political Science rather than 4 See Joyce et al. (1995) for a detailed proposal and study of the investment game. 4

5 Economics. Dwivedi (1967) tries to find out if corruption in developing countries is really an obstruction to economic and political development, or a catalysis in harnessing the process of modernization, showing evidence of its necessity during the maturity process of political systems in developing countries. For the case of Latin America, Seligson (2002), provides evidence supporting the counter argument to the idea that corruption in the developing world enables citizens to overcome intransigent, inefficient bureaucracies while increasing loyalty to the political system. The Mexican National Survey for the Perception of Corruption (GAUSSC, 2006) shows that 40% of the citizens believe that bribery at the citizen-bureaucrat level is necessary. Out of 8 main reasons, and with a weight of 35.3%, the most important motive to bribe is to speed up bureaucratic procedures. Whichever the arguments are for the context of Latin America, it is always necessary to identify the structural factors in the legal and social systems that make this region different from others in terms of incentives in the public administration. In this region of the world, the structure of the legal system to prosecute public servants, the ineffectiveness of the justice system, the monopolistic power of bureaucratic unions, the heterogeneous impact of social cost of corruption, and the never ending administrative procedures are some of the most important structural reasons why recommendations derived from standard studies have not succeeded. A large part of the literature, especially the one based on experimental economics and agent-based models, is highly concerned with the effect in changes of the probability that the supervisor catches a corruption act, taking for granted that the justice system will punish effectively the corrupt agent 5. Furthermore, many of these models assume that the punishment for the corrupt bureaucrats is either being fired or going to jail. From the normative and ethical points of view, these premises might be justifiable and even desirable. Recommendations built on these kind of studies are directed to the Executive power of developing countries, while the Legislature and the Judiciary are left aside and untouched 6. Under this scenario, the only government player is the Executive Power, left with public policy tools and public administration rules as all its strategies. In Latin American countries, the previous situation can be better understood by analyzing the legal system structure rather than just to the economic incentives. Latin America inherited the continental law tradition that was developed in Germany, France, Italy and Spain. Under this approach, the Public Law developed the State of Law, while the British tradition created the Rule of Supremacy of Law. (Entrena Cuesta, 1961) The main difference relies on the legal conditions under which public servants are prosecuted. In the former, bureaucrats are subject to the Administrative Law, which contemplates special courts for solving any issue concerned with the officials performance. Under this system the public servants enjoy great respect, preeminence and their own juridical regime. Under the Rule of Supremacy of Law, public servants are subject to the same tribunals as private individuals; they have the same reputation as any other citizen and do not enjoy of any juridical privilege for being officials 7. Another factor that situates public servants in a privileged position is the strong power possessed by the bureaucratic unions. In Latin American countries it is very common to have a unique bureaucratic union representing all the public servants in a government agency that earn a salary relatively low. The monopolistic condition of these organizations lets them have a high negotiation power against the government directors. In several cases, even if a bureaucrat is spotted in 5 Apparently this approach is product of the success of the application of the Principal-Agent model to multiple supervision problems in the social sciences. 6 In several cases, these the Legislatures or the Judiciary account for higher corruption indicators than the Executive power. 7 See Entrena Cuesta (1961) for details on the difference between State of Law and Rule of Law. 5

6 an illegal activity, her union will negotiate the punishment with the supervisors, which in most of the cases is resolved in a change of position instead of dismissal or jail time. Since most of the bureaucrats represented by the unions receive a relatively low wage, the incentives structure provokes them to behave with conformism 8. Autonomy of corrupted Legislature and the Judiciary power, joint with legal privileges for public servants not only imposes serious barriers for effective prosecution but also has a direct influence on the incentives structure in the public administration. This condition is called the sala model (Riggs, 1964), where sala stands for living room in the administrative Ibero-American jargon. Sala model alludes to the state of personal or familiar privacy which allows discretionary non-administrative behaviours among public servants. Under the conditions of sala, the bureaucracy appears to be a interference factor rather than an administrative one, as well as a source of privileges 9. Under these conditions, the only way that government supervisors can prevent bureaucrats discretion is through the imposition of numerous administrative regulation requirements when conducting a procedure for a citizen. The imposition of many administrative rules, added to the conformism of bureaucrats can derive in extremely long periods to finish an administrative procedure that a citizen may demand. In several occasions, these procedures may seem never ending, making it easier for a citizen to propose a bribe to the bureaucrat to accelerate the process. Another important element to consider is the impact of social cost. Abbink et al. (2002) proposes an experimental model in which there is a social cost that impacts every subject each time there is a bribe transfer. This treatment is presented as an externality, and is highly relevant for the topic. Nevertheless, in developing countries, there is an important issue related to the income distribution and access to oportunities that should be taken into account: due to economic and social disparities, different segments of the population have access to distinct privileges in the political system. If such privileges can be used as means to absorb, transform, evade or transfer the social cost of corruption, then its impact will differ depending on the population subsets affected. The idea is that, even if there is a social cost generated every time there is a bribe transfer, its impact varies in level and homogeneity across the population. It follows that an agent-based model should be able to test hypothesis that involve homogeneous versus heterogeneous impact of the social cost of corruption. This characteristic will affect the learning process of the population, the final level of bribery, and the aggregate welfare of the society. 3 The Model A citizen Requires a public service from the government. The procedure for the service can vary, for example, from obtaining a driving license to a permission of opening a business. The main point is that the citizen will get a benefit from it. Since the citizen has already invested an amount of resources 10 (even the time spent in going to the official s desk), the more time it takes to do the procedure, the less the value of the benefit obtained from it. Every procedure takes an amount 8 A common phenomena in Latin American public administration is the employmania, which consists in the excessive creation of job positions even if they are not necessary. Those positions are created with the intention of distributing the loot obtained after political elections. Employmania generates conformism among the employed ones. It procures a safe life for the aspirants because the employed ones work infinitely less than a craftsman or a laborer: all their goals are reduced to the procuration of a promotion or a retirement that will let a free position for another aspirant. The custom of living from the employs produces conformism because it destroys the necessary talent for invention and perfectibility. (Mora, 1963) 9 See Riggs (1964) for further discussions on the sala model 10 There also exists the possibility that by the time the procedure is done, the service will not be useful anymore. 6

7 of time, which is public information. Citizens know the official time required for the procedure, therefore they can have an idea of the value of the benefit obtained. In order to increase the benefit, they can offer a bribe to the bureaucrats, who are able to reduce the time it takes to do the procedure (at the cost of increasing the chances of being caught). Every time an agreement which involves time reduction is reached, there is a chance that the bureaucrats will be caught and penalized by the authority. This probability varies upon how much the bureaucrat helps the citizen. The relation between the time reduction and the probability of being caught is inversely proportional. That is, a decrease in the procedure time of 10% will increase the chance of being spot by 10%. The basic framework of the model is partly inspired in the game-theoretic-experimental design of Abbink et al. (2002). The game setup has the following elements: (i) (ii) (iii) A population of agents Payoffs for the agents Utility functions Element (i) is of size N and is even. The population is composed of two types of agents: Citizens and Bureaucrats 11. Half of the population is of Citizens and the other half of Bureaucrats. In contrast to Hammond (2000); Situngkir (2004, 2005); Situngkir and Khanafiah (2006) models, which assume a simultaneous prisoner s dilemma framework, this model considers a sequential game because it is more appropiate for games that involve trust and reciprocity (Joyce et al., 1995). The payoffs (ii) of the game are detailed in Figure 1. Utility functions (iii) will be specified in sections 3.1 and 3.2. During period t, each Citizen is matched with a random Bureaucrat, so that they negotiate the bribe. Citizen i proposes a bribe b i,t, next, Bureaucrat j accepts or rejects. If he accepts then he decides the amount of time y j,t to be reduced from the official time x t of procedure. If a bribery agreement is reached, then a lottery is played with probability α t to decide if the government will spot the bribery agreement. After each round, the agents learn from their history of strategies using the learning mechanisms that will be presented and discussed in sections 3.1 and 3.2. Figure 1 provides a general illustration of one round of the sequential game with the corresponding payoffs for each possible outcome, where the upper argument inside the brackets is the Citizen s payoff and the lower one is the Bureaucrat s one (full details of parameters are given in sections 3.1 and 3.2). 3.1 Citizen Agents As observed in Figure 1, the Citizen s utility comes from the function: U i,t (b i,t ; x t,y j,t )=Π i,t δ xt yj,t γ i,t Θ t c i,t b i,t (1) where Π i,t is the benefit for Citizen i obtained from the administrative procedure at period t; δ [0, 1] is the discount factor of the procedure benefit; x t is the official procedure time; y j,t is the reduction in time chosen by Bureaucrat j; γ i,t is 11 From this point on I will refer to the representative agents of the model, starting their names with capital letters. In the case of third person reference, I will use feminine pronouns for the Citizen, masculine ones for the Bureaucrat and neutral pronouns for the Government. 7

8 Figure 1: Sequential game at period t the impact parameter that the total social cost Θ t has on Citizen i. The parameter c i,t is the cost for Citizen i of establishing a negotiation procedure with Bureaucrat j, for bribing him, thus, if she decides not to bribe then the negotiation cost will be zero. Finally, b i,t is the bribe chosen by Citizen i. In the case a bribery agreement has been reached and caught by the government, the administrative procedure is annulled, leaving the Citizen with the burden of the total social cost, the negotiation cost and the spent bribe. The bribe is kept by the Bureaucrat because, compared to a firm s bribe, a citizen s bribe is relatively insignificant, and the cost for the government to trace it is too high. In other words, at the citizen-bureaucrat level, the signal that the Government has about corruption is not the unexplained enrichment of the Bureaucrat, but the fact that he is skipping certain rules when working on administrative procedures. According to Meidinger and Terracol (2002), the two broad classes of predictions in learning models observed in experimental games are the belief-based models and the reinforcement-based ones. Agents who fall in the first class are able to keep track of their history and to build complete contingency plans for all possible outcomes. In the case of Bayesian, sequential and repeated games, the Nash equilibria refinements such as sub-game perfect equilibrium, perfect Bayesian equilibrium and sequential equilibrium are useful to rule out non credible strategies, thus, being able to predict the most realistic Nash equilibrium outcomes. In the case where agents are firms or organization, the assumption of high computation capabilities may be fair to some extent. Such agents are built by smaller components who can perform simpler tasks, but in the aggregate can achieve certain planning and forecasting abilities that a single individual could never do 12. However, for the case in which the agents are single individuals such as citizens and bureaucrats, the assumption of high computation capabilities can be easily challenged. Kaneko (2005) makes a clear statement by pointing out that, for an infinitely repeated game, the 12 The principle or large systems assembled of smaller units capable of doing simple rules and interact with each other. Is one of the basis for the analysis of complex systems in terms of information and computation capabilities. The idea that these aggregated units can produce more complex units with higher capabilities and the need of analyze the dynamics entire system are part of the legacy of John von Neumann s work on self-reproducing automata. (Mirowski, 2002) 8

9 Folk Theorem 13 requires the assumption that players are able to perceive an enormous number of strategies because they have a plan for every possible outcome. For each period t played, the number of information sets Y t grows according to the difference equation Y t =4 t + Y t 1 with Y 0 =1. Thus, in the case that the game is played 5 periods, the number of strategies to be considered by each player is 2 341, which is larger than 2 266, the number of protons and neutrons existing in the entire universe 14. Due to the evident limits in computation capabilities, the learning mechanism used by the Citizens falls into the reinforcement-based class. The learning mechanism in this model is population-wide. This means that agents learn through the success or failure of strategies chosen by clusters across the population through time. In order to capture this idea, I am using the replicator dynamics (Hofbauer and Sigmund, 1998). Vega-Redondo (1996) argues that in real world there are decision problems that are extremely complicated. Therefore, given our limited capabilities of attention, computation and association, we end up turning to the usage of simple rules that have worked well in our past history. Bribery at the level of a citizen is more about culture and inheritance of strategies than of fairly complicated estimations: whenever a citizen attends a government office requiring an administrative procedure, she will not offer a bribe depending on her expectations about the possible responses of the bureaucrat, rather she will take her bribing decision upon the cultural prevalence of bribing as a standard way to get an administrative procedure done. She will have inherited the social success of bribing and reinforce it with the feedback she gets from the process outcome. The replicator dynamics is the best model for this situation since it is not a best reply dynamics. Agents in a replicator system have limited and localized knowledge concerning the system as a whole 15 (Gintis, 2000). The discrete replicator equation used for this model is (2), where Z k,t is the proportion of population who plays a pure bribing strategy k in period t and U k,t (b k ) is the average utility of those Citizens who bribed with k at time t. U k,t (b k ) Z k,t = Z k,t 1 (2) U k,t (b m )Z m,t 1 m The equation determines the proportion of those Citizens who bribe at k by taking their average utility as a fitness measure, which then is divided by the weighed average utility of all the Citizens population and then multiplied by the proportion of those who played k at period t 1. Before proceeding, I must make a clarification regarding the use of the replicator dynamics. This model is a difference equation (or differential in the continuous case) that describes the dynamics of populations of agents that are predetermined to do only one type of strategy (Weibull, 1995). Its origins derive from the demographical studies of species by evolutionary biologists (Smith, 1982). Only those agents that are predetermined to perform the best strategy are the ones that will survive and inherit their success for the next round. It follows to classify this model as one with a top-down construction, since it is predefining the conditions under which the population s dynamics will rely, leaving no room for agent s autonomy, purposefulness, and learning. From this description, it may seem that its usage is a contradiction of this model since it leaves out a bottom-up construction and agents heterogeneity (Axtell, 2000). However, the replicator dynamics it not being used 13 The Folk Theorem states that Pareto optimal outcomes may be achieved in Nash equilibria when a one-shot game is infinitely repeated. (Kaneko, 2005) 14 See Kaneko (2005) for details on this argument 15 Localized knowledge or the system can be understood as the mental model that agents have about the world. 9

10 in the same form as in the traditional models. In this model, each Citizen has a set of strategies (bribing levels) to choose as an offer for the Bureaucrat. At the beginning of the simulation, they select the strategies from an initial uniform probability distribution. After each iteration, the replicator builds a propensity distribution for the chosen strategies of the population. The magnitude of each propensity is directly related to the aggregate payoff of those Citizens who chose the corresponding strategies. Finally, each Citizen updates her strategies probability distribution by constructing one, based on the population frequency distribution. Therefore, the role of the replicator dynamics is not of predetermining a unique strategy for the agents, but of implementing a social learning mechanism. Heterogeneity of Citizens relies on the fact that they randomly choose their strategies based on the social learned probability distribution, leaving room for changing strategies and adapting when the environment affects the payoff of the game. The selection of the replicator dynamics as the social learning mechanisms was motivated by stability properties (Samuelson, 1997) that permit the best replication of bribery emergence under the model s context. for (each citizen i) chooserandomstrategy(strategiespropensities); end; bargainprocess(citizenpopulation, bureaucratspopulation); computepayoffs(); computepopulationpropensities(populationpayoffs); for (each citizen i) updatestrategiespropensities(populationpropensities) end; Algorithm 1: Pseudo-code of the adaptation of the replicator dynamics 3.2 Bureaucrat Agents As seen in Figure 1, the Bureaucrat decision process consists of two stages. Stage I corresponds to the simplest model of a public servant; he only has two choices: accept the bribe or reject it. At stage II, if the bribe was accepted, then the Bureaucrat proceeds to choose the amount y j,t to reduce from the official time x t required for the completion of the procedure. The total learning mechanism implemented by this agent is a hybrid of reinforcement learning 16 and population-wide learning. The best way to explain it is in a backwards fashion. In stage II: y j,t =ỹ j,t x t, where ỹ j,t is the proportion of the official time x t to be reduced by the Bureaucrat. Therefore, the strategy space is [0, 1]. The Bureaucrat has a set of initial propensities for choosing each strategy. At the beginning of the simulation the propensities distribution is uniform. The Bureaucrat also has a probability distribution over the set of pure strategies for ỹ j,t. The distribution function assigns probabilities to each strategy k as shown in (3), where p j,k (t) is the probability for Bureaucrat j at time t of choosing strategy k, and q j,k (t) is the respective propensity. p j,k (t) = q j,k(t) q j,m (t) m (3) 16 In reinforcement learning agents care about the previous payoffs as a feedback from their chosen pure strategies. High feedback augment the probability of choosing such strategy. (Erev and Roth, 1998) 10

11 The obtained feedback is computed at every outcome of the game through the utility function (4) of the Bureaucrat, where w t is the wage b i,t (y t ) is the bribe offered by Citizen i (which is a function if the history of all previous offered bribes y t ) and p t is the penalty for those who are caught in an illegal activity. The parameter a {0, 1} tells if the bribe was accepted or rejected, and f {0, 1} says if the bribing agreement has been caught or not. V (w t ; b i,t (y j,t )) = w t + a b i,t (y t ) f p t (4) Once the utility has been received, the Bureaucrat updates his propensities distribution according to (5) and then proceeds to update his probability distribution over the strategy set using (3). q j,a (t)+v(w t ; b i,t (y j,t )) if a = k q j,a (t + 1) = q j,a (t) if a k (5) It is evident that when choosing ỹ j,t, the Bureaucrats learn only of their own actions. The reason for this behaviour is that bribery acceptance as an average of the Bureaucrats population may be something of common knowledge. However, it is still illegal, therefore, spreading detailed information about the levels of the accepted bribe and the reduced time is risky at the individual level. Thus, Bureaucrats decide the amount of time to reduce from the procedure based only on their own feedback. As mentioned before, the general custom of accepting or rejecting bribes across the Bureaucrats population is common knowledge, hence they are able to use that information to adapt their behaviours according to the population average 17. This is when stage I of Bureaucrats learning is introduced. At this point the agents have to face a simpler decision: accept or reject. Bureaucrats have propensities and probabilities for both strategies in the same fashion as (3) and (5). Both distributions are also updated after receiving feedback, however, every time a Bureaucrat faces this binary decision, he will mimic the average acceptance/rejection propensity. After adopting the same propensities and probabilities as the bureaucrats population average, he will decide his strategy. If he picks to accept the bribe, then he proceeds to stage II, otherwise the outcome is as shown in Figure 1. for (each bureaucrat i) acceptorrejectbribe(bribeoffer, socialacceptancepropensity); if(accept) selecttimereduction(strategiespropensities); end; computepayoffs(); updatesocialacceptancepropensity(); for (each bureaucrat i) updatestrategiespropensities(individualpayoff); end; Algorithm 2: Pseudo-code of Bureaucrats learning mechanism 17 Alignment of an individual agent to the average behaviour of the population is a protection mechanism since the agent can mimetize with the population and avoid exposure of its irregular behaviour. 11

12 Parameter Calibrated Value Π i,t 200 (for every Citizen i) δ t 0.9 (constant for every t) x 0 16 c i,t 2 { (constant for every t) } b i,t 0, 0.5, 1,..., Π i,t δ (xt yt) t w t 37 corruption externality 3 total population 1000 Table 1: Initial calibration 4 Experiment Design and Calibration The experiment design and calibration are aimed to fulfill two objectives. First, we need to replicate the dynamics of bribery as a stable. However, since we are using an agent-based model, we need to design it in such way that both Citizens and Bureaucrats develop bribery as a macroscopic pattern of the system through interactions, learning mechanisms, and strategic decision-making. In the ABM literature, this is known as Pattern-Oriented Modeling (POM) (Grimm, 2005). Although we do not count with clear empirical patterns, I will rely on the intuitive notions and general surveys (GAUSSC, 2006), which give a fairly qualitative picture of the problem. In general, the system should be able to generate a social learning process, which will drive the population to stabilize a natural level of bribery that will endure through time. Second, the model should be robust enough to test different hypothesis. The tested hypothesis is the idea that societies with disparities in the social cost of corruption across the population, bribery remains higher than in those with flat social cost distribution. Furthermore, the former will be better off than the later. The simulation was programmed in Java language, using the Ascape toolkit (Parker, 1998). Initially, the calibration of the experiments was based on the parameters used by Abbink et al. (2002). Their selection is justified on the fact that the model structure of Abbink et al. (2002) is the closest to the one presented in this work. Although the usage of all ready calibrated parameters, this model has structural differences from the one of Abbink et al. (2002). Therefore, some parameters were calculated in a different manner. In order that the model could produce reactive and consistent dynamics, I designed a series of experiments for the calibration of the penalty level and the probability of catching corruption. 4.1 Searching for Consisten Behavior The first exercise was to run several simulations for different probabilities of corruption (α t ), and a penalty equal to the Bureaucrats wage in order to observe if there is a relation between the average bribe levels and α t. I ran 200 experiments for each probability segment as follows: α t =0, 0.1, 0.2,..., 1. The results were not clear as I increased α t, so I changed the search strategy after reaching α t =0.5. Analyzing this result a little closer, it makes sense not to expect a well defined relation between these two variables for the following reasons: 12

13 p Average bribe level SD Average time reduced SD Table 2: Initial sampling The punishments are enforced on the Bureaucrats, not on the Citizens, so even if the Bureaucrats stop accepting bribes, the Citizens can still offer bribes. The externality impact is homogeneously distributed, thus, if a sufficiently large part of the population bribes, the externality will be sufficiently large to overcome the other costs of bribing. These conditions produce an effect of constant costs, even for those Citizens that are not bribing, therefore the marginal outcome difference between strategies can be almost imperceptible when providing feedback to the Citizen s propensities. Hence, the cost produced by bribing does not necessarily guarantee that the population will drop the bribes down to zero. Nevertheless, we should expect a negative relation between α t and the time y t chosen by Bureaucrats. In this case the results were only consistent for the first 3 levels of α t, but not for the rest. The previous results suggests that the segmentation of α t was not an adequate form of sampling the space; therefore, I ran 2, 000 experiments, sampling the interval [0, 1] with 10 decimals of precision. Again, the results were unclear for the case of average bribing levels, and partially clear for the average time reduced. In Figure 2 we should expect a negative relation if the Bureaucrats do not respond accordingly to the Citizens incentives. We could also expect a positive relation in case the reward for bribing was high enough. Additionally, we could expect both types of relations, separated at a critical values of α t. However, the result appears to be random, making it difficult to do any inference about the model. Figure 2 shows a clear negative relation for probabilities α t < 0.2. For the rest of the space, there does not seem to be a clear effect of the punishments on the Bureaucrats incentives structure. This result can be a first sign of a critical value, however, this intuition needs to be confirmed with more experiments. A possible explanation for this behavior is that the penalty imposed by the government is not high enough (by default it is equal to the Bureaucrat s wage). In order to find such a relation, I need to identify which is the range of penalties that allow sensibility to changes in α t. 4.2 Working on Bureaucrats Incentives First, I created a parameter that indicates the iteration at which the average y t undergoes 1%. This parameter is a proxy of how fast or slow is the learning process for the Bureaucrats population. Therefore, if we have low penalties, learning should take more, and with higher penalties it should be faster. It must be noted that there must exist a penalty level, at which the character of learning breaks down, becoming almost instantaneous, which leaves no space for negotiation between Citizens and Bureaucrats or reactive behavior. From the point of view of corruption legislation, excessively high penalties 13

14 Figure 2: Time reduced by bureaucrats in general sampling Penalty Range [37, 1000) [1000, 2000) [2000, 3000) [3000, 4000) [4000, 5000) Variance SD Table 3: Variance of learning time for Bureaucrats are are not a feasible policy for several reasons. Penalties that are too high would turn a bureaucratic job in a risky position, with huge costs in case of personal conflicts with other public servants. It would be hard to think that a bureaucrat union labor would not react to the imposition of this type of penalty. Especially in countries with corruption problems in the judicial system, this kind of penalties is not a suitable measure as a first strategy to fight bribery. I ran 2,000 simulations, sampling penalties in the range (37, 5000) and the entire probabilistic space of α t. As shown in Figure 3, the general trend follows a power law in the learning time for penalties in the range (37, 600), and almost constant for those penalties larger than 600. Table 3 shows the sensibility of Bureaucrats learning time for different penalty ranges. This sensibility is measured in terms of the standard deviation. If a range of penalties has a high variance, it means that there is a higher probability of having different learning times for the same penalty level, but at different probabilities of catching corruption. Thus, those ranges with a higher variance might represent signs of response, or promising areas for finding sensibility to modifications. As shown in Figure 3, the range (37, 1000) is the one with the highest variance, therefore the most promising for finding a sensible parameterization of the penalty level. I ran 1,000 simulation sampling the probabilistic space of α t, and the penalty range (37, 100). The results shown in 4 depict a clear negative relation between the penalty levels and the time it takes for Bureaucrats to decide to reduce y t less than 1%. Furthermore, the model throws the first critical parameter. As observed in Figure 4, the average of the realizations for each penalty level does not follow a consistent declining tendency, until it reaches 14

15 Figure 3: Bureaucrats learning time as penalty increases p <=0.2 >0.2 Average SD E-05 Table 4: Proportion of time reduced penalty levels greater than 50. More specifically, 52 is the critical level of penalty of which Bureaucrats react and learn in a less-randomized fashion to the probability of being caught. Any penalty below that level will result in erratic decision making by Bureaucrats. 4.3 Shrinking the Search Space of α t In order to revise the intuition suggested in section 4.1, that Bureaucrats learn the amount of y t in a coherent way when α t is less than 0.2, I ran 2, 000 experiments sampling the complete probabilistic space and the reduced penalty search space (52, 100). Since there are 48 integers, each penalty level was tried out in approximately 40 experiments. As seen in Figure 5, the y t reduced by Bureaucrats evolves smoothly when penalties are lower. We can identify the penalty level of 70 as a critical value for this characteristic. When penalties become higher than 70, the slope of time reduced by Bureaucrats becomes too steep, leaving those α t < 0.2 without effect on the agents learning process. Under this criteria, we can discard every penalty level above 70 and bound our search space to a new set: (53, 70). About α t =0.2 being a critical value, so far it appears to be correct, since it is the furthest value where α t provokes sensibility in Bureaucrats decision making. I obtained the average percentage of y t for each group of experiments of each penalty level and observed its evolution across the probability space (0, 0.2). As shown in Figure 6, the decision of Bureaucrats of how much time to take off from the administrative procedure is indeed decreasing when the probability of catching corruption is less than 0.2, and it reaches its constant level of almost zero are such critical value. Additionally, Table 4 shows the difference in mean and variance of the proportion of time reduced from the procedure before and after the 15

16 Figure 4: Consistent learning of Bureaucrats critical value α t = Selecting a Penalty Level I ran 2,000 experiments for the 16 integers in the search space for penalties, sampling the probability space (0, 0.2). After obtaining the data I noticed that the evolution of y t could be fitted to a linear and an exponential function, with a high R 2. I proceeded to run regressions for the data of each penalty level, for both the linear and the exponential models, obtaining that in some cases the linear had a better fitness than the exponential, and some other times it was the other way around. Figure 7 depicts that, although the R 2 of most of the regressions was high, there was not a dominant model between the linear and the exponential one, therefore I had to define some criteria to discriminate penalty levels. I defined three domination criteria to filter penalty levels and select the most appropriate one to represent a high penalty: 1. A dominant penalty is one for which the difference between the R 2 of its linear and exponential models is less than A dominant penalty is one for which both of its regression models have R 2 higher than A dominant penalty is one for which the middle point between its highest and lowest R 2 is the highest. Table 5 shows the filtering process of the 16 penalty levels. Level 55 was the only one that passed the three domination criteria. Lets remember that the selection of this parameter follows a long and detailed search process, based on the criteria that seeks for coherent behavior in the learning process of agents. Therefore, level of 55 is the best calibration since it permits consistent, reactive, and purposive behavior of Bureaucrats. 16

17 Figure 5: Evolution of time reduced from administrative procedure Penalty Criteria 1 Criteria 2 Linear R 2 Exp R 2 Criteria Table 5: Filtering process and high penalty selection 17

18 Figure 6: Total evolution of time reduced before threshold Figure 7: Regressions R 2 for penalty levels 18

19 Figure 8: Reduced time distributions with five different probabilities of catching corruption. The blue distribution is the real one, estimated with Kernel Smoothing method. The red distribution corresponds to a normal. α t Mean Variance Skewness Kurtosis K-S p-value Table 6: Statistical moments and K-S test p-value 4.5 Studying Distributions in the α t Space I took 5 values of α t and ran 500 experiments for each one. After obtaining the data of the results I analyzed the distribution of the time reduced by Bureaucrats, and how it fitted a normal probability density function. From the five values, α t =0.15 produced the best distribution. The probability distribution (obtained using a kernel estimation) of the results of time reduced by Bureaucrats for 500 experiments behaves very similar to a normal distribution. This selection was based on the comparison of the first four statistical moments, and a Kolmogorov-Smirnov (K-S) test. The values of α t that passed the K-S test were 0.01, 0.05, and From those three, the last was the one with the smallest variance and skewness. Finally, all the tested distributions were leptokurtic, with 0.05 being the smallest and 0.15 the next one. There is a another important argument for selecting α t =0.15 as the critical value to explore: 0.15 is the only value that passed the K-S test and is located between two values that did not pass it. This tells us that there are some very interesting dynamics 19

20 Parameter Calibrated Value Π i,t 200 (for every Citizen i) δ t 0.9 (constant for every t) x 0 16 c i,t 2 { (constant for every t) } b i,t 0, 0.5, 1,..., Π i,t δ (xt yt) t w t 37 α t 0.15 penalty for bribing 55 corruption externality 3 total population 1000 Table 7: Final calibration Figure 9: Typical run, formation of stable bribery level and this probabilistic space. 5 Results After calibration, the first objective was accomplished: the model replicates the dynamics of a social learning process, in which Citizens repeatedly bargain bribes with Bureaucrats, and eventually, stable levels of bribes emerge and endure through the rest of the simulation. This matches the pattern observed in certain developing regions such as Latin America. Figure 9 shows a typical run of the model, where the evolution of the average bribe level starts at a high value due to the uniform probability distribution of strategies for each Citizen. As the simulation advances, the agents learn from their history of 20

21 Treatment Average Bribe Reduced Time Citizens Average Utility Bureaucrats Average Utility Homogeneous % Heterogeneous % Table 8: Main variables in two treatments selections, payoffs, social transmission, etc. 5.1 Findings I ran 3,000 experiments under a treatment of homogeneous social cost impact, and 3,000 for the heterogeneous case. Homogeneity and heterogeneity is expressed in terms of the distribution of the impact of the social cost across the population. The results are summarized in Table 8. Apparently, heterogeneity accounts for corruption levels. Both societies had exactly the same incentives structure. The society with heterogeneity in social impact had agents that were minimally impacted by the social externality of bribing, therefore, they were willing offer higher bribes and get better discounts in the time required for the procedure, regardless of the impact on the rest of the Citizens. Those agents that were highly impacted by the externality had to bribe, since it was the only way to have a possibility of being better off. Even without bribing, they would incur in the social cost due to the Citizens with lowest impact of the social cost. This situation drove the Citizens to offer a higher level of bribes than the homogeneous society; therefore, Bureaucrats were more generous in their selection of y t. The average utility was higher for both types of agents under the heterogeneous treatment. The opposite situation happened in the homogeneous society, where bribing was socially costly. The previous mechanism is part of the model verification The simulation behaves as expected, and it seems that the search area chosen for calibrating the parameters was the adequate one. However, there are still two issues regarding the robustness of the results. First, we need to verify that the difference between the outcomes of the two treatments is statistically significant. Second, it might be the case that the sample of 3,000 is just a specific cased in which the mechanism of the model happened to work properly. If it was the case that the results are robust, then we should observe that by increasing the size of the sample, the first two statistical moments should show convergence to their real values. We need to verify that the results fall into the last case. 5.2 Robustness The first robustness test is the one of statistical significance of the differences between the means of each variable for each treatment. Table 9 shows that two variables follow a normal distribution and the others do not. For those with normality, I used a paired t-test, and for the other variables, a K-S tests to compare if both variables come from the exact same distribution. As shown in Table 10, all the tests confirmed that every difference between the treatments is statistically significant. For the second test, I computed the mean and variance of the four relevant variables. The computation was done for each size of sample between 2 and 3,000 experiments. If the results are robust, then we should observe stabilization in the parameters as we add more experiments to the sample. According to Figures 10 and 11, as we increase the sample size, the 21

22 Treatment Average Bribe Reduced Time Citizens Average Utility Bureaucrats Average Utility Homogeneous 5.96e e-5 Heterogeneous 1.74e Table 9: P-values for K-S tests of normality Test Average Bribe Reduced Time Citizens Average Utility Bureaucrats Average Utility t-test N/A 0 0 N/A K-S test 3.69e-7 N/A N/A Table 10: P-values for difference in means of treatments mean and variance of each variable behave stable. They seem to converge to a value, and we do not observe the shrinking variance problem. The only exception seems to be the mean of the percentage of time reduced, which appears to exchange positions repeatedly between the two treatments, nevertheless does not show signs of volatility. We can say that these results are coherent and robust enough to draw conclusions from the model. 6 Conclusions The agent-based bribery model with evolutionary learning is one of the first attempts to understand the problem of bribery at the citizen-bureaucrat level, from a bottom-up perspective. It is capable of replicating the notional dynamics under which bribery stable levels are achieved through agents interactions and strategic behavior with computational constraints. Under the adequate calibration (one that produces reactive and well-distributed behaviors), the model is robust enough to provide further insights of the problem, by allowing hypothesis testing. In this work I test the idea that, in societies with disparities in the distribution of the social cost of corruption, bribery remains higher than in those with uniform distributions. Further results were found in the fact that, although both societies produce stable bribery dynamics, utility levels are higher for heterogeneous social cost of corruption. The social interpretation given to this results is that, in the presence of unequal distribution of the social cost of corruption, agents with low impact are willing to take advantage of their position, regardless the third party effects. Eventually, the social cost becomes large enough to incentivize bribery among agents with high impact, since the marginal social cost generated by them is relatively small compared to the marginal potential benefit from bribing. In other words, when the distribution of social costs of corruption is uniform in a population, the marginal cost of bribing becomes higher for the population, while it becomes lower in the opposite case. The results can provide us with some pointers of how to approach the problem of corruption in countries with this type of incentives structures. As explained in this work, the legal structure of certain countries (like several in Latin America) is such that it generates privileges to bureaucrats. This condition, together with the power and inside-corruption of bureaucrats unions, corruption at high political hierarchies, and other external political interests makes it very difficult for governments, to enforce new regulations that can modify the punishments regime and change the incentive structures of agents. The insight provided by this work is founded on the finding that public policies that are aimed to reduce bribery, do not have to start by implementing the hardest task: modifying the public sector incentives structure. A first step can be to 22

23 Figure 10: Evolution of variables means (blue: homogeneous, red: heterogeneous) implement public policies that make it socially costly to bribe. The distribution of the impact of social cost of corruption does not exclusively mean income distribution, but also distribution of access to public services and opportunities of several types of economic and political activities. As long as a society has an unequal distribution of the social cost of corruption, bribery will emerge as a natural mechanism by which agents will be better off. Finally, we can think of the problem of bribery, not as one that will be solely solved by the instant enforcement of government reforms. But as one that needs to be attacked from distant approaches, which will produce third party effects in society. We expect that, starting by implementing feasible public policies, the social system will transmit their impact into the domain of bribery, turning it into a socially expensive activity. Eventually, there society should achieve a state in which it is feasible to implement the political reforms that will change the incentive structure of the public sector. 23

24 Figure 11: Evolution of variables variances (blue: homogeneous, red: heterogeneous) 24