Corruption driven by imitative behavior

Economics Letters 117 (2012) 84–87 Contents lists available at SciVerse ScienceDirect Economics Letters journal homepage: www.elsevier.com/locate/ecolet Corruption driven by imitative behavior Elvio Accinelli a , Edgar J. Sánchez Carrera a,b,∗ a Facultad de Economía, UASLP, Av. Pintores S/N Fraccionamiento Burócratas del Estado, CP 78263, San Luís Potosí SLP, Mexico b Postdoctoral Researcher at the Department of Economics and Statistics, University of Siena, Pzza San Francesco 7, I-53100, Siena (SI), Italy article info Article history: Received 8 June 2011 Received in revised form 20 April 2012 Accepted 25 April 2012 Available online 3 May 2012 abstract One can restructure institutions, but if individual-level motivations for corrupt behavior are not understood, these restructuring may not be effective. We introduce an evolutionary-game modeling to deal with the problem of corruption driven by imitative behavior. © 2012 Elsevier B.V. All rights reserved. JEL classification: C72 C73 D02 K42 P37 Keywords: Corrupt behavior Evolutionary dynamics Imitative behavior Institutions and operations 1. Introduction Corruption is defined as behavior that deviates from implicit or explicit behavioral norms with or without legal and ethical connotations ruled out by institutions (see Mishra, 2006). Acemoglu and Robinson (2005) pointed out that of primary importance for economic performance is the type of institutions in society, since they influence the structure of economic incentives and economic agents’ behavior. The scourge of corruption in Mexico is a nice real example of corrupt behavior driven by imitation, i.e. to be corrupt if others are also corrupt. The tenacity of corruption in Mexico has not kept politicians from promising to eradicate it, since Mexicans correctly identify corruption either as the root of Mexico’s development problems or as the cause of a poverty trap through the cultural behavior of doing as others do (see Wydick, 2008). Corruption occurs at all levels of Mexican society, for instance, the legendary case of the 114 million dollars deposited into Swiss bank accounts by Raul Salinas, brother of the former President Carlos Salinas ∗ Corresponding author at: Department of Economics and Statistics, University of Siena, Pzza San Francesco 7, I-53100, Siena (SI), Italy. Tel.: +39 3392551833; fax: +39 055 0944123. E-mail addresses: elvio.accinell@eco.uaslp.mx (E. Accinelli), carrera.edgar@gmail.com, sanchezcarre@unisi.it, edgar.carrera@uaslp.mx (E.J.S. Carrera). 0165-1765/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2012.04.092 de Gortari, who shortly thereafter fled to exile in Ireland. The money was believed to have came through relationships with Mexican and Colombian drug cartels during his presidential period of administration of the Mexican government. Another case of corruption in Mexico was that in 2004 by Carlos Ahumada, a 40year-old millionaire, who offered million-peso bribes to officials of Mexico’s left-of-center Partido de la Revolución Democrática to obtain lucrative sewer-cleaning contracts in Mexico city.1 Corruption appears at all levels in Mexico as a kind of ‘‘cultural behavior’’, since the word for bribe, mordida, literally means bite, and getting bitten in Mexico is regrettably common. In Mexico, mordida permeates every level of society and institutions in which individuals act because it is a norm, and they just do what others are doing (whether corrupt or not). Bribes in Mexico are common and indeed are often deemed necessary for obtaining business licenses and other types of permit. There is a popular Mexican saying: el que no transa no avanza, i.e., who does not corrupt does not move on. Of course corruption is not a norm of behavior in every country (e.g. New Zealand, Singapore, or Finland; see Transparency International on the Global Corruption Barometer, 2010),2 but 1 See Wydick (2008, 1) from BBC News, 20 October 1998, and Wall Street Journal, 23 June 2004, respectively. 2 Available at: http://www.transparency.org/policy_research/surveys_indices/ gcb/2010. E. Accinelli, E.J.S. Carrera / Economics Letters 117 (2012) 84–87 we wonder what accounts for adopting a corruptive behavior or not. We argue that the answer to this question is influenced by individual expectations driven by imitative behavior about the choices of others around to corrupt or not. We present a novel model to explain why individuals imitate a corrupt behavior, thinking of it as a kind of rational behavior. Rational imitation can be explained as follows. An individual, A, can be said to imitate the behavior of another individual, B, when observation of the behavior of B affects A in such a way that A’s subsequent behavior becomes more similar to the observed behavior of B. An individual can be said to act rationally when the individual, faced with a choice between different courses of actions, chooses the course which is the best with respect to his/her interests, his/her beliefs about possible action opportunities, and the effects of these potential action opportunities (for a survey on the notion of imitation see Sanditov, 2006). In our model, imitation results in individuals performing a spectrum of tasks ‘‘as others do’’, whether corrupt or not. We assume that occasionally each individual in a finite population gets an impulse to revise his/her (pure) strategy choice (either corruption or non-corruption). There are two basic elements for modeling. 1. First, it is a specification of the time rate at which individuals in the population review their current strategy choice, i.e., whether they are currently corrupt individuals or not. This rate may depend on the current performance of the agent’s pure strategy and other aspects of the current population state. 2. Second, it is a specification of the choice probabilities of a reviewing individual. The probability that an i-strategist will switch to some pure strategy j may depend on the current performance of these strategies and other aspects of the current population state, i.e., how large the share of corrupt individuals currently is, and the types of institution in the economy. If these impulses arrive according to independent and identically distributed (i.i.d.) Poisson processes, then the probability of simultaneous impulses is zero, and the aggregate process is also a Poison process. Moreover, the intensity of the aggregate process is just the sum of the intensities of the individual processes. If the population is large, then one may approximate the aggregate process by deterministic flows given by the expected payoffs from corruptive and non-corruptive behaviors. Weibull (1995) and Björnerstedt and Weibull (1996) studied a number of such models, in which those individual may imitate other agents in their player population, and show that a number of payoff-positive selection dynamics, including the replicator dynamics, may be so derived. In particular, if an individual’s revision rate is linearly decreasing in the expected payoff to his/her strategy (or to the individual’s latest payoff realization), then the intensity of each pure strategy’s Poisson process will be proportional to its population share, and the proportionality factor will be linearly decreasing in its expected payoff. If every revising agent selects his/her future strategy by imitating a randomly drawn agent in his/her own player population, then the resulting flow approximation is the replicator dynamics. 2. The model: institutions and corrupt behavior Institutions, according to North (1990), are the rules of the game. This broad definition bundles norms together with institutions and is also favored by Greif (2006): An institution is a system of rules, beliefs, norms and organizations that together generate a regularity of social behavior. However, in this paper we do not consider an ‘‘institution formation’’, since we consider that it is given from some external (to the individual) form which include 85 the whole structure of rules, means of detecting violation, and adjudicating punishments. Institutional quality will refer to the extent to which such external elements are effective in detecting and punishing corrupt activities. Consider an economy populated by types of institutions, g, from individuals, i, where individuals must behave as a corrupt or not, i.e., an individual’s decision type i ∈ {c , nc } is corrupt (c) or not (nc). Let the share of institutions’ type be denoted by the vector g = (gc , gnc ) normalized to one: gc + gnc = 1. Efficient or good institutions use mechanisms to punish corrupt individual behavior, or non-legal activities of its employees, while these mechanisms do not exist in corrupt institutions. Corrupt individuals may obtain an extra income from non-legal activities. Let us assume that the social welfare is measured by a common good and that all individuals in the society have the opportunity to use this good in an equalitarian form. So, if the total welfare in the society is defined by a real positive number, S ∈ R+ , each individual receives the same quantity of welfare from the society, s = S /N, where N is the finite size of the population. However, not all individuals have the same taste for this good. Suppose also that all individuals in the society are engaged in an institution, and assume in addition that all individuals receive the same salary, given by m. The pair (s, m) defines a social state. Hence an individual’s utility is defined by the function Ui : R × R → R, i ∈ {c , nc }, i.e., Ui (s, m) = sαi · mβi , (1) where αi > 0 measures the marginal impact of welfare distribution and βi > 0 measures the marginal utility from monetary payoffs for all behavior i ∈ {c , nc }. Individual utility is affected by a good institution or a bad one. Hence, when a corrupt individual matches a bad institution gc , Ucgc (s, m) = sαc (m + bc )βc , (2) where bc > 0 is the earnings from corrupt activities. When matching a good institution gnc , Ucgnc (s, m) = sαc (m + bc )βc − MPnc (e), (3) where M > 0 is the cost or punishment for the corrupt behavior and Pnc (e) ∈ (0, 1) measures a probability of monitoring corrupt activities or institutional effectiveness in eliminating corruption. On the other hand, the individual utility of non-corrupt behavior is Uncgnc (s, m) = sαnc · mβnc , (4) with αnc > αc meaning that non-corrupt individuals enjoy higher social status because such an individual matches a good gnc institutional type. Moreover, let us continue by considering that, if a non-corrupt individual is matched with a bad institution, then his/her utility function linearly decreases by; i.e. facing such an institution gc ; i.e., Uncgc (s, m) = sαnc · mβnc − Anc , (5) where Anc > 0 measures the disagreement for facing a bad institution. As we see, this parameter indirectly measured the degree of social disapproval towards a corrupt behavior. Then the expected payoff of a corrupt individual is given by E (c /g ) = [sαc (m + bc )βc ]gc + [sαc (m + bc )βc − MPnc (e)]gnc . (6) The expected payoff of a non-corrupt individual is E (nc /g ) = [sαnc mβnc − Anc ]gc + [sαnc mβnc ]gnc . (7) 86 E. Accinelli, E.J.S. Carrera / Economics Letters 117 (2012) 84–87 Rational individuals prefer to be non-corrupt when E (c /g ) < E (nc /g ), and this inequality follows if gc < sαnc · mβnc − sαc (m + bc )βc + MPnc (e) Anc + MPnc (e) = gh , (8) or gc < Uncgnc (s, m) − Ucgnc (s, m) Anc + MPnc (e) = gh . Individuals prefer to be non-corrupt if the share of bad institutions is lower than this threshold value, gc < gh . Note that [αnc + MPnc (e)] > [Uncgnc (s, m) − Ucgnc (s, m)], and so long as the punishment cost for corruption, MPnc (e), plus the disagreement of facing a bad institution, Anc , is large enough, individuals prefer to behave honestly and do not engage in corrupt activities. So here we have an incentive for bad individuals to reject non-corrupt activities when the number of bad institutions declines and they will be better under better institutions. Note also that the intensity of the displeasure that corrupt behavior creates in an honest worker (measured here by Anc ) plays a positive social role in the imitation game, given that the threshold value, than which honest behavior has a higher expected value, decreases. It is natural to think that, in a society in which imitative behavior plays a central role, the degree of social disapproval toward a certain behavior influences the choice of an individual’s behavior. The intensity with which society disapproves of social behavior is partially represented in our model by Anc . 3.1. Corrupt behavior by imitation Assume that individuals do not have complete information for the exact values of E (c /g ), E (nc /g ), and g. Our exercise of imitative behavior is carried out in the following way (see Accinelli et al., 2010). A reviewer individual i under the probability ri (x) ∈ [0, 1] raises the question of whether or not he/she must change his/her current type i ∈ {c , nc }. So ri (x) is the time rate at which individuals review their strategy choice. This probability depends on the actual distribution of the population x and in the benefits associated with the current behavior.3 It is natural to assume that the likelihood with which an individual poses this question to himself/herself depends inversely on the performance of his/her current behavior. Having opted for a change, the individual will adopt a strategy followed by the first person from the population to be encountered (his/her neighbor), i.e., there is a probability pi/j (x) ∈ [0, 1], that a reviewing i-strategist really switches to some pure strategy j ∈ {c , nc }. In a finite population, one may imagine that the reviewing process of an agent are the arrival times of a Poisson process with probability ri (x), and that at each time the agents select a pure strategy according to the probability distribution pi/j (x). Consider independence of switches across agents, and the process of switches from strategy i to strategy j as a Poisson Process with arrival rate xi ri pi/j . Assuming a continuum of agents and by the law of large numbers, we model these aggregate stochastic processes as a deterministic flow. So the evolution of the population will be given by the following dynamic system: ẋnc = rc (x)pc /nc xc − rnc (x)pnc /c xnc ẋc = −ẋnc 3. On the dynamics of corrupt behavior (10) x(t0 ) = (xc (t0 ), xnc (t0 )). Consider that the individual’s population i ∈ {c , nc } compromises a profile distribution x = (xc , xnc ) normalized to one: xc + xnc = 1. Individuals are absolutely rational, so they change their behavior according to the expected payoffs associated with such an adopted behavior i. Assume that at time t = t0 the profile distribution is x(t0 ) = (xc (t0 ), xnc (t0 )), and that the profile distribution of institutions is fixed at g = (gc , gnc ). The evolution of individual’s type i ∈ {c , nc } depends on differences in expected payoffs, and the dynamic flow of individuals must follow the following evolution: ẋnc = [E (nc /g ) − E (c /g )] xnc ẋc = −ẋnc . (9) The share of non-corrupt individuals may increase, decrease, or remain stationary, and this is according to the sign of E (nc /g ) − E (c /g ). In fact the share of the most successful behavior may increase. To simplify the future analysis of this evolution, and to obtain some conclusion, let assume that System (10) represents the interaction between two groups (corruptive and not) of individuals that imitate their neighbors. The righ-hand side of ẋnc is an inflow–outflow model: all those corrupt strategists becoming non-corrupt apart from all those non-corrupt agents that become corrupt. The differential equation (10) can be written as ẋnc = rc pc /nc − xnc (rc pc /nc + rnc pnc /c ). (11) Let us denote A = rc pc /nc + rnc pnc /c and B = rc pc /nc . Hence the next proposition is straightforward. Proposition 1. The share of non-corrupt individuals evolves to a mixed situation which depends on the average probabilities for reviewing rates of imitation and the probabilities of coping behavior, ri pi/j . There is only one situation in which corrupt behavior vanishes, and this is when all non-corrupt individuals do not do a review of their behavior and all corrupt individuals are reviewing their behavior. Proof. By looking at the solution of system (11) B αc = αnc = βc = βnc = 1. xnc (t ) = xnc (0) exp(−Bt ) + In this case, it follows that E (nc /g ) − E (c /g ) = MPnc (e) − gc (1 + MPnc (e)) − bc s, and the threshold value for this case is given by where xnc (0) is the share of non-corruptive individuals at time t = 0. Note that the share of non-corruptive individuals converges to ĝh = MPnc (e) − bc s MPnc (e) + Anc . Then, the share of the non-corrupt individuals increases as long as the share of bad institutions is lower and verifies the inequality gc < ĝh . To get an economy of non-corrupt individuals the share of good institutions should be sufficiently large, and this happens when the welfare distribution s is high and the earnings from corrupt activities bc decreases. B A = rc pc /nc rc pc /nc + rnc pnc /c A , , which is an average time rate at which corrupt individuals are reviewing whether they are doing well or not. Note that xnc (t ) → 1 3 This is the ‘‘behavioural rule with inertia’’ (see Schlag, 1998, 1999) that allows an agent to reconsider his/her action with probability r each round. E. Accinelli, E.J.S. Carrera / Economics Letters 117 (2012) 84–87 87 reviewing rate more often than a non-corrupt individual if and only if E (nc /g ) > E (c /g ). We may assume that individuals do not know the exact value of the expected payoffs; however, they are able to make an approximation of such true values in order to estimate it. Let us denote by Ē (i/g ) the estimators of the true values E (i/g ). The process of imitating successful behaviors exhibits payoff monotonic updating, since each i-strategist changes his/her strategy if and only if Ē (i/g ) < Ē (j/g ), ∀i ̸= j ∈ {c , nc }. To simplify, consider the case αc = αnc = α and βc = βnc = β . Then Eq. (12) becomes ẋnc = −β(Ē (nc /g ) − Ē (c /g ))x2nc + β(Ē (nc /g ) − Ē (c /g ))xnc ẋc = −ẋnc (14) x(t0 ) = (xc (t0 ), xnc (t0 )). Fig. 1. Dynamic flow with time for the share of non-corrupt individuals, xnc , from above and below the threshold B/A. as rnc pnc /c = 0 and this happens when all non-corrupt individuals get stuck in this current behavior while all corrupt individuals must review their current behavior and decide to change from it. A graphic representation of the dynamic flow of non-corrupt individuals is given in Fig. 1. Therefore, the share of non-corrupt individuals, xnc , evolves according to the threshold level given by the initial profile distribution of institutions g = (gc , gnc ), and it always converges to zero if its expected payoff is negative, E (nc /g ) < 0, which may depend positively on how large the share of bad institutions gc is.  However, an individual may change behavior just as a result of dissatisfaction and also because he/she does not know the true value for expected payoffs, E (i/g ). We analyze this situation in the next section. 4. Imitation by dissatisfaction Because of dissatisfaction with current behavior, an individual reviewer must copy the behavior of the first person he/she meets on the street, so pi/j = xj , ∀i ̸= j ∈ {c , nc }. Then, by rearranging terms, the dynamic system (10) takes the form ẋnc = (rnc − rc )x2nc + (rc − rnc )xnc ẋc = −ẋnc , (12) So the share of non-corrupt individuals increases if the inequality Ē (nc /g ) − Ē (c /g ) > 0 is verified. It is natural to assume that the probability P (Ē (c /g ) − Ē (nc /g )) > 0 increases with the difference between the true expected payoffs E (nc /g ) − E (c /g ), i.e., individuals prefer to behave as noncorrupt when E (c /g ) < E (nc /g ), and this inequality follows if the share of bad institutions is lower than the threshold value, gc < gh , given by Eq. (8). 5. Concluding remarks We studied an individual-level approach and tackled the question of why people engage in corrupt exchange. The above model pointed out that corruption increases because of imitation of agents. The approach is novel, and it explains the strategic foundations and evolutionary dynamics of corruption. More generally, it illustrates the importance of thinking about corruption as ‘‘the rules of the game’’: when almost everyone is corrupt, honesty is the deviant behavior. What needs to be explained under those circumstances is not why anyone takes bribes, steals from public officers, or distributes administrative advantages in return for material gain. What needs to be explained is how corruption became the rules of the game, and here it was done by an imitative behavior of the economic agents. Acknowledgments We are grateful to Bruce Wydick for helpful comments, and Eric Maskin for stimulating the research that led to this note. We thank the publisher’s staff for providing language help, writing assistance, and proofreading the article. References and after some little of algebra, we get the following chain of equalities: ẋnc = (rnc − rc )x2nc + (−rnc + rc )xnc = (rnc − rc )xnc (xnc − 1). Note that ẋn ≥ 0 if and only if (rnc (x) − rc (x)) ≤ 0 and xc (t0 ) > 0, which means that the share of non-corrupt individuals increases; when rc > rnc , more corrupt individuals review their behavior, considering a change. Let us consider that, so long as the payoff level of the i-strategist, Ei (·), increases, his/her average reviewing rate, ri (x), will decrease. So, assume that ri is linear in payoff levels; thus the propensity to switch behavior is decreasing in the level of the expected payoff, i.e., ri (x) = αi − βi E (i/g ) ∀i ∈ {c , nc }, αi βi (13) where αi , βi ≥ 0 and ≥ E (i/g ) ensure that ri ∈ [0, 1]. We interpreted αi as a marginal degree of dissatisfaction and βi as the marginal performance of one’s own expected payoff when reviewing the current strategy i. Then a corrupt individual undertakes this Acemoglu, D.S.Johnson, Robinson, J., 2005. Institutions as a fundamental cause of long-run growth. In: Aghion, Philippe, Durlauf, Steven (Eds.), Handbook of Economic Growth. Vol. 1. Elsevier, pp. 385–472. Part A, ch. 06. Accinelli, E., Brida, J., Carrera, E., 2010. Imitative behavior in a two population model. In: Breton, Michële, Szajowski, Krzysztof (Eds.), Advacens in Dynamic Games. 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