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POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS W.C. ABRAM AND K. NORAY

Abstract. We study a two population evolutionary game that models the role of public activism as a deterrent to political corruption. In particular, suppose that politicians can choose whether or not to engage in corruption, lowering the public good in exchange for personal gain, and citizens can choose whether or not to engage in public activism for corruption reform, influencing the rate of detection and severity of punishment of corrupt politicians. We study the Nash equilibria of this game and also conduct static and dynamic evolutionary analyses.

1. Introduction There are many ways to define political corruption. For this paper, let political corruption be the illegal or culturally unacceptable use of resources intended to benefit the public for personal gain. Thus defined, corrupt actions reallocate wealth without creating wealth. Corruption, then, is a special case of rent-seeking, and as such two conclusions from public choice theory apply: (1) corrupt behavior decreases the public good by taking government resources and using them for private benefit, and (2) corruption wastes time and energy that could be used productively (cf. Tullock 1967). Kaldor-Hicks efficiency, then, is hard to achieve in a society with a government that is laden with corruption (Hicks 1939, Kaldor 1939). Economic research has already provided useful insights into how political corruption works. We hope to add to this rich literature. Our purpose is to examine the relationship between public activism and the level of political corruption in a given government. We define public activism as any action intended to reduce corruption in the government by changing public awareness and perception of the issue. There are several reasons why activism may induce a change in the level of corruption in a government. A high level of public activism may pressure the current regime to crack down on corruption. For example, a government may respond to public activism by passing legislation that allocates more resources to monitoring government officials. Societal pressure could also change the prosecution rates of unlawful politicians; attorneys may have a greater incentive to prosecute corrupt politicians if society advocates for corruption reform. Further, activism may increase the severity of punishment for political corruption (e.g. longer jail sentences). Lastly, in a democracy, public activism could change the preferences of voters by informing them about the prevalence of corrupt practices and their deleterious effects. For a corrupt politician, this would reduce the chance of winning re-election. Date: Received: March 30, 2016 / Accepted: This work was partially supported by the Hillsdale College LAUREATES Research Program. 1

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For these economic reasons, and perhaps others, it is plausible that public activism against corruption decreases the expected value of corruption for politicians. Given this, how exactly would activism influence the level of political corruption in society? We attempt to provide insight into this question with a two population, evolutionary game that analyzes the behavior of politicians who have the option to be corrupt or honest and citizens who can choose whether or not to participate in anti-corruption activism. 2. Summary of Previous Work Because our paper contains both economics and evolutionary game theory, there are multiple bodies of work that are relevant. We break our literature summary up accordingly, dealing with the economics literature first and the game theory literature from other disciplines second. While the economics literature informs our theoretical justification for modeling political corruption, game theory models from outside economics bear on the technical contribution our model makes. 2.1. Economics and Corruption. Rose-Ackerman was the first to analyze the economics of corruption in her book Corruption: a Study in Political Economy (Rose-Ackerman 1978). Building on ideas introduced in her book, a substantial economic literature aimed at modeling corruption emerged, including game-theoretic analyses. Though the types of mathematical models economists have developed to understand political corruption vary, it is useful to divide them into two categories: models of "external corruption" and models of "internal corruption" (Levin and Tsirik 1998). External corruption occurs when the participants of a corrupt transaction are in different institutional settings (i.e. a firm owner bribing a public officer). Internal corruption, on the other hand, occurs when the participants of a corrupt transaction are within the same institutional setting (i.e a corrupt politician competing against an honest politician in an election). In this section of the review, we focus on external corruption because the model we develop represents a type of external corruption. In the next section of the review, we shift our focus to internal corruption because other disciplines (e.g. evolutionary biology) predominantly model single populations. A major theme in the corruption literature is monitoring: the act of overseeing politicians to catch and punish the corrupt ones and deter the honest ones. Most of the monitoring literature applies a principal-agent-client model, or some variation thereof (Levin and Tsirik 1998). In this type of model, there is a monitor (the principal) whose objective is to prevent a private citizen (the client) from successfully bribing a public official (the agent). For the sake of simplicity, many of these models assume that the monitor is incorruptible, in which case the solution is straightforward: monitor and punish clients so severely that their expected benefit from corruption is zero. Antioci and Sacco (1995) conclude that a bribe-deterring level of punishment always exists. When the incorruptibility assumption is dropped, curbing corruption is more difficult. As empirical evidence suggests (cf. Silva 1999), adding a monitor, or many monitors, increases the number of corruptible actors, which may actually increase corruption. Many of these models also incorporate risk. Corruption is inherently risky because being detected is costly and there is always a chance that others will refuse to participate in a corrupt transaction. At the cost of searching, agents can moderate the riskiness of a corrupt transaction by acquiring more information. Some, like Antioci and Sacco (1995), incorporate risk

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS3

into their otherwise deterministic models. In their case, they let a known probability value represent the likelihood that a public official will take a bribe, instead of assuming he will accept a bribe with certainty. Bayar (2003), in his doctoral thesis, studied the use of middlemen to reduce the search and information costs of bribery. He shows that whether the client (a citizen) or the agent (a government official) initiates the corrupt exchange, the initiator gains by sharing risk with a middleman who facilitates the exchange. The client, however, will always be best off with no corruption at all. 1 Some papers incorporate public opinion as an element of a politician’s decision problem. Multiple of these papers study voting theory, which implicitly assumes that the voting public disdain political corruption. For example, Evrenk (2011) shows that both clean and corrupt politicians may oppose corruption reform in societies where the politicians are sufficiently forward-looking. This paradoxical result occurs because clean politicians gain more politically from being honest in a political system suffused with corruption than in a corruption-free political system. Similarly, Polo (1998) analyzes a type of corrupt, post-election opportunism: the illegal reallocation of funds in excess of the cost of the public goods that the politician promised on the campaign trail. His model seeks to explain how the level of information politicians have about the political desires of the median voter effect electoral competition and, in turn, the size of the appropriable differential between proposed expected tax revenue and the cost of promised public goods. Based on the Median Voter Theorem (cf. Black 1948), Polo concludes that when politicians have sufficiently accurate information about the median voter’s preferences they will compete heavily for the median vote, which causes the winner’s proposed tax revenue and public good cost to be about the same. Without a differential between the two, there will be no appropriable funds, assuming that the politician honors his or her proposal. However, if politicians don’t have accurate knowledge about the median voter’s preferences, the election is less competitive, which increases the chance that the winner will have proposed a tax rate and public good combination with an appropriable differential between revenue and costs (Polo 1998). This gives the politician access to funds that he or she may use illegally for personal gain. Lastly, Rinaldi and Feichtinger (1998) analyze corruption in democratic societies and base the politician’s utility from corrupt behavior on corruption’s effect on public support, the amount of added personal funds, and the chance of getting caught. The paper’s primary conclusion is that increasing investigation effort can reduce but not eliminate corruption; to construct a corruption-free political regime there need to be institutional changes that sufficiently alter public opinion and expected monetary gains from corrupt behavior (Rinaldi and Feichtinger 1998). Our paper advances the literature on public opinion by analyzing how a politician’s choice changes as public opinion varies. Thus, we treat society as a rational player that chooses how much to advocate for corruption reform.

1Though not directly related to our work, others have demonstrated empirically that information imperfections and search costs affect the incentive to be corrupt. For example, Berninghaus and Haller (2013) studies whether preconceived notions about the prevalence of corruption influenced players’ decisions to be corrupt in a simulation. Despite telling participants the actual proportion of corruption, they determined that the original biased opinions correlate strongly with the players’ choices.

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Economists have often employed evolutionary models to study behavior. Jack Hirshleifer was one of the first to apply evolutionary and dynamic models to explain conflict (cf. 1995; 1999; 2001). Other economists have used evolutionary models to analyze corruption specifically. Most of these models use the replicator dynamic, which assumes that agents imitate the successful behaviors they witness over time (in this context it is easiest to think of as a learning process)2. Standard replicator dynamics are used in Antioci and Sacco (1995), Dawid and Feichtinger (1996), Mishra (2006), Dong and Dulleck (2009), and Mueller (2012). These papers all confirm what Dong and Dulleck (2009) call "conditional corruption": the incentive to be corrupt is proportional to the fraction of society participating in corruption. Explanations for why this happens vary. Dong and Dulleck (2009) suggest that the prevalence of corruption changes the “guilt cost” of being corrupt, though others, such as Accinelli and Sánchez Carrera (2012), attribute this to man’s inherent imitative tendencies. Conclusions about the evolutionary stability of populations where some agents are corrupt differ between models. Dawid and Feichtinger (1996), for instance, study a game in which all mixed population states are unstable, while Mueller (2012) finds that semi-stable mixed solutions can exist. Moving away from traditional imitation models, Bichieri and Rovelli (1995) conduct a repeated game analysis within their evolutionary model. They include a social cost with a magnitude determined by previous and current levels of corruption. Their model subtracts this term from each player’s utility The cost builds over time and, eventually, the dominant strategy becomes complete honesty. There is a sudden switch, described as a "revolution," where all corruption is stamped out (Bichieri and Rovelli 1995). Other creative ideas, like allowing evolutionary updating for certain types of players and not others, or changing payoffs based on institutional settings, have also been applied to dynamic models (cf. Accinelli and Sánchez Carrera 2012). Though many economists have applied the replicator dynamic, few have used other updating methods. Our paper analyzes evolutionary behavior under replicator dynamics and logit dynamics. 2.2. Related Models from Biology. Evolutionary models originated in the biological sciences and were later adopted by other disciplines. For this reason, the models that we develop are similar to models outside of economics. Here we outline the models that are similar to ours and explain the technical contribution of our work. Our model bears a non-trivial resemblance to Robert Axelrod’s seminal work on "the evolution of cooperation" in Axelrod (1984) and Axelrod and Hamilton (1981). In his paper, Axelrod discovers that if agents interact repeatedly over a non-determined amount of time, they may have an incentive to cooperate in situations where they otherwise would not. Specifically, Axelrod analyzes a repeated, two player prisoner’s dilemma game where each player can cooperate or defect in each round. Axelrod finds that, unlike the one-time prisoner’s dilemma game, the repeated version has many cooperative Nash equilibria because players can punish 2Though not obvious, there are empirical and theoretical reasons to think that a politician may imitate others when he or she chooses to be corrupt. A persistent finding in corruption literature is that political corruption reinforces itself. This suggests that politicians may follow the behavior of those around them, reinforcing changes in corruption. For example, Accinelli and Carrera (2012) note that "Mexico is a real nice example of corrupt behavior driven by imitation, i.e. to be corrupt if others are also corrupt..."

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS5

defectors. Analogously, we can represent the relationship between politicians and society as one where these parties can "cooperate," through honest behavior from politicians and no activism from society, or "defect," through corrupt behavior from politicians and high levels of activism from society. Effectively, our paper explores the conditions under which "cooperation" between politicians and society is sustainable under three different behavioral assumptions about how these populations respond to each other through time. Though the analogy between our model and Axelrod’s is illuminating, it is not perfect because our model, unlike Axelrod’s, is multi-populational and the payoff structures of the two games differ slightly. More recently, there have been other evolutionary game theory papers that analyze "corruption." It is important to keep in mind that, though the game-theoretic models presented in this section are structurally similar to ours, exactly what is meant by the term "corruption" can differ between disciplines. For example, when biological papers refer to corruption, they usually mean that, under certain circumstances, monitors choose not to cooperate with others (cf. Úbeda and DuénezGuzmán 2011). This is very different from the political corruption that we study. Keeping that distinction in mind, Úbeda and Duénez-Guzmán (2011) develop a corruption model that is similar to ours. Theirs is a four by four game in which monitors ("policers" in their language) can choose to be corrupt by defecting when they interact with a defector. Like Axelrod (1984), this is a single population, symmetric game. But, unlike Axelrod, members have a choice of four behaviors (just cooperate, just defect, cooperate and punish, defect and punish) instead of two. Úbeda and Duénez-Guzmán conclude that the only way policers have sufficient incentive to coexist with civilians is if policers have more "power" than civilians. This model differs from ours for two reasons. First, their model is a single population game where each type of player can change into every other type of player through time. In our game, the populations of players interacting are distinct and cannot mutate into one another. Because we are interested in the populations of society and politicians as distinct entities where the individuals within them change their behavior, a multipopulational approach is more appropriate. Secondly, we examine replicator dynamics, best-response dynamics, and logit dynamics, whereas Úbeda and Duénez-Guzmán (2011) only analyze the replicator dynamic. 3 Though the application is different than ours, the "battle of the sexes" game, developed by Richard Dawkins (1976) and John Maynard Smith (1982), has a similar structure to ours under certain assumptions (see section 4.1, case 0). Because the battle of the sexes game is important in biology, multiple papers have already explored the evolutionary dynamics of this game under replicator dynamics (cf. Selten 1980; Schuster and Sigmund 1981; Hofbauer and Sigmund 1998; Cressman 2003). Though our model is structurally identical to the battle of the sexes game in the most interesting case, we explore the dynamic behavior under logit dynamics in addition to replicator dynamics, which biologists do not consider.

3An interesting extension of Úbeda and Duénez-Guzmán (2011) is presented in DuénezGuzmán and Sadedin (2012). Here the authors conclude that in the presence of psychological or status-related benefits to being a monitor, society could reasonably travel from an unstable equilibrium of cooperating civilians and monitors to a stable equilibrium of strictly cooperating monitors

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3. The Model Henceforth we study a two population game that models a specific aspect of corruption: the interaction between public activism and political corruption levels. The players of the game are P , the population of politicians, and S, the society or general citizenry. The politician P can behave honestly, H, earn income IH and distribute a public good level GH to society; or engage in corrupt behavior, C, earn a higher income IC and distribute a lower public good level GC to society. If he engages in corruption, the politician incurs a risk of being caught and punished. The higher income and lower public good level for the corrupt politician represents an illegal transfer or embezzlement of public goods for private use by the politician. The society can either engage in activism, A, incurring some cost a for the effort, while increasing the probability of detection as well as the severity of punishment for corruption; or not engage in activism, N , having no effect on the probability of detection or the severity of punishment for corrupt politicians. To summarize, the parameters under consideration are: GH , public good level with honest politician; GC , public good level with corrupt politician; IH , honest politician income; IC , corrupt politician income; δA , probability of detection of corruption in a society with activism; δN , probability of detection of corruption in a society without activism; σA , severity of punishment for corruption in a society with activism; σN , severity of punishment for corruption in a society without activism; a, utility cost to society of activism. We also make the following assumptions: IC > IH , that is, the politician incurs some gain from corruption. GH > GC > 0, that is, political corruption incurs a cost on society. 1 > δA > δN > 0, that is, increased public activism increases the probability of detection of corruption, and there is always some chance of detection and of non-detection. σA > σN , that is, increased public activism increases the severity of punishment for corruption. a > 0, that is, the activism effort incurs a positive cost on society. The first two assumptions, designating that corruption both increases the income of the politician and decreases the public good level, can be justified, as in Polo’s model (Polo 1998), by arguing that the more the politician appropriates funds for personal gain the less revenue remains to supply a public good. Since all the revenue obtained by the government besides that used to pay the public officials is supposed to benefit the public, all extra funds used for personal benefit by a public official must take away from funds for the public good. The logic of the third assumption is that societal activism for corruption enforcement reform will have an effect on the level of monitoring, potentially effecting both the probability of detection of corruption and the intensity of punishment. Let G be the game with payoff bimatrix:

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS7

P C A

H

(1 − δA )(GC − a) + δA (GH − a), IC − δA σA GH − a, IH

S N

(1 − δN )GC + δN GH , IC − δN σN

GH , IH .

Thus, an additional feature of this game is that, should the corrupt politician be detected, the level of public good will be GH . This can be interpreted in several ways: as the redistribution to society of the politicians corrupt wages, as the replacement of the corrupt politician with an honest politician, et cetera. Most generally, the assumption is that society receives a benefit from the corrupt politician’s detection. 4. Nash Equilibrium Analysis We now study the Nash equilibria of the game G. If politicians are always honest, society should never engage in activism, since a > 0 implies that GH > GH − a. (1) This result is intuitive, since it means that in a society free from corruption it is not beneficial to engage in activism against corruption. For society, activism is a strictly better response than non-activism to corruption if (1 − δA )(GC − a) + δA (GH − a) > (1 − δN )GC + δN GH . (2) This will happen if the increased monitoring effectiveness resulting from activism is enough to compensate for the cost of activism. Thus, either activism or nonactivism could be the best response to corruption. For the politician, honesty is a strictly better response than corruption to activism if IH > IC − δA σA . (3) Thus, if the probability of detection δA and severity of punishment σA are sufficiently large, they can offset the income IC − IH > 0 that politicians could have earned by engaging in corrupt behavior. If society does not engage in activism, then the politician should be corrupt if IC − δN σN > IH .

(4)

If the benefit of corruption IC outweighs the expected cost of such behavior then the politician should be corrupt. This, again, may or may not hold. We will show below that G always has a unique mixed strategy Nash equilibrium, which may be in pure strategies or in strictly mixed strategies. 4.1. Pure Strategy Nash Equilibria. The number of pure strategy Nash equilibria (PSNE) depends on the values of the various parameters, and there are several cases. Case 0: No PSNE Since Non-activism is always the best response to Honesty, there will be no PSNE if Corruption is the unique best response to Non-Activism, Activism is the unique

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best response to Corruption, and Honesty is the unique best response to Activism (a matching pennies situation). This will occur if equations (2), (3), and (4) all hold. This is the most interesting case, and we will consider it further below when we discuss dynamics. Case 1: Unique PSNE Depending on the values of the parameters, (A, C), (N, C), or (N, H) may be a unique PSNE, where, for instance, (A, C) refers to the strategy profile (Activism, Corruption). Case 1AC : For (A, C) to be the unique PSNE, we must have IC − δA σA > IH ,

(5)

in which case (4) follows, since δA σA > δN σN . We must also have (2). Note that it is possible for both (2) and (5) to occur if GH is sufficiently larger than GC , a is sufficiently small, and the severity of punishment σA is sufficiently small. Equation (5) may be rearranged to give IC − IH > δA σA ,

(6)

which allows one to see the threshold level of detection δ and punishment severity σ required to deter corruption. We see that less severe punishments can allow corruption to thrive even in a society that engages in activism against corruption. The benefit to continued Activism in this case is that the public benefits when a corrupt politician is caught and punished, since in this case the higher level of public good GH is restored. Case 1NC : For (N, C) to be the unique PSNE, equation (4) must hold, and there must also hold (1 − δN )GC + δN GH > (1 − δA )GC + δA GH .

(7)

Equation (7) will hold if activism is not sufficiently effective at increasing the probability of detecting corruption - δA is not sufficiently larger than δN - or if the level of corruption GC − GH is not sufficiently large to warrant activism efforts. This will occur if (δA − δN )(GC − GH ) < a, (8) that is, if activism is sufficiently costly, the corruption level is sufficiently low and activism is sufficiently ineffective. Interestingly, activism would be a strictly dominated strategy if δA = δN , since activism’s effects on the severity of punishment σA > σN alone do not improve the payoffs of the society player. This model does not account for a “justice" payoff, but only those payoffs which come from improvements to public service when corrupt politicians are prosecuted. Case 1NH : For (N, H) to be the unique PSNE, there must hold IH > IC − δN σN ,

(9)

and in this case (3) must also hold, since δN σN < δA σA . This is a society without corruption, and where activism is not needed to deter corruption. This will occur if δN and σN are sufficiently large, reflecting an effective anti-corruption mechanism in the status quo government. Other Possibilities: Multiple PSNE Two or more PSNE could only occur if (A, C) and (N, H) are both Nash equilibrium strategy profiles, since N is necessarily the unique best response to H. But if (N, H) is a PSNE, then (9) must hold, in which case so must (3). But in this case

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS9

(5) cannot hold, so (A, C) cannot be a PSNE. Thus, it is not possible for there to be two or more PSNE at the same time. Notice here that we are not considering the case when two strategies have equal payoffs, since this should be a probability zero event. If we consider equal payoffs, then two or even three PSNE may occur simultaneously. 4.2. Mixed Strategy Nash Equillibria. Here we study the mixed strategy Nash equilibria of the activism-corruption game G. Mixed strategies have two interpretations: as randomization by individual players or as the prevalence of strategies in society. The latter interpretation is natural to the scenario being studied because each player represents multiple agents. If there is a unique pure strategy Nash equilibrium, there can be no other mixed strategy Nash equilibria. Therefore, we restrict our attention now to Case 0, wherein no PSNE exist, meaning that equations (2), (3), and (4) all hold. Now if a politician is corrupt with probability pC and the society engages in activism with probability pA , then the payoff functions are πP = pC [pA (IC − δA σA ) + (1 − pA )(IC − δN σN )] + (1 − pC )IH

(10)

for the politician and πS = pA (pC [(1 − δA )(GC − a) + δA (GH − a)] + (1 − pC )(GH − a)) + (1 − pA )[pC ((1 − δN )GC + δN GH ) + (1 − pC )GH ]

(11)

for the society. The critical point equation for the politician’s payoff function is 0 = dπp /dpC = pA (IC − δA σA ) + (1 − pA )(IC − δN σN ) − IH ,

(12)

which gives 0 = pA (δN σN − δA σA ) + IC − δN σN − IH ,

(13)

pA = p∗A = (IH − IC + δN σN )/(δN σN − δA σA ).

(14)

so that A necessary condition for this to be a Nash equilibrium component strategy is that the value p∗A of (14) be in the interval (0, 1). This added condition is satisfied when δN σN < IC − IH < δA σA .

(15)

But (15) is guaranteed by (3) and (4), which hold in Case 0. We may perform a similar analysis to obtain p∗C . First, the critical point equation is 0 = dπs /dpA = pC [(1 − δA )(GC − a) + δA (GH − a)] + (1 − pC )(GH − a) (16) − pC [(1 − δN )GC + δN GH ] − (1 − pC )GH . This reduces to 0 = pC (GH − GC )(δA − δN ) − a.

(17)

Lastly, if we solve for pC , we obtain pC = p∗C = a/[(GH − GC )(δA − δN )].

(18)

For p∗C to fall within (0, 1) there must hold 0 < a/[(GH − GC )(δA − δN )] < 1,

(19)

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or 0 < a < (GH − GC )(δA − δN ).

(20)

But this is guaranteed in Case 0 by (2), to which (20) is equivalent. Mixed strategies in this context mean that only a portion of the society engages in activism and, likewise, only a portion of the population of politicians is corrupt. While the proportions that make up the MSNE are dependent on numerous parameters, there are some relationships worth mentioning. Ceteris paribus, p∗C is inversely proportional to the increase in public good level from switching to honesty. This suggests that, given responsiveness of the government to public opinion, there should be less corruption if society tends to gain a high amount of welfare from political honesty. Ceteris paribus, p∗C is also inversely proportional to the increase in monitoring effort due to activism and to the cost of activism. Finally, ceteris paribus, p∗A is directly proportional to the increased expected cost of punishment for corruption attributable to activism. In summary, we have the following theorem. Theorem 4.1. Assuming its payoff bimatrix has unequal entries, the game G has a unique mixed strategy Nash equilibrium, which is either in pure strategies: (A, C), (N, C), or (N, H), or in strictly mixed strategies: pA = (IH − IC + δN σN )/(δN σN − δA σA ), pC = a/[(GH − GC )(δA − δN )] . (21) Proof. This follows immediately from the above discussion.

5. Static Evolutionary Analysis Weibull (1997, Definition 5.1) defines multi-population ESS in the following way: a strategy profile x is an evolutionarily stable strategy (ESS) if for every strategy profile y 6= x there exists a z ∗ ∈ (0, 1) such that for all z ∈ (0, z ∗ ), and for w = zy + (1 − z)x, πi (xi , w−i ) > πi (yi , w−i ) (22) for all players i of the game. For a two-population game, this is equivalent to Definition 3.2.1 of Cressman (1992). Another multi-population analog to ESS, also developed by Swinkels (1992), is the robust against equilibrium entrants (REE) concept. This relaxes the ESS definition by requiring that a profile merely be resistant against other profiles that are best responses to the resulting population after invasion. Formally, a strategy profile x is an REE if there exists z ∗ ∈ (0, 1) such that the below non-inclusion holds for all profiles y 6= x and z ∈ (0, z ∗ ): y∈ / β[zy + (1 − z)x],

(23)

where β is the (set-valued) best response function. If we denote by ΘP the set of all strategy profiles satisfying some property P , letting SN E stand for "strict Nash equilibria" and N E for "Nash equilibria", it is easy to show that ΘSN E = ΘESS ⊆ ΘREE ⊆ ΘN E , (24) see for example Section 5.1 of Weibull (1997). Recall that a strategy profile x is a strict Nash equilibrium if β(x) = {x}, that is, x is the unique best response to itself. Clearly, a strict Nash equilibrium strategy profile must be in pure strategies.

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Theorem 5.1. Consider the game G. (1) If the unique MSNE of G is not in pure strategies, then it is REE, but not ESS. (2) Assume the payoff bimatrix of G has unequal entries. If the unique MSNE of G is in pure strategies, then it is a strict Nash equilibrium, is ESS and is REE. Proof.

(1) Suppose the unique mixed strategy Nash equilibrium strategy profile x∗ of G is not in pure strategies. Then x∗ cannot be a strict Nash equilibrium, since all strict Nash equilibria are in pure strategies, hence x∗ cannot be ESS since ESS strategy profiles are equivalent to strict Nash equilibria. x∗ is, however, vacuously REE, since x∗ is the unique Nash equilibrium and there are therefore no equilibrium entrants to consider. (2) Suppose the unique mixed strategy Nash equilibrium strategy profile x∗ of G is in pure strategies. Since the payoff bimatrix of G has unequal entries, it follows that for any strategy profile y in pure strategies, the best response set β(y) is a singleton set. Then since x∗ is a Nash equilibrium, x∗ ∈ β(x∗ ), which is a singleton set. It follows that β(x∗ ) = {x∗ }, so x∗ is a strict Nash equilibrium. We can conclude from (24) that ΘSN E = ΘESS = ΘREE = ΘN E = {x∗ },

(25)

that is, the unique PSNE in Case 1 is also the unique evolutionarily stable strategy profile and the unique robust against equilibrium entrants strategy profile. This means that the unique PSNE of G in Case 1 is stably resistant to small-scale invasions of players using other strategy profiles. We see that complete honesty and complete corruption within the government can be stable. In Case 0, we see that the unique MSNE is not evolutionarily stable. We will see below that, under the replicator dynamics, this MSNE is nonetheless Lyapunov stable. 6. Evolutionary Dynamics While static evolutionary theory provides some insight, evolutionary dynamics will allow us to understand better the evolution of corruption in a society where activism has some effect on monitoring levels. In this section we apply replicator dynamics to the activism-corruption game G. 6.1. Selection Dynamics. We consider the behavior of the corruption game G under the multi-population replicator dynamics, best-response dynamics, and logit response dynamics. 6.2. Replicator Dynamics. The replicator dynamics were defined by Taylor (1979). When applied to G the replicator dynamics are: p˙A = [πs (A, pC C + (1 − pC )H) − πs (pA A + (1 − pA )N, pC C + (1 − pC )H)]pA ; (26) p˙C = [πp (C, pA A + (1 − pA )N ) − πp (pC C + (1 − pC )H, pA A + (1 − pA )N )]pC ; (27) where for a variable p, p˙ denotes the time-derivative of p. Equation (26) bases the rate of change of the proportion of society that selects the action "Activism"

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on the differential between the expected payoffs within the current population. Likewise, (27) bases the rate of change of the proportion of politicians that select the action "Corrupt" on the relative performance of this strategy compared to the average performance within the current population state. The resulting system of differential equations, after some algebraic manipulation, takes the following form: p˙A = (GH − GC )(δA − δN )pA pC (1 − pA ) − apA (1 − pA ); p˙C = (IC − δN σN − IH )pC (1 − pC ) − (δA σA − δN σN )pA pC (1 − pC ).

(28) (29)

Assuming that the initial condition (pA (0), pC (0)) is in the interior of the unit square [0, 1]2 , the first-order linear system of ordinary differential equations given by (28) and (29) has implicit solution IC −δN σN −IH a pA pC = K > 0, I −δ σ −I a−(G C A A H H −GC )(δA −δN ) (1 − pA ) (1 − pC )

(30)

where K is a positive constant depending upon the initial condition. The stationary points will be the pairs (pA , pC ) satisfying the system of equations: 0 = apA (pA − 1) + (GH − GC )(δA − δN )pA pC (1 − pA )

(31)

and 0 = (δA σA − δN σN )pA pC (pC − 1) + (IC − δN σN − IH )pC (1 − pC ).

(32)

Thus, the possible stationary points are: (1) (2) (3) (4) (5)

pA pA pA pA pA

= 0, pC = 0; = 1, pC = 0; = 0, pC = 1; = 1, pC = 1; = (IC − δN σN − IH )/(δA σA − δN σN ), pC = a/[(GH − GC )(δA − δN )].

The stationary points 1. through 4. are the corners of the phase portrait for G. These must be stationary points since under the replicator dynamics a population without mutants cannot spontaneously generate mutants. In the context of G, if all politicians are corrupt (resp. honest), then all politicians will remain corrupt (resp. honest). Likewise, if all citizens are active (resp. inactive), then all citizens will remain active (resp. inactive). In Case 0, in which there is no PSNE, these four stationary points are all unstable, in fact they are saddles. However, in any of the sub-cases of Case 1, the unique PSNE is asymptotically stable (a nodal sink), while the other three corners of the phase portrait are unstable and can be either nodal sources or saddles. The last stationary point 5. describes the possible mixed strategy Nash equilibrium. This is a stationary point if and only if 0 ≤ pA , pC ≤ 1, which is satisfied in Case 0 but not in Case 1 for the formulas given by 5. This stationary point, if it exists, is Lyapunov stable but not asymptotically stable. In fact, it is a center. Roughly speaking, a stationary point is Lyapunov stable if sufficiently small perturbations do not induce movement away from the stationary point, whereas a stationary point is asymptotically stable if sufficiently small perturbations induce movement back toward the stationary point. For a nice introduction to these and related concepts, see Chapter 6 of Evolutionary Game Theory (Weibull 1997).

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS 13

pC

pA Figure 1. Phase Portrait in Case 1NC under the Replicator Dynamics. The parameters are IC = 3, IH = 2, GC = 3, GH = 4, δN = 0.3, δA = 0.4, σN = 1, σA = 5, and a = 0.3.

A sample phase portrait for G under the replicator dynamics in Case 1NC is given in Figure 1. Of the four stationary points of this phase portrait, only (pA = 0, pC = 1) is asymptotically stable. The other three stationary points are unstable. A similar portrait exists in Case 1AC and Case 1NH, and the behavior in each of these cases is analogous. A phase portrait for G in Case 0 is given in Figure 2. The unique MSNE in Case 0 is in general Lyapunov stable but not asymptotically stable, and is a center. The four corners of the phase portrait are unstable stationary points, and are saddles. We see that in Case 0 there arises a cycling of corruption and activism levels with the potential for very large variations in both. One takeaway from this is that current levels of corruption and public activism do not clearly indicate future levels, even in a closed system. In summary, we have the following theorem: Theorem 6.1. Consider the game G under the replicator dynamics. Assume that the payoff bimatrix of G has unequal entries. (1) If the unique MSNE of G is not in pure strategies, then it is Lyapunov stable but not asymptotically stable, and is a center. (2) If the unique MSNE of G is in pure strategies, then it is asymptotically stable, and is a nodal sink.

14

W.C. ABRAM AND K. NORAY

pC

pA Figure 2. Phase Portrait in Case 0 under the Replicator Dynamics. The parameters are IC = 3, IH = 2, GC = 2, GH = 4, δN = 0.1, δA = 0.3, σN = 1, σA = 5, and a = 0.1. The coordinates of the interior stationary point are (pA , pC ) = (9/14, 1/4) ≈ (0.64, 0.25).

Proof.

(1) If the unique MSNE of G is not in pure strategies, then it must be of the form

pA = (IC − δN σN − IH )/(δA σA − δN σN ), pC = a/[(GH − GC )(δA − δN )]. Since we are trying to show that this stationary point is a center, which is a non-generic type, linearization would not be helpful. Rather, let us refer to the implicit solution (30). Now since we are in Case 0, (15) and (20) necessarily hold, so that (30) is of the form β pα β γ A pC = pα A pC (1 − pA ) (1 − pC ) = K > 0, (1 − pA )−γ (1 − pC )−

(33)

when 0 < pA , pC < 1 and where α, β, γ, are positive constants. That is, the orbits under the replicator dynamics are the level curves of the twoβ γ variable function f (pA , pC ) = pα A pC (1 − pA ) (1 − pC ) . A countour plot for this function computed with α = 2, β = 3, γ = 1, = 5 is given in Figure 3. In general, the level curves of a function of this form are closed, so the orbits under the replicator dynamics must be periodic. It follows that the interior stationary point is a center. A center is necessarily Lyapunov stable, but not asymptotically stable.

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS 15 1.0

0.8

0.6

pC

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

pA Figure 3. Contour plot for f (pA , pC ) = p2A p3C (1 − pA )1 (1 − pC )5 . (2) We will prove this in Case 1AC. Cases 1NC and 1NH are similar. Suppose (pA = 1, pC = 1) is the unique Nash equilibrium of G. From (28) and (29) we can compute the Jacobian matrix J = =

∂ p˙ A ∂pA ∂ p˙ C ∂pA

∂ p˙ A ∂pC ∂ p˙ C ∂pC

(2pA − 1)[a − pC (GH − GC )(δA − δN )] (δA σA − δN σN )pC (1 − pC )

(GH − GC )(δA − δN )pA (1 − pA ) (2pC − 1)[(δA σA − δN σN )pA − (IC − δN σN − IH )]

, (34)

which at the stationarity point (pA , pC ) = (1, 1) becomes a − (GH − GC )(δA − δN ) 0 J= . 0 δA σA − IC + IH

(35)

The Jacobian determinant is therefore det(J) = [a − (GH − GC )(δA − δN )](δA σA − IC + IH ).

(36)

It follows from our assumption of unequal payoffs that the terms of (36) are nonzero, so the Jacobian at (pA , pC ) = (1, 1) is nonsingular. Since this is also an isolated stationary point, it follows that the replicator dynamical system is almost linear at that point. We now need to study the eigenvalues of the Jacobian J: λ1 = a − (GH − GC )(δA − δN ) and λ2 = δA σA − IC + IH . Since (4) holds in Case 1AC, (8) must not hold, or else (pA , pC ) = (0, 1) would be a Nash equilibrium and we would actually be in Case 1NC. Therefore, by our assumption of unequal payoffs, a < (GH − GC )(δA − δN ),

(37)

and it follows that λ1 < 0. On the other hand, it follows immediately from (5) that λ2 < 0 also. Since the dynamical system is almost linear at

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W.C. ABRAM AND K. NORAY

(pA , pC ) = (1, 1) and both eigenvalues of the Jacobian at this stationary point are negative, it follows that this is a nodal sink, and is therefore asymptotically stable. 6.3. Logit Dynamics. Consider now the best response dynamics given by p˙A ∈ BRS (pA , pC ) − pA ;

(38)

p˙C ∈ BRP (pA , pC ) − pC ; (39) where BRS , BRP are set-valued best response functions. Thus, the best response dynamics are given by a system of differential inclusions rather than differential equations. A smooth variant of the best response dynamics are given by the logit dynamics, defined as follows. Let eu1 . (40) L : R2 → [0, 1] be given by L(u1 , u2 ) = u1 e + eu2 Fix > 0. Then the logit dynamics are given by πS (A, pC C + (1 − pC )H) πS (N, pC C + (1 − pC )H) p˙A = L , − pA ; (41) πP (pA A + (1 − pA )N, C) πP (pA A + (1 − pA )N, N ) , − pC . (42) p˙C = L As → 0, the logit dynamics approach the best response dynamics, so they are a smooth approximation thereof. Note that unlike under the replicator dynamics, it is possible for an initial state on the boundary to follow a trajector that leaves the boundary. For instance, if the initial value of pA is 0, but activism is a better response to the initial pC than is non-activism, then pA will follow a non-zero trajectory under either the best response or logit dynamics. A complacent society can spontaneously decide to begin activism efforts if it is rational to do so. We will focus in this section on the logit dynamics. A phase portrait for the logit dynamics in case 1AC is given in Figure 4, while case 0 is given in Figure 5. These figures suggest that the unique MSNE for G should be a nodal sink. In fact, if is sufficiently small that will be the case. Theorem 6.2. Consider the game G. Assume that the payoff bimatrix of G has unequal entries. Under the logit dynamics, the unique MSNE of G is a asymptotically stable. If the unique MSNE is in pure strategies, it is a nodal sink. Otherwise, it is a spiral sink. Proof. If the unique MSNE is in pure strategies, it is not difficult to show that the resulting Jacobian at the PSNE has two negative eigenvalues. It follows immediately that the Nash equilibrium is a nodal sink, and is asymptotically stable. Let us turn now to case 0. In this case, by Theorem 4.1, the unique MSNE is given by pA = (IH − IC + δN σN )/(δN σN − δA σA ), pC = a/[(GH − GC )(δA − δN )] . (43) We can compute the entries of the Jacobian " J=

∂ p˙ A ∂pA ∂ p˙ C ∂pA

∂ p˙ A ∂pC ∂ p˙ C ∂pC

#

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS 17

1.0

0.8

0.6

pC

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

pA Figure 4. Phase Portrait in Case 1AC under the Logit Dynamics with = 0.001. The parameters are IC = 5, IH = 2, GC = 3, GH = 5, δN = 0.2, δA = 0.6, σN = 1, σA = 3, and a = 0.5. at the stationary point (43) as follows. We will denote by fi (for i ∈ N) some expressions that have been compactified to simplify notation. ∂ p˙A = −1; ∂pA

(44)

∂ p˙A ef1 (1 − pa )(GH − GC )(δA − δN ) = > 0; ∂pC (ef2 + ef3 )2

(45)

∂ p˙C −ef4 (1 − pC )(δA σA − δN σN ) = < 0; ∂pA (ef5 + ef6 )2 ∂ p˙C = −1. ∂pC

(46) (47)

If we let b=

∂ p˙A ef1 (1 − pa )(GH − GC )(δA − δN ) = ∂pC (ef2 + ef3 )2

and ef4 (1 − pC )(δA σA − δN σN ) ∂ p˙C = , ∂pA (ef5 + ef6 )2 then it is clear from the above that b, c > 0, and we can write the Jacobian at the MSNE simply as −1 b J= . −c −1 c=−

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W.C. ABRAM AND K. NORAY

1.0

0.8

0.6

pC

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

pA Figure 5. Phase Portrait in Case 0 under the Logit Dynamics with = 0.001. The parameters are IC = 3, IH = 2, GC = 3, GH = 5, δN = 0.2, δA = 0.6, σN = 1, σA = 3, and a = 0.5. The interior stationary point is at coordinates (pA , pC ) = (0.5, 0.625). √ Thus, J has eigenvalues λ = −1 ± 2 bci, which are complex conjugate eigenvalues with negative real part. Since the MSNE is an isolated stationary point, the logit dynamical system is almost linear at that point, and we can conclude that the MSNE is a spiral sink, hence is asymptotically stable. In light of the asymptotic stability of the unique MSNE of G in case 0, it is reasonable to ask whether the analogous result holds for the best response dynamics. In fact, the asymptotic stability of this equilibrium under the best response dynamics has already been established in the work of Brown (1951), and proofs also appear in Rosenmüller (1971), Hofbauer (1995), and Berger (2001). A corollary, which should be of broader interest, is the following. Corollary 6.3. Let H be the "Battle of the Sexes" game of Dawkins (1976). Under the logit or best response dynamics, the unique MSNE of H is a spiral sink, and is asymptotically stable. Proof. From the point of view of this analysis, H is equivalent to case 0 of G. The result then follows by Theorem 6.2. It is well known that under the replicator dynamics, the unique MSNE of H is a center, and is therefore Lyapunov stable but not asymptotically stable. We see here that that behavior is not robust under a change of dynamic, though the

POLITICAL CORRUPTION AND PUBLIC ACTIVISM: AN EVOLUTIONARY GAME THEORETIC ANALYSIS 19

logit and best response dynamics do not have obvious application to such biological examples. 7. Conclusion In this paper we explore the factors that govern the extent to which a society will be active against corruption within their government. Concerning the game G, the main insights are: (1) in all possible cases, there is exactly one Nash equilibrium; (2) if the Nash equilibrium is in pure strategies, it is a nodal sink, is evolutionarily stable, and is asymptotically stable; (3) if the Nash equilibrium is in mixed strategies, the evolutionary behavior depends upon the selection dynamic. In any case, the MSNE is not evolutionarily stable. Under the replicator dynamics, the MSNE is a center and is Lyapunov stable but not asymptotically stable. Under the logit or best response dynamics, the MSNE is a spiral sink and is asymptotically stable. In other words, a given society in our model will be on a path directed towards or oriented around one equilibrium point. If the unique equilibrium is in pure strategies, the populations will either approach complete corruption and non-advocacy, complete corruption and advocacy, or complete honesty and non-advocacy. If the unique equilibrium is in mixed strategies, the populations will orbit periodically about the mixed equilibrium levels of corruption and advocacy. In any case, the singular equilibrium strategy is at least Lyapunov stable. This suggests that cultures of corruption are self-reinforcing. The behavior of the dynamic orbit structure in the mixed strategy Nash equilibrium case indicates that the populations will oscillate between high and low levels of activism and corruption periodically given certain parameter values. One interpretation of this is that there is a time-lag between changes in corruption or activism and the corresponding population adjustments. Our results demonstrate that the similarities between our model and Axelrod’s Evolution of Cooperation game are primarily conceptual. Axelrod analyzes a prisoners dilemma game, which has a fundamentally different structure than our game. Specifically, a prisoner’s dilemma requires that, in a one-time game, the dominant strategy for both players is to defect. In our game activism (defect) can never be a dominant strategy because there is a cost associated with it. In the corruption literature, our results are similar to Úbeda and Duénez-Guzmán (2011). Though theirs is a single population game, they also conclude that stable PSNE and stable MSNE exist. Specifically, they conclude that a population of defecting civilians, a population of corrupt policers, and a mixed population of cooperating civilians and corrupt policers are all stable under certain restrictions. But, there are some key differences as well. In our game we find four possible stable equilibria, one of which only possesses Lyapunov stability under replicator dynamics. Furthermore, our game has a unique Nash equilibrium under each set of parameters, whereas Úbeda and Duénez-Guzmán find cases where two stable equilibria exist under the same parameters. Thus, the comparisons between the models are limited. Lastly, as we mentioned in the introduction, the replicator dynamics of our game are identical to that of the battle of the sexes game in biology. As our game concerns rational agents, we have also conducted analyses using the logit dynamics. This is an extension of the biological literature, and we find novel equilibrium behavior.

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W.C. ABRAM AND K. NORAY

8. Future Work 8.1. Improving the Model. Our model is a good start for understanding the relationship between activism and political corruption, but some of the assumptions we make deserve more examination. For example, in our model, we group all political actors together and assign them the same utility function. In reality, different political actors (bureaucrats, politicians, lobbyists, etc.) all have different incentive structures. Adding populations that represent types of political actors may make our model more realistic and offer insight into other sources of political corruption. Furthermore, the assumption that activism affects political incentives may not hold for nations where the government is tyrannical. A country with no voting and no well-defined judicial system may have negligible δA , σA δN , and σN = 0, rendering activism essentially futile. Tyrannical governments may also punish civilians if they advocate for reform, which is a possibility we do not explore in our model. Furthermore, the study of dynamics deserves more attention. As we have modeled the dynamic behavior, each set of parameters and initial conditions give one, deterministic path, but in reality deviations and shocks are to be expected. Introducing a stochastic perturbation term could give a more realistic dynamic behavior. 8.2. Falsifiability and Testability. In this paper, we do not analyze what parameter values are reasonable. Additionally, measuring the parameters in the real world would be difficult. Thus, testing or falsifying our model will be difficult, due to a sparse availability of reliable data on corruption and activism. Current measures of corruption are unreliable, and, to our knowledge, such metrics would be a boon to economics research generally, and to the study of corruption in particular. From a causal inference perspective, testing the model will have intrinsic identification problems. Because our model seeks to explain the interaction between politicians and society on a national scale, finding a convincing comparison group will be difficult. For example, suppose that activism for corruption reform increases in a country. It will be difficult to determine whether that change causally influenced the level of corruption in that country or if some other change caused activism and corruption to change simultaneously. Empirical work may address this by comparing changes in corruption levels in the country of interest to changes in corruption in a comparable country where activism levels did not change (also called a difference in difference approach). But the key to this empirical strategy is that the two countries are comparable. Most countries are not. Thus, the problem our model represents does not lend itself to a simple empirical analysis. acknowledgements. The authors would like to thank Charles Steele for his careful reading of an early draft of this paper. We would also like to thank the reviewers, whose incisive comments helped us to greatly improve this paper. References [1] Accinelli E, Sánchez Carrera EJ (2012) Corruption Driven by Imitative Behavior. Econom. Lett. 117(1): 84-87. [2] Antioci A, Sacco PL (1995) A Public Contracting Evolutionary Game with Corruption. J. Econom. 61(2): 89-122. [3] Axelrod R (1984) The Evolution of Cooperation. Basic Books Inc., New York. [4] Axelrod R, Hamilton WD (1981). The Evolution of Cooperation. Science 211(27): 1390-1396.

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