Social evolution leads to persistent corruption

Abstract

Cooperation can be sustained by institutions that punish free-riders. Such institutions, however, tend to be subverted by corruption if they are not closely watched. Monitoring can uphold the enforcement of binding agreements ensuring cooperation, but this usually comes at a price. The temptation to skip monitoring and take the institution's integrity for granted leads to outbreaks of corruption and the breakdown of cooperation. We model the corresponding mechanism by means of evolutionary game theory, using analytical methods and numerical simulations, and find that it leads to sustained or damped oscillations. The results confirm the view that corruption is endemic and transparency a major factor in reducing it.

Keywords: cooperation; corruption; evolutionary game theory; punishment; social contract.

Fig. 1.

Strategy dynamics resulting from the social-learning process in the player population and the umpire population. (A and B) Sustained oscillations. Corrupt umpires can invade and take over when players are OCs. Cooperation then gives way to defection, and investments drop drastically. PCs, however, turn the tide. They can coexist with ODs in sufficient amount to make honesty advantageous for the umpires. At this stage, OCs can spread, and the cycle resumes. The shown social-learning process is close to the exploration-limited case (with vanishing exploration rates; see text). The time-averaged ratio of honest-to-corrupt umpires is 2.15, whereas the theoretical value in the exploration-limited case is 2.54. Similar cycles of corruption prevail for a broad range of other exploration rates and imitation rules, see SI Appendix. (C and D) Damped oscillations. For larger populations, higher exploration rates, weaker selection, or higher information costs, the oscillations can be damped, leading to stable levels of persistent corruption (SI Appendix, Figs. S8 and S9). Parameters: b = 1, c = 0.5, f = B = 0.2, h = 0.1, A = 2, and M = 50, N = 10, μ = 0.001, ν = 0.005, and s = 10¹⁰ (A and B), or M = 5,000, N = 1,000, μ = 0.01, ν = 0.05, and s = 0.3 (C and D); thus, in A and B as in C and D, new strategies enter both populations through exploration at the same rate.

Fig. 2.

Backbone of the model. Shown are the replicator dynamics (Methods) on a projection of the four-dimensional state space S = Δ₄ × [0,1]. The simplex to the left is the state space of the player population when all umpires are honest, while that on the right applies when all umpires are corrupt. The colored balls depict fixed points. Apart from the fixed-point edge with x = (1, 0, 0, 0) and y₁ ∈ [0, 1] (shown by the line of small colored balls at the top), the fixed points are isolated; the absorbing state 5 is given by x = (0, 0, h/(f + A), (f + A − h)/(f + A)) and y₁ = 1, whereas the absorbing state 10 is given by x = (0, (f + B − h)/(f + B), h/(f + B), 0) and y₁ = 0. The small arrowheads indicate whether the linearized flow near the corresponding fixed point leads toward that fixed point or away from it, or, technically speaking, whether the corresponding “transversal eigenvalue” is positive or negative (6). If the arrowhead is open, that eigenvalue is 0: The direction of the flow then follows from its nonlinear components. The two block arrows along the top edge and within the bottom plane indicate the locally prevailing trends from honesty to corruption (top) and back (bottom).

Fig. 3.

Strategy dynamics in the large-population limit. Replicator-mutator equations (Methods) describe social learning in large populations of players and umpires. The orbits in A and B have the same starting point (y₁ = 1, x₁ = x₄ = 0.1, x₂ = x₃ = 0.4), but differ in the exploration rates μ = ν of players and umpires (orange, 0.0001; violet, 0.02; blue, 0.07). With increasing exploration rates, the limit cycle shrinks and turns into a stable fixed point reached by damped oscillations. The limit cycle for small exploration rates follows orbits along the edges of the replicator equations (Fig. 2): Depending on whether (c − B)/(A − B) < (c − B − h)/(b − B − f) or not, the limit cycle leaves the edge where all players are OCs to visit the absorbing state 8 or 9, respectively, before moving toward the absorbing state 10 and from there again toward more honest umpires. This is shown for large penalties A = 2 in A and small penalties A = 0.7 in B. The strategy cycles of the frequencies of the four types in the player population (C) and of the two types in the umpire population (D) are shown for the orange limit cycle in A, together with the frequencies of cooperators and defectors in the player population. Other parameters are as in Fig. 1 A and B.