Evolving Righteousness in a Corrupt World

Punishment offers a powerful mechanism for the maintenance of cooperation in human and animal societies, but the maintenance of costly punishment itself remains problematic. Game theory has shown that corruption, where punishers can defect without being punished themselves, may sustain cooperation. However, in many human societies and some insect ones, high levels of cooperation coexist with low levels of corruption, and such societies show greater wellbeing than societies with high corruption. Here we show that small payments from cooperators to punishers can destabilize corrupt societies and lead to the spread of punishment without corruption (righteousness). Righteousness can prevail even in the face of persistent power inequalities. The resultant righteous societies are highly stable and have higher wellbeing than corrupt ones. This result may help to explain the persistence of costly punishing behavior, and indicates that corruption is a sub-optimal tool for maintaining cooperation in human societies.

where each row corresponds to the four strategies in the above order. For conciseness, we will refer to the strategies as cooperator (C), defector (D), cooperative punisher (H), and defecting punisher (K). Throughout this article, we use bold letters to represent non-scalar variables with upper-and lower-case letters corresponding to matrices and vectors, respectively. LikeÚbeda & Duéñez Guzmán (2011), we analyze the model through the continuous time replicator dynamics (Hofbauer & Sigmund, 1998):

Equilibria and Stability
Interior An equilibrium population x in which no strategy has gone to exticntion satisfies Therefore, internal equilibria are normalized solutions to the system Ax = 1 4 , where 1 n is the vector of all ones of size n. Solving the linear system and assuming e 2 = 0, we find that the equilibrium exists if an only if can lie inside the simplex. However, the components have opposite signs and therefore y CDHK cannot lie in the simplex. Therefore, all equilibria must lie in the corners, edges, or faces of the simplex, and, by the exclusion principle (Hardin, 1960), all interior trajectories converge to the simplex boundary. The normalization factor 1/est is so that the sum of the coordinates of y CDHK adds up to one (i.e. so it lies on the simplex). To streamline notation we will ommit this normalization factor, and instead give the values of y as proportional to a vector, using the notation
It is worth noting that for any frequency based analysis of n strategies, we only need to consider n − 1 eigenvalues of the Jacobian, since there will always be an eigenvector that lies outside of the simplex (i.e. the eigenvector v such that v T .1 ̸ = 0). For this reason, whenever we discuss eigenvalues of Jacobians, we will only give those that are associated with eigenvectors laying in the simplex.
The eigenvalues for the corner equilibria z C , z D , z H and z K are

respectively.
It is now possible to analyze local stability of these equilibria. Only z D is always stable, and z C is always unstable (recall that t > r). z H can be stable in the directions pointing towards z D and z K if p > t − r and q > t − r respectively; as discussed above, z H is stable in the direction pointing towards z C if and only if e > 0. Finally, if q + c < s + e, q + c < p and q < s, then z K is stable in the directions of z C , z D and z H respectively.

Edges
For convenience we will use y to represent an equilibrium in a reduced game, much in the same way as we use z. The difference here being that y will have as many components as strategies are in the reduced games, while z is always a vector with 4 components. There is a natural mapping from a y vector to a z vector, simply by inserting zeroes in the components of z that are not in its subindex (i.e. not present in y). For example y DH = (y 1 , y 2 ) corresponds to an internal equilibrium in the reduced game between D and H, and z DH = (0, y 1 , y 2 , 0) corresponds to that same equilibrium, but considered in the whole game.
Existence of internal equilibria in reduced games with two strategies are as in (Úbeda & Duéñez Guzmán, 2011), except for the game between C and H. Conditions for stability are slightly different, and the derivations are explicitly given whenever they differ from (Úbeda & Duéñez Guzmán, 2011). For completeness, we summarize the conditions for existence and stability of all two-strategy games below.
Game Between C and D This case corresponds to the well-known Prisoner's Dilemma.

Game Between C and H This game is characterized by the payoff matrix
which corresponds to a perturbation of the neutral line of stability in the original Corruption Game. An internal equilibrium exists if and only if the solution y CH of the system A CH y CH = 1 2 can lie on the interior of the simplex. However, is the solution to that system, and therefore it cannot lie in the simplex. Consequently, there is no internal equilibrium.

Game Between C and K This game has an interior equilibrium of the form
if and only if q + c > s + e. Equilibrium z CK is stable in the whole game if and only if all these are satisfied That is, if z K is stable from the direction of z D (p > q + c), K is unstable toward z C (q + c > s + e), and e is small.

Game Between D and H This game has an interior equilibrium of the form
if and only if z H is stable from the direction of D(p > t − r). The equilibrium z DH is always unstable in the full game.

Game Between D and K This game has an interior equilibrium of the form
if and only if z K is stable from the direction of z D (p > q + c).
The equilibrium z DK is always unstable in the full game.

Game Between H and K There is an interior equilibrium
The equilibrium z HK is stable if and only if all of the following are satisfied In particular, this implies that q is very close to t − r, we also require e > 0 and p > q. Therefore, we can parametrize the conditions in the above inequalities as so this inequality is automatically satisfied. In other words, both z HK and z CK can be globally stable.

Faces
Here we focus on the three-strategy games, and analyze existence of internal equilibria as well as their stability.
Game without C This game has an internal equilibrium of the form The equilibrium z DHK is stable if and only if the following conditions are satisfied where χ = t − r − q, σ = t − r − s and ∆ is an expression that depends on the parameters of the game.
Observe that since e ≈ 0, then c(p − q)(t − r) + e (s(p − q) − cq) > 0 and therefore the equilibrium is always unstable.
Game without D If an internal equilibrium existed, it would be of the form Which is an interior point when H is stable from K (q > t − r) and e is small enough.
Let α = (c − e)(t − r) + es, and note that cq > α > eq for the equilibrium to exist. The equilibrium z CHK is stable if and only if where β = α − q(q + c − s) and γ = eq(eq − α) (cq − α). Notice that if e > 0, then −4γ > 0 and thus, the discriminant √ β 2 − 4γ is larger than |β|. Therefore, it is impossible for all quantities to be negative at the same time, and then equilibrium z CHK is always unstable.
Game without H This game has an internal equilibrium of the form which, given that e ≈ 0, exists if and only if p > q + c and (p − q)s < c(p − e).
As before, let α = (c − e)(t − r) + es. Equilibrium z CDK is stable if and only if the following inequalities hold Nota that each factor ofγ is positive, thus, the discriminant is larger thañ β, which implies that not all eivengalues can be negative at the same time. Therefore, equilibrium z CDK is always unstable.
Game without K This game corresponds to the Punishment Game and has an internal equilibrium of the form which, given that e ≈ 0, we can assume p > |e| and c > |e|. In this case, (c + e)(p − e) > 0, and the equilibrium exists if and only if −e(c + s) > c(t − r − p). In particular, in order for the equilibrium to exist, if e > 0, then p > t − r, and if e < 0, it requires p to be not too much smaller than t − r.
Equilibrium z CDH is stable if and only if the following condition are satisfied (p − q − e)(c(t − r) + es) < 0 e (c(t − r) + ps) ± √ ∆ < 0 where ∆ is an expression that depends on the parameters of the game. If e > 0, then e (c(t − r) + ps) > 0 which makes equilibrium z CDH always unstable.