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Ligonier Ministries The teaching fellowship of R. Close Your Cart Loading Search Home Learn. Article Obeying God or Man? Sproul Sinclair Ferguson W. Sproul R. Sproul Books That Influenced R. Coram Deo In his commentary Romans, Dr. Passages for Further Study Job Romans — Topics Spiritual Growth , Sanctification. Events At least one such perturbation appears to be a general feature of human psychology. Costly punishment is used to express negative emotion . Expressions of anger result in increased social status and perceived competency  , and aggression enhances perceived popularity and social centrality .
Anger especially enhances status when it is perceived as retaliatory  ,  ; in this situation, observers often respond uncritically to hostile action  ,  and may even assist punishers . In humans, social status is strongly related to several forms of wellbeing, including health  , happiness  , absence of psychological distress  , and income  , as well as evolutionary fitness . The increased status of individuals who express anger at injustice can therefore be interpreted as a small payment to punishers.
More generally, the tendency to punish may have social consequences for the punisher beyond the immediate cost of punishment. Such consequences might be negative or positive. In the original corruption game, corruption stabilized cooperation by effectively offsetting the cost of punishment. There are, however, other ways in which this cost might be offset. A small benefit to punishers in interactions with cooperators, such as the status-payments described, could provide an alternative means to offset the cost of punishment.
Here, we explore how such small benefits to punishers affect the maintenance of cooperation and the evolution of corruption and righteousness. Importantly, these payments avoid most components of a reputation system, where individuals decide whom to cooperate with based on information about past interactions received from others 47 — Such reputation systems require cognitive and social resources that may be unavailable in some systems. Cooperators in our model automatically make small payments to punishers.
For this to work, punishers need only be physically recognizable by cooperators or by a centralized authority. This is biologically plausible for social insects, where punishers are often larger and stronger  ,  , . In human societies, punishers can often be identified by cultural tags such as uniforms even in the absence of individual recognition, and payments can also be conferred via taxation systems without any need for individual observation.
We consider an evolutionary game with four strategies, namely: cooperative non-punisher , defecting non-punisher , cooperative punisher and defecting punisher. The game is defined by the payoff matrix where each row corresponds to the four strategies in the above order. For conciseness, we will refer to the strategies as cooperator , defector , cooperative punisher , and defecting punisher.
Throughout this article, we use bold letters to represent non-scalar variables with upper- and lower-case letters corresponding to matrices and vectors, respectively. For simplicity we assume that. Parameters correspond to the cost experienced by a defecting non-punisher and a defecting punisher when punished. Parameter corresponds to the cost experienced by a punisher when punishing another individual.
To account for payments from cooperators to punishers, we introduce the parameter. For simplicity, we will assume that is very small, in particular. Although payment is made by non-punishers at an individual level during interactions, it is dynamically equivalent to a payment by all non-punishers. However, in the presence of even a small amount of defection which is very biologically and socially realistic , punishers have a smaller payoff than pure cooperators.
This is due to the fact that payments are very small when compared to the costs of punishing. There are two differences between our payoff matrix and the payoff matrix of the Corruption Game  : introduction of parameter , and collapsing the costs of inflicting a punishment on defectors and on defecting punishers into one parameter. The choice to collapse and is to maintain tractability of the model by maintaining the same number of parameters. Moreover, the existence and stability of all equilibria in the Corruption Game was independent from .
Although had a quantitative effect on the size of basins of attraction, parameter was also free in the sense that the relevant dynamics involved , and never alone. Notice that in the special case when the cost of punishing is equal for both types of punishers the Corruption Game corresponds to. The game reduced to only and has a degenerate payoff matrix in the Corruption Game, that is, both strategies have exactly the same payoff.
In the current model, however, the game reduced to the strategies and has a non-degenerate payoff matrix 1 thus avoiding the existence of structurally unstable equilibria. We chose the perturbation in Equation 1 because of its simplicity and because it is zero-sum. Other perturbations can lead to qualitatively different dynamics, but require a significant surplus of payoffs or costs , which is harder to justify biologically.
For humans, could result from increased social status of punishers; for other species, the value of could be either positive or negative. When , righteousness is globally unstable, and the dynamics lead to either defection or corruption, which is qualitatively equivalent to the Corruption Game. Thus, is the only situation which may lead to righteousness. Both cultural and genetic evolution are most commonly studied using replicator dynamics  — .
Note that we are representing the frequencies of strategies in the population by a vector of dimension 4. Therefore, we can geometrically consider all possible population states as elements of the -simplex. Populations consisting of only one strategy would lie at vertices of this tetrahedron see Figure 1. If the condition is satisfied, then the direction pointed by the arrow behaves as a local attractor. While many equilibria at the edges of the simplex may be stable in the reduced games, we reserve filled circles to indicate globally stable equilibria i.
The equilibria of Equation 2 may rest at discrete points in the interior, corners, edges, or faces of the simplex formed by all possible population states. When an equilibrium is stable, it is locally asymptotically stable. We will refer to equilibria by with subindices denoting where the equilibrium lies.
For instance, will correspond to the equilibrium at the vertex , and to an internal equilibrium in the face comprised by the strategies , and. A monomorphic population of defectors is always stable. As is customary, we will call this equilibrium defection. In addition to this equilibrium, two other stable equilibria can exist. One is either or i. Figure 2 shows the conditions for stability of the three main equilibria defection, corruption and righteousness depending on the severity of the punishment towards defectors and corrupt punishers parameters and , respectively.
The total cost of punishing a corrupt punisher determines the stability of the corruption equilibrium. The overall temptation to defect , that is, the difference between the payoff of a defector and a cooperator against a cooperator mediates the stability of the righteousness equilibrium. Righteousness is stable whenever. The white area corresponds to the cases in which defection is the only globally stable equilibrium.
Notice that there is an area where righteousness and corruption intersect, in this region, all three main equilibria are stable. Depicted are representative cases for each of the four areas. While the position of the main equilibria might change and existence of other unstable internal equilibria in some edges might exist for specific parameter combinations, the qualitative dynamics are captured by these depicted cases.
For simplicity, internal equilibria in the faces of the simplex are not drawn. All internal equilibria in the faces are unstable see Appendix. Righteousness is stable either at if or at if ; where depends on see Text S1 , equation S9 , and is denoted by the two-segment line bounding the narrow region under in Figure 2. Note that the regions of stability for righteousness and corruption overlap all through the region delimited by and , as well as through most of the narrow region where is stable see Text S1.
Intuitively, righteousness and corruption are both stable when punishment against non-punishers is larger than the total cost of punishing a punisher, and when the punishment against corrupt punishers is severe i. Recall that denotes a power inequality in favor of defecting punishers, and indicates a case of egalitarian punishments. To estimate the basin of attraction of each of the equilibria, we simulated the dynamics of the system numerically.
All runs were performed with , , , and. We conduct the analysis for the punishment parameters and with values between and in increments of. Given a value of the parameters and , we analyze the dynamics starting close to the simplex corners , and. The corner is not analyzed for it is always stable. For each of these three cases, we take a set of small perturbations of order uniformly around the corresponding corner, and simulate the dynamical system using an Euler scheme until the population is close enough to one of the three main equilibria: defection, corruption or righteousness.
We summarize the proportion of runs that end in each of the three possible equilibria see Figure 3.
The Gift of Righteousness - Exploring Issues in Righteousness
The horizontal axis corresponds to the value of. The vertical axis corresponds to the value of. Isoclines represent the proportion of runs converging to corruption red and righteousness blue. All runs that do not converge to either corruption or righteousness end up in defection white. As expected, whenever power inequalities favor non-punishers , and thus corruption is unstable , the proportion of runs converging to corruption is zero see Figure 2.
In general, as long as , increasing power asymmetries by increasing or decreasing increases the basin of attraction of corruption.
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This is seen in Figure 3 , where the basin of attraction of corruption increases from zero when to close to one when. It is worth noting that when the population starts at righteousness, and both and , then the dynamics always remain at righteousness. Moreover, even when the population starts close to , up to of the runs end up in righteousness Figure 3. This proportion grows as both and grow and is maximal whenever , that is, when there are no power inequalities. This pattern is maintained even when the cost to punish is much larger data not shown. We have explored the effect of a perturbation to the corruption game, namely, small payments such as a slight increase in social status to punishers.
We find that the more egalitarian and harsher the punishments toward defectors and defecting punishers, the more likely the population will maintain cooperation through punishment and keep both corruption and defection at bay. In this scenario, the most likely outcome is a monomorphic population of cooperative punishers righteousness. This shows not only that costly punishment can evolve recall that , but that even when the social investment in punishment a payment of from non-punishing cooperators to punishers diminishes as non-punishers shrink in frequency, punishing still pays off better than defecting.
Moreover, the prevalence of cooperative punishers makes defection by punishers an inviable strategy, even when defecting punishers get more lenient punishments power inequality. Thus, a righteous population can effectively resist the spread of corruption. Righteousness, by stabilizing cooperation and providing a higher payoff to cooperative groups, constitutes a mechanism to shift the scale of selection from an individual to a group level.
Unlike alternative mechanisms to maintain cooperation, such as reputation, righteousness requires no individual recognition or memory. Righteousness does require some ability to discriminate between punishers and non-punishers, but such discrimination can occur without complex cognition; for example, ant punishers are often larger and more aggressive than non-punishers  ,  , .
Because the collective payoff of righteousness is higher than that of alternative outcomes, righteous groups are likely to outcompete those that have converged on defection or corruption. As a result, righteousness is expected to spread either culturally or genetically. This mechanism may explain the observation of righteous punishment in some ant species  and some human societies .
Our results show that a population that starts at or close to all cooperation will either go to defection or corruption, but not to righteousness Figure 3 , top. However, if it goes to corruption, it is possible to destabilize this equilibrium and have the population end up at righteousness in a reliable manner Figure 3 , middle and bottom. Much of the dynamics revolve around the costs imposed by corruption and the appeal of defection. An initially cooperating population that faces invasion by defectors might institute punishment in response.
Even mild punishment , can suppress defection. However, the punishers in this population are susceptible to corruption. In fact, power inequalities that favor corruption are required to keep defectors at bay, and if costs are small, corruption runs rampant. Now that the population is corrupt, it is in a stable situation. Small changes to the costs and punishments will not change the dynamics qualitatively.
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While cooperation can be increased by increasing the cost of corruption, there will always be defecting punishers, and in fact they are needed to prevent defection spreading . However, if cooperators invest even a tiny amount in punishment , there is a possible route from corruption to righteousness.
Conferring increased status on punishers, as occurs among humans, may be one form of such investment. Corruption can be destabilized by making punishments both more egalitarian and harsher and close to or above , see inequality 4. Provided punishments are sufficiently harsh, completely removing power inequalities eliminates corruption, and cooperative punishment will likely spread.
However, complete equality is likely to be unfeasible in human societies. Given that power inequalities cannot be removed completely, a sudden, large change in punishment can still destabilize corruption and stimulate a transition to righteousness. The righteous population is resilient to invasion by both corruption and defection. In fact, righteousness is so stable that once there, a population needs to drop at least one of the punishments or below making defection appealing once again in order for a perturbation to destabilize it see Figure 3 , bottom.
Power inequalities are largely irrelevant to the righteous population; reducing power inequalities is only required initially to destabilize corruption, and lead the dynamics toward righteousness instead of defection. Mandatory payments from cooperators to punishers are justified empirically as discussed at the end of the introduction. However, we can conceive of a scenario in which there exist cooperators that do not make payments to punishers. In this case, we would have a line of neutral stability between the two types of cooperators as well between non-paying cooperators and honest punishers.
Introducing this non-paying cooperator does not change the existence or stability of the righteousness equilibrium in the replicator equation. However, in the presence of noisy dynamics, righteousness could be lost due to drift confirmed via numerical simulation; data not shown. This loss of righteousness is similar to the way tit-for-tat, which is an attractor and a promoter of cooperation in the presence of defection, is lost in noisy dynamics due to the neutral stability with pure cooperators . Intuitively, righteousness is typically capable of eradicating defection from a population, which allows for non-paying cooperators to spread due to drift.
Later, if no paying cooperators remain, the dynamics are governed by the original Corruption Game, and righteousness is lost. Nonetheless, notice that the interpretation of the payment is flexible. For instance, assuming that pure cooperators have a small chance of giving a gift to a punisher instead of being it mandatory does not change the dynamics.
In this way, can be interpreted as the expected payment over many interactions. Our results may help to explain the paradoxical data observed in human societies. In reality, however, the extent of corruption varies markedly between societies and is negatively correlated with several aspects of economic development  —  ,  social wellbeing  —  ,  and cooperation . Crime in general can be considered as defection, but corruption is positively related to other forms of crime .
Field studies of an egalitarian nomadic prestate society, the Turkana, also show that power inequalities are not required for the maintenance of large-scale cooperation via collective punishment of free-riders . One possible explanation for the observed variance and negative impact of corruption is that some societies have transitioned, or are transitioning, from widespread corruption to righteousness the reverse transition being much more difficult, as described above.
Democratization and improved law enforcement may tend to reduce the power inequalities that favor corruption; such change may occur suddenly, facilitating the transition to righteousness, due to new policies or change of government. Because the total societal payoff of righteousness exceeds that of corruption, groups that have attained righteousness are likely to out-compete those that remain corrupt. For example, Mathew and Boyd  suggest that the cooperation generated by collective punishment may explain the dominance of the Turkana over competing groups.
For tractability, game theory models necessarily consider a restricted set of possible strategies. In contrast, humans may use an endless variety of strategies, including maladaptive ones such as antisocial punishment  , .