The similar and upward social comparison heuristic dominates

Initially we consider the effect of social comparison using the image scoring assessment rule, which is the least sophisticated approach that allows observation of evolution without any effects from discriminatory assessment. Keeping other variables constant, we vary the cost-benefit ratio c/b as shown in Fig. 1.

Figure 1: Evolution of social comparison heuristics with image scoring assessment while varying the cost-benefit ratio c/b. The plots represent the relative distribution of heuristics present in the population taken from all generations. The shaded areas are proportional to the frequency of the associated heuristic. Parameters settings are reported in the Methods Section. Full size image

Low c/b ratios, such as 0.1, are typically required for indirect reciprocity to be sustained through image scoring models17, and when the cost-benefit ratio reaches 0.5, they are known to perform quite poorly. Figure 1 reflects this; however in the cost-benefit range where evolution is sustained (i.e., at most c/b = 0.25), the (1, 1, 0) heuristic, representing donation to those with similar or better reputation, clearly dominates. This indicates a potential cycle of donation that is driven by an escalating relative perception.

An individual i who frequently donates will experience an increase in their own reputation, which affects their perception of others relative to themselves. For example, after a reputational increase for i, a third party j who originally had a similar reputation to i is subsequently perceived by i as having lower standing. When i adopts the dominant strategy of donating to those with a similar or better reputation, then j must increase her own reputation (i.e., number of donations) in order to remain eligible to receive donations from i. We note that this dynamic operates within each generation, between selection and reproduction. Social comparison couples individual perception of others to their own standing, and evolution acts on the heuristics governing relative perceptions, rather than on absolute thresholds for the perception of acceptable/unacceptable donation behaviour.

Figure 2 shows the results from Fig. 1 in terms of average payoff per player per generation, where the payoff to an individual adopting a given strategy is the difference between benefit and cost incurred over a generation. For lower cost-benefit ratios (e.g., 0.1, 0.25) that support the emergence of cooperation, the payoff per individual reflects the behaviour in Fig. 1 where the cooperative strategies produce the highest payoff, and in particular the dominant approach of donating to those with similar or higher reputation. When the cost-benefit ratio reaches 0.5 this trend is reversed. The dominant (1, 1, 0) heuristic still produces the highest payoff per individual but with marginal average payoff as compared to lower c/b ratios. Beyond this c/b ratio (i.e., c/b = 0.75), defection becomes rational (Fig. 1c) but yields little positive payoff on average. Here the vast majority of generations are characterised by near zero donations being made.

Figure 2: Average payoff per player per generation for the alternative social comparison strategies, using image scoring assessment while varying the cost-benefit ratio c/b. Parameter settings are consistent with those in Fig. 1. Full size image

Discriminatory assessment rules reinforce the dominant strategy

The evolution of indirect reciprocity under image scoring is known to be susceptible to non-discriminatory assessment rules3,57 and therefore it is valuable to consider the effects of standing and judging2,54 to update reputation (Fig. 3). When generalised to a non-binary representation of reputation and considered in the context of social comparison, standing involves decrementing the reputation of i when i defects in light of a request from a player j with at least the reputation of i. Judging offers greater penalisation than standing by punishing a donor for not further targeting their behaviour, with the reputation of r i decremented when i makes a donation to a less reputable recipient j.

Figure 3: Cooperation from the social comparison strategies using different assessment rules while varying the cost-benefit ratio c/b. Parameter settings are consistent with Fig. 1. “Average cooperation” indicates the frequency of cooperative interaction: the number of donations made as a proportion of the total number of games played in all preceding generations. Full size image

We observe that the discrimination provided by standing and judging exactly represents penalties for actions which are inconsistent with the dominant social comparison rule of donation to a recipient of similar or upward standing. Consequently the social norms provided by standing and judging embody social comparison and this mechanism further supports the evolution of indirect reciprocity, as seen in Fig. 3. In particular standing and judging increase the extent of cooperative behaviour in the population, reaching in excess of 90% for low cost-benefit ratios (e.g., 0.1). The selective effects of discrimination from standing and judging, as compared to image scoring, also significantly extend the range of cost benefit ratio at which cooperation is sustained, for example with both standing and judging reaching nearly 90% cooperation levels with cost-benefit ratios of 0.85. Thus when the cost is relatively high, discrimination becomes influential.

Social comparison provides robustness against errors

We investigate the sensitivity of the social comparison model to errors in both user perception and execution. Perception errors involve inaccuracy in the perceived reputation, modelled by misreading the potential recipient’s reputation with probability p r , in which case an alternative reputation is uniformly selected from another member of the population. This type of error has been a focus for attention in previous studies2, aligned to the effects of gossip and malicious misreporting5. Perception error is known to cause negative effects on discriminatory assessments such as standing58, but exhibiting robustness when error rates are relatively small17.

Results (Fig. 4) are consistent with previously published work applying perception error17. When applying standing and judging for social comparison, evolution is resilient to reasonable error rates such as 5% with similar degradation in the frequency of cooperative interaction evident when the experiment is repeated at a higher error rate (e.g., p r = 10%). Image scoring exhibits similar behaviour under perception error but shows a large degradation in the population’s cooperative behaviour as error level increases.

Figure 4: Effect of perception and execution error on the social comparison strategies. c/b ratio = 0.25. Parameter settings are consistent with Fig. 1. The error rate applied is 5%. Full size image

In contrast to perception error, execution errors represent involuntary human mistakes, which have received less attention3,59. This error represents a failure to execute the intended strategy and has two forms: one-way execution error is applied with probability e to any donation action; two-way execution error is applied with probability e to both donation and defection decisions. Consistent with the published literature17, results from our experiments show that strategies based on social comparison are robust to modest errors of both types (e.g., e = 5%). However, the impact of execution errors on the frequency of donation is generally worse than perception errors, increasing with the error rate. Additionally, the discriminating strategies of standing and judging show almost identical characteristics for both one-way and two-way errors. With perception errors there is a chance that reputation will still be appropriately classified by social comparison, however failure to execute an intended action offers no direct opportunity for evolutionary recovery through rebalancing effects, that is errors leading to increased defection being offset by errors leading to increased cooperation.

For non-discriminating assessment provided by image scoring, the results from the two-way execution error not only exhibits superior cooperation levels as compared to one-way execution error, but the results are comparable to those of standing and judging in terms of the decrease in average cooperation as compared to a zero error state. This is consistent with the observation that two-way execution error may self-compensate through the equal treatment of error in defection and donation3,59 which is more likely to occur when reputation is updated without discrimination, as in image scoring.

The dominant social comparison heuristic provides evolutionary stability

We assess the dominant similar and upward comparison heuristic for evolutionary stability. As with all other strategies, strategies involving social comparison cannot discriminate against duplicitous agents who initially cooperate to encourage the evolution of a prosocial population, with a view to subsequently exploiting the population by free-riding60. However, it is prudent to examine the extent to which discrimination present in the similar and upward comparison heuristic is sufficient to dominate over defectors, including performance in extreme scenarios. This is shown in Fig. 5, where a sub-population adopting the similar and upward comparison heuristic is examined in the presence of defectors. The sub-population adopts this heuristic with assessment through either image scoring, standing or judging. The probability of convergence to zero defectors represents the proportion of cases from 1000 runs.

Figure 5: The ability of a discriminating sub-population adopting the (1, 1, 0) heuristic, to dominate in the presence of defectors. Population size N is fixed at 100. c/b ratio = 0.25. μ = 0. Other parameter settings are consistent with Fig. 1. Error rates in both execution and perception are applied at 5%. The probability of convergence to zero defectors represents the proportion of cases from 1000 runs in which the behaviour is observed. Full size image

Overall, a high proportion of defectors are needed to prevent the evolution of a sub-population that adopts the dominant (1, 1, 0) heuristic. If just 10% of the population apply the similar and upward comparison heuristic while discriminating through standing or judging, then the chance of fully eradicating defectors is 98.7% for standing and 99.2% for judging. Consistent with previous observations made on the lesser evolutionary stability of image scoring3, a much larger sub-population is required (over 40%) to achieve similar levels of performance when image scoring is applied as the assessment rule. These results suggest considerable resilience, particularly when the social norm in standing and judging further reinforces behaviour consistent with similar and upward comparison.

Convergence to zero defectors is relatively rapid even when the initial sub-population adopting the similar and upward comparison heuristic is small. For example, on average, when adopting standing within a sub-population representing 5%, the population converges to zero-defectors within 10.55 generations (SD = 3.45), where each player acts as a potential donor on average 50 times per generation. Under the same conditions judging converges marginally quicker (mean = 10.3, SD = 3.37) and image scoring never converges to a population with zero defectors. With a sub-population of 40 adopting the similar and upward comparison heuristic, there is a greater chance that the population will converge to zero defectors. This occurs more slowly for image scoring (mean = 7.63, SD = 2.24) as compared to standing (mean = 3.61, SD = 0.69) and judging (mean = 3.52, SD = 0.69).

A hetrogeneous population structure can enhance the global cooperation level

We assume a heterogeneous population structure by sub-dividing the population into isolated social groups consistent with the idealised Island Model17. The social groups define the boundaries within which members may donate to others. The global population (N = 100) is structured into g social groups of equal size for g = 2, 3, 4, 5 (when g = 3 the groups are of size 33 and 34). We adopt assessment by image scoring and standing with c/b ratios selected as 0.1 and 0.85 respectively, and execution and perception error rates of 2.5% are applied. These conditions allow the observation of a heterogeneous population when p, the probability of reproduction from the local sub-population rather than the global population, is varied.

Under these parameters the results show that a social group structure can positively affect the evolution of cooperation. This is particularly evident for the less sophisticated image scoring assessment, as compared to standing, where potential increases in cooperation are at best marginal. Figure 6 shows that for image scoring cooperation increases with both the number of social groups and the probability of reproduction within groups p. However, when reproduction is entirely limited to the local population (p = 1), total cooperation levels drop significantly, with smaller groups increasing this effect for both image scoring and standing.

Figure 6: Average cooperation level and percentage of the (1, 1, 0) heuristic from all games in all generations, applying a heterogeneous population with g groups, for g = 1, 2, 3, 4, 5. c/b ratio for image scoring is 0.1. c/b ratio for standing is 0.85. Perception and execution errors are applied, both with a rate of 2.5%. Other parameter settings are consistent with Fig. 1. “Average cooperation” indicates the frequency of cooperative interaction: the number of donations made as a proportion of the total number of games played. Full size image

Contributory to this phenomenon is the small number of possible strategies that social comparison affords, with just eight possible states as compared to 121 for the original image scoring model1. This encourages dominant strategies to readily evolve in small sub-groups, although such dominant strategies may be non-cooperative due to the lower chance of in-group diversity and the effects of genetic drift. However when a small chance of reproduction from the global population is introduced (e.g., p = 0.95), this provides an opportunity to introduce, with high payoff, cooperative strategies into any non-cooperative sub-groups. As found in the previous section of results, only a small number of players with number of the (1, 1, 0) strategy are required to dominate over a defecting population, allowing non-cooperative sub-groups to be dominated. The results in Fig. 6 also reaffirm the correlation between the dominant (1, 1, 0) social comparison heuristic and high cooperation levels.