Agent-based model of social dilemma gamesWe construct an agent-based model here. Consider a well-mixed finite group of N individuals playing social dilemma games. In the game, there are two players: a donor and a recipient. The donor decide whether to donate (cooperate or help, denoted as C) or not (defect or do nothing, denoted as D) to recipients on the basis of their own strategies. As with many previous analytical studies, players have three strategies: to be the first-order free-rider who always chooses D (perfect defector, denoted as Y), the second-order free-rider (free-riding against the ability to eliminate the first-order free-rider) who always chooses C (perfect cooperators, denoted as X), and conditional cooperators adopting a given social norm (discriminator, denoted as Z). All players play two games at each time \(t \in [1,T]\) where time is discrete. Among two games, each player participates in one game as a donor and in another game as a recipient. Every partner and the playing order are randomly determined at each time. Note that a total of 2NT games are played in all periods. In each game, When a donor chooses C, the donor pays the cost c while the recipient receives the benefit b, where \(b>c>0\). If the donor chooses D, nothing happens, that is, neither the donor nor recipient change their payoffs.Even if donors choose C, C might not be executed with a small probability \(e_1\); thus, D is executed instead of C. This is called an implementation error. The implementation error is valid for C only; thus, if donors choose D, D is executed. Each player in the population observes each game with a probability q. In other words, there is normally a mixture of individuals who observe the game and those who do not. There are also observation errors. Any observation content (C or D) is misunderstood with a small probability \(e_2\); thus, the observer would recognize the opposite action for the actual one (either C or D).An observer observing the game updates the assessment or impression (image) of the players privately on the basis of the observation result. Only discriminators (conditional cooperators) can become observers because they use players’ images to decide whether C or D when they become donors. The image is binary, i.e., good (\(=G\)) or bad (\(=B\)). Note that public assessment schemes help make decisions (whether C or D) on the basis of the image (reputation) of the donor and that of the recipient, whereas in private assessment schemes the image (impression) is something they have privately. Thus, it is natural to make it a simple action function (C to G and D to B). In other words, the action rule of discriminators is choose C if a recipient’s image in the mind of the discriminator is G; otherwise, choose D. Let an image that player i has on player j at time t be \(A_{ij}(t)\in \{G,B\}\).Definition of social normsObservers of games have the opportunity to update their image of donors or recipients. How to update the player’s image is an essential point. We introduce a consistency function to formulate this. It is a function for observers to determine whether a game being observed conforms to social norms. If an observer feels that a game is consistent, they do not update the player’s images (but may update the update time to current). However, if the observer feels that the game is not consistent, they update the player’s images to be consistent. This consistency function is defined as$$\begin{aligned} f_C: \hbox {Donor’s image} \{G,B\} \times \hbox {Recipient’s image} \{G,B\} \times \hbox {Donor’s action to Recipient} \{C,D\} \rightarrow \{0,1\} \end{aligned}$$
(1)
where 1 represents “consistent” while 0 not. There are 256 consistency functions (because the total number of cases in the domain is \(2 \times 2 \times 2 = 8\), and there are \(2^8\) ways to assign either 0 or 1 to each case) and numbered in the Supplementary Information (SI). In each simulation run, one consistency function is fixed given and adopted as a social norm. We did not consider any case of multiple consistency functions competing against each other in the simulations.If an observer sees a game as inconsistent with social norms, how do they update a player’s image? Almost all studies to date dealt only with updating donor images. However, this constraint is too strong for private assessment schemes. Here, we consider the possibility updating the images of recipients as well as donors. We consider the following six rules, which include the most common rule (Rule 1).[Rule 1: Prioritizing Recipient’s Image] Prioritize the recipient’s image and update the donor’s image[Rule 2: Prioritizing Donor’s Image] Prioritize the donor’s image and update the recipient’s image[Rule 3: Prioritizing New Image] Prioritize the most recently updated image and update the older image[Rule 4: Prioritizing Good Image] Prioritize good images and update bad images[Rule 5: Prioritizing Bad Image] Prioritize bad images and update good images[Rule 6: Random Updating] Randomly decide what image to updateIn Rules 3–5, the donor and recipient images may have the same priority; thus, if so, one image (either donor or recipient) is selected at random for updating.The total number of possible social norms in this model is \(256 \times 6 = 1536\), where each norm follows one single rule. For example, a social norm called \([f_C^{150},\text{ Rule } 4]\) means that the 150th consistency function under [Rule 4: Prioritizing Good Image] as the image update rule is adopted. We comprehensively analyzed all possible social norms.Simulation results for all social normsOur agent-based simulation results for all social norms indicate that, on the one hand, Rules 2, 3, 5, and 6 cannot maintain cooperation of any consistency function except for 231 (Fig. 1). On the other hand, Rule 1 (hereafter referred to as Donor Updating Rule) and Rule 4 (hereafter referred to as Good Image Prioritizing Rule) have several consistency functions that can maintain cooperation. In other words, what is prioritized when updating the image is an important factor as a social norm for maintaining a cooperative regime. It is interesting to note that in contrast to the Donor Updating Rule, Rule 2 (which favors the donor image and updates the recipient image) fails to maintain cooperation with any consistency function. It might make sense that Rule 6 (Random Updating) does not work, but surprisingly Rule 3 (Prioritizing New Image) does not work either. Rule 5, the rule to update good images in favor of bad images, failed to maintain cooperation for any consistency function. In contrast, the Good Image Prioritizing Rule, the rule to update bad images in favor of good images, was able to maintain cooperation with various consistency functions. These results reveal the necessary conditions for priority rules that can sustain cooperation for several consistency functions. It should be either a rule that prioritizes the recipient image and updates the donor image, or a rule that prioritizes the good image and updates the bad image.Figure 1Rate of maintaining cooperation in all possible social norms. This heat map shows percentage of simulation runs where cooperation rate exceeded \(80\%\) at end of 100th generation for each social norm \([f_C^{x},\text{ Rule } y]\), where horizontal axis represents x and vertical axis represents y. Each social norm was run 10 times with different random seeds. Parameter values were \(N = 100\), \((b,c) = (3,1)\), \((e_1,e_2) = (1\%,1\%)\) and \(q=0.1\).Stable cooperative norms determinedAs the exhaustive results shown in Fig. 1 reveal, only a few social norms can sustain cooperation. We then show in Fig. 2 the results of examining the distribution of the cooperation rate for the social norms where the rate of maintaining cooperation exceeds 0 in Fig. 1. Out of the distribution of cooperation rates in the box plot of Fig. 2, a social norm in which the cooperation rate at the third quartile exceeds \(80\%\) is called a stable cooperative norm (SCN). As shown in this figure, the SCNs under the Donor Updating Rule (Rule 1) are social norms with consistency functions are 166, 167, 182, or 183. SCNs under the Good Image Prioritizing Rule (Rule 4) are those with consistency functions 134, 135, 150, 151, 165, 166, 167, 181, 182, or 183. As can be seen in Fig. 2, there are other social norms that is possible to maintain cooperative systems, but they are not stable due to the effects of randomness of the simulations. Note that social norms that can be stabilized by the Donor Updating Rule can also be stabilized by the Good Image Prioritizing Rule. Therefore, it is suggested that the Good Image Prioritizing Rule is the easiest rule for constructing SCNs.Figure 2Cooperation rates in each social norm. This box plot shows cooperation rates at end of 100th generation for each social norm. Each social norm was run 50 times with different random seeds. Left panel represents results of Donor Updating Rule (Rule 1) and center panel represents results of Good Image Prioritizing Rule (Rule 4), where horizontal axis represents x of \(f_C^{x}\) consistency function. Right panel represents results of consistency function 231 where horizontal axis represents y of \([f_C^{231},\text{ Rule } y]\). Parameter values were as in Fig. 1.Features of SCNs in donor updating ruleNext, we investigated what consistency functions constitute the SCNs. We first consider the Donor Updating Rule. This rule has been typically assumed in almost all previous studies on indirect reciprocity. Ohtsuki and Iwasa13 identified eight norms that can sustain cooperation in public assessment schemes. These norms are known as the Leading Eight, which correspond to consistency functions 148–151 and 180–183. Consistency functions 180–183, called the Leading Four, can maintain cooperative regimes even in private assessment schemes46. In contrast with those studies, our simulations allow only two norms, called L4 (or consistency function 182) and L7 (or Staying, consistency function 183) to be included in SCNs. Why are our simulations more intolerant? This is because our simulations are conducted in finite populations not a mathematical analysis in infinite populations, and we assume that the effects of perturbation originating from mutation and the learning process is not small.In the Donor Updating Rule, two norms (consistency functions 166 and 167) are included in SCNs. These norms have a remarkable feature: they can keep a cooperative regime both in private assessment and in public assessment schemes, but stable populations consist of a mixture of norm-adopters and perfect cooperators, as shown in the SI.Image scoring maintain cooperation in good image prioritizing ruleWe find 10 SCNs in Good Image Prioritizing Rule of all 256 consistency functions. The most important SCN under the Good Image Prioritizing Rule may be consistency function 165. The social norm called \([f_C^{165},\text{ Rule } 1]\) corresponds to Image Scoring, the simplest strategy for indirect reciprocity. The image-updating pattern of Image Scoring is shown in Fig. 3a. As many theoretical analyses including that by Sigmund3 point out, Image Scoring temporally reaches a cooperative regime, but cooperation often breaks down in very long-term simulations. In our simulations, however, \([f_C^{165},\text{ Rule } 4]\) shown in Fig. 3b maintained cooperative regimes. Compared with the normal Image Scoring (Donor Updating Rule), this norm (Good Image Prioritizing Rule) differs in three cases of the updating rule, as shown in Fig. 3a, b. This feature may re-shed light on Image Scoring.Figure 3Configuration of image-updating patterns in social norm \([f_C^{x},{{Rule }} y]\): (a) \((x,y)=(165,1)\), (b) \((x,y)=(165,4)\), (c) \((x,y)=(150,4)\), (d) \((x,y)=(167,4)\). Each panel shows image-updating patterns for all eight cases corresponding to the domain in the definition equation of the consistency function (Eq. 1). In each case, donor’s image is either G or B in left circle, recipient’ image is either G or B in right circle, and donor’s action is either C or D in center arrow. If observer observes one of eight cases, observer recognizes whether case is consistent. If it looks consistent (with green check mark), no image will be updated. If not, either donor’s image or recipient’s image will be updated in accordance with diagram using curved arrows and small circles in each case. Note that (a) corresponds to normal image scoring while (b) represents Image Scoring under the Good Image Prioritizing Rule, and (c) represents L6 (or Kandori, Heider-type) under the good image prioritizing rule.Norms adopting Heider’s balance theory maintain cooperationThe social norm \([f_C^{150},\text{ Rule } 1]\) (or Stern-judging or Kandori) included in Leading Eight cannot maintain cooperation in the private assessment scheme, but social norm \([f_C^{150},\text{ Rule } 4]\) (Fig. 3c) under the Good Image Prioritizing Rule is included in SCNs. This norm corresponds to Heider’s balance theory58. Thus, the interpretation of this norm is updating an image of either the donor or recipient as prioritizing good images if the game in the eyes of an observer is inconsistent with respect to the view point of Heider’s balance theory for the triplet of donor’s image, recipient’s image, and donor’s action.Figure 4 shows a summary of known cooperation norms and SCNs. There are social norms that have similar characteristics and have close consistency function numbers; thus, we group some norms together. The Heider type (consistency function 150) has assessment rules following the Heider’s balance theory for all eight cases of a game. Let us consider another social norm that does not follow the theory for only at most two of the eight cases: defecting to bad recipients by either good and bad donors. These cases have often been analyzed as so-called justified defection. Let us extend the Heider type of allowing assessments for such a justified defection to be freely made and call such extended social norms Heider-like norms. As shown in Fig. 4, consistency functions 134, 135 and 151, which are Heider-like norms, are SCNs under the Good Image Prioritizing Rule. We call Image-Scoring-like norms those where assessments for a game played by bad donors and bad recipients are made freely while the other assessments follow like Image-Scoring norm. Three Image-Scoring-like norms (consistency functions 165, 166, and 167) are SCNs under the Good Image Updating Rule.Figure 4Summary diagram of relationship between SCNs and known cooperation norms. This diagram shows (1) SCNs under Donor Updating Rule, (2) SCNs under Good Image Prioritizing Rule, (3) Leading Eight discovered by Ohtsuki and Iwasa (2004), (4) Leading Four discovered by Okada (2020), (5) Image-Scoring norm and its similar norms (IS-like), and (6) norm consistent with Heider’s balance theory and its similar norms (Heider-like). Diagram number corresponds to consistency function number while number beginning with L in blanket corresponds to number of Leading Eight given by Sigmund (2010). Note that although this diagram shows the SCNs and the Leading Eight correspondences, this correspondence holds “only in the Donor Updating Rule (Rule 1)” and not in Rule 4.Although we defined SCN above, there are many ways to define it. Despite such flexibility, the most stable social norms may be narrowed down to several candidates. In our simulation results, \([f_C^{167},\text{ Rule } 4]\) (Fig. 3d) and \([f_C^{183},\text{ Rule } 4]\) are included as strong candidates for the champion which can maintain cooperative regimes most robustly. Consistency Function 183 corresponds to Staying (L7), and consistency function 167 has characteristics like mixing of Image Scoring (consistency function 165) and Staying. See details in the SI.
Norms prioritizing positive assessments are likely to maintain cooperation in private indirect reciprocity
