Exact cluster dynamics of indirect reciprocity in complete graphs

Abstract

Heider's balance theory emphasizes cognitive consistency in assessing others, as is expressed by ``The enemy of my enemy is my friend.'' At the same time, the theory of indirect reciprocity provides us with a dynamical framework to study how to assess others based on their actions as well as how to act toward them based on the assessments. Well known are the ``leading eight'' from L1 to L8, the eight norms for assessment and action to foster cooperation in social dilemmas while resisting the invasion of mutant norms prescribing alternative actions. In this work, we begin by showing that balance is equivalent to stationarity of dynamics only for L4 and L6 (stern judging) among the leading eight. Stern judging reflects an intuitive idea that good merits reward whereas evil warrants punishment. By analyzing the dynamics of Stern Judging in complete graphs, we prove that this norm almost always segregates the graph into two mutually hostile groups as the graph size grows. We then compare L4 with stern judging: The only difference of L4 is that a good player's cooperative action toward a bad one is regarded as good. This subtle difference transforms large populations governed by L4 to a ``paradise'' where cooperation prevails and positive assessments abound. Our study thus helps us understand the relationship between individual norms and their emergent consequences at a population level, shedding light on the nuanced interplay between cognitive consistency and segregation dynamics.

Figures (10)

• Figure 1: Schematic representation of transitions among balanced configurations under L6 or L4, where $v$, $v'$, and $v"$ are arbitrary elements of $V$. Every arrow in the diagrams denotes a transition induced by a single assessment error, and the dotted arrows exist only in the case of L4. The dashed box containing the transition between the paradise $\{V\}$ and $\{\{v\}, V-\{v\}\}$ is detailed in Fig. \ref{['fig:diagram']}.
• Figure 2: Zoomed-in views of the dashed box in Fig. \ref{['fig:L4state']}, containing the transition between the paradise and $\{\{v\}, V-\{v\}\}$. Self-loops are all omitted for the sake of brevity. Each node represents a set of configurations that are identical up to the permutation of vertices. The paradise and $\{\{v\}, V-\{v\}\}$ are labeled with $0$ and $N$, respectively. The symbols attached to edges are transition probabilities: We have $P_j^+ = (N-j)/N^2$, whereas $P_j^- = (N-j)(j-1)/N^2$ for $j>1$ with $P_1^- = (N-1)/N^2$. The other transition probabilities are $\mu_j = (N-j)/N^2$, $\nu_{j} = j/N^2$, and $S_j = j(N-j)/N^2$.
• Figure 3: Absolute difference of cluster sizes divided by $N$, obtained from $2\times 10^2$ Monte Carlo samples. Each sample starts from a random initial configuration and follows the common social norm with $\epsilon=0$ until arriving at a balanced configuration. We have defined $\eta \equiv \left| n - (N-n) \right|$, assuming that the clusters are of sizes $n$ and $N-n$, respectively. (a) Under L6, the behavior of $\eta/N$ is proportional to $N^{-1/2}$, which is fully consistent with a prediction from the binomial distribution. (b) Under L4, by contrast, $\eta/N$ tends to $1$ as $N$ grows, which implies the emergence of the paradise. The error bars are smaller than the symbol size.
• Figure C1: An example of Fig. 2 when $N=6$. The lower left corner is the paradise, denoted by $0$ in Fig. 2, whereas the upper right corner shows the segregation of a single individual from the others, denoted by $1$ in Fig. 2.
• Figure D1: Schematic diagrams to specify trajectories from a segregated configuration to the paradise. (a) The initial segregation between $A \cup \{a\}$ and $B \cup \{b\}$ is perturbed by $a$'s misjudgment of $b$. (b) If a member of $B$ donates to $b$, he or she will also be regarded as good by $a$. In this way, $a$ can assess $b_1, b_2, \ldots, b_m$ as good. This configuration can change to the paradise as soon as $a$ is chosen as the donor and one of $b_i$’s is chosen as the recipient.