Table 3 Analytical solutions to Eq. (19). h is defined as \(h(p)=p(1-e_1)+(1-p)e_1\) (see Eq. (1)). We employ the convention, \(\prod _{k=1}^{0} \cdot = 1\). From this table, we see that, for SJ norm, neither the error rate in action \(e_1\) nor the error rate in assessment \(e_2\) influences the stationary distribution. For SS and SH, \(e_1\) influences only masses \(q_j\), and \(e_2\) influences masses \(q_j\), means \(\mu _j\), and variances \(\sigma _j^2\). For SC, \(e_1\) influences nothing, but \(e_2\) influences means \(\mu _j\) and variances \(\sigma _j^2\).
From: Reputation structure in indirect reciprocity under noisy and private assessment
Social norm | # of Gaussians used | Mass, \(q_j\) | Mean, \(\mu _j\) | Variance, \(\sigma _j^2\) |
|---|---|---|---|---|
SJ | 1 | 1 | \(\displaystyle \frac{1}{2}\) | \(\displaystyle \frac{1}{4N}\) |
SS | \(\infty\) | \(\displaystyle \frac{\prod _{k=1}^{j-1}(1-h(\mu _k))}{\sum _{\ell =1}^{\infty }\prod _{k=1}^{\ell -1}(1-h(\mu _k))}\) | \(\displaystyle \frac{1-\{-(1-2e_2)\}^{j}}{2}\) | \(\displaystyle \frac{1-(1-2e_2)^{2j}}{4N}\) |
SH | \(\infty\) | \(\displaystyle \frac{\prod _{k=1}^{j-1}h(\mu _k)}{\sum _{\ell =1}^{\infty }\prod _{k=1}^{\ell -1}h(\mu _k)}\) | \(\displaystyle \frac{1-(1-2e_2)^{j}}{2}\) | \(\displaystyle \frac{1-(1-2e_2)^{2j}}{4N}\) |
SC | 2 | \(\displaystyle \frac{1}{2}\) | \(\mu _1=1-e_2, \mu _2=e_2\) | \(\displaystyle \frac{e_2(1-e_2)}{N}\) |
