Notes on the model of Smaldino & Epstein (2015)

Background

In a recent paper, (Smaldino 2015) provide a model for conformist bias that allows individuals to deviate from the mean trait value, \(\bar{x}\), via a “distinctiveness” trait, \(\delta_i\) for individual \(i\). Each individual updates its trait value \(x_i\) via the following recursion \[x_i(t+1) = (1-k) x_i(t) + k \left( \bar{x}(t) + \delta_i \sigma(t) \right)\] where \(\sigma(t)\) is the standard deviation of \(x_i\) at time \(t\) and \(k\) scales the rate of adjustment.

Proof of “conformity” result

Figure 2 of the paper provides numerical evidence that the explosion of the variance in \(x\), \(\sigma^2\), occurs when the variance in \(\delta_i\), which I call \(V_\delta\), is greater than one. Equation D12 provides analytical evidence of this pattern for a special case where \(\delta_i\) takes only two values in the population. This increase in the variance is interpreted as non-conformity.

However, this result holds for any distribution of \(\delta_i\), namely that \(\sigma^2(t) > 0\) as \(t \to \infty\) when \(V_\delta > 1\). To show this mathematically, I will show that the equilibrium point of zero variance is unstable. To begin proving this, I need the recursion for the variance in the trait, \[\label{eq-sig2} \sigma^2(t+1) = (1-k)^2 \sigma^2(t) + 2 k (1-k) \sigma \text{Cov}(x,\delta)(t) + k^2 \sigma^2(t) V_\delta \:,\] which holds for any distribution on \(\delta\). Unlike the case in (Smaldino 2015) where \(\delta\) is a constant, the recursion for \(\sigma^2\) contains the covariance between \(x\) and \(\delta\). Even though this covariance maybe small initially, it may build up over time. Thus, I need the recursion for the covariance as well: \[\label{eq-cov} \text{Cov}(x,\delta)(t+1) = (1-k) \text{Cov}(x,\delta)(t) + k \sigma(t) V_\delta \: .\] The only equilibrium point of these two equations is \((\sigma^2,\text{Cov}(x,\delta))=(0,0)\), which is the conformity outcome. In order to show that this equilibrium point is unstable, I apply standard linear stability analysis for discrete-time systems. This involves calculating the Jacobian matrix, \(J\), from equations \ref{eq-sig2} and \ref{eq-cov}, \[\label{eq-J} J = \begin{pmatrix} (1-k)^2+\frac{k \text{Cov}(x,\delta ) (1-k)}{\sigma}+k^2 V_{\delta } & 2 (1-k) k \sigma \\ \frac{k V_{\delta }}{2 \sigma} & 1-k \\ \end{pmatrix} \: ,\] and determining the conditions under which the eigenvalues of \(J\) have magnitude greater than one. A sufficient condition for the eigenvalues, \(\lambda\), to be greater than one in magnitude is that the characteristic polynomial, \(P(\lambda)\), evaluated at one is negative (this comes from the so-called “Jury condition”). Applied to the Jacobian in \ref{eq-J}, \[P(1) = k^2 \left( (1-k) \left(1-\frac{\text{Cov}(x,\delta )}{\sigma }\right)+1-V_{\delta } \right) \:.\] At the equilibrium point with zero variance (zero standard deviation) and zero covariance, \[P(1) = k^2 ( 1 - V_\delta) \: ,\] which is negative when \[V_\delta > 1 \: .\]

References

  1. P. E. Smaldino, J. M. Epstein. Social conformity despite individual preferences for distinctiveness. Royal Society Open Science 2, 140437–140437 The Royal Society, 2015. Link

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