Basins of attraction and equilibrium selection under different learning rules

Erratum to: J Evol Econ

                    DOI 10.1007/s00191-009-0136-x

As stated in the article, Theorems 2 and 4 are incorrect. They refer to strict equilibrium actions in the limit $\vec{P} \to \vec{\hat{P}}$ when in fact the results only hold for pure, evolutionarily stable strategies (ESS) that are uniform in this limit. The corrected versions along with the definition of a uniformly ESS are included below.

FormalPara Definition

An equilibrium s is a uniformly ESS in the limit as $\vec{P} \to \vec{\hat{P}}$ if there is a punctured neighborhood $\dot{U}(\mathbf{s})$ of s such that for all $\mathbf{s'} \in \dot{U}(\mathbf{s})$ and all $\vec{P} \neq \vec{\hat{P}}$ in some neighborhood of $\vec{\hat{P}}$,

$$ \mathbf{s} \cdot \vec{\pi}(\mathbf{s'}) > \mathbf{s'} \cdot \vec{\pi}(\mathbf{s'}), $$

where the payoff vector $\vec{\pi} = (\pi_1, \ldots, \pi_n)$.

FormalPara Theorem 2

Suppose

$$\lim\limits_{\vec{P} \to \vec{\hat{P}}} \sum\limits_{a} m\left( B(\mathbf{R}, a, \vec{P}) \cap B(\mathbf{B}, a, \vec{P}) \right) = 0.$$

Then every pure, uniformly ESS satisfies the Never an Initial Best Response Property at $\vec{\hat{P}}$ . ¹

FormalPara Proof

Suppose that s is a pure, uniformly ESS such that $m\left(\mathrm{BR}^{-1}(s)\right)$ remains strictly positive in the limit $\vec{P} \to \vec{\hat{P}}$. We will identify a nonvanishing region inside the basins of attraction of s for both replicator dynamics and best response dynamics.

As a pure strategy equilibrium, s can be written as (x _s  = 1, x _− s  = 0) where s is the action always taken in this equilibrium. Let U(s) be a neighborhood of s such that $\dot{U}(\mathbf{s}) = U(\mathbf{s}) \backslash \{\mathbf{s}\}$ satisfies the condition for s to be a uniformly ESS. Let ν = sup _{x ∉ U(s)} x _s . Define the neighborhood $W(\mathbf{s}) \subseteq U(\mathbf{s})$ of all points satisfying x _s  > ν. We have constructed W(s) such that $\mathbf{x} \in \dot{W}(\mathbf{s})$ implies that $\dot{x}_s > 0$ under the replicator dynamics (because by the ESS condition, action s has better than average payoff here) and in turn, $\dot{x}_s > 0$ implies that x remains in W(s).

We now observe that BR^− 1(s) is a convex set because of the linearity of payoffs. Additionally, since s is a pure Nash Equilibrium, s ∈ BR^− 1(s). Thus, BR^− 1(s) and W(s) have positive intersection. By the fact that W(s) is independent of $\vec{P}$ and our hypothesis that BR^− 1(s) is nonvanishing, we conclude that $m\left(W(\mathbf{s}) \cap \mathrm{BR}^{-1}(s)\right)$ remains strictly positive in the limit $\vec{P} \to \vec{\hat{P}}$. Note that by the ESS condition and the linearity of payoffs, we can rule out the possibility that there are multiple best responses anywhere in the interior of BR^− 1(s). For points x in the interior of W(s) ∩ BR^− 1(s), best response dynamics flows to s because BR(x) = {s} and replicator dynamics flows to s because x ∈ W(s).□

FormalPara Theorem 4

Suppose

$$\lim\limits_{\vec{P} \to \vec{\hat{P}}} \sum\limits_{a} m\left( B(\mathbf{OSPP}, a, \vec{P}) \cap B(\mathbf{TD}, a, \vec{P}) \right) = 0.$$

Then every pure, uniformly ESS satisfies the Never an Initial Best Response Property at $\vec{\hat{P}}$.

FormalPara Proof

The proof here mirrors the one for Theorem 2. We construct the neighborhood W(s) in the same way, but with the additional condition that $x_s > 1 - \hat{K}$. We need only show that for $\mathbf{x} \in \mathrm{int}\!\left(W(\mathbf{s}) \cap \mathrm{BR}^{-1}(s)\right)$, both classes of dynamics flow to s. Under one-sided payoff positive dynamics, $\dot{x}_s > 0$ for x ∈ W(s) because action s has an above average payoff, and such a flow cannot leave W(s). Under threshold dynamics, when $\mathbf{x} \in \mathrm{int}\!\left(W(\mathbf{s}) \cap \mathrm{BR}^{-1}(s)\right)$, Eq. 5 applies to all actions other than s because they have payoffs below the $\hat{K}^{{\rm th}}$ percentile. All other actions must have the same negative growth rate, so $\dot{\mathbf{x}} = \alpha (\mathbf{s} - \mathbf{x})$ for some positive constant α.□