Imitation in heterogeneous populations

Abstract

We study a boundedly rational model of imitation when payoff distributions of actions differ across types of individuals. Individuals observe others’ actions and payoffs, and a comparison signal. One of two inefficiencies always arises: (i) uniform adoption, i.e., all individuals choose the action that is optimal for one type but suboptimal for the other, or (ii) dual incomplete learning, i.e., only a fraction of each type chooses its optimal action. Which one occurs depends on the composition of the populations and how critical the choice is for different types of individuals. In an application, we show that a monopolist serving a populations of boundedly rational consumers cannot fully extract the surplus of high-valuation consumers, but can sell to consumers who do not value the good.

Appendices

Appendix 1: Proof of necessity in Proposition  1

We define the probability space $(\hat{\varOmega },\hat{{\mathscr {F}},}\hat{\mu })$ with $\hat{\varOmega }=X^{2}\times \varDelta$, $\hat{{\mathscr {F}}}=2^{X^{2}\times \varDelta }$, and the probability measure induced by $\mu _{\tau c,\tau ^{\prime }d}$, which we call the environment. The (marginal) distribution functions $\mu _{\tau c}$, $\mu _{\tau 'd}$ and $\mu _{\tau \tau 'd}$ are derived from $\mu _{\tau c,\tau ^{\prime }d}$ in the usual way. Thus, the payoff distribution of a type $\tau$ individual who chooses c is $\mu _{\tau c}$ and the probability of the event $\{x\in D\}$, with $D\subseteq X$, is denoted by $\mu _{\tau c}(D)$; and if D is a singleton $\{x\}$, with $x\in X$, then its probability is denoted by $\mu _{\tau c}(x)$. The analogous notation conventions apply to the distributions of the comparison signal and the payoff distribution of a type $\tau '$ individual who chooses d. Necessity in Proposition 1 is argued using the following lemmata.

Lemma 5

Suppose L is payoff-ordering. Then, $\pi _{\tau d}=\pi _{\tau c}$ implies $L_{cd}(\tau ,\tau ^{\prime })=\frac{1}{2}$ for all different actions c, d and types $\tau ,\tau '\in T$.

Proof

Consider the environment induced by $\mu _{\tau c,\tau ^{\prime }d}$ such that $\pi _{\tau d}=\pi _{\tau c}$ and assume $L_{cd}(\tau ,\tau ^{\prime })<\frac{1}{2}$ (the argument for the case $L_{cd}(\tau ,\tau ^{\prime })>\frac{1}{2}$ is analogous). We will now consider a different environment induced by a slightly different probability mass, $\widetilde{\mu }_{\tau c,\tau ^{\prime }d}$. Suppose that payoffs and comparison signals, in both environments, are independent. The modified version of $\mu _{\tau d}$, denoted by $\widetilde{\mu }_{\tau d}$, is such that for any set $I\subset X{\setminus }\{1\}$ we have $\widetilde{\mu }_{\tau d}(I)=(1-\varepsilon )\mu _{\tau d}(I)$ and $\widetilde{\mu }_{\tau d}(1)=\mu _{\tau d}(1)+\varepsilon \mu _{\tau d}(X{\setminus }\{1\})$ for some $\varepsilon \in (0,1]$. Thus, $\widetilde{\pi }_{\tau d}=(1-\varepsilon )\pi _{\tau d}+\varepsilon$. The modified version of $\mu _{\tau c}$, denoted by $\widetilde{\mu }_{\tau c}$, is such that for any $I\subset X{\setminus }\{0\}$ we have $\widetilde{\mu }_{\tau c}(I)=(1-\varepsilon )\mu _{\tau c}(I)$ and $\widetilde{\mu }_{\tau c}(0)=\mu _{\tau c}(0)+\varepsilon \mu _{\tau c}(X{\setminus }\{0\})$. Thus, $\widetilde{\pi }_{\tau c}=(1-\varepsilon )\pi _{\tau c}$. The modified expected value of the comparison signal of a type $\tau$ individual who observes a type $\tau '$ individual choosing d, denoted by $\widetilde{\pi }_{\tau \tau ^{\prime }d}$, is

$$\begin{aligned} \widetilde{\pi }_{\tau \tau ^{\prime }d}&=(1-\varepsilon )\pi _{\tau d}+\varepsilon -\pi _{\tau ^{\prime }d}\\&=(1-\varepsilon )\left( \pi _{\tau d}-\pi _{\tau ^{\prime }d}\right) +\varepsilon (1-\pi _{\tau ^{\prime }d})\\&=(1-\varepsilon )\pi _{\tau \tau ^{\prime }d}+\varepsilon (1-\pi _{\tau ^{\prime }d}), \end{aligned}$$

where $\pi _{\tau \tau ^{\prime }d}$ is the expected value of the comparison signal in the initial environment. Suppose that the distribution of this comparison signal in the modified environment is given by a compounded distribution which weights with probabilities $1-\varepsilon$ and $\varepsilon$ the distribution of the comparison signal in the initial environment and a degenerate distribution which assigns all the probability to $1-\pi _{\tau ^{\prime }d}$, respectively. In all the other respects, the modified and initial environments are the same. Let $\widetilde{L}_{cd}(\tau ,\tau ^{\prime })$ denote the expected value of the probability of switching to d when $i\in \tau$ chooses c, observes $j\in \tau ^{\prime }$ who chooses d and the comparison signal, in the modified environment. Then, $\widetilde{L}_{cd}(\tau ,\tau ^{\prime })$ can be written as a continuous function of $\varepsilon$ over the domain [0, 1] and when $\varepsilon =0$, $\widetilde{L}_{cd}(\tau ,\tau ^{\prime })=L_{cd}(\tau ,\tau ^{\prime })<\frac{1}{2}$. In order to see this, notice that

$$\begin{aligned} \widetilde{L}_{cd}(\tau ,\tau ^{\prime })= & {} \int \int \int L(x,y,\delta )d\widetilde{\mu }_{\tau c}(x)d\widetilde{\mu }_{\tau \tau 'd}(\delta )d\widetilde{\mu }_{\tau 'd}(y)\\= & {} \int \int \left[ \varepsilon L(0,y,\delta )+(1-\varepsilon )\int L(x,y,\delta )d\mu _{\tau c}(x)\right] d\widetilde{\mu }_{\tau \tau 'd}(\delta )d\mu _{\tau 'd}(y)\\= & {} \int \left\{ (1-\varepsilon )\int \left[ \varepsilon L(0,y,\delta )+(1-\varepsilon )\int L(x,y,\delta )d\mu _{\tau c}(x)\right] d\mu _{\tau \tau 'd}(\delta )\right. \\&+\left. \varepsilon \left[ \varepsilon L(0,y,1-\pi _{\tau 'd})+(1-\varepsilon )\int L(x,y,1-\pi _{\tau 'd})d\mu _{\tau c}(x)\right] \right\} d\mu _{\tau 'd}(y), \end{aligned}$$

where the right-hand side is a polynomial function of $\varepsilon$. Thus, for small enough $\varepsilon$, $\widetilde{L}_{cd}(\tau ,\tau ^{\prime })<\frac{1}{2}$ and, since $\widetilde{\pi }_{\tau d}>\widetilde{\pi }_{\tau c}$, L is not payoff-ordering. $\square$

Corollary 2

If L is payoff-ordering, $x,y\in [0,1]$, $\delta \in [-1,1]$, and $x=y+\delta$, then $L(x,y,\delta )=\frac{1}{2}$.

Proof

Consider $x,y,\delta$ which satisfy the hypothesis and an environment in which $\mu _{\tau c}(x)=\mu _{\tau d}(y+\delta )=\mu _{\tau ^{\prime }d}(y)=\mu _{\tau \tau ^{\prime }d}(\delta )=1$.³⁴ In this environment, $L_{cd}(\tau ,\tau ^{\prime })=L(x,y,\delta )$, and thus, Lemma 5 implies $L(x,y,\delta )=\frac{1}{2}$. $\square$

Lemma 6

If L is payoff-ordering, then $L(x,y,\delta )$ is an affine transformation of $y+\delta -x$ for all $x,y\in [0,1]$ and $\delta \in [-1,1]$.

Proof

Consider arbitrary $x,y\in [0,1]$ and $\delta \in [-1,1]$. Consider an environment such that, with probability $\frac{1}{2}$, a first event occurs in which the payoff received by type $\tau$ when she chooses c is x, the payoff received by type $\tau ^{\prime }$ individual when she chooses d is y, and the comparison signal observed by a type $\tau$ individual when she observes the type $\tau ^{\prime }$ individual is $\delta$. Otherwise, a second event occurs in which the payoff received by a type $\tau$ individual when she chooses c is y, the payoff received by a type $\tau ^{\prime }$ individual when she chooses d is x, and the comparison signal observed by a type $\tau$ individual when she observes a type $\tau ^{\prime }$ individual is $-\delta$. It follows that $\pi _{\tau c}=\pi _{\tau d}=\frac{x+y}{2}$. If L is payoff ordering, then $L_{cd}(\tau ,\tau ^{\prime })=\frac{1}{2}$, i.e.,

$$\begin{aligned} \frac{1}{2}L(x,y,\delta )+\frac{1}{2}L(y,x,-\delta )=\frac{1}{2}. \end{aligned}$$

(11)

Now, consider a second environment that differs from the previous one only in that the first event is replaced by two events that occur with probability $\frac{1}{2}\frac{y+\delta -x+2}{4}$ and $\frac{1}{2}\left( 1-\frac{y+\delta -x+2}{4}\right)$, respectively. In the first of these events, the payoff received by a type $\tau$ individual when she chooses c is 0, the payoff received by a type $\tau ^{\prime }$ individual when she chooses d is 1, and the comparison signal observed by a type $\tau$ individual when she observes a type $\tau ^{\prime }$ individual is 1. In the second of these events, the payoff received by a type $\tau$ individual when she chooses c is 1, the payoff received by a type $\tau ^{\prime }$ individual when she chooses d is 0, and the comparison signal observed by a type $\tau$ individual when she observes a type $\tau ^{\prime }$ individual is $-1$. As in the previous environment, $\pi _{\tau c}=\pi _{\tau d}$ and, if L is payoff-ordering, then $L_{cd}(\tau ,\tau ^{\prime })=\frac{1}{2}$, i.e.,

$$\begin{aligned}&\frac{1}{2}\frac{y+\delta -x+2}{4} L(0,1,1)+\frac{1}{2}\left( 1-\frac{y+\delta -x+2}{4}\right) L(1,0,-1)\nonumber \\&\quad +\,\frac{1}{2}L(y,x,-\delta )=\frac{1}{2}. \end{aligned}$$

(12)

Subtracting (11) from (12), we obtain

$$\begin{aligned} L(x,y,\delta )=L(1,0,-1)+\left( L(0,1,1)-L(1,0,-1)\right) \left( \frac{1}{2}+\frac{y+\delta -x}{4}\right) . \end{aligned}$$

It follows that L is a linear function of $y+\delta -x$, and thus, from Corollary 2, L must satisfy $L(x,y,\delta )=\frac{1}{2}+\beta (y+\delta -x)$ for some real number $\beta$.

Finally, consider $x,y\in [0,1]$ and $\delta \in [-1,1]$ such that $y+\delta >x$ and the environment F such that $\mu _{\tau c}(x)=\mu _{\tau d}(y+\delta )=\mu _{\tau ^{\prime }d}(y)=\mu _{\tau \tau ^{\prime }d}(\delta )=1$. Since $L_{cd}(\tau ,\tau ^{\prime })=L(x,y,\delta )=\frac{1}{2}+\beta (y+\delta -x)$, payoff-ordering implies that $\beta >0$.   $\square$

To close the proof of Proposition 1, since the range of L is [0, 1], $x,y\in [0,1]$, and $\delta \in [-1,1]$, we also need $\beta \le \frac{1}{4}$.

Appendix 2: Formal derivation of the dynamic system

Probability space The populations is modeled as an index probability space $(W,{\mathscr {W}},\lambda )$ where W is the continuum of individuals, ${\mathscr {W}}$ is a $\sigma -$algebra of subsets of W, and $\lambda$ is a super-atomless measure; in particular, $\lambda (A)=\alpha$ and $\lambda (B)=1-\alpha$.³⁵ Time is indexed by $t\in {\mathbb {R}}^{+}$ with the Borel $\sigma -$algebra denoted by ${\mathscr {B}}$. The probability space that we use to model the random aspects of the imitation process is $(\varOmega ,{\mathscr {F}},P)$.³⁶

Dynamical system Our analysis is concerned with the fraction of type A individuals choosing a, $p(\omega ,t)$, and the fraction of type B individuals choosing b, $q(\omega ,t)$ for all $\omega \in \varOmega$ and $t\in {\mathbb {R}}^{+}$. In order to analyze the paths $\left( p(\omega ,t),q(\omega ,t)\right) _{t\in {\mathbb {R}}^{+}}$, we define the state function $\rho :W\times \varOmega \times {\mathbb {R}}^{+}\rightarrow \varSigma$ where $\rho (i,\omega ,t)$ is the state of individual i in the state of the world $\omega$, at time t. We also define the sampling function $\pi :W\times \varOmega \times {\mathbb {R}}^{+}\rightarrow W\cup \{J\}$ specifying the individual $\pi (i,\omega ,t)$ that the individual i samples at time t in the state of the world $\omega$, where $\pi (i,\omega ,t)=J$ means that individual i does not sample any other individual at time t in the state of the world $\omega$. We assume that $\pi (i,\omega ,t)\ne J$ implies $\pi (\pi (i,\omega ,t),\omega ,t)=i$; that is, individuals sample each other.

The fraction of individuals whose state is $\sigma$ at time t in the state of the world $\omega$, denoted by $p_{\sigma }(\omega ,t)$, is given by $p_{\sigma }(\omega ,t)=\lambda \left( \left\{ i\in W:\rho (i,\omega ,t)=\sigma \right\} \right)$ for all $\sigma \in \varSigma$, $\omega \in \varOmega$, and $t\in {\mathbb {R}}^{+}$. Therefore, we have the following version of Eq. (2), that express directly the state-of-the-world dependence of the fractions of individuals making optimal choices:

$$\begin{aligned} \alpha p(\omega ,t)=\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }(\omega ,t)\quad {\text { and }}\quad (1-\alpha )q(\omega ,t)=\sum _{\sigma \in \varSigma :\tau =B,c=b}p_{\sigma }(\omega ,t). \end{aligned}$$

(13)

Differential equations of the system We now provide the differential equations governing the fractions of the populations making their optimal choices within each type.

Proposition 4

Fix the initial fractions of the populations in each individual state at $\left( p_{\sigma }(0)\right) _{\sigma \in \varSigma }\in \hat{\varDelta }(\varSigma )$. There exists a Fubini Extension $(W\times \varOmega ,{\mathscr {W}}\boxtimes {\mathscr {F}},\lambda \boxtimes P)$ ³⁷ in which the state and sampling function $(\rho ,\pi )$ are defined, such that: (i) $(p(\omega ,t),q(\omega ,t))$ is deterministic almost surely; and (ii) for $P-$almost all $\omega \in \varOmega$, $(p(\omega ,\cdot ),q(\omega ,\cdot )):{\mathbb {R}}^{+}\rightarrow [0,1]^{2}$ is the solution of the system of differential equations

$$\begin{aligned} \alpha \dot{p}&= \alpha (1-p)\left[ \alpha p+(1-\alpha )(1-q)\right] L_{ba}(A)\nonumber \\&\quad -\alpha p\left[ \alpha (1-p)+(1-\alpha )q\right] L_{ab}(A) \end{aligned}$$

(14)

$$\begin{aligned} (1-\alpha )\dot{q}&= (1-\alpha )(1-q)\left[ (1-\alpha )q+\alpha (1-p)\right] L_{ab}(B)\nonumber \\&\quad -(1-\alpha )q\left[ (1-\alpha )(1-q)+\alpha p\right] L_{ba}(B), \end{aligned}$$

(15)

with initial condition³⁸

$$\begin{aligned} (p(\omega ,0),q(\omega ,0))=\left( \alpha ^{-1}\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }(0),(1-\alpha )^{-1}\sum _{\sigma \in \varSigma :\tau =B,c=b}p_{\sigma }(0)\right) . \end{aligned}$$

Before providing the proof of Proposition 4, we need to introduce the following definition, which is adapted from Duffie et al. (2016).

Definition 2

Consider a pair of a state and a sampling function $(\rho ,\pi )$ defined on a Fubini Extension $(W\times \varOmega ,{\mathscr {W}}\boxtimes {\mathscr {F}},\lambda \boxtimes P)$, $\left( p_{\sigma }(0)\right) _{\sigma \in \varSigma }\in \hat{\varDelta }(\varSigma )$, and $\left( \left( \varsigma _{\sigma \sigma '}(\sigma '')\right) _{\sigma ''\in \varSigma }\right) _{(\sigma ,\sigma ')\in \varSigma ^{2}}$ with $\left( \varsigma _{\sigma \sigma '}(\sigma '')\right) _{\sigma ''\in \varSigma }\in \varDelta (\varSigma )$ for all $(\sigma ,\sigma ')\in \varSigma ^{2}$. The pair $(\rho ,\pi )$ is said to be a continuous time dynamical system with independent random sampling and independent random state-changing with parameters ³⁹ $\left( p_{\sigma }(0)\right) _{\sigma \in \varSigma }$ and $\left( \left( \varsigma _{\sigma \sigma '}(\sigma '')\right) _{\sigma ''\in \varSigma }\right) _{(\sigma ,\sigma ')\in \varSigma ^{2}}$ (denoted by DS) if: (i) $\left( p_{\sigma }(\omega ,t)\right) _{\sigma \in \varSigma }$ is deterministic almost surely with given initial conditions $\left( p_{\sigma }(\omega ,0)\right) _{\sigma \in \varSigma }=\left( p_{\sigma }(0)\right) _{\sigma \in \varSigma }$; (ii) for $\lambda -$almost every $i\in W$, $\rho (i,\cdot ,\cdot )$ is a continuous time Markov chain in $\varSigma$ with transition intensity

$$\begin{aligned} R_{\sigma \sigma ''}(\omega ,t)=\sum _{\sigma '\in \varSigma }p_{\sigma '}(\omega ,t)\varsigma _{\sigma \sigma '}(\sigma ''), \end{aligned}$$

(16)

for all two different states $\sigma$ and $\sigma ''$ in $\varSigma$, and $R_{\sigma \sigma }(\omega ,t)=-\sum _{\sigma '\in \varSigma {\setminus }\{\sigma \}}R_{\sigma \sigma '}(\omega ,t)$ for all $\sigma \in \varSigma$, $\omega \in \varOmega$, and $t\in {\mathbb {R}}^{+}$; (iii) $\rho (i,\omega ,t)$ is $\left( {\mathscr {W}}\boxtimes {\mathscr {F}}\right) \otimes {\mathscr {B}}-$measurable; and (iv) for $\lambda -$almost all $i\in W$, $\rho (i,\cdot ,t)$ and $\rho (j,\cdot ,t)$ are independent for $\lambda -$almost all $j\in W$.

Now, we are ready to provide the proof of Proposition 4.

Proof

From Corollary 2 in Duffie et al. (2016), there exists a Fubini Extension $(W\times \varOmega ,{\mathscr {W}}\boxtimes {\mathscr {F}},\lambda \boxtimes P)$ such that the pair of state and sampling function $(\rho ,\pi )$, defined on $(W\times \varOmega ,{\mathscr {W}}\boxtimes {\mathscr {F}},\lambda \boxtimes P)$, is a DS with parameters $\left( p_{\sigma }(0)\right) _{\sigma \in \varSigma }\in \hat{\varDelta }(\varSigma )$ and $\left( \left( \varsigma _{\sigma \sigma '}(\sigma '')\right) _{\sigma ''\in \varSigma }\right) _{(\sigma ,\sigma ')\in \varSigma ^{2}}$ with $\left( \varsigma _{\sigma \sigma '}(\sigma '')\right) _{\sigma ''\in \varSigma }\in \varDelta (\varSigma )$ for all $(\sigma ,\sigma ')\in \varSigma ^{2}$.⁴⁰ From their Corollary 1, for $P-$almost all $\omega$, $\left( p_{\sigma }(\omega ,t)\right) _{\sigma \in \varSigma }$ is the solution of the system of differential equations

$$\begin{aligned} \dot{p}_{\sigma }(\omega ,t)=&\sum _{\sigma ''\in \varSigma {\setminus }\{\sigma \}}p_{\sigma ''}(\omega ,t)\sum _{\sigma '\in \varSigma }p_{\sigma '}(\omega ,t)\varsigma _{\sigma ''\sigma '}(\sigma )\nonumber \\&-p_{\sigma }(\omega ,t)\sum _{\sigma ''\in \varSigma {\setminus }\{\sigma \}}\sum _{\sigma '\in \varSigma }p_{\sigma '}(\omega ,t)\varsigma _{\sigma \sigma '}(\sigma '') \end{aligned}$$

(17)

for all $\sigma \in \varSigma$, with $\left( p_{\sigma }(\omega ,0)\right) _{\sigma \in \varSigma }=\left( p_{\sigma }(0)\right) _{\sigma \in \varSigma }$. Therefore,⁴¹

$$\begin{aligned} \alpha \dot{p} =&\sum _{\sigma \in \varSigma :\tau =A,c=a}\dot{p}_{\sigma }\nonumber \\ =&\sum _{\sigma \in \varSigma :\tau =A,c=a}\sum _{\sigma ''\in \varSigma {\setminus }\{\sigma \}}p_{\sigma ''}\sum _{\sigma '\in \varSigma }p_{\sigma '}\varsigma _{\sigma ''\sigma '}(\sigma )\nonumber \\&-\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }\sum _{\sigma ''\in \varSigma {\setminus }\{\sigma \}}\sum _{\sigma '\in \varSigma }p_{\sigma '}\varsigma _{\sigma \sigma '}(\sigma '') \end{aligned}$$

(18)

If $\tau =A$ and $\tau ''=B$, then $\varsigma _{\sigma ''\sigma '}(\sigma )=\varsigma _{\sigma \sigma '}(\sigma '')=0$ for all $\sigma '\in \varSigma$. Therefore, in both the minuend and subtrahend, the second summation is over $\{\sigma ''\in \varSigma :\tau ''=A\}$; thus,

$$\begin{aligned} \alpha \dot{p} =&\sum _{\sigma \in \varSigma :\tau =A,c=a}\sum _{\sigma ''\in \varSigma :\tau ''=A}p_{\sigma ''}\sum _{\sigma '\in \varSigma }p_{\sigma '}\varsigma _{\sigma ''\sigma '}(\sigma )\\&-\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }\sum _{\sigma ''\in \varSigma :\tau ''=A}\sum _{\sigma '\in \varSigma }p_{\sigma '}\varsigma _{\sigma \sigma '}(\sigma '')\\ =&\sum _{\sigma \in \varSigma :\tau =A,c=a}\sum _{\sigma ''\in \varSigma :\tau ''=A,c''=b}p_{\sigma ''}\sum _{\sigma '\in \varSigma }p_{\sigma '}\varsigma _{\sigma ''\sigma '}(\sigma )\\&-\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }\sum _{\sigma ''\in \varSigma :\tau ''=A,c''=b}\sum _{\sigma '\in \varSigma }p_{\sigma '}\varsigma _{\sigma \sigma '}(\sigma ''), \end{aligned}$$

where the second equality is obtained canceling the terms appearing in the minuend and subtrahend out. Next, since $c=a$ and $c''=b$ we have $\varsigma _{\sigma ''\sigma '}(\sigma )=0$ whenever $c'=b$ and $\varsigma _{\sigma \sigma '}(\sigma '')=0$ whenever $c'=a$. Therefore, in the minuend, the third summation is over $\{\sigma '\in \varSigma :c'=a\}$, and in the subtrahend, the third summation is over $\{\sigma '\in \varSigma :c'=b\}$. Thus,

$$\begin{aligned} \alpha \dot{p}= & {} \sum _{\sigma \in \varSigma :\tau =A,c=a}\sum _{\sigma '':\tau ''=A,c''=b}p_{\sigma ''}\sum _{\sigma '\in \varSigma :c'=a}p_{\sigma '}\varsigma _{\sigma ''\sigma '}(\sigma )\nonumber \\&-\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }\sum _{\sigma ''\in \varSigma :\tau ''=A,c''=b}\sum _{\sigma '\in \varSigma :c'=b}p_{\sigma '}\varsigma _{\sigma \sigma '}(\sigma ''). \end{aligned}$$

(19)

The minuend in the right-hand-side of (19), denoted by M, may be written as

$$\begin{aligned} M= & {} \sum _{\sigma ''\in \varSigma :\tau ''=A,c''=b}\sum _{\sigma '\in \varSigma :c'=a}p_{\sigma ''}p_{\sigma '}\left( \sum _{\sigma :\tau =A,c=a}\varsigma _{\sigma ''\sigma '}(\sigma )\right) \\= & {} \sum _{\sigma ''\in \varSigma :\tau ''=A,c''=b}\sum _{\sigma '\in \varSigma :c'=a}p_{\sigma ''}p_{\sigma '}L(x'',x',\delta ''_{\tau '}), \end{aligned}$$

where we have used the definition of $\left( \left( \varsigma _{\sigma \sigma '}(\sigma '')\right) _{\sigma ''\in \varSigma }\right) _{(\sigma ,\sigma ')\in \varSigma ^{2}}$ in (1) and the fact that

$$\begin{aligned} \sum _{\sigma :\tau =A,c=a}\mu _{Aa}(x)\mu _{AAb}(\delta _{A})\mu _{ABb}(\delta _{B})=1. \end{aligned}$$

Therefore,

$$\begin{aligned} M= & {} \sum _{\sigma ''\in \varSigma :\tau =A,c=b}\left( \sum _{\sigma '\in \varSigma :\tau '=A,c'=a}p_{\sigma ''}p_{\sigma '}L(x'',x',\delta ''_{A})+\sum _{\sigma '\in \varSigma :\tau '=B,c'=a}p_{\sigma ''}p_{\sigma '}L(x'',x',\delta ''_{B})\right) \\= & {} \alpha (1-p)\alpha p\sum _{x''\in X,\delta _{A}''\in \varDelta }\left( \sum _{x'\in X}\mu _{Ab}(x'')\mu _{Aa}(x')\mu _{AAa}(\delta _{A}'')L(x'',x',\delta ''_{A})\right) \\&+\,\alpha (1-p)(1-\alpha )(1-q)\sum _{x''\in X,\delta _{B}''\in \varDelta }\left( \sum _{x'\in X}\mu _{Ab}(x'')\mu _{Ba}(x')\mu _{ABa}(\delta _{B}'')L(x'',x',\delta ''_{B})\right) \\= & {} \alpha (1-p)\left[ \alpha p+(1-\alpha )(1-q)\right] L_{ba}(A), \end{aligned}$$

where the second equality follows from the fact that

• 1. $\sum _{\sigma \in \varSigma :\tau =A,c=a,x=x_{0}}p_{\sigma }=\alpha p\mu _{Aa}(x_{0})$,

• 2. $\sum _{\sigma \in \varSigma :\tau =A,c=b,x=x_{0},\delta _{A}=\delta }p_{\sigma }=\alpha (1-p)\mu _{Ab}(x_{0})\mu _{AAa}(\delta )$,

• 3. $\sum _{\sigma \in \varSigma :\tau =B,c=a,x=x_{0}}p_{\sigma }=(1-\alpha )(1-q)\mu _{Ba}(x_{0})$, and

• 4. $\sum _{\sigma \in \varSigma :\tau =A,c=b,x=x_{0},\delta _{B}=\delta }p_{\sigma }=\alpha (1-p)\mu _{Ab}(x_{0})\mu _{ABa}(\delta )$,

for all $x_{0}\in X$ and $\delta \in \varDelta$.⁴²

Similarly, the subtrahend in (19) can be written as

$$\begin{aligned}&\sum _{\sigma \in \varSigma :\tau =A,c=a}p_{\sigma }\sum _{\sigma ''\in \varSigma :\tau ''=A,c''=b}\sum _{\sigma '\in \varSigma :c'=b}p_{\sigma '}\varsigma _{\sigma \sigma '}(\sigma '')\\&\quad =\alpha p\left[ \alpha (1-p)+(1-\alpha )q\right] L_{ab}(A) \end{aligned}$$

thus,

$$\begin{aligned} \alpha \dot{p}=\alpha (1-p)\left[ \alpha p+(1-\alpha )(1-q)\right] L_{ba}(A)-\alpha p\left[ \alpha (1-p)+(1-\alpha )q\right] L_{ab}(A). \end{aligned}$$

The analogous argument yields (15). $\square$

Appendix 3: Proof of Theorem  1

Proof

First, we establish asymptotic stability. Define the Jacobian matrix of $(\dot{p},\dot{q})$

$$\begin{aligned} J(p,q):=\begin{bmatrix}\dot{p}_{1}(p,q)&\quad \dot{p}_{2}(p,q)\\ \dot{q}_{1}(p,q)&\quad \dot{q}_{2}(p,q) \end{bmatrix}, \end{aligned}$$

where $\dot{p}_{i}(p,q)$ and $\dot{q}_{i}(p,q)$ denote the corresponding partial derivatives of $\dot{p}(p,q)$ and $\dot{q}(p,q)$ with respect to their $i^{th}$ arguments, i.e., with respect to p or q. A rest point $(p^{*},q^{*})$ is asymptotically stable if the real part of the eigenvalues of $J(p^{*},q^{*})$ are negative (see, e.g., Sydsaeter et al. 2008, Theorems 6.8.1 and 7.5.1). This is equivalent to $Det(J(p^{*},q^{*}))>0$ and $Tr(J(p^{*},q^{*}))<0$, where $Det(J(p^{*},q^{*}))$ and $Tr(J(p^{*},q^{*}))$ are the determinant and trace of $J(p^{*},q^{*})$, respectively. Consider first (1, 0). We have $Tr(J(1,0))=2D-(1+U)-\alpha (U+D-1)<0$. Next, $Det(J(1,0))=U(1-2D)+\alpha (U+D-1)>0$ is equivalent to $\alpha >\frac{U(1-2D)}{U+D-1}=\overline{\alpha }$. An analogous calculation holds for $\,(0,1)$. Now, consider $(\hat{p},\hat{q})$. Note that

$$\begin{aligned} Tr(J(\hat{p},\hat{q}))=\frac{2U(1-U)(\alpha +(1-2D)^{2})+2D(1-D)(2-\alpha )-1}{(2D-1)(2U-1)}<0. \end{aligned}$$

Next,

$$\begin{aligned}&{ Det}(J(\hat{p},\hat{q}))\\&\quad =\frac{(\alpha (U+D-1)-(1-U)(2D-1))(U(2D-1)-\alpha (U+D-1))}{(2U-1)(2D-1)}>0 \end{aligned}$$

if $\alpha >\frac{(1-U)(2D-1)}{(U+D-1)}=\underline{\alpha }$ and $\alpha <\frac{U(2D-1)}{(U+D-1)}=\overline{\alpha }$.

In order to prove that the asymptotically stable points are global attractors, notice that $\dot{p}_{2},\dot{q}_{1}<0$, hence $\lim _{t\rightarrow \infty }(p(t),q(t))\in RP$ for all paths (see, e.g., Theorem 3.4.1 in Hofbauer and Sigmund 1998). It is easy to verify that all $(p,q)\in RP{\setminus }\{(p^{*},q^{*})\}$ are saddle-path stable with no stable arm in $[0,1]^{2}$. Hence, the system always converges to the asymptotically stable point. $\square$

Appendix 4: Proofs Section 4

1.1 Dynamical systems with biased sampling

In this subsection, we briefly sketch how to derive (9)–(10). The general version of Eq. (16), when we allow for general intensities, is

$$\begin{aligned} R_{\sigma \sigma ''}(\omega ,t)=\sum _{\sigma '\in \varSigma }\theta _{\sigma \sigma '}((p_{\sigma '''}(\omega ,t))_{\sigma '''\in \varSigma })\varsigma _{\sigma \sigma '}(\sigma '') \end{aligned}$$

and the corresponding version of Eq. (17) is⁴³

$$\begin{aligned} \dot{p}_{\sigma }=&\sum _{\sigma ''\in \varSigma {\setminus }\{\sigma \}}p_{\sigma ''}\sum _{\sigma '\in \varSigma }\theta _{\sigma ''\sigma '}((p_{\sigma '''})_{\sigma '''\in \varSigma })\varsigma _{\sigma ''\sigma '}(\sigma )\nonumber \\&-p_{\sigma }\sum _{\sigma ''\in \varSigma {\setminus }\{\sigma \}}\sum _{\sigma '\in \varSigma }\theta _{\sigma \sigma '}((p_{\sigma '''})_{\sigma '''\in \varSigma })\varsigma _{\sigma \sigma '}(\sigma ''). \end{aligned}$$

(20)

Then, (20) and analogous derivations to those following Eq. (17) in “Appendix 2” yield (9)–(10).

1.2 Proofs of stable equilibria

First, we provide the proof of Lemma 2.

Proof of Lemma 2

(i) We use the determinant and trace of the Jacobian matrix of the system (3)–(4). Recall that (p, q) is asymptotically stable if and only if $Det(J(p,q))>0$ and $Tr(J(p,q))<0$. Now, $Det(J(1,0))=U-D+\alpha _{B}D(1-2U)+\alpha _{A}(2D-1)(1-U))>0$ if and only if $\alpha _{A}>\overline{\alpha }_{A}(\alpha _{B})$. $\overline{\alpha }_{A}(\alpha _{B})\ge 1$ for all $\alpha _{B}\ge \frac{1-D}{D}$, so if $\alpha _{A}>\overline{\alpha }_{A}(\alpha _{B})$, then $\alpha _{B}<\frac{1-D}{D}$. Finally, if $\alpha _{B}<\frac{1-D}{D}$, then $Tr(J(1,0))=\alpha _{A}-1-U(1+\alpha _{A})+D(1+\alpha _{B})<0$. (ii) is established analogously. (iii) follows from (i) and (ii) and the fact that $\overline{\alpha }_{A}(\alpha _{B})>\left[ \overline{\alpha }_{B}^{-1}\right] (\alpha _{B})$ for all $\alpha _{B}\in (0,1)$. $\square$

Proof of Corollary 1

Part (i) follows by observing that $\overline{\alpha }_{A}(\alpha _{B})>1$ for all $\alpha _{B}>\frac{1-D}{D}$, and if $U>D$, then $\overline{\alpha }_{A}(\alpha _{B})<0$ for all $\alpha _{B}<\frac{U-D}{D(2U-1)}$. An analogous argument proves (ii). $\square$

Proof of Proposition 2

Since $\dot{p}_{2}(p,q),\dot{q}_{1}(p,q)<0$ the system always converges to a rest point as $t\rightarrow \infty$ (see, e.g., Theorem 3.4.1 in Hofbauer and Sigmund 1998). If $\alpha _{A}>\overline{\alpha }_{A}(\alpha _{B})$, then, by Lemma 3, $RP=\{(1,0),(0,1)\}$. By Lemma 2, (0, 1) is not asymptotically stable. Furthermore, (0, 1) has no stable arm in $[0,1]^{2}$. Hence, the system converges to (1, 0). Analogously if $\alpha _{A}<\left[ \overline{\alpha }_{B}^{-1}\right] (\alpha _{B})$, then the system converges to (0, 1). If $\left[ \overline{\alpha }_{B}^{-1}\right] (\alpha _{B})<\alpha _{A}<\overline{\alpha }_{A}(\alpha _{B})$, then by Lemma 3, $RP=\{(1,0),(0,1),(\widetilde{p},\widetilde{q})\}$, for some $(\widetilde{p},\widetilde{q})\in (0,1)^{2}$. By Lemma 2, (1, 0) and (0, 1) are not asymptotically stable and, furthermore, have no stable arm in $[0,1]^{2}$. Hence, the system converges to $(\widetilde{p},\widetilde{q})$. $\square$

Appendix 5: Proofs of Section 5

Proof of Proposition 3

Since $G(r)=0$ for all $r\ge \varphi$ and G is continuous on $[0,\varphi ]$, it attains its maximum somewhere in this interval. Both Q and G are twice continuously differentiable on $(\underline{r}(\varphi ,\alpha ),\overline{r}(\varphi ,\alpha ))$. Let $G^{\prime }(x^{+})$ and $G^{\prime }(x^{-})$ (corresp. $Q^{\prime }(x^{+})$ and $Q^{\prime }(x^{-})$) denote the right and left derivatives, respectively, of G (corresp. Q) for all $x\in (\underline{r}(\varphi ,\alpha ),\overline{r}(\varphi ,\alpha ))$.

Suppose $k\ge \overline{r}(\varphi ,\alpha )$. Then, $G(r)<0$ for all $r\in [0,\overline{r}(\varphi ,\alpha ))$ and $G(r)=0$ for all $r\ge \overline{r}(\varphi ,\alpha )$. Hence, in this case, the monopolist withdraws from the market.

Suppose $k<\overline{r}(\varphi ,\alpha )$. Then,

$$\begin{aligned} G^{\prime }(\overline{r}(\varphi ,\alpha )^{-})&=Q(\overline{r}(\varphi ,\alpha ))+(\overline{r}(\varphi ,\alpha )-k)Q^{\prime }(\overline{r}(\varphi ,\alpha )^{-})\\&=(\overline{r}(\varphi ,\alpha )-k)Q^{\prime }(\overline{r}(\varphi ,\alpha )^{-})<0. \end{aligned}$$

Hence, in this case $r^{*}<\overline{r}(\varphi ,\alpha )$. From Lemma 4, $G^{\prime }(\underline{r}(\varphi ,\alpha )^{+})>0$ if $k>\hat{k}$. Hence, if $k>\hat{k}$, then $r^{*}>\underline{r}(\varphi ,\alpha )$. This proves that $r^{*}\in (\underline{r}(\varphi ,\alpha ),\overline{r}(\varphi ,\alpha ))$ if $k\in (\hat{k},\overline{r}(\varphi ,\alpha ))$. In order to see that in this case, $r^{*}$ is the unique solution to $G^{\prime }(r)=0$, first note that $G(r)<0$ for $r<k$, and hence, $r^{*}\ge k$. Next, for any k, r such that $0\le k\le r\in (\underline{r}(\varphi ,\alpha ),\overline{r}(\varphi ,\alpha ))$, we have that

$$\begin{aligned} G^{\prime \prime }(r)=2Q^{\prime }(r)+(r-k)Q^{\prime \prime }(r)<0. \end{aligned}$$

If $Q^{\prime \prime }(r)\le 0$, this is obvious. Note that if $k=0$ then

$$\begin{aligned} G^{\prime \prime }(r)=2Q^{\prime }(r)+rQ^{\prime \prime }(r)=-\frac{3(1-\alpha )\varphi }{2(\varphi -r)^{3}}<0 \end{aligned}$$

for $r<\varphi$. Thus, if $Q^{\prime \prime }(r)\ge 0$ then $G^{\prime \prime }(r)=2Q^{\prime }(r)+(r-k)Q^{\prime \prime }(r)\le 2Q^{\prime }(r)+rQ^{\prime \prime }(r)<0$. G is therefore strictly concave on $(k,\overline{r}(\varphi ,\alpha ))$ which together with $G^{\prime }(k)>0$ implies that $r^{*}$ is the unique solution to $G^{\prime }(r)=0$.

Suppose $k\le \hat{k}$. Then, from Lemma 4, $G^{\prime }(\underline{r}(\varphi ,\alpha )^{+})<0$ and G is strictly concave on $(\underline{r}(\varphi ,\alpha ),\overline{r}(\varphi ,\alpha ))$ given $k\le \hat{k}<\underline{r}(\varphi ,\alpha )$. Since $G^{\prime }(\underline{r}(\varphi ,\alpha )^{+})<0$, we have $G^{\prime }(r)<0$ for all $r\in (\underline{r}(\varphi ,\alpha ),\overline{r}(\varphi ,\alpha ))$. Finally, since G is increasing on $[0,\underline{r}(\varphi ,\alpha ))$, we obtain $r^{*}=\underline{r}(\varphi ,\alpha )$. $\square$