The evolution of control in the digital economy

Abstract

Control over digital transactions has steadily risen in recent years, to an extent that puts into question the Internet’s traditional openness. To investigate the origins and effects of such change, the paper formally models the historical evolution of digital control. In the model, the economy-wide features of the digital space emerge as a result of the endogenous adaptation (co-evolution) of users’ preferences (culture) and platform designs (technology). The model shows that: a) in the digital economy there exist two stable cultural-technological equilibria: one with intrinsically motivated users and low control; and the other with purely extrinsically motivated users and high control; b) before the opening of the Internet to commerce, the emergence of a low-control-intrinsic-motivation equilibrium was favored by the specific set of norms and values that formed the early culture of the networked environment; and c) the opening of the Internet to commerce can indeed cause a transition to a high-control-extrinsic-motivation equilibrium, even if the latter is Pareto inferior. Although it is too early to say whether such a transition is actually taking place, these results call for a great deal of attention in evaluating policy proposals on Internet regulation.

Appendices

Appendix A

Proof

of Lemma 1 The derivative of Eqs. 4 with respect to a gives us the following best-response function for EI- and PE-users when paired with a generic designer j: a _{E I, j} = ϕ + λ(1 − t) and a _{P E, j} = ϕ. By substituting away for t, we obtain the best-response level of a reported in the lemma. □

Proof

of Proposition 1 {E I, L} is proven to be Nash equilibrium as long as: (a) (ϕ + λ)²/2 > ϕ ²/2, and (b) q(ϕ + λ) − γ η k > q ϕ − δ/2. Condition (a) is self-explained. Condition (b) reduces to $\delta >2(\gamma \eta k - q\lambda )=\underline {\delta }$. Similarly, {P E, H} is a Nash equilibrium as long as: (c) ϕ ²/2 > ϕ ²/2 − μ and (d) q ϕ − γ k < q ϕ − δ/2. Condition (c) is self-explained. Condition (d) reduces to $\delta <2\gamma k=\overline {\delta }$. For 0 < η < 1, $\underline {\delta }<\overline {\delta }$ is always true. It follows that: (i) when $\delta >\overline {\delta }$ condition (b) is satisfied but not condition (d), hence {E I, L} is the only Nash equilibrium; (ii) when $\delta <\underline {\delta }$ condition (d) is satisfied but not condition (b), hence {P E, H} is the only Nash equilibrium; and (iii) when $\underline {\delta }<\delta <\overline {\delta }$ conditions (b) and (d) are simultaneously satisfied, hence both {E I, L} and {P E, H} are Nash equilibria. Corollary 1.1 follows from the fact that two necessary conditions for {P E, L} and {E I, H} to be Nash equilibria are that PE is a best-response to L and EI is a best response to H, but this is impossible because it would violate conditions (a) and (c) above. Corollary 1.2 follows directly from points (i), (ii) and (iii) above. □

Proof

of Proposition 2 For any λ > 0, a necessary and sufficient condition for {P E, H} to be Pareto efficient is that q(ϕ + λ) − γ η k < q ϕ − δ/2, which reduces to $\delta <\overline {\delta }$. Otherwise, {E I, L} Pareto dominates {P E, H}. This, together with the results of Proposition 1, implies that: (i) if $\delta <\overline {\delta }$, then {P E, H} is Pareto efficient and it is also the only Nash equilibrium of the game; (ii) if $\delta >\overline {\delta }$, then {E I, L} is Pareto dominant and it is also a Nash equilibrium. Points (i) and (ii), together with the fact that for $\underline {\delta }<\delta <\overline {\delta }$ both {E I, L} and {P E, H} are Nash equilibria, prove the proposition. □

Proof

of Proposition 3 The five cultural-technological equilibria are derived by simply solving the system (10)–(11) for ${\Delta }\omega _{EI}^{\tau }=0$ and ${\Delta }\omega _{L}^{\tau }=0$. The proof in this case is omitted. The asymptotic properties of each equilibrium are derived by analyzing the Jacobian Matrix J(ω _{E I} , ω _L ) associated with system (10)–(11), which takes the following form:

$$J=\left( \begin{array} {ll} (1-2\omega_{EI})\left[\omega_{L}\left( \frac{\lambda^{2}}{2}+\phi\lambda+\mu \right)-\mu \right] &\quad\quad\quad\,\, (\omega_{EI}-\omega_{EI}^{2})\left( \frac{\lambda^{2}}{2}+\phi\lambda+\mu \right) \\ \quad\quad\,\,\,(\omega_{L}-{\omega_{L}^{2}})\left[q\lambda+\gamma k (1-\eta)\right] & (1-2\omega_{L})\left\lbrace\omega_{EI} \left[q\lambda+\gamma k (1-\eta)\right]+\frac{\delta}{2}-\gamma k \right\rbrace \end{array} \right)$$

At {0, 0}, we have

$$J=\left( \begin{array}{cc} -\mu & 0 \\ 0 & \frac{\delta}{2}-\gamma k \end{array} \right)$$

from which it follows that

$$ Tr(J)=-\mu+\frac{\delta}{2}-\gamma k \;\;\;\;\;\;\;\; and \;\;\;\;\;\;\;\; Det(J)=-\mu\left( \frac{\delta}{2}-\gamma k\right) $$

(16)

Since Tr (J) < 0 and Det (J) > 0 for any δ < 2γ k, {0, 0} is asymptotically stable.At {1, 0}, we have

$$J=\left( \begin{array}{cc} \mu & 0 \\ 0 & q\lambda-\gamma\eta k+\frac{\delta}{2} \end{array} \right)$$

from which it follows that

$$ Tr(J)=\mu+ q\lambda -\gamma\eta k+\frac{\delta}{2} \;\;\;\;\;\;\;\; and \;\;\;\;\;\;\;\; Det(J)=\mu \left( q\lambda -\gamma\eta k+\frac{\delta}{2}\right) $$

(17)

Since Tr (J) > 0 and Det (J) > 0 for any δ > 2(γ η k − q λ), {1, 0} is unstable.At {0, 1}, we have

$$J=\left( \begin{array}{cc} \frac{\lambda^{2}}{2}+\phi\lambda & 0 \\ 0 & -\frac{\delta}{2}+\gamma k \end{array} \right)$$

from which it follows that

$$ Tr(J)=\frac{\lambda^{2}}{2}+\phi\lambda -\frac{\delta}{2}+\gamma k \;\;\;\;\;\;\;\; and \;\;\;\;\;\;\;\; Det(J)=\left( \frac{\lambda^{2}}{2}+\phi\lambda\right)\left( \gamma k-\frac{\delta}{2}\right) $$

(18)

Since Tr (J) > 0 and Det (J) > 0 for any δ < 2γ k, {0, 1} is unstable.At {1, 1}, we have

$$J=\left( \begin{array}{cc} -\frac{\lambda^{2}}{2}-\phi\lambda & 0 \\ 0 & -q\lambda+\gamma\eta k -\frac{\delta}{2} \end{array} \right)$$

from which it follows that

$$ \begin{array}{l} Tr(J)=-\frac{\lambda^{2}}{2}-\phi\lambda -q\lambda +\gamma\eta k -\frac{\delta}{2} \;\;\;\;\;\;\;\; and \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; Det(J)=-\left( \frac{\lambda^{2}}{2}+\phi\lambda\right) \left( \gamma\eta k -q\lambda -\frac{\delta}{2}\right) \end{array} $$

(19)

Since Tr (J) < 0 and Det (J) > 0 for any δ > 2(γ η k − q λ), {1,1} is asymptotically stable.At $\lbrace \omega _{EI}^{*},\omega _{L}^{*} \rbrace$, we have

$$J=\left( \begin{array} {cc} 0 & \frac{\left( 2\gamma k-\delta\right)\left[2q\lambda-2\gamma\eta k+\delta\right]}{4\left[q\lambda+\gamma k(1-\eta)\right]^{2}} \left( \frac{\lambda^{2}}{2}+\phi\lambda+\mu\right) \\ \frac{2\mu\lambda(\lambda+2\phi)}{\left[\lambda(\lambda+2\phi)+2\mu\right]^{2}} \left[ q\lambda+\gamma k(1-\eta)\right] & 0 \end{array}\right)$$

from which it follows that

$$ \begin{array}{l} Det(J)=- \frac{\left( 2\gamma k-\delta\right)\left[2q\lambda -2\gamma\eta k+\delta\right]}{4\left[q\lambda +\gamma k(1-\eta)\right]^{2}} \left( \frac{\lambda^{2}}{2}+\phi\lambda +\mu\right) \;\;. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad. \;\; \frac{2\mu\lambda(\lambda +2\phi)}{\left[\lambda(\lambda +2\phi)+2\mu\right]^{2}} \left[ q\lambda +\gamma k(1-\eta)\right] \end{array} $$

(20)

Since Det (J) < 0 for any δ > 2(γ η k − q λ), $\lbrace \omega _{EI}^{*},\omega _{L}^{*} \rbrace$ is a saddle. □

Proof

of Proposition 4 From Definition 3 and the value of $\omega _{EI}^{*}$ and $\omega _{L}^{*}$ reported in Proposition 3 it follows that:

• μ ≥ ψ(2γ k − δ)/2[δ + 2(q λ − γ k η] ⇔

  $$ r_{01}=\omega_{EI}^{*}=\frac{2\gamma k-\delta}{2\left[q\lambda +\gamma k(1-\eta)\right]}\;\;\;\;and\;\;\;\;r_{10}=1-\omega_{L}^{*}=\frac{\psi}{\psi +2\mu} $$

  (21)

• μ < ψ(2γ k − δ)/2[δ + 2(q λ − γ k η] ⇔

  $$ r_{01}=\omega_{L}^{*}=\frac{2\mu}{\psi +2\mu}\;\;\;\;and\;\;\;\;r_{10}=1-\omega_{EI}^{*}=\frac{\delta + 2(q\lambda -\gamma k\eta)}{2\left[q\lambda +\gamma k(1-\eta)\right]} $$

  (22)

where ψ = λ(λ + 2ϕ). According to Definition 5, E ₀ is SSS if and only if r ₁₀ < r ₀₁. Simple algebra shows that, given Eqs. 20 and 21, the latter condition holds if and only if k < [ψ(2q λ + δ) + 2μ δ]/2γ(2μ + η ψ) = k ^∗. The second part of the proposition follows directly from Proposition 2. □

Appendix B

Payoffs in Table 1

Let us indicate with U _{i, j} the utility of an i-type user when matched with a j-type designer, and with π _{j, i} the return to an j-type designer when matched with a i-type user. Moreover, let us write a _{i, j} as the best-response level of a for an i-type user when matched with a j-type designer. Given Eqs. 1 and 2 we have:

$$ U_{EI,j}=[\phi +\lambda(1-t)]a_{EI,j}-\frac{a_{EI,j}^{2}}{2}-\mu t \;\;\;\; , \;\;\;\; U_{PE,j}=\phi a_{PE,j}-\frac{a_{PE,j}^{2}}{2} $$

(23)

$$ \pi_{L,i}=q a_{i,L} -\gamma\eta(\lambda)k \;\;\;\; , \;\;\;\; \pi_{H,i}=q a_{i,H}-\frac{\delta}{2} $$

(24)

where η(λ) takes the following form:

$$ \eta(\lambda)=\left\{\begin{array}{ll} 1, & \text{if } i=PE \\ & \\ \eta, & \text{if } i=EI \end{array}\right. $$

(25)

By replacing into Eqs. 23 and 24 the value for a _{i, j} reported in Lemma 1, and substituting away for t (i.e. replacing t = 0 and t = 1 for a match with an L- and a H-type designer respectively), we obtain the following results:

$$ U_{EI,L}=\frac{(\phi +\lambda)^{2}}{2} \;\;\;\; , \;\;\;\; U_{EI,H}=\frac{\phi^{2}}{2}-\mu \;\;\;\; , \;\;\;\; U_{PE,L}=U_{PE,H}=\frac{\phi^{2}}{2} $$

(26)

$$ \pi_{L,EI}=q (\phi +\lambda) -\gamma\eta k \;\;\;\; , \;\;\;\; \pi_{L,PE}=q\phi -\gamma k \;\;\;\; , \;\;\;\; \pi_{H,EI}=\pi_{H,PE}=q\phi -\frac{\delta}{2} $$

(27)

Replicator equations

The systems of replicator equations represented by Eqs. 10 and 11 is obtained as follows. Let’s write the probability that an agent (user and designer) of type i switches to type j at time τ as $p_{ij}^{\tau }$. Given the updating process described above we have:

$$ p_{ij}^{\tau}=\left\{\begin{array}{ll} \beta \left( V_{j}^{\tau}-V_{i}^{\tau}\right), & \text{if }V_{j}^{\tau}>V_{i}^{\tau} \\ & \\ 0, & \text{if }V_{j}^{\tau}\leq V_{i}^{\tau} \end{array}\right. $$

(28)

for i, j = E I, P E and i≠j in the case of users and i, j = L, H and i≠j in the case of designers. On this basis, the expected fractions of EI-users in period τ + 1 is given by:

$$ \omega_{EI}^{\tau + 1}= \omega_{EI}^{\tau}-\omega_{EI}^{\tau} (1-\omega_{EI}^{\tau})\alpha \sigma_{PE}\beta(V_{PE}^{\tau}-V_{EI}^{\tau})+(1-\omega_{EI}^{\tau})\omega_{EI}^{\tau} \alpha \sigma_{EI}\beta(V_{EI}^{\tau}-V_{PE}^{\tau}) $$

(29)

where σ _{P E} and σ _{E I} are two binary functions such that σ _{P E} = 1 if $V_{PE}^{\tau }>V_{EI}^{\tau }$ and is zero otherwise, σ _{E I} = 1 if $V_{EI}^{\tau }\geq V_{PE}^{\tau }$ and is zero otherwise, and σ _{P E} + σ _{E I} = 1. Equation 29 reads as follows: the expected fraction of EI-users at τ+1 is given by the fraction of EI-users at τ (first term), minus the fraction of EI-users who are paired with an PE-user and switch their type (second term), plus the fraction of PE-users who are paired with an EI-user and switch their type (third term). Similarly, the expected fractions of L-designers in period τ + 1 is given by:

$$ \omega_{L}^{\tau + 1}= \omega_{L}^{\tau}-\omega_{L}^{\tau} (1-\omega_{L}^{\tau})\alpha \sigma_{H}\beta(V_{H}^{\tau}-V_{L}^{\tau})+(1-\omega_{L}^{\tau})\omega_{L}^{\tau} \alpha \sigma_{L}\beta(V_{L}^{\tau}-V_{H}^{\tau}) $$

(30)

where σ _H = 1 if $V_{H}^{\tau }>V_{L}^{\tau }$ and is zero otherwise, σ _L = 1 if $V_{L}^{\tau }\geq V_{H}^{\tau }$ and is zero otherwise, and σ _H + σ _L = 1. Subtracting $\omega _{I}^{\tau }$ and $\omega _{L}^{\tau }$ from both sides of Eqs. 29 and 30 respectively and rearranging we get Eqs. 10 and 11.

Stochastic dynamical system

In the stochastic environment described in Section 4, the expected fraction of EI-users in period τ + 1 is given by

$$ \begin{array}{l} \omega_{EI}^{\tau + 1}= [\omega_{EI}^{\tau}-\omega_{EI}^{\tau} (1-\omega_{EI}^{\tau})\alpha \sigma_{PE}\beta(V_{PE}^{\tau}-V_{EI}^{\tau})+\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(1-\omega_{EI}^{\tau})\omega_{EI}^{\tau} \alpha \sigma_{EI}\beta(V_{EI}^{\tau}-V_{PE}^{\tau})]\chi_{u}^{\tau}+\nu_{EI}^{\tau}(1-\chi_{u}^{\tau}) \end{array} $$

(31)

where σ _{P E} and σ _{E I} are two binary functions such that σ _{P E} = 1 if $V_{PE}^{\tau }>V_{EI}^{\tau }$ and is zero otherwise, σ _{E I} = 1 if $V_{EI}^{\tau }\geq V_{PE}^{\tau }$ and is zero otherwise, and σ _{P E} + σ _{E I} = 1, and where

$$ \chi_{u}=\frac{n_{u}^{\tau}}{n_{u}^{\tau}+\varepsilon s_{u}^{\tau}}=\frac{1}{1+\varepsilon\rho_{u}} $$

(32)

is a normalizing factor that varies according to the number of new users who enter into the economy. The part of Eq. 31 inside the square brackets refers to the inside population and reads as follows: the expected fraction of EI-users at τ+1 is given by the fraction of EI-users at τ (first term), minus the fraction of EI-users who are paired with an PE-user and switch their type (second term), plus the fraction of PE-users who are paired with an EI-user and switch their type (third term). Once such updating process is completed, $s_{u}^{\tau }$ new users enter the economy with probability ε. The fraction of EI-users at the beginning of next period is thus given by the updated fraction of EI-users normalized by the new size of the users’ population (i.e. multiplication by $\chi _{u}^{\tau }$), plus the fraction of EI-users that are included in the set of new entrants (i.e. $\nu _{EI}^{\tau }(1-\chi _{u}^{\tau })$). Similarly, the expected fractions of L-designers in period τ + 1 is given by:

$$ \begin{array}{l} \omega_{L}^{\tau + 1}= [\omega_{L}^{\tau}-\omega_{L}^{\tau} (1-\omega_{L}^{\tau})\alpha \sigma_{H}\beta(V_{H}^{\tau}-V_{L}^{\tau})+\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(1-\omega_{L}^{\tau})\omega_{L}^{\tau} \alpha \sigma_{L}\beta(V_{L}^{\tau}-V_{H}^{\tau})]\chi_{d}^{\tau}+\nu_{L}^{\tau}(1-\chi_{d}^{\tau}) \end{array} $$

(33)

where

$$ \chi_{d}=\frac{n_{d}^{\tau}}{n_{d}^{\tau}+\varepsilon s_{d}^{\tau}}=\frac{1}{1+\varepsilon\rho_{d}} $$

(34)

where σ _H = 1 if $V_{H}^{\tau }>V_{L}^{\tau }$ and is zero otherwise, σ _L = 1 if $V_{L}^{\tau }\geq V_{H}^{\tau }$ and is zero otherwise, and σ _H + σ _L = 1. Subtracting $\omega _{EI}^{\tau }$ and $\omega _{L}^{\tau }$ from both sides of Eqs. 31 and 33, respectively, we get:

$$ {\Delta}\omega_{EI}^{\tau}= \omega_{EI}^{\tau} (1-\omega_{EI}^{\tau})\alpha\beta(V_{EI}^{\tau}(\omega_{L}^{\tau})-V_{PE}^{\tau}(\omega_{L}^{\tau}))\chi_{u}+(1-\chi_{u})(\nu_{EI}^{\tau}-\omega_{EI}^{\tau}) $$

(35)

$$ {\Delta}\omega_{L}^{\tau}= \omega_{L}^{\tau} (1-\omega_{L}^{\tau})\alpha\beta(V_{L}^{\tau}(\omega_{EI}^{\tau})-V_{H}^{\tau}(\omega_{EI}^{\tau}))\chi_{d}+(1-\chi_{d}) (\nu_{L}^{\tau}-\omega_{L}^{\tau}) $$

(36)

Equations 35 and 36 represent a system of differential equations which describes how the distribution of types $\lbrace \omega _{EI}^{\tau }, \omega _{L}^{\tau } \rbrace$ evolves over time. The main difference with the system composed of Eqs. 10 and 11 is that this time there are also some stochastic components represented by variables χ _u , χ _d , $\nu _{EI}^{\tau }$ and $\nu _{L}^{\tau }$. The latter are the sources of exogenous variation that make a transition between the basins of attraction of the two stable equilibria E ₀ and E ₁ possible.