The impact of strategic agents in two-sided markets

Abstract

This paper introduces strategic agents into a two-sided market. We model strategic agents as advertisers who can invest in ad quality, which in turn affects readers, platforms and themselves. When platforms can choose prices on both the reader side and the advertiser side, strategic agents intensify competition, leading to lower prices and profits for the platforms. However, when prices on the reader side are restricted to be zero, equilibrium prices on the advertiser side can be higher or lower under strategic agents. We also investigate the interplay between strategic agents and platform asymmetry, and analyze platforms’ incentive to invest in content quality. Our results suggest that strategic agents increase the degree of platform asymmetry and make it harder for the disadvantaged platform to compete. However, strategic agents may raise or lower platforms’ incentive to invest in content quality.

Appendices

Appendices

Estimating group externality parameters

We start with the benchmark passive agents model. Recall that equations (1) and (2) in the main text characterize the marginal advertiser and marginal reader respectively. Let $n_1^a=\frac{N_1^a}{N_1^a+N_2^a}$ and $n_1^r=\frac{N_1^r}{N_1^r+N_2^r}$. Since $N_1^j+N_2^j=1$ (unit mass of agents on either side), we have $n_i^j=N_i^j$. Then

$$\begin{aligned} n_1^a= & {} \frac{1}{2t}\left[ t + \rho (N_1^r- N_2^r) -(a_1-a_2)\right] , \end{aligned}$$

(A.1)

$$\begin{aligned} n_1^r= & {} \frac{1}{2t}\left[ t + \gamma (N_1^a- N_2^a)-(p_1-p_2)\right] . \end{aligned}$$

(A.2)

With data on sales $N_i^j$, market share $n_i^j$ and prices ($p_i$ and $a_i$), equations (A.1) and (A.2) can be used to estimate group externality parameters $\rho$ and $\gamma$ respectively.²¹

Next, we move on to the case of strategic agents. Recall that equations (7) and (9) characterize the advertiser and reader utilities, and equations (8) and (10) report the marginal advertiser $x^a$ and marginal reader $x^r$ respectively. Since $N_1^j+N_2^j=1$, we have $x^j=n_1^j$. Then

$$\begin{aligned} n_1^a= & {} \frac{1}{2t}[ t + \rho (N_1^r-N_2^r)+ \underbrace{\frac{\rho ^2}{2\lambda } \left[ \left( N_1^r\right) ^2 - \left( N_2^r\right) ^2\right] }_{\text{Endogenous ad quality effect}} -(a_1-a_2)], \end{aligned}$$

(A.3)

$$\begin{aligned} n_1^r= & {} \frac{1}{2t}[t + \gamma (N_1^a - N_2^a)- \underbrace{\frac{\gamma \cdot \rho }{\lambda } \cdot (N_1^r \cdot N_1^a - N_2^r \cdot N_2^a)}_{\text{Endogenous ad quality effect}} - (p_1-p_2)]. \end{aligned}$$

(A.4)

Different from equation (A.1) in the benchmark model, in equation (A.3), $n_1^a$ is now linear-quadratic (rather than linear) in $N_i^r$. This can also be seen with general $f(\rho ,\kappa _i,N_i^r)$ and $c(\kappa )$. $\kappa ^*$ in general will be a function of $N_i^r$. Once the $\kappa _i^*$ value is substituted, $n_i^a$ will not be linear in $N_i^r$ anymore. In addition, $\rho$ appears twice in equation (A.3), including in the form of $\frac{\rho ^2}{2}$. Note that the endogenous ad quality term disappears in the limiting case of $\lambda \rightarrow +\infty$. Similarly, equation (A.4) differs from (A.2) in the benchmark model. Combining results for both the advertiser and reader sides, we have the following proposition.

Proposition A1

(Estimating group externality parameters) Ignoring advertisers’ strategic investment decisions will lead to wrong econometric models for estimating the group externality parameters on both the advertiser and reader sides.

Intuitively, when ad quality affects the reader and advertiser side, one needs to know the cost and benefit of having higher quality ad to the advertiser, in order to recover the equilibrium ad quality. One can then substitute the optimal ad quality to obtain the revised econometric model such as (A.3) and (A.4) for estimation. In our specific model, given equations (A.3) and (A.4), one needs to construct additional explanatory variables $(N_1^r)^2-(N_2^r)^2$ and $N_1^r \cdot N_1^a - N_2^r \cdot N_2^a$. Together with $N_1^r-N_2^r$ and $N_1^a-N_2^a$, which are also used under passive agents model, we have 4 explanatory variables and in turn 4 coefficients (after normalizing $t=\frac{1}{2}$ so $\frac{1}{2t}=1$):

$$\begin{aligned} Coeff_1 = \rho , \quad Coeff_2 = \frac{\rho ^2}{2\lambda }, \quad Coeff_3 = \gamma ,\quad Coeff_4 = -\frac{\gamma \cdot \rho }{\lambda }. \end{aligned}$$

The first two coefficients come from equation (A.3) while the last two come from equation (A.4). However, there are only 3 model primitives to be estimated: $\gamma$, $\rho$ and $\lambda$. Thus the econometric model is over-identified. In fact, there is a constraint linking the 4 coefficients in the form of $\frac{Coeff_2}{Coeff_1}=-\frac{1}{2}\cdot \frac{Coeff_4}{\cdot Coeff_3}$. This constraint needs to be imposed in the estimation.

Proof of propositions

Proof of Proposition 1

Recall that the marginal advertiser and reader are,

$$\begin{aligned} x^a= & {} \frac{1}{2t}\left[ t + \rho (N_1^r-N_2^r)+ \frac{\rho ^2}{2 \lambda } \left[ \left( N_1^r\right) ^2 - \left( N_2^r\right) ^2\right] -(a_1-a_2)\right] ,\\ x^r= & {} \frac{1}{2t}\left[ t + \gamma (N_1^a - N_2^a)- \frac{\gamma \cdot \rho }{\lambda } \cdot (N_1^r \cdot N_1^a - N_2^r \cdot N_2^a) - (p_1-p_2)\right] . \end{aligned}$$

Note that $N_1^j=x^j$ and $N_2^j=1-x^j$, $j=a,r$. Substituting them into the $x^a$ and $x^r$ expressions above, we can solve the stand-alone $N_i^j$. We then substitute $N_i^j$ into platform i’s profit maximization problem,

$$\begin{aligned} \max _{p_i, a_i} \quad \pi _i = (p_i- f_r) \cdot N_i^r + (a_i-f_a) \cdot N_i^a,\quad i=1,2. \end{aligned}$$

Solving firms’ FOCs, we can obtain the equilibrium prices,

$$\begin{aligned} p_i=t-\rho +\frac{1}{2} \frac{\rho }{\lambda } (\gamma -\rho )+f_r,\quad a_i=t-\gamma +\frac{1}{2} \frac{\rho \gamma }{\lambda }+f_a, \end{aligned}$$

In the equilibrium, each platform earns a profit of

$$\begin{aligned} \pi _i=t+\frac{1}{2}\frac{\gamma \rho }{\lambda }-\frac{1}{2}\gamma -\frac{1}{2}\rho -\frac{1}{4}\frac{\rho ^2}{\lambda } \quad i=1,2. \end{aligned}$$

In the symmetric equilibrium, $N_i^r=\frac{1}{2}$ and all advertisers choose $\kappa ^*=\frac{\rho }{2\lambda }$. $\square$

Proof of Proposition 2

(i) Recall that equilibrium prices in the benchmark passive agents model and the strategic agents model are,

$$\begin{aligned} p_i^{passive}= & {} t-\rho +f_r,\quad a_i^{passive}=t-\gamma +f_a,\\ p_i^{strategic}= & {} t-\rho +\frac{1}{2} \rho (\gamma -\rho )+f_r,\quad a_i^{strategic}=t-\gamma +\frac{1}{2} \rho \gamma +f_a. \end{aligned}$$

It is easy to see that

$$\begin{aligned} p_i^{strategic} - p_i^{passive} = \frac{1}{2} \rho (\gamma -\rho )< 0,\quad a_i^{strategic} - a_i^{passive} = \frac{1}{2} \rho \gamma < 0, \end{aligned}$$

because $\gamma <0$ and $\rho >0$. That is, equilibrium prices on both sides are lower under strategic agents relative to the benchmark case.

(ii) With lower prices on both sides of the market, advertisers and readers must be better of at the cost of platforms. In addition, higher ad quality further benefits the advertisers and readers. $\square$

Proof of Proposition 3

We start with same platform profit functions as when platforms are free to chooses both $p_i$ and $a_i$. We then substitute $p_i=0$, and take derivative with respect to $a_i$ only to solve for equilibrium $a_i$. This is done for passive agents and strategic agents respectively.

(i) Under passive agents, solving the FOC ($\frac{\partial \pi _i}{\partial a_i}=0$), we can obtain

$$\begin{aligned} a_i=\frac{1}{2}+2\gamma (f_r-\rho )+f_a, \quad i=1,2. \end{aligned}$$

Platform i’s profit ($i=1,2$) is

$$\begin{aligned} \pi _i= & {} (0-f_r)\cdot \frac{1}{2} + (a_i-f_a)\cdot \frac{1}{2}\\= & {} \frac{1}{4}+\gamma (f_r-\rho )-\frac{1}{2}f_r. \end{aligned}$$

(ii) Under strategic agents, solving the FOC leads to

$$\begin{aligned} a_i= & {} \frac{(4(f_r-\rho )\gamma +1)\lambda ^2 -\rho \gamma (2f_r-1)\lambda + \gamma \rho ^3}{2\lambda (\lambda + \gamma \rho )} + f_a,\\ \pi _i= & {} \frac{(4(f_r-\rho )\gamma +1)\lambda ^2 -\rho \gamma (2f_r-1)\lambda + \gamma \rho ^3}{4\lambda (\lambda + \gamma \rho )} -\frac{1}{2}f_r , \quad i=1,2. \end{aligned}$$

(iii) It suffices to show that $a_i$ can go up or down when $f_r=0$. When $f_r=0$, it can be shown that

$$\begin{aligned} \Delta a_i\equiv a_i^{strategic}-a_i^{passive}= \frac{1}{2}\frac{\gamma \rho ^2(\rho +4\lambda \gamma )}{\lambda (\lambda +\gamma \rho )}. \end{aligned}$$

Note that $\lambda +\gamma \rho >0$ is required for SOC to hold, but $\rho +4\lambda \gamma$ can be positive or negative. With $\gamma <0$, $\Delta a_i>0$ if and only if $\rho +4\lambda \gamma <0$. SOCs are satisfied if $|\gamma |$ and $\rho$ are sufficiently small. $\square$

Proof of Proposition 4

Having an alternative form of strategic agents has no impact on the passive agents model. For strategic agents, recall that $\kappa _i^*=\frac{N_i^r}{\lambda }$. Substituting it into advertiser and reader utility functions, and solving for marginal advertiser and marginal reader, we can obtain

$$\begin{aligned} x^a= & {} \frac{1}{2} + \rho (N_1^r-N_2^r)+ \frac{1}{2\lambda } \left[ \left( N_1^r\right) ^2 - \left( N_2^r\right) ^2\right] -(a_1-a_2),\\ x^r= & {} \frac{1}{2} + \gamma (N_1^a - N_2^a)- \frac{1}{\lambda } \cdot (N_1^r \cdot N_1^a - N_2^r \cdot N_2^a) - (p_1-p_2). \end{aligned}$$

Using the fact that $N_1^j=x^j$ and $N_2^j=1-N_1^j$, $j=r,a$, we can solve the stand-alone $x^r$ and $x^a$, and in turn platform profit $\pi _i$. Next, we proceed to solve for the equilibrium, depending on whether prices on the reader side are restricted to be zero.

(i) Case 1: Platforms are free to choose prices on both sides.

Platform i solve the following problem,

$$\begin{aligned} \max _{p_i,a_i} \quad \pi _i=(a_i-f_a)N_i^a + (p_i-f_r)N_i^r,\quad i=1,2. \end{aligned}$$

Solving the FOCs, we can obtain

$$\begin{aligned} p_i=\frac{1}{2}-\rho -\frac{1}{\lambda }+f_r,\quad a_i=\frac{1}{2}-\gamma -\frac{1}{2\lambda }+f_a. \end{aligned}$$

Recall the following equilibrium prices from the corresponding passive agents model,

$$\begin{aligned} p_i=\frac{1}{2}-\rho +f_r,\quad a_i=\frac{1}{2}-\gamma +f_a. \end{aligned}$$

It is straightforward to verify that both $p_i$ and $a_i$ are lower under strategic agents, relative to the prices under passive agents.

(ii) Case 2: Zero prices on the reader side

Now platform i solve the following problem,

$$\begin{aligned} \max _{p_i=0,a_i} \quad \pi _i=(a_i-f_a)N_i^a + (p_i-f_r)N_i^r,\quad i=1,2. \end{aligned}$$

Solving the FOCs, we can obtain

$$\begin{aligned} a_i=\left[ \frac{1}{2}+2\gamma (f_r-\rho )+f_a \right] + \frac{1}{2}\frac{(-2\gamma -2\rho -4\gamma \rho )\lambda -1}{\lambda (\lambda -1)} \end{aligned}$$

Recall the following equilibrium prices from the corresponding passive agents model,

$$\begin{aligned} a_i=\frac{1}{2}+2\gamma (f_r-\rho )+f_a, \quad i=1,2. \end{aligned}$$

Then

$$\begin{aligned} \Delta a_i=a_i^{strategic}-a_i^{passive}=\frac{1}{2} \frac{(-2\gamma -2\rho -4\gamma \rho )\lambda -1}{\lambda (\lambda -1)}. \end{aligned}$$

$\Delta a_i$ can take either sign depending on the values of $\gamma$, $\rho$ and $\lambda$. For example if $\gamma =0$, $\rho =0$ and $\lambda =2$, then $\Delta a_i= -0.25$. However, if $\gamma =-0.2$, $\rho =0$ and $\lambda =10$, then $\Delta a_i= 0.16$. $\square$