Platform-Mediated Competition

I present here the baseline model. Alternative payoff structures are explored in the extensions in Section 2. Agent $\ell\in[0,1]$ on side $K\in\{F,I\}$ is characterized by a vertical type $v_{K}^{\ell}$ and a horizontal type $\sigma_{K}^{\ell}\geq 0$. The platform’s objective function will be of the form

The components of (1) will be discussed below. What is important to note at present is that the platform’s payoff depends on the sum of some aggregate payoffs coming from the firm side and the individual side. If the platform’s objective is utilitarian welfare maximization then $U^{F}$ and $U^{I}$ will correspond to the utilities of firms and individuals respectively. However there are many other objectives of the form given in (1). One such setting, in which agents are privately informed about their vertical type, will be discussed in Section 1.5. Further examples will be illustrated in the applications of Section 3. For simplicity, in what follows I will refer to $U^{F}$ and $U^{I}$ as firm and individual payoffs respectively. Later, when discussing individual welfare I will be careful to differentiate between the true utilities of agents and the payoffs that are relevant for the firm’s objective in (1).

The vertical type $v_{K}^{\ell}$ determines the value that $\ell$ attaches to matches with agents on the other side of the market, while the horizontal type $\sigma_{K}^{\ell}$, which I also refer to as salience, represents how important $\ell$ is to agents on the other side. The be precise, for an individual $i$ on side $I$ the payoff of being matched with a set $s_{I}(i)\subseteq[0,1]$ of firms is

where $|s_{I}(i)|$ is the salience weighted size of the set $s_{I}(i)$, given by

Payoffs on the firms side are similar, but account for competition effects. This means that the payoff of a firm depends on the entire matching, not just their own matching set. The payoff to firm $j$ with matching set $s_{F}(j)$ when individuals have matchings $S_{I}=\{(s_{I}(i),i)\}_{i\in[0,1]}$ is given by

where

and $h$ is non-negative. The function $h$ captures competition effects. It can be thought of as depending on the exogenous ($\sigma_{I}^{i}$) and endogenous ($|s_{I}(i)|$) components of individual salience.⁴⁴4$h$ need not be a function of $|s_{I}|$. The general results presented below apply as long as $h$ is continuous (in an appropriate sense) in $s_{I}$ and independent of the vertical types $v^{j}_{F}$ for $j\in s_{I}$. For example, $h$ could be a function of $\lambda(s_{b})$ rather than $|s_{I}|$. In most applications $h$ will be decreasing; individuals are less valuable to firms if they are matched with many other firms. However I will not assume that this is always the case.

In the model described above, neither firms nor individuals care about the vertical types of the agents they are matched with. The interpretation is the an agent’s vertical type is a private taste parameter that describes their value for matches. I will discuss settings in which this assumption is natural. On the other hand, in some situations we may derive the vertical type from characteristics of the agent, for example the cost function of a firm that is also choosing prices, that may be relevant for agents on the other side. In Section 3.2 I explore payoffs of this form. The main qualitative conclusions of the model will be the same in both cases.

For the time being, assume that there is no horizontal differentiation; $\sigma_{I}^{i}=\sigma_{I}^{k}$ for all $i,k$ and $\sigma_{F}^{j}=\sigma_{F}^{l}$ for all $j,l$. If there is horizontal differentiation, the results of this section will hold conditional on horizontal types.

I will refer to the order specified in the definition of Order-Supermodularity as the supermodular order. Clearly, Supermodularity is a special case of Order-Supermodularity in which the order is given by type.

For the remainder of this section assume that Supermodularity holds. All results extend immediately to other supermodular orders.

I now turn to establishing the aforementioned threshold structure of matching sets. One might be tempted to apply a similar proof as that of Lemma 1. To see why this does not work, suppose $g_{F}$ is increasing and concave. Fix an individual, and suppose that they are matched with a firm with type $v^{\prime}$ but not with a type $v^{\prime\prime}$ firm, where $v^{\prime\prime}>v^{\prime}$. Simply dropping the low type firm from and adding the high type firm to the individual’s set clearly does not change the individual’s payoff, but will not necessarily improve payoffs on the firm side. This is because despite having a higher vertical type, the $v^{\prime\prime}$ firm will also have a higher quality matching set, by Lemma 1. Then by concavity this means precisely that the marginal change in match quality for the $v^{\prime\prime}$ firm is lower than for the $v^{\prime}$ firm. The proof of the following proposition modifies the switching argument to accommodate this case.

The previous section established that optimal matchings have a threshold structure when there is no horizontal differentiation. With horizontal differentiation, the same structure continues to hold conditional on horizontal types; each individual $i$’s matching set will be characterized by a threshold function $v^{*}_{I}(\sigma_{F},i)$ such that the individual matches with a firm of type $v_{F}$ and salience $\sigma_{F}$ if and only if $v_{F}\geq v^{*}_{I}(\sigma_{F},i)$. Similarly for firms. In this section I will be interested in the features of optimal threshold functions. In particular, I will identify when this functions are increasing/decreasing.

The platform’s problem simplifies greatly when $U_{F}(v,\cdot)$ is affine. In this case the firm’s payoffs can be written as $U_{F}(v,x)=a^{F}(v)+b^{F}(v)\cdot x$. If supermodularity holds then we can make a change of variables so that without loss of generality $b^{F}(v)=v$. Then the platform’s objective function can be written as

This integral can be maximized point-wise for each individual. This is true even when match qualities are constrained to be monotone in agent type, as will be the case when types are private infromation, since Lemma 1 tells us that these constraints will not bind. For simplicity, assume all primitive functions are differentiable. Consider the objective

The marginal effect of adding a firm of type $v^{F}$ and salience $\sigma^{F}$ to the matching set of individual $i$ is

The function $v^{*}_{F}(\sigma_{F},i)$ will be decreasing (increasing) if for $\sigma_{F}^{\prime\prime}>\sigma_{F}^{\prime}$ the following single crossing property holds: if the marginal benefit in (3) is positive (negative) for $\sigma_{F}^{\prime}$ then it is positive (negative) for $\sigma_{F}^{\prime\prime}$. Assume $h$ is decreasing it its first and third arguments (If $h$ is not decreasing in $\sigma^{F}$ then the threshold functions may be non-monotone). The term in parentheses in (3), $U^{I}_{2}(v_{i}^{I},|s_{I}(i)|)+\int_{s_{I}(i)}v^{F}_{j}\cdot h_{1}(|s_{I}(i)|,\sigma^{I}_{i},\sigma^{F}_{j})\ d\lambda(j)$, is the key determinant of the slope of $v^{*}_{F}(\sigma_{F},i)$. The first part of this expression is the marginal benefit to the individual of increasing the size of their matching set. The second part is the inframarginal cost to all firms matched with this individual of increasing the size of the matching set. If the sum of these two is positive, it means that the benefit to the individual outweighs the externality imposed on firms. But then this individual should be matched with all firms that have positive values $v^{F})j$. Moreover, the individual should be matched with firms with negative values only if $\sigma^{F}_{j}$ is large enough, so that the benefit the individual is sufficient to outweigh both the cost to the new firm and the inframarginal cost to all other firms. If on the other hand $U^{I}_{2}(v_{i}^{I},|s_{I}(i)|)+\int_{s_{I}(i)}v^{F}_{j}\cdot h_{1}(|s_{I}(i)|,\sigma^{I}_{i},\sigma^{F}_{j})\ d\lambda(j)<0$ then the individual should never match with negative value firms, and should only match with positive value firms if their salience is not too high. These observations are summarized in the following Lemma.

Lemma 2 has some interesting implications. If any firms have positive values then high-value, low-salience firms will be matched with the largest set of individuals (in the set inclusion sense). If no firms have positive values then the largest matching sets will instead go to high-value, high-salience firms.

Suppose that firms vertical types are their private information, while horizontal types are known to the platform. For example, the vertical type may reflect a firm’s marginal cost, while its horizontal type is the attractiveness of its product to consumers. Below, I will discuss more extensively the platform’s problem when types are private information. Suppose that the platform make monetary transfers with firms, and that firms payoffs are quasi-linear in money. In this context it is easy to see, from the usual Myerson (1981) argument, that the objective of a revenue maximizing platform will look the same as 1, except that the firms’ vertical types $v^{F}_{j}$ will be replaced with their “virtual values” $\varphi(v^{F}_{j},\sigma^{F}_{j})=v^{F}_{j}-\frac{1-Q_{F}(v^{F}_{j}|\sigma^{F}_{j})}{q_{F}(v^{F}_{j}|\sigma^{F}_{j})}$. Virtual values may be negative even if all true values are positive. As a result, the second best matching, in which firm types must be elicited, and the first best matching in which they are known may be qualitatively different: in the first best, conditional on values, low-salience firms will always be matched with a superset of the individuals matched with high-salience firms. In the second best the reverse may hold: for low enough values, high-salience firms are matched with a superset of the individuals that low-salience firms with the same value are matched with.

Throughout this section, assume that there is no horizontal differentiation on either side. I will be interested in how the optimal matching changes when some of the firms become “more prominent”, in a sense I will make precise. Intuitively, there are two effects of a platform placing greater weight on the payoffs of firm $j$. For one, it should increase the size firm $j$’s’ matching set. This effect is easy to see. On the other hand, it should change the other firms’ matching sets to increase the value of firm $j$’s matching. If $h$ is decreasing this means reducing the size of the other firms’ matching sets. How these two effects interact is not obvious in general. However there will be cases in which the effect on the matching structure can be identified. I explore such cases in this section.

By “more prominent” I mean that the marginal value of increasing the quality of a firm’s matching increases.

To further understand the changes in the matching, is important to first clarify the necessary conditions for optimality of a matching. Assume that there are $N<\infty$ firms and a continuum of individuals. The argument can be easily modified for the case of finitely many individuals. For notational simplicity, assume that Supermodularity holds (all results go through under Order-supermodularity). Label the firms in order of types, with firm 1 being the highest type and firm $N$ the lowest.

If there is no horizontal differentiation $h(n)$ is the endogenous salience of an individual matched with $n$ firms. Let $v_{I}^{*}(j)$ be the threshold type for firm $j$’s matching set. If firm $j$ marginally increases the size of its matching set by lowering the cut-off it benefits from a larger matching set, but affects the endogenous salience of the newly added individuals, who are included in the matching sets of all higher types. Suppose that firm $j$ adds a marginal individual to its matching set. Assume that there is not another firm with the exact same matching set as $j$, and that $U^{I}$ is continuous in its first argument. Using the fact that the optimal matching has the threshold structure described in Proposition 1, the FOC for such a change is given by⁸⁸8If another firm has the same matching set we just need to modify the term $[h(j-1)-h(j)]$ in equation (4).

with equality if $v^{*}_{I}(j)$ is interior.

The first term in (4) is the marginal benefit to firm $j$, the second is the inframarginal cost to all firms matched with the newly added individual, and the last is the direct benefit to the new individual.

The assumption of concavity of $U_{F}(v,\cdot)$ is used throughout the proof, but it is not necessary. This can be seen most easily by appealing to continuity of the objective function. The strict version of the comparative statics result of Proposition 2 holds when $U_{F}(v,\cdot)$ is affine, and given appropriate continuity assumptions Berge’s theorem implies that it will continue to hold if $U_{F}(v,\cdot)$ is perturbed slightly to be strictly convex. Thus it is not clear exactly what the role of concavity is in the result. To try to build some intuition, consider an increasing differences change to the payoffs of firm 1. At the new optimal matching, the sum of payoffs for all other firms must be smaller (otherwise it would have been worthwhile to change their matchings before the change in firm 1’s payoffs). Roughly speaking, concavity of $U_{F}(v,\cdot)$ implies that it is optimal to spread the reduction in payoffs across all these firms. This intuition is incomplete, and it would be valuable to know how far convexity of $U_{F}(v,\cdot)$ can be pushed.

Let $v^{*}_{I}(k)$ be the threshold individual type defining firm $k$’s matching set. If all firms with lower types than $k$ receive smaller sets then all individuals with types higher then $v^{*}_{I}(k)$ receive smaller matching sets.

If an individual’s utility is given entirely by $U^{I}$, and is increasing in match size, then Corollary 3 implies that all individuals with types above $v^{*}_{I}(k)$ are weakly worse off following the change in $k$’s payoffs. Even if this is not the case, for example if there are transfers between individuals and the platform, it may be possible to determine the welfare effects of such a change. I return to this question in the next section.

It is easy to see from the proof of Proposition 2 that the result can be extended to increasing changes in the payoffs of multiple firms.

Proposition 2 does not specify what happens to the matching sets for higher types than the one for which there is an increasing differences change. This will in general depend on the function $h$. This is because it is not clear what the effect of the described changes is on the value of the original matching sets for such types. The fact that all $j>k$ receive smaller matching sets benefits types $m<k$. However these types are also hurt by the fact that $k$ receives a larger matching set.

If $U^{F}(v,\cdot)$ is affine then comparative statics are much easier to identify. In this case, we can write $U^{F}(v,x)=\alpha^{F}(v)+\beta^{F}(v)\cdot x$ In this case, we can look at the problem entirely from the point of view of an individual, and the problem separates across individuals. In other words, the objective function can be written as

An increasing differences change in payoffs here corresponds to an increase in $\beta^{F}(v)$. The following Lemma is immediate.

In this case we can also identify what happens when there is an increasing differences change to the payoffs of the lowest type. The following result is immediate given the threshold structure of matching sets.

Propositions 1 and 2 will be particularly interesting when there is a continuum of individuals who have private information about their type which must be elicited by the platform. Assume that Supermodularity holds on side $I$ (the arguments extend directly to Order-supermodularity). In this setting the usual Myerson (1981) argument implies that in any incentive compatible mechanism the payoff of individual $i$ can be written as

Suppose that the conditions of Proposition 2 are satisfied, and that there is a increasing differences change in the payoffs of firm $1$. Corollary 3 says that all individuals with types above $v^{*}_{I}(1)$ receive smaller matching sets. Under Supermodularity, $U^{F}_{1}(v,x)$ is increasing in $x$. Then if $V(v^{*}_{I}(1))$ does not increase, the envelope condition in (7) implies that all individuals will be worse off. This will be the case if $v^{*}_{I}(1)=\bar{v}^{I}$, which in turn will hold whenever both $U^{F}(v^{F}_{1},\cdot)$ and $U^{I}(\underline{v},\cdot)$ (or the variant of payoffs used in the platforms problem, if it is not the same as (1)) are increasing.

In order to apply Lemma 7 it need not be the case that the platform’s objective exactly corresponds to that in (1). So long as the platform objective satisfies the conditions of Proposition 2.

In many settings individual payoffs are quasi-linear and the platform wants to maximize transfers from individuals. Normalize the outside option of the lowest type, $U(\underline{v},0)$, to zero. If individual payoffs are given by $u^{I}(v,|s_{I}(v)|)-t(v)$ then the sum of transfers from all individuals is given by

Under Supermodularity, the matching mechanism is incentive compatible for individuals if and only if individual match sizes are increasing in types and payments are constructed to satisfy the envelope condition in (7). Proposition 1 will hold provided the integrand in (8) is Supermodular. Moreover, under this condition the monotonicity constraint will not bind.⁹⁹9Without this condition Proposition 1 will hold, i.e. the optimal matching will still have a threshold structure, but the proof of Proposition 2 does not go through.