Instrumental Variation in Mediator Take-up Costs

• Instrumental variation in mediator take-up costs is defined as using exogenous shifts, like policy changes or randomized designs, to measure fees mediators extract in market platforms.
• The approach reveals how capacity constraints and collusion vulnerabilities in auctions, such as sponsored search, shape equilibrium mediator fees.
• Analytical and empirical models illustrate that variations in auction design and surplus distribution directly influence the efficiency and profitability of mediation.

Instrumental variation in mediator take-up costs refers to the use of exogenous variation—typically generated through policy changes, randomization, or market features—as an identification strategy to understand, estimate, and rationalize the fees, frictions, or barriers associated with the adoption of mediators in economic, technological, and experimental systems. This concept appears in various literatures, both in causal mediation analysis (where the mediator is an endogenous outcome between treatment and final outcome) and in mechanism and market design (where mediator "take-up" is operationalized as the decision of agents to participate in or contract with a for-profit mediation service). The notion is central to robust inference about mechanisms through which interventions operate in the presence of selection, cost, or strategic behavior, and is especially relevant for platform markets, sponsored search, field experiments, and policy analysis.

1. Market Forces and the Role of Capacity and Collusion

Mediators emerge in marketplaces to address frictions created by capacity constraints and the vulnerability of prevailing mechanisms to collusion. In the sponsored search market, finite slot availability and substantial heterogeneity in click-through rates (CTR) for top placements create competing demands among advertisers, resulting in potential oversubscription or exclusion of certain advertisers. Standard revenue-ranked mechanisms (RBR), such as generalized second-price (GSP) auctions, are also known to be collusion-vulnerable: groups of agents may coordinate bids to reduce prices, sharing the resulting surplus. This market structure introduces opportunities for for-profit mediators to aggregate demand and coordinate bidding or participation, increasing the overall efficiency and providing an explicit basis for modeling mediator take-up costs as a fraction of generated surplus (0711.0259).

Mediators exploit these inefficiencies by acting as entrepreneurial entities that buy access in the primary market (e.g., securing ad slots) and redistribute surplus to affiliated advertisers via sub-auctions or bid coordination, collecting a service fee as the “take-up cost.” The degree of market scarcity, the fragmentation of capacity, and the susceptibility of the mechanism to collusion together instrumentally determine the extent and form of take-up fees in equilibrium.

2. Competing Models of Capacity Expansion and Mediation

Two canonical models are employed to paper mediator take-up costs in capacity-constrained environments (0711.0259):

• Mediator-based capacity provision: Independent for-profit mediators create additional slots (e.g., by attracting user traffic to alternative platforms or channels), run sub-auctions among their clients, and capture a fraction α of the total surplus (reduction in advertiser costs net of mediator fees). Empirically and theoretically, this approach can simultaneously increase both auctioneer revenue and social efficiency, enabling higher mediator take-up costs justified by positive-sum gains.
• Auctioneer-based capacity expansion: The auctioneer itself forks one or more slots (e.g., adding sub-pages or subsidiary listings with diminished CTR determined by a "fitness factor" f). The analysis of revenue and efficiency in this scenario reveals that, under certain distributions of CTR and bidder values, excessive capacity can lower average per-impression prices, compressing the surplus available for mediator extraction and thus influencing equilibrium take-up costs. The sufficiency of revenue-enhancement from added capacity is established via explicit formulas:

$$R_0 = \sum_{j=1}^K (\gamma_j - \gamma_{j+1}) j s_{j+1}; \quad R = \sum_{j=1}^{K+L-1} (\tilde{\gamma}_j - \tilde{\gamma}_{j+1}) j s_{j+1}$$

with value-of-capacity gain $(R - R_0)/R_0$ determined by the reordering of CTRs.

Take-up costs are instrumentally sensitive to these market design features. When additional capacity is high-quality or the division of surplus is efficient, mediators can justify higher fees; in contrast, dilute expansion depresses the surplus, and take-up costs must fall accordingly.

3. Game-Theoretic Foundations and Strategic Mediation

The game-theoretic analysis formalizes the process by which mediators coordinate agents (M-bidders) to achieve improved payoffs in mechanisms that lack collusion-resistance. In typical sponsored search auctions, advertisers (with values $v_i$, quality scores $e_i$) bid for ranked positions, with equilibrium determined by symmetric Nash equilibrium (SNE) conditions:

$u_j = \gamma_j (e_j v_j - r_{j+1})$

where $r_j$ is the bid for position $j$.

Mediators can strategically set bids for affiliated clients, equalizing them across positions to suppress intra-group competition and maximizing payoffs, contingent on maintaining incentive compatibility for outside bidders (I-bidders). The minimum required value of the group bid $r$ (for M-bidders) is derived to ensure no I-bidders wish to deviate into M-bidder slots:

$$r \geq \max_{j \geq L+1} \left\{ \left(1 - \frac{\gamma_j}{\gamma_1} \right) e_j v_j + \frac{\gamma_j}{\gamma_1} r_{j+1} \right\}$$

The mediator collects a take-up fee equal to an $\alpha$-fraction of the gain in payoff:

$$U_M = \alpha \sum_{j=1}^{l-1} (r_{j+1} - r) \gamma_j$$

This take-up cost directly scales with the strategic surplus available from coordinated bidding and is diminished if outside bidders or the auctioneer modify the rules to reduce collusion-vulnerability.

Notably, attempts by auctioneers to clamp down on mediator-driven surplus via collusion-proof designs (posted pricing, etc.) are shown to come at the expense of auctioneer revenue, making such defenses unattractive in most practical settings (0711.0259).

4. Revenue, Efficiency, and Surplus-Driven Variation in Take-up Costs

The take-up cost charged by mediators hinges on surplus generated relative to prevailing (non-mediated) equilibria, with crucial dependence on market parameters:

• Revenue: Expressed pre- and post-mediation or capacity expansion via

$$R_0 = \sum_{j=1}^K (\gamma_j - \gamma_{j+1}) j s_{j+1}, \quad R = \sum_{j=1}^{K+L-1} (\tilde{\gamma}_j - \tilde{\gamma}_{j+1}) j s_{j+1}$$

The difference $(R-R_0)$ underpins the surplus eligible for mediator extraction.

• Social efficiency: Quantified as

$$E_0 = \sum_{j=1}^K \gamma_j s_j, \quad E = \sum_{j=1}^{K+L-1} \tilde{\gamma}_j s_j$$

Mediation always improves social efficiency when employing the mediator-based model, but efficiency gains under auctioneer-based models only accrue for fitness factors $f$ above critical thresholds.

• Variation with market conditions: When capacity is tightly constrained and/or collusion-vulnerability is acute, mediators can deliver and extract greater incremental surplus, raising justified take-up costs (higher $\alpha$). If abundant low-value capacity is added (large $L$, high $f$), per-bidder surplus is compressed, lowering both the market for mediation and the price mediators can command.

Take-up costs are thus endogenous and instrumentally determined by the capacity profile, equilibrium structure, and market fragility, with boundary cases identified via mathematical theorems (e.g., Theorem 1, sufficient revenue improvement conditions).

5. Heterogeneity, Empirical Illustration, and Policy Implications

Numerical examples from the paper (e.g., Examples 1 and 2) illustrate how variations in CTR hierarchy, advertiser values, and added slot quality jointly affect the absolute and marginal value available to mediation. In cases where values and CTRs are well-separated, adding capacity may depress prices to such an extent that mediation yields minimal aggregate surplus, lowering the viability of for-profit mediation.

Mediators capture take-up fees only to the extent that they expand the effective surplus beyond what individual advertisers could independently achieve. Heterogeneity in advertiser values, differential vulnerability to collusion, and the structure of CTRs all affect the magnitude and division of this surplus.

From a policy perspective, the findings explain the resilience and evolution of for-profit mediators in digital market ecosystems and delineate the limits of mechanism design in suppressing or controlling their influence. Instrumental variation—in the sense of market-wide or designed exogenous changes in capacity or rules—serves as the essential lever in both empirical measurement and policy calibration of mediator take-up costs.

6. Synthesis and Broader Context

Instrumental variation in mediator take-up costs encapsulates the relationship between structural features of a system (e.g., slot allocation, auction format, information regime) and the strategic or realized costs of mediation. The concept provides both a theoretical and practical framework for evaluating when, where, and how much surplus mediators can extract, and what factors moderate this process. It is foundational for the analysis of intermediation in sponsored search, ad auctions, and more broadly any system where entry and coordination by mediators may enable or undermine efficiency and revenue objectives (0711.0259).

The approach is generalizable beyond sponsored search to a wide array of domains—including facility location with mediators (Babaioff et al., 2015), bilateral trade with mediation fees (Fan et al., 15 Oct 2024), and policy settings where quasi-experimental designs yield exogenous variation in mediator adoption or costs (Hogan-Hennessy, 7 Aug 2025). In each, the structure and magnitude of take-up costs are fundamentally mediated by the instrumented market environment.