# Sellers’ Pay-for-Ranking Strategies on Platforms

Patrick Maillé, Bruno Tuffin

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Patrick Maillé, Bruno Tuffin. Sellers’ Pay-for-Ranking Strategies on Platforms. 2025. hal-05110534

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# Sellers' Pay-for-Ranking Strategies on Platforms

Patrick Maillé

IMT Atlantique, IRISA, UMR CNRS 6074

F-35700 Rennes, France

patrick.maille@imt.fr

Bruno Tuffin

Inria, Univ. Rennes, IRISA

F-35700 Rennes, France

bruno.tuffin@inria.fr

***Abstract***—Digital platforms shape users' choices and opinions through the ranking of the displayed items, the top-ranked ones being more likely to be considered than the next ones. While we expect the ranking to be based on some sort of relevance, there is a suspicion that platforms could alter their ranking to promote some content for financial or economic reasons, among other possibilities. We consider a model of a ranking platform implementing revenue-maximizing rankings, and analyze the strategy of sellers paying to be ranked higher for a better exposition, thereby increasing sales. We investigate the impact of such strategic behaviors on revenues, overall click rates for all actors, and global relevance perceived by users.

***Index Terms***—Ranking platforms, search neutrality, pricing, game theory

## I. INTRODUCTION

In ICT (for Information and Communication Technologies), platforms on the Internet play a crucial role, shaping the way we apprehend the world or influencing us in our choices. This is the case for instance of search engines, Google being the most well known one due to its market power [1]; in the list of links proposed as a response to a query, the first ones are by far the most likely to be looked at. Retail platforms such as Amazon also propose items when a request is typed, with the first suggestions more likely to be selected. Other types of actors with similar behaviors include hotel booking platforms, social networks or news websites (that may also influence opinions by putting forward some content [2]). It is therefore of primary importance to investigate the placement strategies of such (mostly revenue-oriented) platforms, and their impact on society. Our purpose here is to study the difference of outputs and their consequences, between a platform choosing “relevance” as the ranking/placement of items *versus* “revenue” (i.e., biasing their ranking for monetary benefits), to see if any bias can lead to less social satisfaction.

Our goal is therefore to design a model where a platform can offer to rank higher content/items from the actors–that we will call sellers from now on but can just be entities wishing to influence opinions in the case of news websites– offering emoluments to the platform. Thanks to this, “sellers” could attract more attention, the platform earn more money, at least in the short term, but users end up with less relevant informations. It is typically what is feared about search engines potentially having payments influence their ranking of organic links (we are not talking about sponsored links here) [4], or some brands paying to be ranked ahead of competitors on selling platforms, leading to unfair competition and possible

misjudgment from purchasers. Hotel booking websites are for example accused of promoting hotels paying the largest commission¹. For simplicity, in our model each seller will present a single item, so the two terms item or seller will be used indifferently throughout the paper. We aim at using a model for the platform where its action is the ranking strategy (relevance-based or revenue-maximizing), and determine the optimal strategy as well as the consequences on all actors. The platform would be tempted to favor items yielding revenue (through direct payments or because itself is selling the item [6], [12], [14]), but also should be aware that not providing the most relevant results could lead users to leave the platform for an alternative [5].

The platform model we use is based on the one in [7], which studies the optimal ranking strategy of a platform mixing long-term benefits (ranking according to relevance brings more visits, meaning more revenue) and short-term ones (ranking according to the immediate revenue a click could bring to the platform). The model shows a simple optimal ranking strategy combining immediate revenue from a click and the relevance of the results. This has been used in [8] to investigate the relations between search neutrality, that is, search engines being regulated (or not) to rank based on relevance only, and innovation, and in [10] where several such platforms in competition are considered to analyze the impact of regulation, in particular the Digital Markets Act (DMA) passed in Europe [3].

In the previous model and its extensions, item sellers on the platform are not part of the game: the revenue they bring to the platform is fixed. But analyzing the specific situation where they can be strategic and pay the platform to get ranked higher deserves some attention. Including those actors as players adds a level to the game representing interactions among all actors; we are going to analyze that extended game to study the impact of strategically-behaving sellers.

The remainder of the paper is organized as follows. Section II presents the model, summarizing the results from [7] in a first part, and then in a second part explaining the novelty, that is, sellers participating to the game by potentially paying to be ranked higher. Section III looks at two sellers in competition at this level of the game. Section IV compares the outputs of three situations: when there are two sellers in

¹https://www.theguardian.com/business/2018/jun/28/hotel-booking-sites-cma-investigation-areas-of-concern
competition, when a single seller is strategic, and the neutral situation where no seller can pay. Section V concludes the paper.

## II. MODEL

We first recall the model in [7] and the platform optimal ranking strategy, in its basic formulation (for a general formulation the reader is advised to see [7]).

### A. Model of a strategic platform from [7], [8]

Let us provide a version of the model in [7] for a query at a platform, that will be used and extended in the next subsection.

Consider a (random) query at the considered platform as a random vector

$$Y := (M, R_1, G_1, \dots, R_M, G_M),$$

where

* $M$ is the (random) number of displayed sellers/items for the query;

* For each displayed seller indexed by $i \in \{1, \dots, M\}$,

- $R_i$ is the estimated relevance factor for Seller $i$ relative to the query

- $G_i$ is the expected revenue for the platform if the link for Seller $i$ is clicked (a revenue which may come from associated ads, a percentage of the sale, etc.).

The ranking of each seller has an impact on its visibility and its probability to be clicked. We assume that the probability to be clicked for Seller $i$ (also called the *click-through rate* or CTR) when ranked at position $k$ is the product $\theta_k \psi(R_i)$, taking into account separately a relevance effect of Seller $i$ through some non-decreasing function $\psi$ and a (fixed) position factor $\theta_k$, with $\theta_1 \ge \theta_2 \ge \dots \ge 0$. This models the fact that top positions in the ranking have higher visibility: indeed one can interpret $\theta_k$ as the probability that an item at position $k$ is seen, and $\psi(R_i)$ as the conditional probability of a click on Item $i$ given that item is seen. Note that a more general CTR framework is considered in [7], but such a product form is quite standard in the literature, for example when considering search engines [9].

From this general framework, the platform has to choose the ranking rule, that is a permutation

$$\pi(Y) = (\pi_1(Y), \dots, \pi_M(Y))$$

of the $M$ links for each request $Y$, where $\pi_i(Y)$ represents the position of Seller $i$.

A platform (then called *neutral*) ranking sellers based on relevance only will display the sellers in decreasing order of $R_i$; but it could also rank based on the short-term revenue, that is, decreasing order of $\psi(R_i)G_i$. Paper [7] takes into account the more subtle effect that users may not use the system as much if dissatisfied with its output: if the perceived relevance decreases (e.g., due to a bias based on revenue) some users may move to a competing platform, hence a tradeoff for the platform between short-term revenue and total

number of visits. To simplify the notations, we denote by $\tilde{R}_i := \psi(R_i)R_i$ and $\tilde{G}_i := \psi(R_i)G_i$ the CTR-weighted relevance and (platform) revenue for Seller $i$, respectively, and let

$$r := \mathbb{E}_Y \left[ \sum_{i=1}^{M} \theta_{\pi_i(Y)} \tilde{R}_i \right] \quad \text{and} \quad g := \mathbb{E}_Y \left[ \sum_{i=1}^{M} \theta_{\pi_i(Y)} \tilde{G}_i \right] \qquad (1)$$

be the respective average relevance and average revenue per query, for a given ranking rule $\pi$. Typically, a platform seeks to maximize a function of the form

$$\lambda(r)(g + \beta)$$

where

* $\lambda(\cdot)$ is an increasing function representing the (relevance-dependent) rate of visits to the platform;

* $\beta$ is the average revenue coming from visits to the platform, independently of the displayed content. It corresponds to displayed sponsored content revenues for search engines for example (as opposed to the organic content supposed to be ranked by relevance), or to ads displayed on the platforms. Having no such “external” revenue per visit just means taking $\beta = 0$, in which case all platform gains come from showing among the results some sellers with positive revenue $G$..

It is shown in [7] that there is an optimal ranking policy called an LO-$\rho$ policy, standing for *linear ordering policy with weight $\rho$*. Such a policy simply ranks sellers by decreasing order of $\tilde{R}_i + \rho\tilde{G}_i$ for some value of the scalar coefficient $\rho \ge 0$, whose optimal value (that exists and is unique under mild conditions) will be denoted by $\rho^*$. Determining $\rho^*$ is easy by a stochastic optimization procedure such as fixed-point iteration combined with simulation [7].

### B. Extending the model: sellers paying to be ranked higher

In [7], the gain $G_i$ representing a direct or indirect revenue by the platform from clicked Seller $i$ is associated to the search but is fixed. In contrast, we study in this paper the situation where the sellers can play strategically and pay the platform to be ranked higher and thereby increase revenue.

To model this situation, we consider that a seller $s$ offers to pay the platform a deterministic price $g_s$ each time it is clicked through the platform’s ranking. Denoting by $v_s$ the seller’s willingness-to-pay for a click, and by $c_s$ the resulting mean CTR for $s$, the utility that Seller $s$ extracts from visits to the platform can be written as

$$U_s = \lambda(r)c_s(v_s - g_s) \qquad (2)$$

Note that while the CTR $c_s$ increases with $g_s$ because of the platform adapting its rankings, the bias introduced is at the expense of result relevance, hence $r$ decreases with $g_s$.

We end up with a Stackelberg game [11] with sellers playing first, before the platform, but playing strategically by anticipating what the subsequent decision of the platform will be. More precisely, the order of decisions is as follows:
1) Sellers choose the amount they pay to the platform ($g_s$ for Seller $s$) for each click;

2) The platform ranks according to the revenue-maximizing policy described in [7].

Since sellers anticipate the decision of the platform, the game is analyzed by backward induction. Note that the seller choice is independent of the query, i.e., Seller $s$ offers the same price $g_s$ for a click, regardless of the search term leading to it.

In what follows, we compare the relevance, query rate, platform gain, visits to all sellers, and gain by the (strategic) sellers with respect to the (neutral) scenario where sellers are not allowed to pay the platform. Due to the complexity of the model, we carry out a numerical study, through simulations.

### C. Simulation setting

We will first consider the case where there is a game between two sellers competing for a higher ranking by paying the platform, and then compare the outcome with the situation where a single seller is playing, and with the neutral case where no subsidy is accepted by the platform. For all our numerical studies in the next sections the variables/parameters will be chosen as follows:

* Relevances uniformly distributed over [0,1] (and all independent).

* $G_i = 0$ for every Seller $i$ not playing strategically by paying to improve its ranking. Each playing Seller $s$ will choose a price $g_s$. Therefore with respect to [7], gains are deterministic, hence the use of lowercase letters. Recall that $G_i = 0$ for all sellers would mean a ranking based on relevance only, and is considered to be a neutral behavior of the platform.

The values of the position-related CTR components $\theta_k$ for $1 \le k \le 10$ are displayed in Table I and we assume $\theta_k = 0$ $\forall k \ge 11$. For simplicity, we also consider $\psi(r) = 1 \forall r$, i.e., click probability only depends on position, so that those values are the actual CTRs considered. We also consider the rate of

<table><caption>Table I</caption><tr><td>θ<sub>1</sub></td><td>θ<sub>2</sub></td><td>θ<sub>3</sub></td><td>θ<sub>4</sub></td><td>θ<sub>5</sub></td></tr><tr><td>0.398</td><td>0.187</td><td>0.102</td><td>0.072</td><td>0.051</td></tr><tr><td>θ<sub>6</sub></td><td>θ<sub>7</sub></td><td>θ<sub>8</sub></td><td>θ<sub>9</sub></td><td>θ<sub>10</sub></td></tr><tr><td>0.044</td><td>0.030</td><td>0.021</td><td>0.019</td><td>0.016</td></tr></table>

POSITION-RELATED CTR VALUES USED IN THE PAPER, TAKEN FROM [13]

request function $\lambda(r) = r$. Unless said otherwise, we will consider $M = 10$ sellers, hence a simplified deterministic case.

## III. TWO SELLERS PLAYING A NON-COOPERATIVE GAME

We start by investigating the situation with two sellers– indexed by 1 and 2–playing strategically in a non-cooperative game and paying the platform, while the other 8 sellers do not pay the platform. Both playing sellers make their decisions anticipating the policy of the platform at the lower level of the Stackelberg game. Note that we limit ourselves to two strategic sellers in this preliminary work, for which determining the Nash equilibrium is already a difficult numerical task.

We consider a heterogeneous case with different willingness-to-pay (valuation) per click for the sellers: $v_1 = 15$ and $v_2 = 12$.

Before looking at the Nash equilibrium, we aim to be convinced that a seller has a unique best strategy maximizing its utility. Figure 1 displays Seller 1 utility as in (2) in terms of the price $g_1$ paid to be ranked higher when $\beta = 1$. The

<figure><img src="image_2.png" /><figcaption>Figure 1. Revenue $U_1$ of Seller 1 when varying $g_1$, for $g_2 = 0$</figcaption></figure>

figure suggests that the utility $U_1$ is a unimodal function of the price $g_1$: it first increases with $g_1$ when for small prices the benefit from better ranking exceeds the extra cost, and then decreases when the combined effects of a reduced visit rate (due to a lower relevance $r$) and a larger price per click outweigh the gain from a better position. Hence a unique price $g_1^*$ maximizing Seller 1's utility.

To better understand the performance metrics (with $g_2 = 0$ fixed), Figure 2 displays the platform average relevance and gain, Seller 1's average position and CTR, and the ranking parameter $\rho^*$ selected by the platform when $g_1$ varies. Having a single seller paying a subsidy moderately influences the average relevance (decreasing from around 0.7 to 0.66), but yields a platform revenue increasing with $g_1$ as expected. Average Seller 1's position and CTR improve with $g_1$ as expected, but the gain with this respect is smaller and smaller for any extra "unit" of $g_1$, illustrating that paying more when already ranked first is useless. Finally, $\rho^*$ decreases with $g_1$, something not obvious at first sight, the platform having more interest to focus on gains due to visits. All the results above are obtained with a Monte Carlo estimation of $r$ and $g$ in (1) with a sample size $10^6$.

Now, Figure 3 displays the best-response curves of the two paying-for-clicks sellers in terms of the strategy of the opponent paying seller. A Nash equilibrium (that is, a profile of strategies from which no seller has an interest to move unilaterally) is an intersection of those best-response curves. The figure highlights the interactions among both sellers: the best amount $g_s$ (for $s = 1, 2$) that a seller should offer depends on the amount offered by the competing seller, more precisely best-responses are increasing in the price of the opponent. Note however that when the price set by the opponent gets
<figure><img src="image_3.png" /><figcaption>Figure 2. Outputs in terms of the price $g_1$ paid per click by Seller 1, for a fixed $g_2 = 0$</figcaption></figure>

large, it is not worth anymore raising one's price further to be ranked above the opponent, and is illustrated when the best-response curve crosses the diagonal: this occurs for Seller 1 when $g_2$ is about 6, and for Seller 2 when $g_1$ is about 5. Furthermore, since the best-response to a seller offering 0 is non-negative (actually, strictly positive in the figure), having continuous best-responses would guarantee the existence of at least one Nash equilibrium. In Figure 3 there is actually a unique equilibrium, as well as in all cases we studied numerically.

We display in Figure 4 the situation at that equilibrium, in terms of the platform's average revenue $\beta$ per visit (from ads or sponsored links), for several values of the total number

<figure><img src="image_4.png" /><figcaption>Figure 3. Best-response prices-per-click of each strategic seller</figcaption></figure>

$M$ of sellers to display. The estimation of $r$ and $g$ in (1) is here with $n = 10^5$ for computational time reasons. We do not display the request rate $\lambda$ of the platform, since it is the same as the relevance ($\lambda(r) = r$).

From the strategic sellers' point of view, a higher $\beta$ makes it harder to affect rankings by paying, because the platform will prefer maintaining a high visit rate by giving more weight to relevance. Hence a higher visibility (CTR) for non-strategic sellers. When the number $M$ of displayed sellers is getting larger, the utility of strategic sellers has the tendency to be larger too: competition helps, as more seller diversity improves the average relevance of results (hence attracting visits), and small strategic payments to the platform gets strategic sellers high-visibility spots. Notice that the opposite holds for non-paying sellers: they do not benefit from the gain in average relevance because they get ranked behind paying sellers, and their average CTR decreases when the degree of competition $M$ increases. For the other curves, as $\beta$ increases, platform revenue, relevance, and arrival rates increase, as expected.

## IV. COMPARING THE OUTPUTS OF SEVERAL SCENARIOS

We aim here at comparing the outputs of our model in three scenarios:

* when two sellers compete on the price they pay the platform as in the previous section,

* when a single seller is strategic and pays the platform to be ranked better,

* when no seller is allowed to pay, the so-called neutral situation where the platform ranks based on relevance with $G_i = 0 \ \forall i$.

To highlight the combined effects of the scenario, the ranking-independent revenue per visit $\beta$, and the number $M$ of sellers, we display in Figure 5 the outputs of the game in terms of $\beta$ when $M = 10$, and in terms of $M$ in Figure 6 when $\beta = 1$.

From the curves of seller utilities, being a single strategic seller leads to more revenue (but at the expense of other sellers). When $M$ is large enough, each of the two potentially strategic sellers prefer to pay for ranking, even through a game, than being in a neutral scenario. However, when $M$ is small we
<figure><img src="image_5.png" /><figcaption>Figure 4. Outputs at the equilibrium of the game when two sellers are strategic, in terms of $\beta$ for different values of the number $M$ of sellers</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5. Outputs in terms of $\beta$ for $M = 10$ sellers, when two sellers, only one, or no seller strategically pays for better ranking</figcaption></figure>

observe that both of them would prefer the neutral case, which we can interpret as a Prisoner's dilemma situation: whatever the other seller is doing, it is in each one's interest to pay, but both engaging in the pricing game leads to high prices paid for a limited gain in CTR, and in the end a lower utility that with a neutral platform. This phenomenon only happens for $M$ small: in that case the CTR of each seller in a neutral situation is already high, while paying for $M$ large ensures a significantly improved CTR.

As pointed out previously, the revenue of a player decreases with $\beta$ since more emphasis is put on advertisement revenue, so that payments have to be higher to impact the platform

rankings. For the same curves in terms of $M$, the revenue of a strategic seller (more obviously here) increases with $M$. As can be seen from the curves of relevances, the larger the number $M$ of sellers, the larger the relevance as expected, but the relative values also increase in favor of the neutral scenario, which can attract the attention of a regulator.

## V. CONCLUSIONS

The model presented in this paper focuses on possible biases on ranking platforms, that would be affected by payments from the ranked entities (here, sellers). We quantified the impact of a single strategic seller on the ranking of a revenue-maximizing
<figure><img src="image_7.png" /><figcaption>Figure 6. Outputs in terms of the number *M* of sellers for a ranking-independent revenue *β* per visit to the platform, when two sellers, only one, or no seller strategically pays for better ranking</figcaption></figure>

platform and on the other actors, as well as the impact of two
sellers acting strategically.

We highlighted that offering sellers the possibility of paying
to influence the platform's ranking can lead to a Prisoner's
dilemma when the number of sellers is low: strategic sellers
can indeed be incentivized to engage in a noncooperative game
whose equilibrium is worse for all of them that a neutral
situation (no payments allowed).

Future work can include extensions of the model for gains
to the platform: in this paper the platform makes revenue from
advertisements and direct payments from strategic sellers, but
there could also be some indirect gains to the platform from

displaying some sellers, as in the original model of revenue-
maximizing rankings [7]. Another extension could consider
payments to the platform that would vary with the query, as
some queries are more likely than others to lead to a sale.

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