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A Game-Theoretic Perspective on the Recommender (Eco-)System

Omer Madmon

PhD Candidate @Technion, Data Science

�

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A Game-Theoretic Perspective on the Recommender (Eco-)System

Omer Madmon

PhD Candidate @Technion, Data Science

�

Economic Theory

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Economic theory at PyData? 🤔

Main interest: strategic behavior in the presence of AI\ML
This talk: strategic behavior in recommendation systems

�On the Convergence of No-Regret Dynamics in Information Retrieval Games with Proportional Ranking Functions. Omer Madmon, Idan Pipano, Itamar Reinman, Moshe Tennenholtz, ICLR 2025.

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Game theory

The study of mathematical models of strategic interactions

Mathematical models

Real-world use cases

Complex enough to capture real-world scenarios

Simple enough to be analytically solvable

"All models are wrong, but some are useful“

��George Box, 1976

Game theory is the study of mathematical models of strategic interactions among rational decision-makers. By rational, we mean that each decision-maker tries to maximize their own utility — but crucially, that utility depends not only on their own actions, but also on the actions of others.

Game theory has applications across almost every discipline — from economics and social sciences to computer science and even biology. What all these fields have in common is the same core practice: taking a real-world situation and translating it into an abstract, simplified mathematical model. This allows us to analyze it, extract meaningful insights, and uncover the underlying principles driving behavior.

This modeling always involves a tradeoff. On one hand, we want our model to be simple enough to analyze mathematically. On the other, it needs to be complex enough to capture the key aspects of real-world behavior. Finding that “sweet spot” is really the art of game-theoretic modeling, and it’s what we, as researchers, aim to achieve.

So, as the great George Box once said (and as you probably hear in almost any data science talk): “all models are wrong, but some are useful”. This is true not only for statistical modelling, but also for game-theoretic modelling.

With that being said, let’s return to recommendation systems and figure out what it has to do with game theory and strategic behavior.

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Recommendation systems

Any recommendation system induces a dynamic ecosystem, �with various types of players, each with its own incentives

Recommender systems are literally everywhere — they shape the way we consume and create content across so many domains: search engines, entertainment platforms, social media, retail, travel, and much more.

At their core, recommender systems aim to match end-users with the most relevant content. But users aren’t the only players here. Alongside them are content creators, who benefit from reaching a wide audience and put significant effort into producing content that gets visibility.

What we end up with is a dynamic ecosystem with different types of entities — or players — each driven by its own goals and incentives.

While there are many economic aspects we could explore in the context of the recommender ecosystem, in this talk we’ll focus on one in particular: stability and long-term behavior. To illustrate what I mean, let’s look at a very simplified example.

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Stability and long-term dynamics

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Stability and long-term dynamics

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Stability and long-term dynamics

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Stability and long-term dynamics

When that happens, it can create problems for everyone involved in the ecosystem — from the platforms and publishers to the users themselves.

The main result behind that, is that such content fluctuations create both monitoring overhead and a high degree of uncertainty, which could be harmful to the entire economic ecosystem.

From the publishers’ perspective, this constant need to adapt their content requires continuous effort and creates a lot of uncertainty about future returns.

From the users’ perspective, instability creates uncertainty about the type and quality of content they can expect to find on the platform.

Instability is also problematic from the platform’s perspective. Platforms need to continuously monitor supply and demand dynamics in order to estimate expected revenue.

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Research question

How can we ensure that long-term dynamics of strategic publishers will converge to a stable state?
We take a game-theoretic approach

Model the one-shot interaction (incentives)
Model the long-term behavior (dynamics)
Define stability criteria (desired outcome)

So, once we understand why stability in content dynamics is so important, the following key research question arises:

How can we, as system designers, ensure that the long-term dynamics of strategic publishers will converge to a stable state?

To approach it from a game-theoretic perspective, we break the problem down into three main components:

First, we model the one-shot interaction between publishers who compete for exposure. This includes defining who the players are, what are their actions, and what objectives they try to maximize.

Second, we model how publishers behave over time, which lets us understand how strategic content creation evolves and what the long-term dynamics look like.

Finally, we define what we actually mean by a stable state — and what kinds of guarantees we want the recommendation mechanism to provide in terms of stability.

Together, these three elements give us a framework to reason about how recommendation systems should be designed to ensure stability, in the presence of strategic publishers.

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The one-shot game

RecSys

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User preferences

.

Exposure gain

Production cost

.

Let’s start by describing the one-shot interaction.

At each stage of the game, all publishers simultaneously create their documents. Since this is a theoretical model, we represent each document simply as a vector in a high-dimensional embedding space.

In that same space, we also represent the preferences of users, which are drawn from a commonly known distribution.

The recommendation system takes as input the embeddings of all the submitted documents, together with the user preferences, and produces an exposure rate for each publisher — essentially, the share of user attention their content receives. (We’ll talk about how exactly this is determined in just a moment.)

Each publisher then aims to maximize its utility, which consists of two components. The first is the expected exposure rate — the average exposure the publisher receives across the entire user population.

The second component is what we call the integrity term: a cost associated with producing content that deviates from the publisher’s “ideal” topic — the one they would naturally write about if there were no competition.

The trade-off between these two objectives is controlled by a parameter we call λ (lambda).

Now that we’ve defined the players’ incentives, let’s look more closely at how the recommendation system decides these exposure rates.

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The recommendation mechanism

As we discussed, the recommendation system takes the documents and the user preferences as input.

The first step is to compute the distance between each document and the user in the embedding space — under the assumption that more relevant documents are located closer to the user.

Next, the system transforms these distances into relevance scores by applying a function we call the activation function, denoted by 𝑔. This function should be decreasing — so that smaller distances result in higher scores.

Finally, the exposure rate assigned to each publisher is simply proportional to these scores.

For instance, if we choose this exponential activation function, 𝑔(𝑑)=𝑒^(−𝑑), then the exposure function simply becomes the softmax function.

Now, an important thing to keep in mind is that in our model, the design choice of the recommendation system is fully captured by this activation function 𝑔. Different choices of 𝑔 lead to different incentives for the publishers — and, as a result, to different behavior patterns in the ecosystem.

Our main goal, then, will be to characterize the class of activation functions that can provide stability guarantees on publishers’ behavior over the long run.

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Game-theoretic modeling

Model the one-shot interaction (incentives)
Model the long-term behavior (dynamics)
Define stability criteria (desired outcome)

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Long-term dynamics

The second component of our model focuses on long-term dynamics — in other words, how publishers behave as time goes on.

There are many ways to model long-term strategic behavior, but in this work we adopt a classical assumption from algorithmic game theory: as time progresses, publishers aim to minimize a quantity called regret.

Regret is defined as the difference between two components. The first is the hypothetical utility a publisher could have achieved if they had known all the other publishers’ actions in advance — of course, this is purely theoretical, since they don’t actually have that information when making decisions.

The second is the utility they actually achieved in practice. The difference between these two — “what I could have done if I knew the future” minus “what I actually did” — is the regret. If you’re familiar with multi-armed bandits, it’s essentially the same notion.

So, we assume that publishers aim to minimize their regret, and in fact, we assume that they also succeed in doing so. More formally, we adopt the assumption that the regret of each publisher's grows slowly as time progresses. Notice that in order to keep our model general and capture a wide range of behavioural patterns, we do not impose any assumption on how the regret is being kept low, but we only assume that it happens – one way or another. When this property holds, we say that the system follows no-regret dynamics.

A recent motivation for studying no-regret dynamics in the context of content creation comes from this great paper, in which the authors showed that the behavior of LLMs can often be studied and analyzed using the regret minimization framework. Since LLMs are being used extensively for competitive content creation, it is quite natural to adopt the no-regret model to capture the long-term dynamics of content modifications under competition.

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Game-theoretic modeling

Model the one-shot interaction (incentives)
Model the long-term behavior (dynamics)
Define stability criteria (desired outcome)

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Stability criteria

Nash equilibrium (NE): No player can benefit from deviating from its action, given that all other players are not deviating themselves

At this point, we’ve described what a single interaction in the recommendation ecosystem looks like, and we’ve defined how publishers behave over time. The final piece of the puzzle is to ask: what do we actually want to happen in the long run? In other words, what does a stable state look like, and when can we say that a recommendation system is stable?

In game theory, there’s a well-established and widely accepted notion of stability: the Nash equilibrium, proposed by John Nash back in the 1950s.

A Nash equilibrium is a state in which no player can improve their outcome by changing their action — given that all other players stick to theirs. In simpler terms, once the system reaches this point, there’s no reason to move away from it – so it’s a stable state.

In our context, the players are the publishers, and their actions are the documents. So, if the system reaches an equilibrium, the corpus of content itself becomes stable — and that’s exactly the outcome we’re aiming for.

Now, recall that the choice of the system designer is captured by the activation function 𝑔. So, We say that an activation function as stable if any no-regret dynamic in the game is guaranteed to converge to a Nash equilibrium.

The key word here is any: No matter how publishers achieve low regret, as long as they do, the system will eventually settle into a stable state. This is what makes this guarantee so powerful.

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Game-theoretic modeling

Model the one-shot interaction (incentives)
Model the long-term behavior (dynamics)
Define stability criteria (desired outcome)

💡We are now ready to state our result…

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Main result

Q: Can we characterize a simple and intuitive set of stable activation functions?

We are now ready to present our main result.

Recall that our goal is to understand what makes a recommendation mechanism stable in the presence of strategic publisher behavior. In our framework, this translates into a simple but fundamental question: Can we characterize a simple and intuitive class of activation functions that guarantee stability?

It turns out the answer is surprisingly elegant — and it’s captured by our main theorem: Concavity is all you need. Under mild conditions, any concave activation function is stable.

This result means that if our primary concern is ensuring stability, then it’s enough to design the recommendation mechanism so that publishers’ incentives are shaped by a concave function.

Just as a reminder – a concave function that satisfies the following condition: for any two points, the chord between these two point always lies below the function itself – as demonstrated in this figure.

�Importantly, concavity is a very broad condition — which means we have a lot of freedom in choosing the exact form of the activation function. This opens the door to exploring empirically how different concave activations affect the overall dynamics of the ecosystem.

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A case study: recommender system design

Congrats! 🎉 You’ve just joined the recommender systems team at your favourite online platform
After a smooth onboarding, the Monetization Lead pulls you aside:

“Users love the content, but publishers are unhappy and might jump ship to a competitor.”

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A case study: recommender system design

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A case study: recommender system design

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A case study: recommender system design

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A final remark:�The role of theory in AI research and development�

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Thank you for listening!