In online platforms, recommender systems are responsible for directing users to relevant contents. In order to enhance the users engagement, recommender systems adapt their output to the reactions of the users, who are in turn affected by the recommended contents. In this work, we study a tractable analytical model of a user that interacts with an online news aggregator, with the purpose of making explicit the feedback loop between the evolution of the users opinion and the personalised recommendation of contents. More specifically, we assume that the user is endowed with a scalar opinion about a certain issue and seeks news about it on a news aggregator: this opinion is influenced by all received news, which are characterized by a binary position on the issue at hand. The user is affected by a confirmation bias, that is, a preference for news that confirm her current opinion. The news aggregator recommends items with the goal of maximizing the number of users clicks (as a measure of her engagement): in order to fulfil its goal, the recommender has to compromise between exploring the users preferences and exploiting what it has learned so far. After defining suitable metrics for the effectiveness of the recommender systems (such as the click-through rate) and for its impact on the opinion, we perform both extensive numerical simulations and a mathematical analysis of the model. We find that personalised recommendations markedly affect the evolution of opinions and favor the emergence of more extreme ones: the intensity of these effects is inherently related to the effectiveness of the recommender. We also show that by tuning the amount of randomness in the recommendation algorithm, one can seek a balance between the effectiveness of the recommendation system and its impact on the opinions.
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