Advance publication December 01, 2025
This article presents the hypothesis that using aggregation functions in mass-deliberation systems — summarizing participants' viewpoints and showing aggregated, representative viewpoints to them — may under certain conditions induce a discursive shift. Specifically, game theoretic incentives may encourage participants to strategically submit extreme proposals.
Using examples, including median aggregation and "k-means"-like clustering, we demonstrate that some aggregation functions can reward strategically extreme proposals. We suspect that a violation of the triangle inequality when measuring distances of viewpoints is relevant for this effect.
This mechanism could distort participants' perception of the viewpoint landscape and might lead to unstable system dynamics (divergence), where ever more extreme positions are proposed and democratic decision-making breaks down. We discuss implications for algorithmic and human moderation, as well as for LLM-based moderation systems.
Opinion polarization in online deliberation systems has been linked to phenomena such as filter bubbles or hate speech, and has also been described using models of argument exchange between participants as well as psychological models, such as bounded confidence models. [Keijzer2022, Mäs2013, KurahashiNakamura2016, Hahn2024]
In this article, we would like to take a different approach by looking at the possibility of polarization and even extremism being caused by game-theoretic effects due to the presentation of aggregated participant opinions presented back to the participants during the deliberation process — in online systems as well as traditional media.
Aggregation functions, also called aggregate functions, are mechanisms that process an input of variable size to a fixed size output. These functions play a key role in online deliberation systems, particularly mass deliberation systems, where it's impossible that all participants read everyone else's content.
For example, LiquidFeedback presents alternative proposals ("initiatives") using a proportional ranking created by analyzing the sets of supporting participants. [PLF] Taking the first N prominently displayed proposals, this can be understood as an aggregation function that takes all proposals (along with supporting participants) on a given issue as input and creates an overview by showing a limited number N of proposals, that ought to represent the current state of discussion.
But also other approaches such as presenting the N most recently published proposals in a (possibly personalized) timeline, can be seen as an aggregation function.
It should be noted that such contributions can be comprised of more than just a proposal, but may also contain collection of arguments to support it. We will just refer to them as "viewpoints" in the following to emphasize this fact. Even if these "viewpoint" aggregation functions are not used in a (final) voting procedure, they are technically equivalent to (multi-winner) voting tallying mechanisms, with the participants' viewpoints being the candidates, and where the winners are the output of the aggregation function.
Aggregation functions play a key role in deliberation processes as they can be used to give other participants an indication of which other viewpoints are presently being discussed or supported. That way, they can also influence other participants' opinions.
Using the terminology of social choice theory for voting rules, an aggregation function fulfills neutrality if a change of the opinions (votes in social choice) by permuting the candidates (in our case possible viewpoints) will cause the same permutation in the output. [May1952]
For example, if all supporters of a viewpoint A will now instead support viewpoint B, and all supporters which previously supported viewpoint B will now support viewpoint A, then, if only viewpoint A was part of the aggregation function's output prior to the swap, viewpoint B should be part of the representative output after the swap instead.
In other words: Only the supporting votes for a specific viewpoint must affect the outcome for that viewpoint in the aggregate.
This also means: If we consider the actual contents of proposals/viewpoints for the aggregation, and if that consideration has an impact on the aggregation function's output, then we violate neutrality according to its very definition.
When it comes to moderation of deliberation, some argue that with the demand of a "neutral" or "factual" moderation, only the content — and not at who is supporting it — should be considered. While there is often no specific definition presented, this notion of neutrality is, as argued above, quite the opposite of neutrality as defined in the context of social choice theory.
Neutrality as defined by social choice theory demands not to look at the contents but only at who is bringing them forward, e.g. through endorsements. The additional criterion of anonymity (not to be confused with secret voting) can still require that all participants are treated equal (i.e. that a permutation of the participants should not change the result). [May1952]
Keeping in mind that considering proposals' contents violates neutrality, there may still be use-cases where violating neutrality might be advantageous. One reason is that a system that fulfills neutrality can only judge about viewpoints based on the participants' input. In turn, systems which violate neutrality may require less input from the participants, which might possibly render a deliberation process easier or even more fair.
However, care must be taken that the violation of neutrality doesn't introduce an unfair treatment in practice. Ideally, the mechanisms used to inspect the content of a viewpoint and to judge upon them would be known by all participants and be supported by at least a majority of them and be subject to constant re-evaluation and continuous ratification.
Where viewpoints are presented in a natural language (like "English" or "French"), content inspection as part of the deliberation framework would either require human moderators or, in case of automated moderation, large language models (LLMs) or comparable algorithms. At least LLMs are difficult to judge about by the participants, possibly making well-informed ratification impossible.
However, we can also consider systems in which viewpoints are constrained such that their content is domain-specific. This content could be a distribution of resources (like a budget plan), for example, or a single (scalar) number. While the latter is a drastic simplification, it may serve at least as a hypothetical use-case in order to observe some interesting game-theoretic effects that can be caused by specific aggregation functions.
We assume that all viewpoints consist of a single number (e.g. a temperature to be achieved in a room or a duration in minutes how long a certain activity should take place). For now, we don't necessarily require a metric (i.e. a distance function) on the viewpoint space (e.g. all possible temperatures). However, we require a notion what is smaller or greater (or left and right on a projected line), i.e. a linear order of all possible viewpoints. Given that linear order, we further assume that if a viewpoint A is somewhere between another viewpoint B and a participant's most preferred viewpoint C, then that participant would prefer A over B.
If every participant can state their most preferred viewpoint (e.g. their most-preferred temperature), which we will call "peak-preference" in the following, then, under the assumptions above, we can define an aggregation function.
This aggregation function takes the peak-preferences of all participants as input and can return a set of representative viewpoints as an output. We will refer to representative viewpoints shortly as "representatives".
For a single representative, the median of each participant's peak-preference is taken. (Note that a tie-breaker may be needed in case of an even number of participants, which will be disregarded here.)
For N representatives with N > 1, the quantiles can be used, e.g. the 25%, 50%, and 75% quantile, such that one representative is a viewpoint where 25% of the other viewpoints are smaller, one representative is the median, and one representative is a viewpoint where 75% of the other viewpoints are smaller.
Such an aggregation function has certain advantages. First of all, the algorithm works when each participant only states their peak-preference and each participant states a different peak-preference. In case of neutrality, it would otherwise not be possible to pick any representatives other than randomly choosing one, at least under the assumption that all participants should be treated equally as well (property of "anonymity" as mentioned above). So violating neutrality here clearly brings an advantage.
Another advantage — under the previously stated assumptions — is that there is no incentive for participants to lie about their peak-preference (criterion of "strategyproofness"). In particular, a participant won't get an advantage by stating an extreme viewpoint as their most preferred option because the median and quantiles are invariant to actual distances between viewpoints and only consider their linear order.
It should be noted, however, that the 1-dimensional median (let alone quantiles) cannot be easily generalized for problems of higher dimensions, e.g. viewpoints consisting of a location on a map, or higher dimensional spaces where several parameters are being discussed at the same time. While there are various generalizations of the median, such as the Tukey median or the geometric median, particularly the property of strategyproofness can get lost in higher dimensions, depending on the assumed preference model of the participants. [Zhou1989]
We will now consider a different aggregation function which takes a look at the magnitude of how much viewpoints differ from each other. For that purpose, we need a metric [Andersson2022], which assigns any two distinct viewpoints a positive number which reflects how much these two viewpoints differ. For example, when viewpoints consist of a desired temperature, that metric could be the absolute (i.e. positive) difference in °C or Kelvin.
This allows us to build a different class of aggregation functions such as the following.
For a single representative, we find a value that minimizes the sum of all squared distances (where the distance is given by the metric) to each participant's peak-preference.
For determining N representatives with N > 1, we can generalize the approach as follows: We choose N representatives such that the sum of all squared distances (given by the metric) from each participant's most preferred viewpoint to the nearest representative is minimized.
Note that the representatives don't need to directly correspond to a particular participants' peak-preference but can also be a value that lies in between.
The "k-means clustering" method, a widely used algorithm, is a heuristic approach to solve this problem by approximation in reasonable time. [MacQueen1967, Reddy2014] In the following we will not consider computational feasibility though, but instead assume an optimal algorithm.
Let's look at the following example data, where the viewpoints are a temperature (e.g. desired room temperature) and we assume to have 27 participants with the following peak preferences:
Participants #1 through #9: 11°C, 12°C, ..., 19°C.
Participants #10 through #18: 21°C, 22°C, ..., 29°C.
Participants #19 through #27: 31°C, 32°C, ..., 39°C.
As a metric, we choose the absolute difference between two temperatures divided by 1°C, e.g. the distance from 18°C to 13°C (or vice versa) would be 5.
In case of a single representative being determined, the described mechanism then corresponds to finding the arithmetic mean (i.e. average). If we calculate 3 representative temperatures instead, then the mechanism leads to the following result:
15°C, 25°C, 35°C.
These values represent the various peak-preferences in a seemingly reasonable way.
However, as part of the deliberation, participants may state dishonest peak-preferences. Let's assume all participants except for the participant who reported 32°C are being honest, but the participant who reported 32°C is now willing to report a different peak-preference on the room temperature with the sole intent to create an aggregation of opinions that is more close to his true peak-preference of 32°C.
If that participant reports 55°C instead of 32°C, i.e. an extremely high temperature, the aggregate function's output would now be (approximately):
17.86°C, 32.75°C, 55°C.
Here, the solution to the optimization problem of minimizing the sum of squared distances to the nearest representative is to pick 17.86°C as an average of all reported peak-preferences between 11°C and 25°C, to pick 32.75°C as an average of all reported peak-preferences between 26°C and 39°C, and to pick the extreme value of 55°C as a third representative. (Any other choices would result in a larger sum of squares.)
Observing those three representative values — and keeping in mind that less than 4% of the participants changed their reported opinion — the calculated representatives now give the false impression that a temperature of 32.75°C is located in the middle ground of the preferred viewpoints.
Assuming that the reported aggregate has an influence on the other participants during deliberation, this could lead to a strategic advantage for the misreporting participant, e.g. if participants tend to orient their own preference on the perceived middle ground of the other participants.
Interestingly, the aggregate will be different if the participant who prefers 32°C reported a smaller value than 55°C, such as 54°C, in which case we get the following representatives:
15°C, 25.6°C, 38.25°C.
Here, the solution to the optimization problem is to pick 15°C as an average of all reported peak-preferences between 11°C and 19°C, 25.6°C as an average of all reported peak-preferences between 21°C and 31°C, and 38.25°C as an average of all reported peak-preferences between 33°C and 54°C.
Arguably, this result may be less favorable for the participant who stated the dishonest peak-preference.
We conclude that at least in this numerical example, the aggregation function can provide an incentive to report extreme preferences (here 55°C) in order to create a shifted impression of what the current state of the deliberation is.
We can modify the previous approach by not minimizing the sum of squared distances, but by minimizing the sum of the absolute (non-squared) distances measured by a metric to the nearest representative.
This roughly corresponds to the "k-medians clustering" method, except that we may use other metrics as a distance function than the one that is usually used for k-medians (which is the "L1 norm", also called "Manhattan distance"). [Reddy2014] The objective function for this optimization problem has already been stated by J. MacQueen in 1967 when generalizing the k-means clustering method to general metric spaces. [MacQueen1967, Cardot2011]
If we apply the previous numerical example data of 27 participants and also use the same metric as in the previous example (absolute difference between two temperatures divided by 1°C), then we can see that changing the reported peak-preference from 32°C to 54°C or 55°C has a slight impact on the output of the aggregation function only:
15°C, 25°C, 36°C.
The dishonest participant doesn't get a clear advantage with regard to representation: The average of the three representatives changed only slightly and the closest representative (previously 35°C, now 36°C) moved even farther away from the participant's true peak-preference of 32°C.
While there are some impossibilities regarding strategyproofness in higher-dimensional problems [Zhou1989], we conjecture that this method provides less incentives to report extreme peak-preferences.
Note that if we measure viewpoint distances using a metric, then the following properties must be fulfilled (as they are required for a distance function to be a metric [Andersson2022]):
Specifically the triangle inequality is relevant, because otherwise one could just create a new metric by squaring it and effectively ending up with example aggregation function II, which has been shown to potentially foster extreme values. The squared Euclidean distance is not a metric because it fails the triangle inequality, but the non-squared Euclidean distance is.
Nonetheless, different metrics (fulfilling the triangle inequality) could still be used and some of them may be more suitable than others. For example, one could
However, note that the discrete metric results in a trivial case where the representatives are simply chosen by the number of participants reporting them as a peak-preference and would, as such, result in a tie if all participants report distinct peak-preferences. Moreover, such a trivial mechanism would be vulnerable to burying minority viewpoints as discussed in [PLF].
Looking at example aggregation function II and the case where one participant reports 55°C as a false, extreme peak-preference, we can also observe a misrepresentation of the participants.
Taking 17.86°C, 32.75°C, and 55°C as the aggregate, we notice that 17.86°C is closest to the reported peak-preferences of 14 participants, 32.75°C is closest to the reported peak-preferences of 12 participants, and 55°C is closest to the reported peak-preference of only 1 participant. Therefore, the participant stating an extreme opinion gets effectively over-represented.
It should also be noted that even better suited aggregation functions, such as example aggregation function III, may still lead to a misrepresentation of participants. However, these might be less prone to participants being inclined to state extreme viewpoints.
The misrepresentation is particularly interesting as the aggregation function fulfills the anonymity criterion, i.e. it is invariant to a permutation of the participants and no participant is per-se treated different than any other participant. However, by inspecting and judging about the participants' specific peak-preferences (which is possible due to violation of neutrality), this results in an effective misrepresentation of the participants where the participant who reports an extreme opinion gets strongly over-represented in the aggregate.
This effect could be compensated by including in the aggregate the respective number of participants (a headcount) backing up each representative viewpoint. Considering the numbers used in example II, the result could be presented as:
17.86°C (14 participants), 32.75°C (12 participants), 55°C (1 participant).
The extent to which a simple presentation of these headcounts as a number or bar graph (or using other methods) has a psychological impact on the participants in the deliberation process, however, remains an open question.
Having shown potential game-theoretic effects caused by formalized aggregation functions, we may raise the question whether such effects could also exist when we assume a human moderator that looks at all brought up viewpoints, their supporters, and uses a content-based way of aggregating the participants' opinions during deliberation.
Also here, a moderator might be inclined to give a minority that expresses an extreme opinion an over-representation — not because of favoring extremes per-se, but because of trying to level out the distances of viewpoints in a fair manner (using a perceived metric) such that big distances to the nearest representative are avoided (similar to minimizing the sum of squared distances to the nearest representative, but using intuition rather than calculation). In other words: Moderators might be inclined to give several similar viewpoints which are supported by a huge number of participants less representation (due to their similarity in content) than an extreme viewpoint with less support.
Participants could learn that stating extreme viewpoints can give an advantage in being noticed in the deliberation process. Thus, the benevolent intent of creating a "neural" or "fair" overview on the opinion landscape could have the following adverse effects:
Artificial intelligence using large language models has been suggested as a means of improving and upscaling online deliberation processes systems by creating summaries of a huge number of participant content.
When these AI systems inspect the content of viewpoints in order to produce an output, then the same considerations regarding aggregation by human moderators apply.
Depending on the specific aggregation mechanism, the existence and use of such AIs, even if performed by individuals using a self-chosen AI, game-theoretic effects could create incentives to propose and support extreme ideas, as this might result in a favorable representation of one's true peak-preference in the AI's aggregate output.
Because the use of such AIs may happen at each participant's discretion, it would be very difficult to regulate this effect.
We have shown that aggregation functions may provide incentives to state extreme opinions, at least in some cases. A particular example has been presented (example II), where — from a game-theoretic view — a participant can gain a direct advantage by reporting extreme opinions if their goal is to shift the perceived space of the discussed viewpoints towards their favored opinion. Aggregations created by large language models (LLMs) might create similar effects, depending on how their content aggregation works.
Further multi-disciplinary research is needed to formally characterize the strategic vulnerabilities of aggregation functions creating multiple representative viewpoints with the aim to provide an overview on the collective viewpoint landscape. Quantifiable estimations of the described effects would be of further interest, as well as considerations of potentially existing counter-effects and evaluation of the psychological impact of opinion aggregates presented to participants in a deliberation process.
If participants orient their own opinion based on the presented aggregate in some way, this could lead to system instabilities (divergence) where more and more extreme ideas are being proposed, and where democratic processes break down. A formal and more comprehensive model would, however, be required to prove or disprove this conjecture and to determine the specific parameters which lead to divergence.
As a mitigation strategy for potential system instabilities, the inspection of viewpoints could be avoided altogether by strictly treating all viewpoints interchangeably (such as done by LiquidFeedback), or, if content-based aggregate functions are being used, using suitable functions that do not provoke extreme viewpoints or cause divergence of the dynamic system. With example III, we propose considering a specific class of aggregation functions, which might be less vulnerable to such instabilities. Notwithstanding, we would like to emphasize once more that any use of such functions is a violation of neutrality.
It would further be interesting to investigate when and under which conditions content-based aggregation by a human moderator could lead to the same game-theoretic effects as algorithmic opinion aggregation methods.
The authors acknowledge support by the European Union under the Horizon Europe Perycles project (Participatory Democracy that Scales).
Funded by the European Union. Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or European Research Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
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