Title: Strategic Cost Selection in Participatory Budgeting
Venue: NeurIPS 2025
Accepted: yes
Keywords: Participatory Budgeting

Social Choice

Game Theory

Equilibria
Abstract: We study strategic behavior of project proposers in the context of approval-based
participatory budgeting (PB). In our model we assume that the votes are fixed and
known and the proposers want to set as high project prices as possible, provided
that their projects get selected and the prices are not below the minimum costs of
their delivery. We study the existence of pure Nash equilibria (NE) in such games,
focusing on the AV/Cost, Phragmen, and Method of Equal Shares rules. We also
provide an experimental study of cost selection on real-life PB election data.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a \"meta game\" of participatory budgeting: when all the preference is known, what is a stable pricing for each alternatives if their makers are self-interested? It use a game theoretical approach to address this question and analyze equilibria under different participatory budgeting rules and restricted or general preferences domains. Results vary from case to case. For example, the most popular alternative takes all in BasicAV, while rules that emphasizes more on fairness (such as MES) may have more averaged equilibrium. There are also many negative results where a stable outcome does not exists. The paper also provides experiements that supports the theoretical findings.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nThe paper provides a very interesting perspective to examine PB mechanisms and games, different from the previous ones which focus on the vote itself. This inspires us to look into the problem more thoroughly. The theoretical characterizations are clean and complete, characterizing a wider range of common rules. I don't check most of the proofs, but some of them seem non-trivial to me. \n\nWeaknesses:\nThe paper indeed has many assumptions and restrictions in its modelling, which it also discusses in the introduction. I am generally satisfied with the second one (project do not affect approvals). Yet for the first one, I will expect more explanations. For example, it remains unclear to me whether a 100% percent sure Ground Truth versus a partial knowledge /noisy signal makes a different in the existence of equilibria.\n\nQUESTIONS:\n1. How will exact ground truth / noisy information on voters' preferences affect the existence of an equilibrium? Do you at least have some guesses or conjectures?\n\n2. In remark 2.1, you mentioned that you complement MES by Phragmen rules to use up the budget as much as possible. Will this complementary process affect the equilibrium?\n\n3. In the second paragraph of the intro, there are sentences like \"Among 25% instances, half of them XXX; in another 50%, more than 30% YYY\". This kind of expression looks counterintuitive to me. Why not count the properties based on the whole dataset? \n\n4. I would like to hear more potential applications or insights this paper could give on future research or voting practices.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper evaluates several Participatory Budgeting (PB) algorithms based on the equilibrium induced by these rules when strategic proposers act rationally. Authors also provide extensive experimental analysis on publicly available datasets. The important results in the paper are:\n\n1) MES with cost utilities always admit a Pure Nash Equilibrium (PNE). The resultant project selection under this rule leads to a larger number of project approvals. \n\n 2) The BasicAV algorithm also admits PNE under which a single largest project is approved.  \n\n3) Other rules, such as AV/Cost, Phragmen and  MES with approval utilities, do not guarantee PNE.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: The paper is well written and easy to follow. The problem studied in the paper is rooted in real-world applications. Authors also do a commendable job in providing real-world instances of the settings studied in the paper. The mathematical analysis and experimental analysis are rigorous. \n\nWeaknesses: I do not find any significant flaws in the paper. However, intuitive explanations of some of the results in the paper will improve its readability, especially in Section 4.4. Further weaknesses include a lack of robustness analysis and a lack of practical relevance of the proposed solution in the real world.\n\nQUESTIONS:\n1. I could not fully understand the end goal here. While equilibrium analysis in the paper is mostly predictive analysis, it will help if the authors comment on the prescriptive angle of the analysis as well. As a proposer deciding how to set project prices, what information do I need for different selection rules, and how realistic is it to obtain and/or process the information? \n\n2. Can authors comment on the rules that induce the desired equilibrium point in the Nash strategy (or dominant strategy) profile? In other words, do there exist mechanisms that induce the desired equilibrium outcome?\n\n3. The paper lacks robustness studies. How robust are the equilibrium notions to miscalculation by a single or/few agents?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the following game-theoretical problem related to participatory budgeting (PB): \nThe input consists of a PB instance with approval preferences where each project has a distinct proposer and minimum cost (delivery cost) and the proposers are the players such that the utility of a player is equal to the suggested cost minus the delivery cost if the  project is winning; otherwise the utility is zero.\n\nThe paper is interested in finding out whether there exits a strategy profile (i.e., cost per project that is at least the delivery cost) that is a Nash equilibrium for each some commonly studied PB rules.\n\nThe authors show the following\n(1) For approval voting (AV) with a fixed tie-breaking order, there is an NE where the profile with the most approvals reports the whole budget as its cost and is the only winner. \n\n(2) For AV/Cost, i.e., approval voting where the approvals are divided by the project cost, they show that if the minimum delivery costs over all the projects are less than the budget, then all the projects are funded. More generally, a NE always exists for some tie-breaking rule. \n\n(3) Phragmén-rule may not admit a NE. \n\n(4)There is a polynomial-time algorithm that computes a near-unique NE for MES with cost utilities, whereas for MES with approval utilities, an NE may not exist.\n\n(5) The authors show experimentally that under AV, the game converges to NE where the winner takes the whole budget. For AV/Cost, Phragmén, and MES with approval utilities, projects report costs that are proportional to their support, while MES with cost utilities has a few projects who report higher costs.\n\nSTRENGTHS AND WEAKNESSES:\nStrength: The paper appears technically sound. The assumptions and setting are described clearly. The paper studies a quite interesting problem and presents an interesting angle for judging different PB rules. \n\nWeakness: It's not clear why the player's utility is precisely how much money the project additionally brings. For instance, it could be that the proposers care first and foremost about getting the project funded to the smallest feasible amount, and the extra money after that is far less important – so something akin to the extended utility function of Remark 3.1.\n\nSecondly, the arguments against cost-interval games in PB (Section 6) sees not very convincing as well. I think your example of “voter approves building a bike path for at most $25000 when others do so only if it costs above $75000” is not any more unrealistic than assuming that the proposer is indifferent between their project being approved for the minimum feasible amount and it not being approved at all. Honestly, the former seems plausible to me because people can consider the same project to have a different worth. Even if we agree this is unrealistic, one could study how the restriction that does not allow this behaves. This is naturally outside the scope of the project, but it could be discussed as a possible avenue of future work.\n\nDetailed remarks:\nLine 12: “One natural question that the proposers face is … ?” This sentence is not a question, so it should not have a question mark.\n\nLines 121 and 131: Tie breaking is named in one case but not in the other, also the first time it is not entirely clear whether $\\succ$ is given or obtained from somewhere.\n\nLine 137: \"have cost(p) funds..\" What does it mean?\n\nLine 154: Are plurality preferences not a special case of party list preferences? Would be good to note this.\n\nSection 3: Please formally define NE. Also, to connect better, one should say that the strategy profile c will set the cost function for the PB rule. \n\nLines 172-173: The utility function needs justification. \n\nLine 174: minimal -> minimum? \n\nLine 176: “In particular, the proposers do not receive funds equal to the utility values.” I do not understand this sentence. Are the utilities not assumed to be the amount of money the projects are allocated?\n\nDefinition 4.1. The profile ap is unique, right? It could be emphasized in the definition.\nTable 1: The double circle in the legend does not seem to be used.\n\nLine 212: Why are you interested Definition 4.1. It would be great to discuss a bit more instead of referring to some other papers.\n\nExample 4.3: The example has tie-breaking p2 > p1. But later you said p1 > p2. Please justify. Also, please state directly how a AV/Cost-NE with your original tie-breaking looks like.\n\nFigures 1 and 2: The text is the figures are too tiny to read. Please don't abuse the font sizes in such extend. \n\nLine 273: “… there are instances with no equilibria, even for very small instances” clumsy.\n\nQUESTIONS:\nLine 28: \"This happens, e.g., ...\": Without knowing more details, this sentence reads unconvincing. For instance, if the price of a project is too unrealistic, it would lose votes. Could you elaborate more why the rules you mentioned encourage submitting more expensive proposals?\n\nIf you have any further justification for your model and/or against the models you decided not to study in detail, I would like to hear it.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis submission concerns strategic aspects of project proposal in the standard model of approval-voting-based participatory budgeting. There are a number of standard preference aggregation rules -- drawn from both theory and practice -- which take in project (voter) approval sets and costs, along with an overall budget, and output a deterministic budget-feasible subset of projects (possibly making use of a prescribed tie-breaking rule or order). \n\nAtop this setting, the submission proposes a game-theoretic formulation of the process from the project proposer's perspective. Each project has a minimum cost (analogous to a reservation price), and proposers can specify a project cost above this minimum. If the project is funded, the proposer's utility is the difference between their specified cost and the minimum; if unfunded, then $0$. The task is then to establish for which combinations of rules, restricted approval profile settings, conditions on the minimum cost, and assumptions about the tie-breaking rule used this game has pure-strategy Nash equilibria (PSNEs), and what they are. \n\nThe submission finally analyzes a few actual participatory budgeting instances, comparing the reported project costs to individually computed best response project costs according to the specified game. It also heuristically attempts to identify PSNEs in these instances, and compares the PSNE found with the reported costs in the profile.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\t- Overall the submission is well-organized and well-written.\n\t- I found the case made in the introduction that there is a large amount of strategic cost selection in practice to be quite compelling. \n\t- This overall inquiry seems careful, thorough, and more or less complete.\n\nWeaknesses:\n\t- At a high level, some of the positive results are unsurprising. For rules that iteratively build the winning slate and are insensitive to project cost, it makes perfect sense that each project should maximize its reported cost, conditioned on not 'falling off' of the end of the budget.\n\t- There is perhaps some tension between the case made in the introduction,  the experimental findings, and the credibility of the model. If strategic behavior is evident, why is it empirically so far from best response or equilibrium? If it is because of incomplete information in practice, are Nash equilibria the right way to model this phenomenon?\n\t- I found the notational choices somewhat confusing. Should $c(p_i)$ on line 173.5 in fact be $c_i$? If $c$ and $d$ are both project-dimensional cost vectors, why not make their `type' and notation symmetric? \n\t- I also found some of the technical language -- particularly the order of quantifiers -- around results relating to tie-breaking unclear. For instance, does (b) in the statement of Corollary 4.6 mean 'for all A/D TBOs' ... or 'there exists an A/D TBO...'?\n\t- More discussion of why rules with PSNEs are normatively preferable would be helpful. This is especially true since the assumption that all votes are common knowledge is quite strong. When are these PSNEs such that they can be played without perfectly knowing all voters' approved projects beforehand?\n\nQUESTIONS:\n1. Why did you choose this particular form of proposer utility (line 173.5)? Since all rules are deterministic, this is a full-information setting, and you're after pure strategies, it seems like you might need only (i) that proposers are indifferent between being funded below the minimum and receiving no funding, and (ii) for c > d utility is monotonically increasing. Does this check out? If so, I think it strengthens the case for this model.\n\n2. In Section 6 and the discussion following Theorem 6.1, the more general model you contrast against seems to allow voters to have both different minimum and maximum thresholds for project costs, outside of which they will not vote for the project. An intermediate choice (and perhaps a more reasonable one for value-conscious voters) might be that they share a common minimum equal to the project's minimum cost, and have different maximum thresholds. Does Theorem 6.1 remain true in this intermediate model, or do some rules still admit PSNEs? \n\n3. For what proportion of runs did your heuristic PSNE-finding algorithm fail to converge? For the rules where you do not prove there is a unique  PSNE, did you find evidence of multiple distinct PSNEs? Multiple substantially different PSNEs?\n\n\nPossible corrections:\n\tan NE exist -> exists (233)\n\talways is -> is always (282)\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-08"}