arXiv subjects

Soumyarup Sadhukhan

Publications and source records attributed to Soumyarup Sadhukhan.

12 recordsLinked to original sources

arXiv2026

Envy-Free Chore Allocation with Unit Subsidies under Negative Dichotomous Valuations

We study the allocation of indivisible chores with subsidies under negative dichotomous valuations, where the marginal disutility of every chore is either zero or one. This is the chore analogue of the dichotomous goods model of Barman et al. (2022), for which a subsidy of at most one unit per agent is known to suffice for envy-freeness. Our main contribution is to show that the same optimal guarantee holds for chores under no structural assumption beyond binary marginals. For every instance with negative dichotomous valuations, there exists a complete allocation $\mathcal{A}$ and a subsidy vector $p\in\{0,1\}^n$ with $\sum_{i\in N} p_i\le n-1$ such that $(\mathcal{A},p)$ is envy-free. Moreover, $\mathcal{A}$ is EF1 even before subsidies are paid. The subsidy bound is tight, and both the allocation and the subsidies can be computed in polynomial time in the value-oracle model. Unlike in the corresponding goods setting, a natural incremental extension approach does not work for chores. Our proof instead exploits key properties of an envy-free partial allocation and its associated equality graph to complete the allocation and determine which agents require subsidies. As a further extension of the main theorem, we establish the same optimal unit-subsidy guarantee for objective mixed instances with signed binary marginals, where every item is a good for all agents or a chore for all agents; these valuations need not be additive. The key step is to prove that the goods-side extension argument of Barman et al.~\cite{BKNS22} remains valid even when the previously allocated bundles contain chores. Finally, even for negative dichotomous chore valuations, we show that the unit-subsidy guarantee cannot, in general, be maintained if Pareto optimality is also required.

cs.GT↗

arXiv2026

Strategy-Proof Probabilistic Social Choice Correspondences under Conditional Expected Utility

We study unanimous and strategy-proof probabilistic social choice correspondences (PSCCs), where the selected set of alternatives is interpreted as an interim outcome, and agents evaluate sets using conditional expected utility. We analyze two preference domains introduced by Barbera et al. (2001): the conditionally expected utility consistent (CEUC) domain and the conditionally expected utility consistent with equal probabilities (CEUCEP) domain. Our results characterize all unanimous and strategy-proof PSCCs on these domains and identify cases when randomization enlarges the class of admissible rules. On the CEUC domain, every unanimous and strategy-proof PSCC is a random dictatorship, showing that randomization over sets yields no additional flexibility. In contrast, the CEUCEP domain admits a richer family of unanimous and strategy-proof PSCCs. For at most three agents, these rules are precisely the random bi-dictatorial rules, which are convex combinations of bi-dictatorial rules introduced in Feldman (1980). For four or more agents, the characterization depends on the number of alternatives. When there are exactly three alternatives, the class expands to the larger family of coalition-weighted rules. Thus, randomization enlarges the class of strategy-proof correspondences in the three-alternative case, producing rules that are not convex combinations of deterministic strategy-proof correspondences. However, for four or more alternatives, the class of unanimous and strategy-proof probabilistic correspondences again collapse to random bi-dictatorships.

econ.TH↗

arXiv2025

A characterization of strategy-proof probabilistic assignment rules

We study the classical probabilistic assignment problem, where finitely many indivisible objects are to be probabilistically or proportionally assigned among an equal number of agents. Each agent has an initial deterministic endowment and a strict preference over the objects. While the deterministic version of this problem is well understood, most notably through the characterization of the Top Trading Cycles (TTC) rule by Ma (1994), much less is known in the probabilistic setting. Motivated by practical considerations, we introduce a weakened incentive requirement, namely SD-top-strategy-proofness, which precludes only those manipulations that increase the probability of an agent's top-ranked object. Our first main result shows that, on any free pair at the top (FPT) domain (Sen, 2011), the TTC rule is the unique probabilistic assignment rule satisfying SD-Pareto efficiency, SD-individual rationality, and SD-top-strategy-proofness. We further show that this characterization remains valid when Pareto efficiency is replaced by the weaker notion of SD-pair efficiency, provided the domain satisfies the slightly stronger free triple at the top (FTT) condition (Sen, 2011). Finally, we extend these results to the ex post notions of efficiency and individual rationality. Together, our findings generalize the classical deterministic results of Ma (1994) and Ekici (2024) along three dimensions: extending them from deterministic to probabilistic settings, from full strategy-proofness to top-strategy-proofness, and from the unrestricted domain to the more general FPT and FTT domains.

econ.TH↗

arXiv2025

The Game of Graph Nim on graphs with four edges

This work is concerned with the study of the Game of Graph Nim -- a class of two-player combinatorial games -- on graphs with $4$ edges. To each edge of such a graph is assigned a positive-integer-valued edge-weight, and during each round of the game, the player whose turn it is to make a move selects a vertex, and removes a non-negative integer edge-weight from each of the edges incident on that vertex, such that the remaining edge-weight on each of these edges is a non-negative integer, and the total edge-weight removed during a round is strictly positive. The game continues for as long as the sum of the edge-weights remaining on all edges of the graph is strictly positive, and the player who plays the last round wins. An initial configuration of edge-weights is considered winning if the player who plays the first round wins the game, whereas it is defined as losing if the player who plays the second round wins. In this paper, we characterize, almost entirely, all winning and losing configurations for this game on all graphs with precisely $4$ edges each. Only one such graph defies our attempt to fully characterize the winning and losing configurations of edge-weights on its edges -- we are still able to provide a significant set of partial results pertaining to this graph.

math.CO↗

arXiv2025

On Probabilistic Assignment Rules

We study the classical assignment problem with initial endowments in a probabilistic framework. In this setting, each agent initially owns an object and has strict preferences over the entire set of objects, and the goal is to reassign objects in a way that satisfies desirable properties such as strategy-proofness, Pareto efficiency, and individual rationality. While the celebrated result by Ma (1994) shows that the Top Trading Cycles (TTC) rule is the unique deterministic rule satisfying these properties, similar positive results are scarce in the probabilistic domain. We extend Ma's result in the probabilistic setting, and as desirable properties, consider SD-efficiency, SD-individual rationality, and a weaker notion of SD-strategy-proofness -- SD-top-strategy-proofness -- which only requires agents to have no incentive to misreport if doing so increases the probability of receiving their top-ranked object. We show that under deterministic endowments, a probabilistic rule is SD-efficient, SD-individually rational, and SD-top-strategy-proof if and only if it coincides with the TTC rule. Our result highlights a positive possibility in the face of earlier impossibility results for fractional endowments (Athanassoglou and Sethuraman (2011)) and provides a first step toward reconciling desirable properties in probabilistic assignments with endowments.

econ.TH↗

arXiv2025

Percolation games on rooted, edge-weighted random trees

Consider a rooted Galton-Watson tree $T$, to each of whose edges we assign, independently, a weight that equals $+1$ with probability $p_{1}$, $0$ with probability $p_{0}$ and $-1$ with probability $p_{-1}=1-p_{1}-p_{0}$. We play a game on a realization of this tree, involving two players and a token that is allowed to be moved from where it is currently located, say a vertex $u$ of $T$, to any child $v$ of $u$. The players begin with initial capitals that amount to $i$ and $j$ units respectively, and a player wins if either she is the first to amass a capital worth $κ$ units, where $κ\in \mathbb{N}$ is prespecified, or she is able to move the token to a leaf vertex, from where her opponent cannot move it any farther, or her opponent's capital is the first to dwindle to $0$. This paper is concerned with analyzing the probabilities of the three possible outcomes such a game may culminate in, as well as with finding conditions under which the expected duration of the game is finite. Of particular interest to us is the exploration of criteria that guarantee the probability of draw in such a game to be $0$. The theory we develop is further supported by observations obtained via computer simulations, providing a deeper insight into how the above-mentioned probabilities behave as the underlying parameters and / or offspring distributions are allowed to vary. We include in this paper conjectures pertaining to the behaviour of the probability of draw in our game (including a phase transition phenomenon, in which the probability of draw goes from being $0$ to being strictly positive) as the parameter-pair $(p_{0},p_{1})$ is varied suitably while keeping the underlying offspring distribution of $T$ fixed.

math.PR↗

arXiv2023

Phase transition in percolation games on rooted Galton-Watson trees

We study the bond percolation game and the site percolation game on the rooted Galton-Watson tree $T_χ$ with offspring distribution $χ$. We obtain the probabilities of win, loss and draw for each player in terms of the fixed points of functions that involve the probability generating function $G$ of $χ$, and the parameters $p$ and $q$. Here, $p$ is the probability with which each edge (respectively vertex) of $T_χ$ is labeled a trap in the bond (respectively site) percolation game, and $q$ is the probability with which each edge (respectively vertex) of $T_χ$ is labeled a target in the bond (respectively site) percolation game. We obtain a necessary and sufficient condition for the probability of draw to be $0$ in each game, and we examine how this condition simplifies to yield very precise phase transition results when $χ$ is Binomial$(d,π)$, Poisson$(λ)$, or Negative Binomial$(r,π)$, or when $χ$ is supported on $\{0,d\}$ for some $d \in \mathbb{N}$, $d \geqslant 2$. It is fascinating to note that, while all other specific classes of offspring distributions we consider in this paper exhibit phase transition phenomena as the parameter-pair $(p,q)$ varies, the probability that the bond percolation game results in a draw remains $0$ for all values of $(p,q)$ when $χ$ is Geometric$(π)$, for all $0 < π\leqslant 1$. By establishing a connection between these games and certain finite state probabilistic tree automata on rooted $d$-regular trees, we obtain a precise description of the regime (in terms of $p$, $q$ and $d$) in which these automata exhibit ergodicity or weak spatial mixing.

math.PR↗

arXiv2023

The spread of an epidemic: a game-theoretic approach

We introduce and study a model stemming from game theory for the spread of an epidemic throughout a given population. Each agent is allowed to choose an action whose value dictates to what extent they limit their social interactions, if at all. Each of them is endowed with a certain amount of immunity such that if the viral risk/exposure is more than that they get infected. We consider a discrete-time stochastic process where, at the beginning of each epoch, a randomly chosen agent is allowed to update their action, which they do with the aim of maximizing a utility function that is a function of the state in which the process is currently in. The state itself is determined by the subset of infected agents at the beginning of that epoch, and the most recent action profile of all the agents. Our main results are concerned with the limiting distributions of both the cardinality of the subset of infected agents and the action profile as time approaches infinity, considered under various settings (such as the initial action profile we begin with, the value of each agent's immunity etc.). We also provide some simulations to show that the final asymptotic distributions for the cardinality of infected set are almost always achieved within the first few epochs.

math.PR↗

arXiv2022

Characterization of Group-Fair Social Choice Rules under Single-Peaked Preferences

We study fairness in social choice settings under single-peaked preferences. Construction and characterization of social choice rules in the single-peaked domain has been extensively studied in prior works. In fact, in the single-peaked domain, it is known that unanimous and strategy-proof deterministic rules have to be min-max rules and those that also satisfy anonymity have to be median rules. Further, random social choice rules satisfying these properties have been shown to be convex combinations of respective deterministic rules. We non-trivially add to this body of results by including fairness considerations in social choice. Our study directly addresses fairness for groups of agents. To study group-fairness, we consider an existing partition of the agents into logical groups, based on natural attributes such as gender, race, and location. To capture fairness within each group, we introduce the notion of group-wise anonymity. To capture fairness across the groups, we propose a weak notion as well as a strong notion of fairness. The proposed fairness notions turn out to be natural generalizations of existing individual-fairness notions and moreover provide non-trivial outcomes for strict ordinal preferences, unlike the existing group-fairness notions. We provide two separate characterizations of random social choice rules that satisfy group-fairness: (i) direct characterization (ii) extreme point characterization (as convex combinations of fair deterministic social choice rules). We also explore the special case where there are no groups and provide sharper characterizations of rules that achieve individual-fairness.

cs.GT↗

arXiv2022

Nash Welfare Guarantees for Fair and Efficient Coverage

We study coverage problems in which, for a set of agents and a given threshold $T$, the goal is to select $T$ subsets (of the agents) that, while satisfying combinatorial constraints, achieve fair and efficient coverage among the agents. In this setting, the valuation of each agent is equated to the number of selected subsets that contain it, plus one. The current work utilizes the Nash social welfare function to quantify the extent of fairness and collective efficiency. We develop a polynomial-time $\left(18 + o(1) \right)$-approximation algorithm for maximizing Nash social welfare in coverage instances. Our algorithm applies to all instances wherein, for the underlying combinatorial constraints, there exists an FPTAS for weight maximization. We complement the algorithmic result by proving that Nash social welfare maximization is APX-hard in coverage instances.

cs.GT↗

arXiv2022

Achieving Envy-Freeness with Limited Subsidies under Dichotomous Valuations

We study the problem of allocating indivisible goods among agents in a fair manner. While envy-free allocations of indivisible goods are not guaranteed to exist, envy-freeness can be achieved by additionally providing some subsidy to the agents. These subsidies can be alternatively viewed as a divisible good (money) that is fractionally assigned among the agents to realize an envy-free outcome. In this setup, we bound the subsidy required to attain envy-freeness among agents with dichotomous valuations, i.e., among agents whose marginal value for any good is either zero or one. We prove that, under dichotomous valuations, there exists an allocation that achieves envy-freeness with a per-agent subsidy of either $0$ or $1$. Furthermore, such an envy-free solution can be computed efficiently in the standard value-oracle model. Notably, our results hold for general dichotomous valuations and, in particular, do not require the (dichotomous) valuations to be additive, submodular, or even subadditive. Also, our subsidy bounds are tight and provide a linear (in the number of agents) factor improvement over the bounds known for general monotone valuations.

cs.GT↗

arXiv2022

Probabilistic Fixed Ballot Rules and Hybrid Domains

We study a class of preference domains that satisfies the familiar properties of minimal richness, diversity and no-restoration. We show that a specific preference restriction, hybridness, has been embedded in these domains so that the preferences are single-peaked at the "extremes" and unrestricted in the "middle". We also study the structure of strategy-proof and unanimous Random Social Choice Functions on these domains. We show them to be special cases of probabilistic fixed ballot rules (introduced by Ehlers, Peters, and Storcken (2002)).

econ.TH↗