Adversarial risk analysis (ARA) provides decision-theoretic arguments to manage uncertainty in competitive decision-making environments. This paper introduces efficient algorithmic approaches to approximate ARA solutions in multi-stage games, covering both sequential and simultaneous settings, through augmented probability simulation. Two examples concerning international piracy and air combat illustrate the proposed methodology.
To make a choice in the presence of multiple criteria, we generally use an aggregation function which determines, for each alternative, the balance of its strengths and weaknesses and its overall evaluation. The aggregation function uses weights to adapt the model to the decision-maker's value system, by specifying the importance of the criteria and possibly their interactions. In this paper, we propose a noise-tolerant active learning method for these parameters, which not only effectively reduces the indeterminacy of the weights to identify an optimal or near-optimal decision among a given set of alternatives, but also simultaneously determines a predictive model of preferences capable of making relevant choices for the decision-maker on new instances. These outcomes are achieved by leveraging a general disagreement-based active learning approach that is theoretically guaranteed to be tolerant to noisy answers. The proposed method applies to various weighted aggregation functions, linear or not, classically used in decision theory.
Conditional preference networks (CP-nets) have received significant attention for modeling preferences over combinations of features. However, dominance testing, the problem of inferring from a CP-net whether one outcome is always preferred over another, is NP-hard, requiring exponential time in practice. In this paper, we introduce the use of independent relaxed subproblems as a heuristic for faster inference from CP-nets. We show how to relax the constraints in the conditional preference tables and partition the network, yielding smaller subproblems that can be solved easily. The lengths of these solutions can then be added together to provide a heuristic for informed search algorithms such as A* and applied to dominance testing in CP-nets. We prove that the resulting additive heuristic function is admissible and consistent, guaranteeing optimality and completeness. We show from experiments on randomly generated binary and multivalued CP-nets that our method performs better than the state of the art, and can further be combined with other pruning techniques, resulting in greatly improved performance for dominance testing.
Electoral control is a scenario where an election chair changes the structure of an election by actions such as adding or deleting either candidates or voters with the goal of either making a favorite candidate win or precluding a despised candidate's victory. Much work has been done on the computational complexity of controlling elections for single-winner voting rules, yet much less work on the control complexity for multiwinner voting rules which aim at electing not only a single winner but a winning committee of candidates. Meir et al. [20] initiated the investigation of electoral control for multiwinner voting rules, including single nontransferable vote (SNTV) and bloc voting. We study these two rules with respect to control by adding, deleting, or replacing candidates.
In Subset Sum Game as studied by Pieterse and Woeginger [Theory of Computing Systems, 2021], two players alternatingly fill a common knapsack each with items from a private collection. The goal of Player A is to reach a value of at least T_ , whereas Player B may follow different strategies. Subset Sum Game is NP-complete and solvable in pseudopolynomial time if Player B greedily selects the biggest available item in each turn; the game is PSPACE-complete, however, if Player B plays a hostile strategy where the only aim is to avoid that Player A wins. We continue the study of the game with these two strategies for Player B. First, we provide a faster pseudopolynomial-time algorithm for a greedy Player B and show that the problem with a hostile Player B is fixed-parameter tractable with respect to the knapsack capacity C. Moreover, we study the influence of further parameters such as T_ , the number of rounds in the game, and the number of different numbers in the input on the complexity of the problem. Second, we consider a further variant of the game, called Protective Subset Sum Game, where Player A additionally has the goal that Player B reaches a value of at least T_ . In a nutshell, we show that most algorithms for the nonprotective variant can be transferred to Protective Subset Sum Game.
The global transition from manned to automated vehicles is anticipated to occur incrementally. As such, interactions between automated driving systems (ADS) and manned vehicles motivate related decision-support research. This manuscript develops a novel modeling framework based on adversarial risk analysis focusing on lane-changing maneuvers. An empirical evaluation is provided within a simulated environment serving to validate the modeling approach and solution methodology under a specified traffic scene. Additional model extensions to alternative traffic scenes and different driver-rationality assumptions are provided. In so doing, we showcase the potential for decision theory to manage ADS behavior in heterogeneous traffic. This research also highlights the need for an ADS to prudently balance computational resources between perception and decision tasks.
The mathematical models of many real-world decision-making problems contain two levels of optimization. In these models, one of the optimization problems appears as a constraint of the other one, called follower and leader, respectively. These problems are known as bilevel optimization problems (BOPs) in mathematical programming and are widely studied by both classical and evolutionary optimization communities. The nested nature of these problems causes many difficulties such as non-convexity and disconnectedness for traditional methods, and requires a huge number of function evaluations for evolutionary algorithms. This paper proposes a fully Bayesian optimization approach, called FB-BLO. We aim to reduce the necessary function evaluations for both upper and lower level problems by iteratively approximating promising solutions with Gaussian process surrogate models at both levels. The proposed FB-BLO algorithm uses the other decision-makers' observations in its Gaussian process model to leverage the correlation between decisions and objective values. This allows us to extract knowledge from previous decisions for each level. The algorithm has been evaluated on numerous benchmark problems and compared with existing state-of-the-art algorithms. Our evaluation demonstrates the success of our proposed FB-BLO algorithm in terms of both effectiveness and efficiency.
Additive preference representation is standard in Multiple Criteria Decision Analysis, and learning such a preference model dates back from the UTA method [11]. In this seminal work, an additive piece-wise linear model is inferred from a learning set composed of pairwise comparisons. In this setting, the learning set is provided by a single Decision-Maker (DM), and an additive model is inferred to match the learning set. We extend this framework to the case where (i) multiple DMs with heterogeneous preferences provide part of the learning set, and (ii) the learning set is provided as a whole without knowing which DM expressed each pairwise comparison. Hence, the problem amounts to inferring a preference model for each DM and simultaneously "discovering" the segmentation of the learning set. In this paper, we show that this problem is computationally difficult. We propose a mathematical programming based resolution approach to solve this Preference Learning and Segmentation problem (PLS). We also propose a heuristic to deal with large datasets. We study the performance of both algorithms through experiments using synthetic and real data.
We consider the complexity of finding envy-free allocations for the class of graphical valuations. Graphical valuations were introduced by Christodoulou et al. [14] as a structured class of valuations that admit allocations that are envy-free up to any item(EFX). These are valuations where every item is valued by two agents, lending a (simple) graph structure to the utilities, where the agents are vertices and are adjacent if and only if they value a (unique) common item. Finding envy-free allocations for general valuations is known to be computationally intractable even for very special cases: in particular, even for binary valuations, and even for identical valuations with two agents. We show that, for binary graphical valuations, the existence of envy-free allocations can be determined in polynomial time. In contrast, we also show that allowing for even slightly more general utilities {0,1,d} leads to intractability even for graphical valuations. This motivates other approaches to tractability, and to that end, we exhibit the fixed-parameter tractability of the problem parameterized by the vertex cover number of the graph when the number of distinct utilities is bounded. We also show that, all graphical instances that admit EF allocations also admit one that is non-wasteful. Since EFX allocations are possibly wasteful, we also address the question of determining the price of fairness of EFX allocations. We show that the price of EFX with respect to utilitarian welfare is one for binary utilities, but can be arbitrarily large {0, 1, d} valuations. We also show the hardness of deciding the existence of an EFX allocation which is also welfare-maximizing and of finding a welfare-maximizing allocation within the set of EFX allocations.
Compiling the votes of a subelectorate is a well-known problem in computational social choice. The goal is to store the information contained in the votes cast by a subelectorate in a space-efficient way, such that when the rest of the votes become available, the winners can be accurately ascertained. This problem has been studied for single-winner voting rules. We provide a comprehensive compilation complexity landscape for several ordinal and approval-based multi-winner voting rules.
The problem of k -tiered coalition formation games ( k -TCFGs) has been considered for ranking members of a stochastic, intransitive round robin tournament, with the restriction that the ordering must have exactly k nonempty ranks for some integer k . As with other coalition formation games, an outcome of a k -TCFG may be evaluated for its stability, using the notions of Nash stability or core stability. An outcome is Nash stable if no one agent can move to a more preferable position, either by forming its own coalition or joining an existing one. An outcome is core stable if no set of agents can form a new coalition such that all agents in the set benefit. Previous research on k -TCFGs has focused on preferences derived from matchups, and has indicated that, under these matchup-oriented preferences, core stable outcomes may be significantly easier to find than Nash stable outcomes. However, the extent of this trend has not been explored. Here, we prove that for a key subset of k -TCFGs with matchup-oriented preferences, there is always at least one core stable partition. We include an illustration of the difference between Nash stabilizability and core stabilizability on an example game. We introduce a preference notation that can be used to represent any preference framework for k -TCFGs, and prove that under the subset of k -TCFGs which this notation can represent within polynomial space, the problem of determining if a game has a Nash stable list is NP-complete.
A tournament on n agents is a complete oriented graph with the agents as vertices and edges that describe the win-loss outcomes of the ( [ n; 2 ]) matches played between each pair of agents. The winner of a tournament is determined by a tournament rule that maps tournaments to probability distributions over the agents. We want these rules to be fair (choose a high-quality agent) and robust to strategic manipulation. Prior work has shown that under minimally fair rules, manipulations between two agents can be prevented when utility is nontransferable but not when utility is completely transferable. We introduce a partially transferable utility model that interpolates between these two extremes using a selfishness parameter λ . Our model is that an agent may be willing to lose on purpose, sacrificing some of her own chance of winning, but only if the colluding pair’s joint gain is more than λ times the individual’s sacrifice. We show that no fair tournament rule can prevent manipulations when λ < 1 . We computationally solve for fair and manipulation-resistant tournament rules for λ = 1 for up to 6 agents. We conjecture and leave as a major open problem that such a tournament rule exists for all n. We analyze the trade-offs between “relative” and “absolute” approximate strategyproofness for previously studied rules and derive as a corollary that all of these rules require λ≥ (n) to be robust to manipulation. We show that for stronger notions of fairness, non-manipulable tournament rules are closely related to tournament rules that witness decreasing gains from manipulation as the number of agents increases.
In dynamic programming and reinforcement learning, the policy for the sequential decision making of an agent in a stochastic environment is usually determined by expressing the goal as a scalar reward function and seeking a policy that maximizes the expected total reward. However, many goals that humans care about naturally concern multiple aspects of the world, and it may not be obvious how to condense those into a single reward function. Furthermore, maximization suffers from specification gaming, where the obtained policy achieves a high expected total reward in an unintended way, often taking extreme or nonsensical actions. Here we consider finite acyclic Markov Decision Processes with multiple distinct evaluation metrics, which do not necessarily represent quantities that the user wants to be maximized. We assume the task of the agent is to ensure that the vector of expected totals of the evaluation metrics falls into some given convex set, called the aspiration set. Our algorithm guarantees that this task is fulfilled by using simplices to approximate feasibility sets and propagate aspirations forward while ensuring they remain feasible. It has complexity linear in the number of possible state-action-successor triples and polynomial in the number of evaluation metrics. Moreover, the explicitly non-maximizing nature of the chosen policy and goals yields additional degrees of freedom, which can be used to apply heuristic safety criteria to the choice of actions. We discuss several such safety criteria that aim to steer the agent towards more conservative behavior.
A tournament organizer must select one of n possible teams as the winner of a competition after observing all ( [ n; 2 ]) matches between them. The organizer would like to find a tournament rule that simultaneously satisfies the following desiderata. It must be Condorcet-consistent (henceforth, CC), meaning it selects as the winner the unique team that beats all other teams (if one exists). It must also be strongly non-manipulable for groups of size k at probability α (henceforth, k-SNM-α ), meaning that no subset of ≤ k teams can fix the matches among themselves in order to increase the chances any of it’s members being selected by more than α . Our contributions are threefold. First, wee consider a natural generalization of the Randomized Single Elimination Bracket rule from [18] to d-ary trees and provide upper bounds to its manipulability. Then, we propose a novel tournament rule that is CC and 3-SNM-1/2 , a strict improvement upon the recent work of [7] who proposed a CC and 3-SNM-31/60 rule. Finally, we initiate the study of reductions among tournament rules.
Simple Stochastic Games (SSGs) were introduced by Anne Condon in 1990, as the simplest version of Stochastic Games for which there is no known polynomial-time algorithm. Condon showed that Stochastic Games are polynomial-time reducible to SSGs, which in turn are polynomial-time reducible to Stopping Games. SSGs are games where all decisions are binary and every move has a random outcome with a known probability distribution. Stopping Games are SSGs that are guaranteed to terminate. There are many algorithms for SSGs, most of which are fast in practice, but they all lack theoretical guarantees for polynomial-time convergence. The pursuit of a polynomial-time algorithm for SSGs is an active area of research. This paper is intended to support such research by making it easier to study the graphical structure of SSGs. Our contributions are: (1) a generating algorithm for Stopping Games, (2) a proof that the algorithm can generate any game, (3) a list of additional polynomial-time reductions that can be made to Stopping Games, (4) an open source generator for generating fully reduced instances of Stopping Games that comes with instructions and is fully documented, (5) a benchmark set of such instances, (6) and an analysis of how two main algorithm types perform on our benchmark set.
Efficient information dissemination is crucial for supporting critical operations across domains like disaster response, autonomous vehicles, and sensor networks. This paper introduces a Multi-Agent Reinforcement Learning (MARL) approach as a significant step forward in achieving more decentralized, efficient, and collaborative information dissemination. We propose a Partially Observable Stochastic Game (POSG) formulation for information dissemination empowering each agent to decide on message forwarding independently, based on the observation of their one-hop neighborhood. This constitutes a significant paradigm shift from heuristics currently employed in real-world broadcast protocols. Our novel approach harnesses Graph Convolutional Reinforcement Learning and Graph Attention Networks (GATs) with dynamic attention to capture essential network features. We propose two approaches, L-DyAN and HL-DyAN, which differ in terms of the information exchanged among agents. Our experimental results show that our trained policies outperform existing methods, including the state-of-the-art heuristic, in terms of network coverage as well as communication overhead on dynamic networks of varying density and behavior.
In recent years, the study of various models and questions related to Liquid Democracy has been of growing interest among the community of Computational Social Choice. A concern that has been raised, is that current academic literature focuses solely on static inputs, concealing a key characteristic of Liquid Democracy: the right for a voter to change her mind as time goes by, regarding her options of whether to vote herself or delegate her vote to other participants, till the final voting deadline. In real life, a period of extended deliberation preceding the election-day motivates voters to adapt their behaviour over time, either based on observations of the remaining electorate or on information acquired for the topic at hand. By adding a temporal dimension to Liquid Democracy, such adaptations can increase the number of possible delegation paths and reduce the loss of votes due to delegation cycles or delegating paths towards abstaining agents, ultimately enhancing participation. Our work takes a first step to integrate a time horizon into decision-making problems in Liquid Democracy systems. Our approach, via a computational complexity analysis, exploits concepts and tools from temporal graph theory which turn out to be convenient for our framework.
The issue of manipulation in the stable marriage game is well-known and have been studied for many decades. The question of weighted manipulation where each manipulative action is charged individually and the question is to decide if a favorable outcome can be attained within a pre-specified budget has only recently been considered by Boehmer et. al [SAGT’20]. They considered several manipulative action such as reordering a preference list, or swapping a pair of agents that are consecutive in a preference list, adding an agent, or deleting an agent or deleting a pair of agents from each others’ preference list, but the cost of each manipulative action was one irrespective of the agent who was undertaking the manipulative action. In this paper, we extend that model and consider arbitrary cost functions, and study an additional question where given a manipulative action and an agent, the goal is to use the manipulative action such that that person can be matched under a stable matching and the cost of the manipulative action(s) is no more than the budget. We formally address some questions raised by Boehmer et. al and in the process show that in this extended model, all problems under consideration are intractable and even exhibit parameterized hardness. The most intriguing aspect of the analysis is that we are able to identify a common underlying structural properties that makes each of the problems hard despite the fact that the manipulative action undertaken and/or the desired outcomes are very different from each other. Moreover, Boehmer et. al’s work revealed several dichotomies–be it classical or parameterized–and therefore it is apriori not obvious why in the weighted setting all problems must be W[1]-hard with respect to the combined parameter of budget and the number of unmatched vertices in a stable matching before manipulation. We discuss our analysis by way of presenting a metagadget that is at the heart of each hardness result, and show how to enrich it in different ways to yield hardness result for each of the individual problems.
After introducing a new distance measure of a preference profile to single-peakedness, directly derived from the very definition of single-peaked preferences by Black [ 4 ], we undertake a brief comparison with other popular distance measures to single-peakedness. We then tackle the computational aspects of the optimization problem raised by the proposed measure, namely we show that the problem is NP-hard and we propose an integer programming formulation. Finally, we carry out numerical tests on real and synthetic voting data. The obtained results show the interest of the proposed measure, but also shed new light on the advantages and drawbacks of some popular distance measures.
We introduce a new parameter which we call the nonmanipulative vote-deficit (NMVD) for single-winner voting rules. In particular, the NMVD of a voting rule at an election is the minimum number of votes needed to be added to transform this election into a nonmanipulable one yet without changing the winner. A voting rule has a bounded NMVD if the NMVDs of this rule at all elections are bounded from above by a constant. We show that the prevalent voting rules Borda, Plurality with Runoff, and Maximin have bounded NMVDs. In addition, we show that the NMVD of r-Approval, r-Veto, and Bucklin at every election can be bounded by a function of the number of candidates. For Copeland ^α , though that in general the NMVDs at elections cannot be bounded by a function of the number of candidates, we show that many special elections are still expected to have small NMVDs. Many of our results are tight.
