Analysing Irresolute Multiwinner Voting Rules with Approval Ballots via SAT Solving

**Read paper on the following link:** https://ifaamas.org/Proceedings/aamas2022/pdfs/p263.pdf **Abstract:** We simulate voting rules for multiwinner elections in a model of judgment aggregation that distinguishes between rationality and feasibility constraints. These constraints restrict the structure of the individual judgments and of the collective outcome computed by the rule, respectively. We extend known results regarding the simulation of single-winner voting rules to the multiwinner setting, both for elections with ordinal preferences and for elections with approval-based preferences. This not only provides us with a new tool to analyse multiwinner elections, but it also suggests the definition of new judgment aggregation rules, by generalising some of the principles at the core of well-known multiwinner voting rules to this richer setting. We explore this opportunity with regards to the principle of proportionality. Finally, in view of the computational difficulty associated with many judgment aggregation rules, we investigate the computational complexity of our embeddings and of the new judgment aggregation rules we put forward.

On the Constitutional Choice of a Democratic Voting Rule: An Experimental Approach

Robustness among multiwinner voting rules

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the $\ell^1$-norm is not strictly convex.

Approval ballots provide an opportunity for agents to make a comment about every candidate, without incurring the overhead of determining a full ranking on the set of candidates; they are very natural for many practical settings. We study the computational complexity of the committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outliers render the committee selection problem intractable for approval, net approval, and minisum approval voting rules. We next study the parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter namely the number of candidates outside the committee); and the number of outliers (and its dual parameter namely the number of non-outliers). For approval, net approval, and minisum approval voting rules, we provide a dichotomous result, which resolves the parameterized complexity of this problem for all subsets of the above five natural parameters considered (by showing either fixed parameter tractability or W[1]-hardness for all subsets of parameters).

Proportional representation (PR) is often discussed in voting settings as a major desideratum. For the past century or so, it is common both in practice and in the academic literature to look towards the single transferable vote (STV) rule as the solution for achieving PR. Some of the most prominent electoral reform movements around the globe are pushing for the adoption of STV. It has been termed a major open problem to design a voting rule that satisfies the same PR properties as STV and better monotonicity properties. In this paper, we first present a taxonomy of proportional representation axioms for general weak order preferences, some of which generalise and strengthen previously introduced concepts. We then present a rule called the expanding approvals rule (EAR) that satisfies properties stronger than the central PR axiom satisfied by STV, can handle indifferences in a convenient and computationally efficient manner, and also satisfies better candidate monotonicity properties. In view of this, our proposed rule seems to be a compelling solution for achieving proportional representation in voting settings.

In the late 19th century, Swedish mathematician Edvard Phragmén proposed a load-balancing approach for selecting committees based on approval ballots. We consider three committee voting rules resulting from this approach: two optimization variants—one minimizing the maximum load and one minimizing the variance of loads—and a sequential variant. We study Phragmén ’s methods from an axiomatic point of view, focusing on properties capturing proportional representation. We show that the sequential variant satisfies proportional justified representation, which is a rare property for committee monotonic methods. Moreover, we show that the optimization variants satisfy perfect representation. We also analyze the computational complexity of Phragmén ’s methods and provide mixed-integer programming based algorithms for computing them.

When Do Conflicting Parties Share Political Power? An Experimental Study

A story in Aristotle's Politics is often cited as the first published discussion of a financial derivative, an instrument whose value is derived from other assets or events. In Greece of the fifth century B.C., the philosopher Thales, believing six months in advance that the spring weather would be exceptionally favorable to olive growth, negotiated the right to rent oil presses for the next harvest at a bargain rate from press owners who wanted to hedge their bets against the possibility of a poor harvest. In today's parlance, Thales purchased a call option. When spring brought a lush olive harvest, Thales rented the machines to others at a much higher rate, making a tidy profit to support his philosophical inquiries. Options and other financial derivatives have become extremely popular in recent years, for better or worse. Overinvestment in derivatives of questionable value has led to notorious financial disasters such as the bankruptcy of Orange County, California, and the collapse of Barings Bank. Warren Buffett famously described derivatives as "financial weapons of mass destruction" in the Berkshire Hathaway Annual Report of 2002. Nonetheless, derivatives are undoubtedly here to stay; in 2002, trade in derivatives worldwide was estimated at $100 trillion or more, equaling or exceeding the total value of "real" equities. Readers of SIAM Review can take a special interest in derivatives because their analysis is an arena in which mathematics has had a huge impact on finance. In particular, the 1970s papers of Black and Scholes (1973) and Merton (1976) began a flood of research on options pricing that continues to this day. This issue's Survey and Review paper focuses on the special case of a spread option, a derivative that is based on the difference in value of two assets. Suppose you have some money to invest and the choice between two stocks to buy. You suspect that one will do much better than the other over the next few years, but you don't know which one. An ideal situation might be if you could buy one of them and hold it for a certain length of time, and then have the option of trading it in for the other stock if it turns out that that one did better. Another way to achieve the same benefit would be if someone agreed to pay you the difference in value between the two stocks at some specified time if you discover with hindsight that you made the wrong decision. Of course no one will offer you such an option free of charge, and so the mathematical question is how to price this option. This question is even more challenging if the option has a strike price that must be paid in order to exercise it. While spread options aren't publicly traded on arbitrary pairs of stocks, a variety of complex spread options are available in many markets, and are particularly popular in energy markets. Energy prices are extremely volatile compared to other commodities, a fact that most consumers are well aware of and that greatly affects producers, distributors, and speculators. Spread options are an important hedging and speculation tool, and are available in many different forms. For example, options may be based on the difference between the prices of two different forms of oil (the crack spread), between the prices of oil and electricity (the spark spread), or between the prices of a single commodity at two different times (calendar spreads) or locations (locational spreads). In the following article by Carmona and Durrleman, section 2 is devoted to a detailed description of these and other spread options that are used in practice. The authors then summarize the classical Black--Scholes pricing paradigm for options based on a single commodity, and the multivariate extensions needed for the analysis of spread options. The underlying stochastic differential equations can be solved by various approaches, and several approximations are illustrated and compared. Particular difficulties associated with energy markets are discussed throughout and are a main focus of the latter sections. Financial mathematics is as exciting growth area for applied mathematics, and we hope this paper will introduce readers to some challenging problems of current interest in this field.

Suppose you are on a mobile device with no keyboard (e.g., a cell or PDA). How can you enter text quickly? T9? Graffiti? This demo will show how language modeling can be used to speed up data entry, both in the mobile context, as well as the desk-top. The Wild Thing encourages users to use wildcards (*). A language model finds the k-best expansions. Users quickly figure out when they can get away with wildcards. General purpose trigram language models are effective for the general case (unrestricted text), but there are important special cases like searching over popular web queries, where more restricted language models are even more effective.

We study the complexity of several manipulation and control problems for six prevalent approval based multiwinner voting rules. We show that these rules generally resist the proposed strategic types. In addition, we also give fixed-parameter tractability results for these problems with respect to several natural parameters and derive polynomial-time algorithms for certain special cases.

We present a unifying framework for understanding and developing SAT-based decision procedures for Satisfiability Modulo Theories (SMT). The framework is based on a reduction of the decision problem to propositional logic by means of a deductive system. The two commonly used techniques, eager encodings (a direct reduction to propositional logic) and lazy encodings (a family of techniques based on an interplay between a SAT solver and a decision procedure) are identified as special cases. This framework offers the first generic approach for eager encodings, and a simple generalization of various lazy techniques that are found in the literature.

We describe an incremental algorithm for computing interpolants for a pair ϕ A , ϕ B of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕ A ∧ ϕ B , and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube.

Tensor networks are an important tool in condensed matter physics. In this paper, we study the task of tensor network non-zero testing (\tnit): Given a tensor network $T$, does $T$ represent a non-zero vector? We show that \tnit~is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of \tnit~on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic $k$-SAT problem on $D$-dimensional qudits is in NP for $k\in O(\log n)$ and $D\in O(1)$. This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic $k$, (2) constant $D$, and (3) for arbitrary interaction graphs.

The outsourcing of chip designs for fabrication has raised concerns regarding the protection of Intellectual Property (IP) from an untrustworthy foundry. Logic locking is a design-for-security technique that has the potential to thwart attacks from such an adversary. On the other hand, the notions of approximate and probabilistic computing have been popularized due to their low energy consumption characteristics and their potential application in error-tolerant frameworks. Prior work has looked into and exposed the vulnerability of logic-locked circuits using concepts of Boolean Satisfiability (SAT), but mostly from the perspective of deterministic designs. Despite existing attack frameworks not being directly applicable, we show in this work that circuits exhibiting probabilistic behavior also face the same threat. We propose StatSAT, an attack methodology incorporating statistical techniques into the existing SAT attack, that can overcome the hurdles imposed by the probabilistic behavior. Our attack results show that the adversary is capable of unlocking the circuit to an extent good for all practical purposes.