Likelihood of voting outcomes with generalized IAC probabilities

After determining all supporting profiles with any number of voters for any specified three-candidate pairwise majority vote outcome, a new, large class of “octahedral” probability distributions, motivated by and including IAC, is introduced to examine various three-candidate voting outcomes involving majority vote outcomes. Illustrating examples include computing each distribution’s likelihood of a majority vote cycle and the likelihood that the Borda Count and Condorcet winners agree. Surprisingly, computations often reduce to a simple exercise of finding the volumes of tetrahedrons.     

An experimental study of voting rules and polls in three-candidate elections

We report the results of elections conducted in a laboratory setting, modelled on a threecandidate example due to Borda. By paying subjects conditionally on election outcomes, we create electorates with (publicly) known preferences. We compare the results of experiments with and without non-binding pre-election polls under plurality rule, approval voting, and Borda rule. We also refer to a theory of voting “equilibria,” which makes sharp predictions concerning individual voter behavior and election outcomes. We find that Condorcet losers occasionally win regardless of the voting rule or presence of polls. Duverger's law (which asserts the predominance of two candidates) appears to hold under plurality rule, but close three-way races often arise under approval voting and Borda rule. Voters appear to poll and vote strategically. In elections, voters usually cast votes that are consistent with some strategic equilibrium. By the end of an election series, most votes are consistent with a single equilibrium, although that equilibrium varies by experimental group and voting rule.     

Choice from a Three-Element Set: Some Lessons of the 2000 Presidential Campaign in the United States

We consider the behavior of four choice rules—plurality voting, approval voting, Borda count, and self-consistent choice—when applied to choose the best option from a three-element set. It is assumed that the two main options are preferred by a large majority of the voters, while the third option gets a very small number of votes and influences the election outcome only when the two main options receive a close number of votes. When used to rate the main options, Borda count and self-consistent choice contain terms that allow both for the strength of preferences of the voters and the rating of the main candidates by voters who vote for the third option. In this way, it becomes possible to determine more reliably the winner when plurality voting or approval voting produce close results.     

Scoring rule and majority agreements for large electorates with arbitrary preferences

This paper examines elections among three candidates when the electorate is large and voters can have any of the 26 nontrivial asymmetric binary relations on the candidates as their preference relations. Comparisons are made between rule-λ rankings based on rank-order ballots and simple majorities based on the preference relations. The rule-λ ranking is the decreasing point total order obtained when 1, λ and 0 points are assigned to the candidates ranked first, second and third on each voter's ballot, with 0 ? λ ? 1.Limit probabilities as the number of voters gets large are computed for events such as ‘the first-ranked rule-λ candidate has a majority over the second-ranked rule-λ candidate’ and ‘the rule-λ winner is the Condorcet candidate, given that there is a Condorcet candidate’. The probabilities are expressed as functions of λ and the distribution of voters over types of preference relations. In general, they are maximized at λ = 1/2 (Borda) and minimized at λ = 0 (plurality) and at λ = 1 for any fixed distribution of voters over preference types. The effects of more indifference and increased intransitivity in voter's preference relations are analyzed when λ is fixed.     

Almost disjoint large subsets of semigroups

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick, central, syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many pairwise disjoint central sets. We extend this result to a large class of semigroups (including (N,+)) by showing that if S is a semigroup in this class which has cardinality κ then any central set can be partitioned into κ many pairwise disjoint central sets. We also show that for this same class of semigroups, if there exists a collection of μ almost disjoint subsets of any member S, then any central subset of S contains a collection of μ almost disjoint central sets. The same statement applies if “central” is replaced by “thick”; and in the case that the semigroup is left cancellative, “central” may be replaced by “piecewise syndetic”. The situation with respect to syndetic sets is much more restrictive. For example, there does not exist an uncountable collection of almost disjoint syndetic subsets of N. We investigate the extent to which syndetic sets can be split into disjoint syndetic sets.     

Applications of bulk queues to group testing models with incomplete identification

A population of items is said to be “group-testable”, (i) if the items can be classified as “good” and “bad”, and (ii) if it is possible to carry out a simultaneous test on a batch of items with two possible outcomes: “Success” (indicating that all items in the batch are good) or “failure” (indicating a contaminated batch). In this paper, we assume that the items to be tested arrive at the group-testing centre according to a Poisson process and are served (i.e., group-tested) in batches by one server. The service time distribution is general but it depends on the batch size being tested. These assumptions give rise to the bulk queueing model M/G^(m,M)/1, where m and M(>m) are the decision variables where each batch size can be between m and M. We develop the generating function for the steady-state probabilities of the embedded Markov chain. We then consider a more realistic finite state version of the problem where the testing centre has a finite capacity and present an expected profit objective function. We compute the optimal values of the decision variables (m, M) that maximize the expected profit. For a special case of the problem, we determine the optimal decision explicitly in terms of the Lambert function.     

Pricing and volume discounting for a dominant retailer with uncertain manufacturing cost information

A product costs the manufacturer c/unit to produce; the retailer sells it at p/unit to the consumers. The retail-market demand volume V varies with p according to a given demand curve D_p. How would or should the “players” (i.e., the manufacturer and the retailer) set their prices? In contrast to many studies that assume a dominant manufacturer implementing the “manufacturer-Stackelberg” (“[mS]”) game, this paper examines how a dominant retailer should operate when his knowledge of c is imperfect. We first derive optimal decisions (some of them counter-intuitive) for the dominant retailer when he is restricted to choosing between [rS] (retailer-Stackelberg) and [mS]. Second, we propose a “reverse quantity discount” scheme that the dominant retailer (i.e., the downstream player) can offer to the manufacturer (note that the standard discount scheme is offered by the upstream player). We show that this discounting scheme is quite effective compared to the considerably more complicated though nevertheless theoretically optimal “menu of contracts.” We also reveal a largely overlooked function of discounting; i.e., discounting enables an “ignorant” but dominant player to usurp the earnings attributable to the knowledge of the dominated player. Finally, we also show that discounting works well when the demand curve is linear, but becomes ineffective when the demand curve is iso-elastic – a result echoing the conclusions of some earlier related works.     

Unitarily achievable zero patterns and traces of words in A and A

Our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices: which 0-patterns may be achieved for any given n-by-n complex matrix via some unitary similarity of it. To this end, certain restrictions on “achievable” 0-patterns are mentioned, both positional and, more important, on the maximum number of achievable 0’s. Prior results fitting this general question are mentioned, as well as the “first” unresolved pattern (for 3-by-3 matrices!). In the process a recent question is answered.A closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of Specht’s theorem about which pairs of complex matrices are unitarily similar, which seems not widely known to matrix theorists. In the process, we mention the number of words necessary for small size matrices.     

Condorcet winner configurations of linear networks

We consider the problem of locating a fixed number of facilities along a line to serve n players. We model this problem as a cooperative game and assume that any locational configuration can be eventually disrupted through a strict majority of players voting for an alternative configuration. A solution of such a voting location problem is called a Condorcet winner configuration. In this article, we state three necessary and one sufficient condition for a configuration to be a Condorcet winner. Consequently, we propose a fast algorithm which enables us to verify whether a given configuration is a Condorcet winner, and can be efficiently used also for computing the (potentially empty) set of all Condorcet winner configurations.     

Minmax regret location–allocation problem on a network under uncertainty

We consider a robust location–allocation problem with uncertainty in demand coefficients. Specifically, for each demand point, only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility. The optimal solution of this problem is determined by the “minimax regret” location, i.e., the point that minimizes the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. For the case where the demand points are vertices of a network we show that the robust location–allocation problem can be solved in O(min{p, n − p}n³m) time, where n is the number of demand points, p (p < n) is the fixed number of demand points that must be served by the new service and m is the number of edges of the network.     

An evolving framework for describing student engagement in classroom activities

Student engagement in classroom activities is usually described as a function of factors such as human needs, affect, intention, motivation, interests, identity, and others. We take a different approach and develop a framework that models classroom engagement as a function of students’ conceptual competence in the specific content (e.g., the mathematics of motion) of an activity. The framework uses a spatial metaphor—i.e., the classroom activity as a territory through which students move—as a way to both capture common engagement-related dynamics and as a communicative device. In this formulation, then, students’ engaged participation can be understood in terms of the nature of the “regions” and overall “topography” of the activity territory, and how much student movement such a territory affords. We offer the framework not in competition with other instructional design approaches, but rather as an additional tool to aid in the analysis and conduct of engaging classroom activities.     

A stochastic and asymmetric-information framework for a dominant-manufacturer supply chain

Consider a dominant manufacturer wholesaling a product to a retailer, who in turn retails it to the consumers at $p/unit. The retail-market demand volume varies with p according to a given demand curve. This basic system is commonly modeled as a manufacturer-Stackelberg ([mS]) game under a “deterministic and symmetric-information” (“det-sym-i”) framework. We first explain the logical flaws of this framework, which are (i) the dominant manufacturer-leader will have a lower profit than the retailer under an iso-elastic demand curve; (ii) in some situations the system’s “correct solution” can be hyper-sensitive to minute changes in the demand curve; (iii) applying volume discounting while keeping the original [mS] profit-maximizing objective leads to an implausible degenerate solution in which the manufacturer has dictatorial power over the channel. We then present an extension of the “stochastic and asymmetric-information” (“sto-asy-i”) framework proposed in Lau and Lau [Lau, A., Lau, H.-S., 2005. Some two-echelon supply-chain games: Improving from deterministic–symmetric-information to stochastic-asymmetric-information models. European Journal of Operational Research 161 (1), 203–223], coupled with the notion that a profit-maximizing dominant manufacturer may implement not only [mS] but also “[pm]”—i.e., using a manufacturer-imposed maximum retail price. We show that this new framework resolves all the logical flaws stated above. Along the way, we also present a procedure for the dominant manufacturer to design a profit-maximizing volume-discount scheme using stochastic and asymmetric demand information.     

Universal cycles of classes of restricted words

It is well known that Universal cycles (U-cycles) of k-letter words on an n-letter alphabet exist for all k and n. In this paper, we prove that Universal cycles exist for several restricted classes of words, including non-bijections, “equitable” words (under suitable restrictions), ranked permutations, and “passwords”. In each case, proving the connectedness of the underlying de Bruijn digraph is a non-trivial step.     

Beyond Cartesian limits: Leibniz's passage from algebraic to “transcendental” mathematics

This article deals with Leibniz's reception of Descartes' “geometry.” Leibnizian mathematics was based on five fundamental notions: calculus, characteristic, art of invention, method, and freedom. On the basis of methodological considerations Leibniz criticized Descartes' restriction of geometry to objects that could be given in terms of algebraic (i.e., finite) equations: “Descartes's mind was the limit of science.” The failure of algebra to solve equations of higher degree led Leibniz to develop linear algebra, and the failure of algebra to deal with transcendental problems led him to conceive of a science of the infinite. Hence Leibniz reconstructed the mathematical corpus, created new (transcendental) notions, and redefined known notions (equality, exactness, construction), thus establishing “a veritable complement of algebra for the transcendentals”: infinite equations, i.e., infinite series, became inestimable tools of mathematical research.     

A generalized Isserlis theorem for location mixtures of Gaussian random vectors

In a recent paper (Michalowicz et al., 2011), Michalowicz et al. provide an extension of the Isserlis theorem to the case of a Rademacher location mixture of a Gaussian vector. This theorem is known to physicists under the name of Wick’s theorem. We generalize here this result to the case of any location mixture of Gaussian vector; we also provide an example of the extension of the Isserlis theorem to a “scale-location” mixture of Gaussian, namely, the d-dimensional generalized hyperbolic distribution.     

Representations of rational Cherednik algebras of rank one in positive characteristic

In this paper, we classify the irreducible representations of the rational Cherednik algebras of rank 1 in characteristic p>0. There are two cases. One is the “quantum” case, where “Planck's constant” is nonzero and generic irreducible representations have dimension pr, where r is the order of the cyclic group contained in the algebra. The other is the “classical” case, where “Planck's constant” is zero and generic irreducible representations have dimension r.     

Generalized convex set-valued maps

The aim of this paper is to show that under a mild semicontinuity assumption (the so-called segmentary epi-closedness), the cone-convex (respectively, cone-quasiconvex) set-valued maps can be characterized in terms of weak cone-convexity (respectively, weak cone-quasiconvexity), i.e., the notions obtained by replacing in the classical definitions the conditions of type “for all x,y in the domain and for all t in ]0,1[…” by the corresponding conditions of type “for all x,y in the domain there exists t in ]0,1[….”     

Tau-functions,Grassmannians and rank one conditions

In integrable systems, specifically the KP hierarchy, there are functions known as “tau-functions”, closely related to the Schur polynomials in terms of which they are often written. Although they are generally viewed as the solutions to a collection of nonlinear PDEs, in this note they will equivalently be characterized by a quadratic difference equation. Sato's theorem associates tau-functions to the points of a Grassmann manifold. To make that amazing theorem clear to non-experts, we will first show an analogous (but easily understood) example of a linear ODE and its solution from a flow on the xy-plane. In each case the solution is created via a flow generated by a certain linear operator. The question we pose is this: “What other operators could have been used to generate solutions in the same way?” Although the answer is well known in the ODE case, the question in the nonlinear case is the main result of our new paper. We will state the result and discuss its relationship to the “trend” of writing tau-functions in terms of matrices satisfying certain rank one conditions. The elucidation of a geometric interpretation of the Hirota bilinear difference equation (HBDE) is a key feature of the proof and will be briefly described.     

An application of cooperative game among container terminals of one port

In this paper a two-stage game that involves three container terminals located in Karachi Port in Pakistan is discussed. In the first stage, the three terminals have to decide on whether to act as a singleton or to enter into a coalition with one or both of the other terminals. The decision at this stage should presumably be based on the predicted outcome for the second stage. The second stage is here modelled as a Bertrand game with one outside competitor, the coalition and the terminal in Karachi Port (if any) that has not joined the coalition. Furthermore, three partial and one grand coalition among the three terminals at Karachi Port are investigated. The concepts of “characteristic function” and “core” are used to analyse the stability of these coalitions and this revealed that one combination does not satisfy the superadditivity property of the characteristic function and can therefore be ruled out. The resulting payoffs (profits) of these coalitions are analysed on the basis of “core”. The best payoff for all players is in the case of a “grand coalition”. However, the real winner is the outsider (the terminal at the second port) which earns a better payoff without joining the coalition, and hence will play the role of the “orthogonal free-rider”.