Smooth approximations by continuous choice-functions

We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake we prove that the theory of densely ordered sets with generic predicates is ℵ0- categorical. A model of the theory and a particular continuous choice-function are constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a byproduct, other functions with surprising properties can be constructed.

Linear and Utilitarian Choice Functions: Revisiting Myerson’s Theorem

Myerson proved that every linear and weakly Paretian choice function is utilitarian. We revisit his model and result for the two-person case and supplement it with an “only if” direction. That is, we characterize the class of linear and weakly Paretian two-person choice functions. It turns out that these are the utilitarian functions with an “egalitarian tie-breaking.”

Necessary and sufficient conditions for pairwise majority decisions on path-connected domains

In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.

Specific unknowns: A case study of epistemic indefinites in Cantonese

This paper concerns how languages bundle an existential claim and>an ignorance inference in a nominal expression. I present a case study on epistemic indefinites (EIs) in Cantonese and show that Cantonese EIs have a different morphological makeup (m + zi + WH ‘not + know + WH’), when compared to other more discussed EIs. I suggest that the ignorance component associated with mzi-WH is a conventional implicature and that m-zi obtains an adnominal usage via grammaticalization. It denotes a choice function that comes with an ignorance component that is inherited from the predicative meaning of m-zi.

Path-Independent Consideration

In the context of choice with limited consideration, where the decision-maker may not pay attention to all available options, the consideration function of a decision maker is path-independent if her choice cannot be manipulated by the presentation of the choice set. This paper characterizes a model of choice with limited consideration with path independence, which is equivalent to a consideration function that satisfies both the attention filter and consideration filter properties from Masatlioglu et al. (2012) and Lleras et al. (2017), respectively. Despite the equivalence of path-independent consideration with the consideration structures from these two papers, we show that, to have a choice with limited consideration that is path-independent, satisfying both axioms on the choice function that characterize choice limited consideration with attention and consideration filters unilaterally (from Masatlioglu et al. (2012) and Lleras et al. (2017)) is necessary but not sufficient.

Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a choice model in which each consumer selects a utility-maximizing bundle subject to a private valuation function, and study the complexity of the resulting optimization problem. Our main result is an exact algorithm for additive k -demand valuations, under a model of vertical differentiation in which customers agree on the relative value of each pair of items but differ in their absolute willingness to pay. For valuations that are vertically differentiated but not necessarily additive k -demand, we show how to obtain constant approximations under a “well-priced” condition, where each product’s price is sufficiently high. We further show that even for a single customer with known valuation, any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS and that no FPTAS exists even when the valuation is succinctly representable.

Local‐global equivalence in voting models: A characterization and applications

The paper considers a voting model where each voter's type is her preference. The type graph for a voter is a graph whose vertices are the possible types of the voter. Two vertices are connected by an edge in the graph if the associated types are “neighbors.” A social choice function is locally strategy‐proof if no type of a voter can gain by misrepresentation to a type that is a neighbor of her true type. A social choice function is strategy‐proof if no type of a voter can gain by misrepresentation to an arbitrary type. Local‐global equivalence (LGE) is satisfied if local strategy‐proofness implies strategy‐proofness. The paper identifies a condition on the graph that characterizes LGE. Our notion of “localness” is perfectly general. We use this feature of our model to identify notions of localness according to which various models of multidimensional voting satisfy LGE. Finally, we show that LGE for deterministic social choice functions does not imply LGE for random social choice functions.

A modified choice function hyper-heuristic with Boltzmann function

Hyper-heuristics are a subclass of high-level research methods that function in a low-level heuristic research space. Their aim objective is to improve the level of generality for solving combinatorial optimization problems using two main components: a methodology for the heuristic selection and a move acceptance criterion, to ensure intensification and diversification [1]. Thus, rather than working directly on the problem's solutions and selecting one of them to proceed to the next step at each stage, hyper-heuristics operates on a low-level heuristic research space. The choice function is one of the hyper-heuristics that have proven their efficiency in solving combinatorial optimization problems [2–4]. At each iteration, the selection of heuristics is dependent on a score calculated by combining three different measures to guarantee both intensification and diversification for the heuristics choice process. The heuristic with the highest score is therefore chosen to be applied to the problem. Therefore, the key to the success of the choice function is to choose the correct weight parameters of its three measures. In this study, we make a state of the art in hyper-heuristic research and propose a new method that automatically controls these weight parameters based on the Boltzmann function. The results obtained from its application on five problem domains are compared with those of the standard, modified choice function proposed by Drake et al. [2,3].