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Register a new userMultinomial logit processes and preference discovery: inside and outside the black box
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We provide two characterizations, one axiomatic and the other neuro-computational, of the dependence of choice probabilities on deadlines, within the widely used softmax representation [ p_{t}left( a,Aright) =dfrac{e^{frac{uleft( aright) }{lambda left( tright) }+alpha left( aright) }}{sum_{bin A}e^{frac{uleft( bright) }{lambda left( tright) }+alpha left( bright) }}% ] where $p_{t}left( a,Aright) $ is the probability that alternative $a$ is selected from the set $A$ of feasible alternatives if $t$ is the time available to decide, $lambda$ is a time dependent noise parameter measuring the unit cost of information, $u$ is a time independent utility function, and $alpha$ is an alternative-specific bias that determines the initial choice probabilities reflecting prior information and memory anchoring. Our axiomatic analysis provides a behavioral foundation of softmax (also known as Multinomial Logit Model when $alpha$ is constant). Our neuro-computational derivation provides a biologically inspired algorithm that may explain the emergence of softmax in choice behavior. Jointly, the two approaches provide a thorough understanding of soft-maximization in terms of internal causes (neurophysiological mechanisms) and external effects (testable implications).
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We study multinomial logit bandit with limited adaptivity, where the algorithms change their exploration actions as infrequently as possible when achieving almost optimal minimax regret. We propose two measures of adaptivity: the assortment switching cost and the more fine-grained item switching cost. We present an anytime algorithm (AT-DUCB) with $O(N log T)$ assortment switches, almost matching the lower bound $Omega(frac{N log T}{ log log T})$. In the fixed-horizon setting, our algorithm FH-DUCB incurs $O(N log log T)$ assortment switches, matching the asymptotic lower bound. We also present the ESUCB algorithm with item switching cost $O(N log^2 T)$.
In the early 1990s, after its Jupiter flyby, the Ulysses spacecraft identified interstellar dust in the solar system. Since then the in-situ dust detector on board Ulysses continuously monitored interstellar grains with masses up to 10e-13 kg, penetrating deep into the solar system. While Ulysses measured the interstellar dust stream at high ecliptic latitudes between 3 and 5 AU, interstellar impactors were also measured with the in-situ dust detectors on board Cassini, Galileo and Helios, covering a heliocentric distance range between 0.3 and 3 AU in the ecliptic plane. The interstellar dust stream in the inner solar system is altered by the solar radiation pressure force, gravitational focussing and interaction of charged grains with the time varying interplanetary magnetic field. The grains act as tracers of the physical conditions in the local interstellar cloud (LIC). Our in-situ measurements imply the existence of a population of big interstellar grains (up to 10e-13 kg) and a gas-to-dust-mass ratio in the LIC which is a factor of > 2 larger than the one derived from astronomical observations, indicating a concentration of interstellar dust in the very local interstellar medium. Until 2004, the interstellar dust flow direction measured by Ulysses was close to the mean apex of the Suns motion through the LIC, while in 2005, the data showed a 30 deg shift, the reason of which is presently unknown. We review the results from spacecraft-based in-situ interstellar dust measurements in the solar system and their implications for the physical and chemical state of the LIC.
In this paper, we consider the contextual variant of the MNL-Bandit problem. More specifically, we consider a dynamic set optimization problem, where in every round a decision maker offers a subset (assortment) of products to a consumer, and observes their response. Consumers purchase products so as to maximize their utility. We assume that the products are described by a set of attributes and the mean utility of a product is linear in the values of these attributes. We model consumer choice behavior by means of the widely used Multinomial Logit (MNL) model, and consider the decision makers problem of dynamically learning the model parameters, while optimizing cumulative revenue over the selling horizon $T$. Though this problem has attracted considerable attention in recent times, many existing methods often involve solving an intractable non-convex optimization problem and their theoretical performance guarantees depend on a problem dependent parameter which could be prohibitively large. In particular, existing algorithms for this problem have regret bounded by $O(sqrt{kappa d T})$, where $kappa$ is a problem dependent constant that can have exponential dependency on the number of attributes. In this paper, we propose an optimistic algorithm and show that the regret is bounded by $O(sqrt{dT} + kappa)$, significantly improving the performance over existing methods. Further, we propose a convex relaxation of the optimization step which allows for tractable decision-making while retaining the favourable regret guarantee.
We propose a theoretical framework under which preference profiles can be meaningfully compared. Specifically, given a finite set of feasible allocations and a preference profile, we first define a ranking vector of an allocation as the vector of all individuals rankings of this allocation. We then define a partial order on preference profiles and write $P geq P^{}$, if there exists an onto mapping $psi$ from the Pareto frontier of $P^{}$ onto the Pareto frontier of $P$, such that the ranking vector of any Pareto efficient allocation $x$ under $P^{}$ is weakly dominated by the ranking vector of the image allocation $psi(x)$ under $P$. We provide a characterization of the maximal and minimal elements under the partial order. In particular, we illustrate how an emph{individualistic} form of social preferences can be $trianglerighteqslant$-maximal in a specific setting. We also discuss how the framework can be further generalized to incorporate additional economic ingredients.
Inspired by Ramseys theorem for pairs, Rival and Sands proved what we refer to as an inside/outside Ramsey theorem: every infinite graph $G$ contains an infinite subset $H$ such that every vertex of $G$ is adjacent to precisely none, one, or infinitely many vertices of $H$. We analyze the Rival-Sands theorem from the perspective of reverse mathematics and the Weihrauch degrees. In reverse mathematics, we find that the Rival-Sands theorem is equivalent to arithmetical comprehension and hence is stronger than Ramseys theorem for pairs. We also identify a weak form of the Rival-Sands theorem that is equivalent to Ramseys theorem for pairs. We turn to the Weihrauch degrees to give a finer analysis of the Rival-Sands theorems computational strength. We find that the Rival-Sands theorem is Weihrauch equivalent to the double jump of weak K{o}nigs lemma. We believe that the Rival-Sands theorem is the first natural theorem shown to exhibit exactly this strength. Furthermore, by combining our result with a result of Brattka and Rakotoniaina, we obtain that solving one instance of the Rival-Sands theorem exactly corresponds to simultaneously solving countably many instances of Ramseys theorem for pairs. Finally, we show that the uniform computational strength of the weak Rival-Sands theorem is weaker than that of Ramseys theorem for pairs by showing that a number of well-known consequences of Ramseys theorem for pairs do not Weihrauch reduce to the weak Rival-Sands theorem. We also address an apparent gap in the literature concerning the relationship between Weihrauch degrees corresponding to the ascending/descending sequence principle and the infinite pigeonhole principle.
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