Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

Subjective optimality in finite sequential decision-making

Many decisions in life are sequential and constrained by a time window. Although mathematically derived optimal solutions exist, it has been reported that humans often deviate from making optimal choices. Here, we used a secretary problem, a classic example of finite sequential decision-making, and investigated the mechanisms underlying individuals’ suboptimal choices. Across three independent experiments, we found that a dynamic programming model comprising subjective value function explains individuals’ deviations from optimality and predicts the choice behaviors under fewer and more opportunities. We further identified that pupil dilation reflected the levels of decision difficulty and subsequent choices to accept or reject the stimulus at each opportunity. The value sensitivity, a model-based estimate that characterizes each individual’s subjective valuation, correlated with the extent to which individuals’ physiological responses tracked stimuli information. Our results provide model-based and physiological evidence for subjective valuation in finite sequential decision-making, rediscovering human suboptimality in subjectively optimal decision-making processes.

Secretary Problem: Graphs, Matroids and Greedoids

In the paper, the generalisation of the well-known “secretary problem” is considered. The aim of the paper is to give a generalised model in such a way that the chosen set of the possible best k elements have to be independent of all previously rejected elements. The independence is formulated using the theory of greedoids and in their special cases—matroids and antimatroids. Examples of some special cases of greedoids (uniform, graphical matroids and binary trees) are considered. Applications in cloud computing are discussed.

From Arms to Trees: Opportunity Costs and Path Dependence and the Exploration-Exploitation Tradeoff

The literature on the exploration-exploitation tradeoff has anchored on the n-armed bandit problem as its canonical formal representation. This structure, however, omits a fundamental property of evolutionary dynamics. Contrary to a bandit formulation, foregoing an opportunity may negate the possibility of engaging in that opportunity in the future, not just modifying the beliefs about the attractiveness of engaging in that opportunity. Thus, the bandit structure only incorporates path dependence with respect to beliefs and not with regard to capabilities as our usual conceptions of dynamics of learning and capabilities would suggest. Furthermore, the consideration of opportunity cost is rather static and does not address the dynamic unfolding of opportunity structures. The nature of path dependence and opportunity costs are used to frame many of our existing conceptualizations of search processes and firm dynamics, including bandit models, real options, pivoting, the “secretary problem,” and “island” models of firm diversification. The discussion points to the need to develop canonical models of what evolutionary biologists’ term phylogenetic trees and opens up a set of new questions, such as what is the degree of parallelism of trajectories that is possible within an organization, what is the fecundity of different trajectories in terms of likelihood of branching possibilities arising, and how are these latent branching opportunities accessed?

On a Competitive Secretary Problem with Deferred Selections

We study the secretary problem in multi-agent environments. In the standard secretary problem, a sequence of arbitrary awards arrive online, in a random order, and a single decision maker makes an immediate and irrevocable decision whether to accept each award upon its arrival. The requirement to make immediate decisions arises in many cases due to an implicit assumption regarding competition. Namely, if the decision maker does not take the offered award immediately, it will be taken by someone else. We introduce a novel multi-agent secretary model, in which the competition is explicit. In our model, multiple agents compete over the arriving awards, but the decisions need not be immediate; instead, agents may select previous awards as long as they are available (i.e., not taken by another agent). If an award is selected by multiple agents, ties are broken either randomly or according to a global ranking. This induces a multi-agent game in which the time of selection is not enforced by the rules of the games, rather it is an important component of the agent's strategy. We study the structure and performance of equilibria in this game. For random tie breaking, we characterize the equilibria of the game, and show that the expected social welfare in equilibrium is nearly optimal, despite competition among the agents. For ranked tie breaking, we give a full characterization of equilibria in the 3-agent game, and show that as the number of agents grows, the winning probability of every agent under non-immediate selections approaches her winning probability under immediate selections.

Strong Algorithms for the Ordinal Matroid Secretary Problem

In the ordinal matroid secretary problem (MSP), candidates do not reveal numerical weights, but the decision maker can still discern if a candidate is better than another. An algorithm [Formula: see text] is probability-competitive if every element from the optimum appears with probability [Formula: see text] in the output. This measure is stronger than the standard utility competitiveness. Our main result is the introduction of a technique based on forbidden sets to design algorithms with strong probability-competitive ratios on many matroid classes. We improve upon the guarantees for almost every matroid class considered in the MSP literature. In particular, we achieve probability-competitive ratios of 4 for graphic matroids and of [Formula: see text] for laminar matroids. Additionally, we modify Kleinberg’s utility-competitive algorithm for uniform matroids in order to obtain an asymptotically optimal probability-competitive algorithm. We also contribute algorithms for the ordinal MSP on arbitrary matroids.