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Regular environmental conditions allow for the evolution of specifically adapted responses, whereas complex environments usually lead to conflicting requirements upon the organism's response. A relevant instance of these issues is bacterial chemotaxis, where the evolutionary and functional reasons for the experimentally observed response to chemoattractants remain a riddle. Sensing and motility requirements are in fact optimized by different responses, which strongly depend on the chemoattractant environmental profiles. It is not clear then how those conflicting requirements quantitatively combine and compromise in shaping the chemotaxis response. Here we show that the experimental bacterial response corresponds to the maximin strategy that ensures the highest minimum uptake of chemoattractants for any profile of concentration. We show that the maximin response is the unique one that always outcompetes motile but nonchemotactic bacteria. The maximin strategy is adapted to the variable environments experienced by bacteria, and we explicitly show its emergence in simulations of bacterial populations in a chemostat. Finally, we recast the contrast of evolution in regular vs. complex environments in terms of minimax vs. maximin game-theoretical strategies. Our results are generally relevant to biological optimization principles and provide a systematic possibility to get around the need to know precisely the statistics of environmental fluctuations.

Fractional factorial designs are widely used in various scientific investigations and industrial applications. Level permutation of factors could alter their geometrical structures and statistical properties. This article studies space-filling properties of fractional factorial designs under two commonly used space-filling measures, discrepancy and maximin distance. When all possible level permutations are considered, the average discrepancy is expressed as a linear combination of generalized word length pattern for fractional factorial designs with any number of levels and any discrepancy defined by a reproducing kernel. Generalized minimum aberration designs are shown to have good space-filling properties on average in terms of both discrepancy and distance. Several novel relationships between distance distribution and generalized word length pattern are derived. It is also shown that level permutations can improve space-filling properties for many existing saturated designs. A two-step construction procedure is proposed and three-, four-, and five-level space-filling fractional factorial designs are obtained. These new designs have better space-filling properties, such as larger distance and lower discrepancy, than existing ones, and are recommended for use in practice. Supplementary materials for this article are available online.

We study the implementation of maximin rational expectations equilibrium (MREE). When each agent's interim information is only his private signal, we show that each non-revealing MREE is implementable. This is not true for a partially revealing MREE or a fully revealing MREE. However, if each agent learns both his private signal and the market price in the interim, then each partially revealing and fully revealing MREE is implementable.

In a partition model, we show that each maximin individually rational and ex ante maximin efficient allocation of a single good economy is implementable as a maximin equilibrium. When there are more than one good, we introduce three conditions. If none of the three conditions is satisfied, then a maximin individually rational and ex ante maximin efficient allocation may not be implementable. However, as long as one of the three conditions is satisfied, each maximin individually rational and ex ante maximin efficient allocation is implementable. Our work generalizes and extends the recent paper of de Castro et al. (Games Econ Behav 2015. doi: 10.1016/j.geb.2015.10.010).

Efficient designs are in high demand in practice for both computer and physical experiments. Existing designs (such as maximin distance designs and uniform designs) may have bad low-dimensional projections, which is undesirable when only a few factors are active. We propose a new design criterion, called uniform projection criterion, by focusing on projection uniformity. Uniform projection designs generated under the new criterion scatter points uniformly in all dimensions and have good space-filling properties in terms of distance, uniformity and orthogonality. We show that the new criterion is a function of the pairwise L₁-distances between the rows, so that the new criterion can be computed at no more cost than a design criterion that ignores projection properties. We develop some theoretical results and show that maximin L₁-equidistant designs are uniform projection designs. In addition, a class of asymptotically optimal uniform projection designs based on good lattice point sets are constructed. We further illustrate an application of uniform projection designs via a multidrug combination experiment.

This paper proposes a new mechanism for combinatorial assignment—for example, assigning schedules of courses to students—based on an approximation to competitive equilibrium from equal incomes (CEEI) in which incomes are unequal but arbitrarily close together. The main technical result is an existence theorem for approximate CEEI. The mechanism is approximately efficient, satisfies two new criteria of outcome fairness, and is strategyproof in large markets. Its performance is explored on real data, and it is compared to alternatives from theory and practice: all other known mechanisms are either unfair ex post or manipulable even in large markets, and most are both manipulable and unfair.

Sliced Latin hypercube designs (SLHDs) have important applications in designing computer experiments with continuous and categorical factors. However, a randomly generated SLHD can be poor in terms of space-filling, and based on the existing construction method that generates the SLHD column by column using sliced permutation matrices, it is also difficult to search for the optimal SLHD. In this article, we develop a new construction approach that first generates the small Latin hypercube design in each slice and then arranges them together to form the SLHD. The new approach is intuitive and can be easily adapted to generate orthogonal SLHDs and orthogonal array-based SLHDs. More importantly, it enables us to develop general algorithms that can search for the optimal SLHD efficiently.

In the classical cake-cutting problem, strategy-proofness is a very costly requirement in terms of fairness: for n=2 it implies a dictatorial allocation, whereas for n≥3 it implies that one agent receives no cake. We show that a weaker version of this property recently suggested by Troyan and Morril (J Econ Theory 185:104970, 2019) is compatible with the fairness property of proportionality, which guarantees that each agent receives 1/n of the cake. Both properties are satisfied by the leftmost-leaves mechanism, an adaptation of the Dubins–Spanier moving knife procedure. Most other classical proportional mechanisms in the literature are obviously manipulable, including the original moving knife mechanism and some other variants of it.

In multi‐agent reinforcement learning (MARL) tasks, the state‐action value, commonly referred to as the Q $Q$ ‐value, can vary among agents because of their individual rewards, resulting in a Q $Q$ ‐vector. Determining an optimal policy is challenging, as it involves more than just maximizing a single Q $Q$ ‐value. Various optimal policies, such as a Nash equilibrium, have been studied in this context. Algorithms like Nash Q‐learning and Nash Actor‐Critic have shown effectiveness in these scenarios. This paper extends this research by proposing a deep Q‐networks algorithm capable of learning various Q $Q$ ‐vectors using Max, Nash, and Maximin strategies. We validate the effectiveness of our approach in a dual‐arm robotic environment, a representative human cyber‐physical systems (HCPS) scenario, where two robotic arms collaborate to lift a pot or hand over a hammer to each other. This setting highlights how incorporating MARL into HCPS can address real‐world complexities such as physical constraints, communication overhead, and dynamic interactions among multiple agents. This paper investigates multi‐agent reinforcement learning, where agents possess individual Q $Q$ ‐values, forming a Q $Q$ ‐vector. We introduce a deep Q‐networks algorithm that learns Q $Q$ ‐vectors using Max, Nash, and Maximin strategies. The proposed method is validated in a dual‐arm robotic environment.

The Poisson–Gamma model is obtained as a generalization of the Poisson model, when Gamma distributed block effects are assumed for Poisson count data. We show that optimal designs for estimating linear combinations of the model parameters coincide for the case of known and unknown parameters of the Gamma distribution. To obtain robust designs regarding parameter misspecification we determine standardized maximin D -optimal designs for a binary and a continuous design region. For standardized maximin c -optimality we show that the optimal designs for the Poisson–Gamma and Poisson model are equal and derive optimal designs for both models.