The Worst-Case Weighted Multi-Objective Game with an Application to Supply Chain Competitions

In this paper, we propose a worst-case weighted approach to the multi-objective n-person non-zero sum game model where each player has more than one competing objective. Our “worst-case weighted multi-objective game” model supposes that each player has a set of weights to its objectives and wishes to minimize its maximum weighted sum objectives where the maximization is with respect to the set of weights. This new model gives rise to a new Pareto Nash equilibrium concept, which we call “robust-weighted Nash equilibrium”. We prove that the robust-weighted Nash equilibria are guaranteed to exist even when the weight sets are unbounded. For the worst-case weighted multi-objective game with the weight sets of players all given as polytope, we show that a robust-weighted Nash equilibrium can be obtained by solving a mathematical program with equilibrium constraints (MPEC). For an application, we illustrate the usefulness of the worst-case weighted multi-objective game to a supply chain risk management problem under demand uncertainty. By the comparison with the existed weighted approach, we show that our method is more robust and can be more efficiently used for the real-world applications.     

Stability in negotiation games and the emergence of cooperation

Consider a two-player game in which each player contributes a costly resource to the common good of the pair. For such contests, the Nash equilibrium contribution, x*, is one for which neither player can increase its pay-off by unilaterally altering its contribution from x*. We study an elaboration of this game, which allows the players to exchange x-offers back and forth in a negotiation phase until they converge to a final pair of contributions, x1 and x2. A significant feature of such negotiation games, hitherto unrecognized, is the existence of a set of neutrally stable equilibrium points in negotiation phase space. To explore the long-term evolutionary outcome of such games, we simulate populations containing various mixtures of negotiation strategies and, contrary to previous results, we often find convergence to a contribution that is more cooperative than the Nash equilibrium. Mathematical analysis suggests why this might be happening, and provides a novel and robust explanation for cooperation, that negotiation can facilitate the evolution of cooperative behaviour.     

Improving probabilistic infectious disease forecasting through coherence

With an estimated $10.4 billion in medical costs and 31.4 million outpatient visits each year, influenza poses a serious burden of disease in the United States. To provide insights and advance warning into the spread of influenza, the U.S. Centers for Disease Control and Prevention (CDC) runs a challenge for forecasting weighted influenza-like illness (wILI) at the national and regional level. Many models produce independent forecasts for each geographical unit, ignoring the constraint that the national wILI is a weighted sum of regional wILI, where the weights correspond to the population size of the region. We propose a novel algorithm that transforms a set of independent forecast distributions to obey this constraint, which we refer to as probabilistically coherent. Enforcing probabilistic coherence led to an increase in forecast skill for 79% of the models we tested over multiple flu seasons, highlighting the importance of respecting the forecasting system’s geographical hierarchy.     

Some game-theoretical aspects of parasitism and symbiosis

Given a two-person continuous non-constant sum game, a “solution” can be arrived at even in ignorance of the pay-off functions as each player tries to maximize his own pay-off by trial and error. The solution of the game considered here is either the intersection of two “optimal lines” (the case of the stable equilibrium) or parasitism of one player upon the other (the case of unstable equilibrium). The introduction of secondary satisfactions (linear combinations of the pay-offs) affects both the position and the stability of the equilibrium and thus the resulting primary satisfactions (the pay-offs). These effects of the matrix of transformation are investigated. In particular, it is shown that the social optimum, which is also the solution of the game where collusion is allowed, is attained if and only if the columns of the matrix are identical, and that this solution is generally stable. Some suggested connections with biological situations are discussed.     

Minimally colored trees

Each edge in a weighted colored tree has a nonnegative weight corresponding to the colors of its incident vertices. The sum of these weights is the weight of the tree. Algorithms of O(n) are known to find minimal colorings, that is, to assign colors from a given finite set to the vertices of a tree so as to minimize the weight of the tree. In this paper generating functions are used to find the number of minimal colorings and the average weight of each edge over such colorings, also using O(n) operations. Applications to evolutionary trees are given.     

Particle swarm optimizer for weighting factor selection in intensity-modulated radiation therapy optimization algorithms

In inverse treatment planning of intensity-modulated radiation therapy (IMRT), the objective function is typically the sum of the weighted sub-scores, where the weights indicate the importance of the sub-scores. To obtain a high-quality treatment plan, the planner manually adjusts the objective weights using a trial-and-error procedure until an acceptable plan is reached. In this work, a new particle swarm optimization (PSO) method which can adjust the weighting factors automatically was investigated to overcome the requirement of manual adjustment, thereby reducing the workload of the human planner and contributing to the development of a fully automated planning process. The proposed optimization method consists of three steps. (i) First, a swarm of weighting factors (i.e., particles) is initialized randomly in the search space, where each particle corresponds to a global objective function. (ii) Then, a plan optimization solver is employed to obtain the optimal solution for each particle, and the values of the evaluation functions used to determine the particle’s location and the population global location for the PSO are calculated based on these results. (iii) Next, the weighting factors are updated based on the particle’s location and the population global location. Step (ii) is performed alternately with step (iii) until the termination condition is reached. In this method, the evaluation function is a combination of several key points on the dose volume histograms. Furthermore, a perturbation strategy – the crossover and mutation operator hybrid approach – is employed to enhance the population diversity, and two arguments are applied to the evaluation function to improve the flexibility of the algorithm. In this study, the proposed method was used to develop IMRT treatment plans involving five unequally spaced 6 MV photon beams for 10 prostate cancer cases. The proposed optimization algorithm yielded high-quality plans for all of the cases, without human planner intervention. A comparison of the results with the optimized solution obtained using a similar optimization model but with human planner intervention revealed that the proposed algorithm produced optimized plans superior to that developed using the manual plan. The proposed algorithm can generate admissible solutions within reasonable computational times and can be used to develop fully automated IMRT treatment planning methods, thus reducing human planners’ workloads during iterative processes.     

Linear Superposition and Prediction of Bacterial Promoter Activity Dynamics in Complex Conditions

Bacteria often face complex environments. We asked how gene expression in complex conditions relates to expression in simpler conditions. To address this, we obtained accurate promoter activity dynamical measurements on 94 genes in E. coli in environments made up of all possible combinations of four nutrients and stresses. We find that the dynamics across conditions is well described by two principal component curves specific to each promoter. As a result, the promoter activity dynamics in a combination of conditions is a weighted average of the dynamics in each condition alone. The weights tend to sum up to approximately one. This weighted-average property, called linear superposition, allows predicting the promoter activity dynamics in a combination of conditions based on measurements of pairs of conditions. If these findings apply more generally, they can vastly reduce the number of experiments needed to understand how E. coli responds to the combinatorially huge space of possible environments.     

Evolving cell models for systems and synthetic biology

This paper proposes a new methodology for the automated design of cell models for systems and synthetic biology. Our modelling framework is based on P systems, a discrete, stochastic and modular formal modelling language. The automated design of biological models comprising the optimization of the model structure and its stochastic kinetic constants is performed using an evolutionary algorithm. The evolutionary algorithm evolves model structures by combining different modules taken from a predefined module library and then it fine-tunes the associated stochastic kinetic constants. We investigate four alternative objective functions for the fitness calculation within the evolutionary algorithm: (1) equally weighted sum method, (2) normalization method, (3) randomly weighted sum method, and (4) equally weighted product method. The effectiveness of the methodology is tested on four case studies of increasing complexity including negative and positive autoregulation as well as two gene networks implementing a pulse generator and a bandwidth detector. We provide a systematic analysis of the evolutionary algorithm’s results as well as of the resulting evolved cell models.     

Weighted Wilcoxon-type smoothly clipped absolute deviation method

Summary .  Shrinkage-type variable selection procedures have recently seen increasing applications in biomedical research. However, their performance can be adversely influenced by outliers in either the response or the covariate space. This article proposes a weighted Wilcoxon-type smoothly clipped absolute deviation (WW-SCAD) method, which deals with robust variable selection and robust estimation simultaneously. The new procedure can be conveniently implemented with the statistical software R . We establish that the WW-SCAD correctly identifies the set of zero coefficients with probability approaching one and estimates the nonzero coefficients with the rate   n ^−1/2  . Moreover, with appropriately chosen weights the WW-SCAD is robust with respect to outliers in both the x and y directions. The important special case with constant weights yields an oracle-type estimator with high efficiency in the presence of heavier-tailed random errors. The robustness of the WW-SCAD is partly justified by its asymptotic performance under local shrinking contamination. We propose a Bayesian information criterion type tuning parameter selector for the WW-SCAD. The performance of the WW-SCAD is demonstrated via simulations and by an application to a study that investigates the effects of personal characteristics and dietary factors on plasma beta-carotene level.     

A new method for the re-implementation of threshold logic functions with cellular neural networks

A new strategy is presented for the implementation of threshold logic functions with binary-output Cellular Neural Networks (CNNs). The objective is to optimize the CNNs weights to develop a robust implementation. Hence, the concept of generative set is introduced as a convenient representation of any linearly separable Boolean function. Our analysis of threshold logic functions leads to a complete algorithm that automatically provides an optimized generative set. New weights are deduced and a more robust CNN template assuming the same function can thus be implemented. The strategy is illustrated by a detailed example.     

Statistical mechanical treatment of α-helices and extended structures in proteins with inclusion of short- and medium-range interactions

A statistical mechanical model of protein conformation with medium-range interactions between theith and (i+k)^th residues (k<-4) is presented. Two two-state models, an α-helix-coil and an extended-structure-coil model, are formulated using the same form of the partition function, but the two models are applied independently to predict the locations of α-helical, extended, and coil segments; in the relatively few cases (<2%) where the predictions from the two models are in conflict, the prediction is scored as an incorrect one. Two independent sets of statistical weights (one set for each model) are derived to describe the interactions between the 20 amino acid residues for each range of interactionk; they are evaluated by minimizing an objective function so that the probability profiles for the α-helix or extended structure, respectively, in proteins computed from these statistical weights correlate optimally with the experimentally observed native conformations of these proteins. Examination of the resulting statistical weights shows that those for the interactions between hydrophobic residues and between a hydrophobic and a hydrophilic residue have reasonable magnitudes compared to what would be expected from the spatial arrangements of the side chains in the α-helix and the extended structure, and that those for the α-helix-coil model correlate well with experimentally determined values of the Zimm-Bragg parameterss and σ of the helix-coil transition theory. From the point of view of a method to predict the conformational states (i.e., α-helix, extended structure, and coil) of each residue, the statistical weights (as inall empirical prediction schemes) depend very much on the proteins used for the data base, since the presently available set of proteins of known structure is still too small for very high predictability; as a result, the correctness of the prediction is not very good for proteins not included in the data base. However, the correctness of the prediction, at least for the 37 proteins utilized as the data base in this study, is 91% and 87% for the α-helix-coil and the extended-structure-coil models, respectively; further, 79% of all the residues are predicted correctly when both the α-helix-coil and extended-structure-coil models are applied independently.     

Theorems on the invasion process in stage-structured populations with density-dependent dynamics

 Several theoretical results on evolution in stage-structured models with density-dependent dynamics are obtained. The first one concerns the invadable condition of mutant types when ‘density’ is considered as the weighted sum of the population densities at each stage. In the second and third results, it is shown that the invadability is equivalent to the increase of the weighted sum at equilibrium when certain conditions are satisfied. It is also shown that the sensitivity for the dominant eigenvalue is proportional to that for the weighted sum at equilibrium. Finally, a theorem on the coexistence of a wild and the mutant types is obtained and the condition for coexistence is discussed. A part of these results are the extended theorems of what Charlesworth (1971, 1980, 1994) has already obtained. Received: 26 October 1996     

Exacerbating the Tragedy of the Commons: Private Inefficient Outcomes and Peer Effect in Experimental Games with Fishing Communities

Economic Experimental Games have shown that individuals make decisions that deviate down from the suboptimal Nash equilibrium. However, few studies have analyzed the case when deviation is above the Nash equilibrium. Extracting from above the Nash equilibrium is inefficient not only socially but also privately and it would exacerbate the tragedy of the commons. That would be the case of a race to the fish when stocks are becoming depleted or driver behavior on a highly congested road. The objective of this study is to analyze private inefficient extraction behavior in experimental games and to associate the type of player and the type of player group with such inefficient outcomes. To do this, we carried out economic experimental games with local coastal fishermen in Colombia, using a setting where the scarcity of the resource allows for an interior Nash equilibrium and inefficient over-extraction is possible. The state of the resource, the type of player and the composition of the group explain, in part, this inefficient behavior.     

Kendall's and Spearman's correlation coefficients in the presence of a blocking variable

The use of a weighted sum of Kendall's taus or a weighted sum of Spearman's rhos for testing association in the presence of a blocking variable is discussed. In a Monte Carlo study the two are shown to have essentially the same power with the optimal choice of weights. In the presence of ties, the weighted sum of Spearman's rhos is preferred because its variance has a much simpler form. An example is given from the field of radiation therapy.     

Quantifying the relative importance of experimental data points in parameter estimation

Background

Ordinary differential equations (ODEs) are often used to understand biological processes. Since ODE-based models usually contain many unknown parameters, parameter estimation is an important step toward deeper understanding of the process. Parameter estimation is often formulated as a least squares optimization problem, where all experimental data points are considered as equally important. However, this equal-weight formulation ignores the possibility of existence of relative importance among different data points, and may lead to misleading parameter estimation results. Therefore, we propose to introduce weights to account for the relative importance of different data points when formulating the least squares optimization problem. Each weight is defined by the uncertainty of one data point given the other data points. If one data point can be accurately inferred given the other data, the uncertainty of this data point is low and the importance of this data point is low. Whereas, if inferring one data point from the other data is almost impossible, it contains a huge uncertainty and carries more information for estimating parameters.

Results

G1/S transition model with 6 parameters and 12 parameters, and MAPK module with 14 parameters were used to test the weighted formulation. In each case, evenly spaced experimental data points were used. Weights calculated in these models showed similar patterns: high weights for data points in dynamic regions and low weights for data points in flat regions. We developed a sampling algorithm to evaluate the weighted formulation, and demonstrated that the weighted formulation reduced the redundancy in the data. For G1/S transition model with 12 parameters, we examined unevenly spaced experimental data points, strategically sampled to have more measurement points where the weights were relatively high, and fewer measurement points where the weights were relatively low. This analysis showed that the proposed weights can be used for designing measurement time points.

Conclusions

Giving a different weight to each data point according to its relative importance compared to other data points is an effective method for improving robustness of parameter estimation by reducing the redundancy in the experimental data.

     

Optimal sibship selection for genotyping in quantitative trait locus linkage analysis

In this paper we present a novel method for selecting optimally informative sibships of any size for quantitative trait locus (QTL) linkage analysis. The method allocates a quantitative index of potential informativeness to each sibship on the basis of observed trait scores and an assumed true QTL model. Any sample of phenotypically screened sibships can therefore be easily rank-ordered for selective genotyping. The quantitative index is the sibship's expected contribution to the non-centrality parameter. This expectation represents the weighted sum of chi(2) test statistics that would be obtained given the observed trait values over all possible sibship genotypic configurations; each configuration is weighted by the likelihood of it occurring given the assumed true genetic model. The properties of this procedure are explored in relation to the accuracy of the assumed true genetic model and sibship size. In comparison to previous methods of selecting phenotypically extreme sibships for genotyping, the proposed method is considerably more efficient and is robust with regard to the specification of the genetic model.     

A statistical approach to computer processing of cryo-electron microscope images: virion classification and 3-D reconstruction

The scattering density of the virus is represented as a truncated weighted sum of orthonormal basis functions in spherical coordinates, where the angular dependence of each basis function has icosahedral symmetry. A statistical model of the image formation process is proposed and the maximum likelihood estimation method computed by an expectation-maximization algorithm is used to estimate the weights in the sum and thereby compute a 3-D reconstruction of the virus particle. If multiple types of virus particle are represented in the boxed images then multiple 3-D reconstructions are computed simultaneously without first requiring that the type of particle shown in each boxed image be determined. Examples of the procedure are described for viruses with known structure: (1). 3-D reconstruction of Flockhouse Virus from experimental images, (2). 3-D reconstruction of the capsid of Nudaurelia Omega Capensis Virus from synthetic images, and (3). 3-D reconstruction of both the capsid and the procapsid of Nudaurelia Omega Capensis Virus from a mixture of unclassified synthetic images.     

One-third rules with equality: Second-order evolutionary stability conditions in finite populations

The one-third law of evolutionary dynamics [Nowak et al. 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 246-650] describes a robustness criterion for evolution in a finite population: If at an A-frequency of 1/3, the fitness of an A player is greater (smaller) than the fitness of a B player, then a single A mutant that appears in a population of otherwise all B has a fixation probability greater (smaller) than the neutral threshold 1/N, the inverse population size. We examine the case where at an A-frequency of 1/3, the fitness of an A player is exactly equal to the fitness of a B player. We find that in this case the relative magnitude of the cross payoffs matters: If the payoff of A against B is larger (smaller) than the payoff of B against A, then a single A mutant has a fixation probability larger (smaller) than 1/N. If the cross payoffs coincide, we are in the special case of a partnership game, where the deviation cost from an inefficient equilibrium is exactly balanced by the potential gain of switching to the payoff dominant equilibrium. We show that in this case the fixation probability of A is lower than 1/N. Finally, we illustrate our findings by a language game with differentiated costs of signals.     

Efficient social contracts and group selection

We consider the Stag Hunt in terms of Maynard Smith’s famous Haystack model. In the Stag Hunt, contrary to the Prisoner’s Dilemma, there is a cooperative equilibrium besides the equilibrium where every player defects. This implies that in the Haystack model, where a population is partitioned into groups, groups playing the cooperative equilibrium tend to grow faster than those at the non-cooperative equilibrium. We determine under what conditions this leads to the takeover of the population by cooperators. Moreover, we compare our results to the case of an unstructured population and to the case of the Prisoner’s Dilemma. Finally, we point to some implications our findings have for three distinct ideas: Ken Binmore’s group selection argument in favor of the evolution of efficient social contracts, Sewall Wright’s Shifting Balance theory, and the equilibrium selection problem of game theory.