Avoiding unfairness of Owen allocations in linear production processes

This paper deals with cooperation situations in linear production problems in which a set of goods are to be produced from a set of resources so that a certain benefit function is maximized, assuming that resources not used in the production plan have no value by themselves. The Owen set is a well-known solution rule for the class of linear production processes. Despite their stability properties, Owen allocations might give null payoff to players that are necessary for optimal production plans. This paper shows that, in general, the aforementioned drawback cannot be avoided allowing only allocations within the core of the cooperative game associated to the original linear production process, and therefore a new solution set named EOwen is introduced. For any player whose resources are needed in at least one optimal production plan, the EOwen set contains at least one allocation that assigns a strictly positive payoff to such player.     

Constrained games as complementary eigenvalue problems

A class of two-person nonzero sum games where the strategy choices are constrained in some form for each player is analyzed here to show the equivalent nonlinear programs which must be solved for the Cournot-Nash equilibrium. This equilibrium solution is shown in appropriate cases to lead to complementary eigenvalue problems, which have applications in normal solutions of stochastic LP models and optimal design problems in linear regression theory.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control

Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the value function. By choosing the Lagrange multiplier equal to its nonanticipative constraint value the usual stochastic (nonanticipative) optimal control and optimal cost are recovered. This suggests a method for solving the anticipative control problems by almost sure deterministic optimal control. We obtain a PDE for the “cost of perfect information” the difference between the cost function of the nonanticipative control problem and the cost of the anticipative problem which satisfies a nonlinear backward HJB SPDE. Poisson bracket conditions are found ensuring this has a global solution. The cost of perfect information is shown to be zero when a Lagrangian submanifold is invariant for the stochastic characteristics. The LQG problem and a nonlinear anticipative control problem are considered as examples in this framework     

LQG量测反馈最优控制的精细积分

对于线性二次型高斯(LQG)量测反馈最优控制问题,提出了精细积分解法。根据分离性原理,LQG控制问题可以分成为最优状态反馈控制问题以及最优状态估计问题,即:离线计算的两套黎卡提微分方程的求解以及状态向量的时变微分方程的在线积分解。该算法不仅适用于求解二点边值问题及其相应的黎卡提微分方程,也适用于求解状态估计的时变微分方程。精细积分高精度的特点,对控制和估计都是有利的。数值算例表明了算法的高精度及有效性。     

Variational inequality formulation for the games with random payoffs

We consider an n-player non-cooperative game with random payoffs and continuous strategy set for each player. The random payoffs of each player are defined using a finite dimensional random vector. We formulate this problem as a chance-constrained game by defining the payoff function of each player using a chance constraint. We first consider the case where the continuous strategy set of each player does not depend on the strategies of other players. If a random vector defining the payoffs of each player follows a multivariate elliptically symmetric distribution, we show that there exists a Nash equilibrium. We characterize the set of Nash equilibria using the solution set of a variational inequality (VI) problem. Next, we consider the case where the continuous strategy set of each player is defined by a shared constraint set. In this case, we show that there exists a generalized Nash equilibrium for elliptically symmetric distributed payoffs. Under certain conditions, we characterize the set of a generalized Nash equilibria using the solution set of a VI problem. As an application, the random payoff games arising from electricity market are studied under chance-constrained game framework.     

Design of Norm-Bounded and Sector-Bounded LQG Controllers for Uncertain Systems

The design of linear-quadratic-Gaussian (LQG) controllers is addressed for uncertain linear time-invariant systems that are either norm bounded or belong to a given sector. The system matrices are assumed to be affine functions of parameters confined to a convex polytopic region in the parameter space. Methods are developed for designing controllers which satisfy certain norm or sector conditions and are simultaneously optimal in the LQG sense. The resulting controllers provide robust stability as well as optimal performance.     

David G. Luenberger,Linear and nonlinear programming, 2nd edition,Addison-Wesley,Reading (1984), xv + 492 pages, $35.50

This paper considers a stochastic version of the linear continuous type knapsack problem in which the cost coefficients are random variables. The problem is to find an optimal solution and an optimal probability level of the chance constraint. This problem P₀ is first transformed into a deterministic equivalent problem P. Then a subproblem with a positive parameter is introduced and a close relation between P and its subproblem is shown. Further, an auxiliary problem of the subproblem is introduced and a direct relation between P and the auxiliary problem is derived through a relation connecting the subproblem and its auxiliary problem. Fully utilizing these relations, an efficient algorithm is proposed that finds an optimal solution of P in at most O(n⁴) computational time where n is the number of decision variables. Finally, further research problems are discussed.     

On an application of game theory to a problem of testing statistical hypothesis

Consider an-dimensional random vector with known covariance matrix. The expectation values of its single components may take arbitrary values subject to the restriction that their sum is a prescribed positive constant. Now choose a linear combination of these components, take its expectation value and divide this by the square root of its variance. This quotient, which is of importance in some problems of test theory serves as the pay-off function of a two-person zero-sum game. Player I wants to maximize the quotient by forming suitable linear combinations and player II wants to minimize it by choosing appropriate expectation values of the single components of the random vector subject to the restriction stated above. It is shown that the game possesses an essentially unique equilibrium point. In the more complicated case, when the strategies of the second player are confined to non-negative expectation values of the random vector's components, there is also an essentially unique equilibrium point of the game. It coincides with that one of the unconstrained case if and only if the row sums of the random vector's covariance matrix are all nonnegative.     

Embracing the giant component

Consider a game in which edges of a graph are provided a pair at a time, and the player selects one edge from each pair, attempting to construct a graph with a component as large as possible. This game is in the spirit of recent papers on avoiding a giant component, but here we embrace it. We analyze this game in the offline and online setting, for arbitrary and random instances, which provides for interesting comparisons. For arbitrary instances, we find that the competitive ratio (the best possible solution value divided by best possible online solution value) is large. For “sparse” random instances the competitive ratio is also large, with high probability (whp); If the instance has ¼(1 + ε)n random edge pairs, with 0 < ε ≤ 0.003, then any online algorithm generates a component of size O((log n)^3/2) whp , while the optimal offline solution contains a component of size Ω(n) whp . For “dense” random instances, the average‐case competitive ratio is much smaller. If the instance has ½(1 ? ε)n random edge pairs, with 0 < ε ≤ 0.015, we give an online algorithm which finds a component of size Ω(n) whp . © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Static search games played over graphs and general metric spaces

We define a general game which forms a basis for modelling situations of static search and concealment over regions with spatial structure. The game involves two players, the searching player and the concealing player, and is played over a metric space. Each player simultaneously chooses to deploy at a point in the space; the searching player receiving a payoff of 1 if his opponent lies within a predetermined radius r of his position, the concealing player receiving a payoff of 1 otherwise. The concepts of dominance and equivalence of strategies are examined in the context of this game, before focusing on the more specific case of the game played over a graph. Methods are presented to simplify the analysis of such games, both by means of the iterated elimination of dominated strategies and through consideration of automorphisms of the graph. Lower and upper bounds on the value of the game are presented and optimal mixed strategies are calculated for games played over a particular family of graphs.     

Pricing and volume discounting for a dominant retailer with uncertain manufacturing cost information

A product costs the manufacturer c/unit to produce; the retailer sells it at p/unit to the consumers. The retail-market demand volume V varies with p according to a given demand curve D_p. How would or should the “players” (i.e., the manufacturer and the retailer) set their prices? In contrast to many studies that assume a dominant manufacturer implementing the “manufacturer-Stackelberg” (“[mS]”) game, this paper examines how a dominant retailer should operate when his knowledge of c is imperfect. We first derive optimal decisions (some of them counter-intuitive) for the dominant retailer when he is restricted to choosing between [rS] (retailer-Stackelberg) and [mS]. Second, we propose a “reverse quantity discount” scheme that the dominant retailer (i.e., the downstream player) can offer to the manufacturer (note that the standard discount scheme is offered by the upstream player). We show that this discounting scheme is quite effective compared to the considerably more complicated though nevertheless theoretically optimal “menu of contracts.” We also reveal a largely overlooked function of discounting; i.e., discounting enables an “ignorant” but dominant player to usurp the earnings attributable to the knowledge of the dominated player. Finally, we also show that discounting works well when the demand curve is linear, but becomes ineffective when the demand curve is iso-elastic – a result echoing the conclusions of some earlier related works.     

Stability analysis of linear parabolic systems and removement of singularities in substructure: Static feedback

We analyze stability property of a class of linear parabolic systems via static feedback. Stabilization via static feedback scheme is most difficult and challenging when both actuators and observation weights admit spillovers. This arises typically in the boundary observation-boundary feedback scheme. We propose a simple static feedback law containing a parameter γ, and enhance the stability property or achieve (slightly) stabilization. In some situations, the evolution of the substructure of finite dimension contains singularities regarding γ. We show that these singularities are removed as long as the dimension is not large.     

Bifurcation analysis of a non-cooperative differential game with one weak player

We study a bifurcation problem for a system of two differential equations in implicit form. For each value of the parameter θ, the solution yields a pair of Nash equilibrium strategies in feedback form, for a non-cooperative differential game. When θ=0, the second player has no power to influence the dynamics of the system, and his optimal strategy is myopic. The game thus reduces to an optimal control problem for the first player. By studying the bifurcation in the solutions to the corresponding system of Hamilton-Jacobi equations, one can establish existence and multiplicity of solutions to the differential game, as θ becomes strictly positive.     

On the approximability of adjustable robust convex optimization under uncertainty

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.     

An empirical study of the complexity and randomness of prediction error sequences

We investigate a population of binary mistake sequences that result from learning with parametric models of different order. We obtain estimates of their error, algorithmic complexity and divergence from a purely random Bernoulli sequence. We study the relationship of these variables to the learner’s information density parameter which is defined as the ratio between the lengths of the compressed to uncompressed files that contain the learner’s decision rule. The results indicate that good learners have a low information density ρ while bad learners have a high ρ. Bad learners generate mistake sequences that are atypically complex or diverge stochastically from a purely random Bernoulli sequence. Good learners generate typically complex sequences with low divergence from Bernoulli sequences and they include mistake sequences generated by the Bayes optimal predictor. Based on the static algorithmic interference model of [18] the learner here acts as a static structure which “scatters” the bits of an input sequence (to be predicted) in proportion to its information density ρ thereby deforming its randomness characteristics.     

From the Schrödinger problem to the Monge–Kantorovich problem

The aim of this article is to show that the Monge–Kantorovich problem is the limit, when a fluctuation parameter tends down to zero, of a sequence of entropy minimization problems, the so-called Schrödinger problems. We prove the convergence of the entropic optimal values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the cluster points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Γ-convergence which we didn?t find in the literature; these Γ-convergence results which are interesting in their own right are also proved.     

Suboptimality of M-step back strategies in Bayesian games

We consider the Bayes optimal strategy for repeated two player games where moves are made simultaneously. In these games we look at models where one player assumes that the other player is employing a strategy depending only on the previousm-move pairs (as discussed in Wilson, 1986). We show that, under very unrestrictive conditions, such an assumption is not consistent with the assumption of rationality of one's opponent. Indeed, we show that by employing such a model a player is implicitly assuming that his opponent is not playing rationally,with probability one. We argue that, in the context of experimental games, thesem-step back models must be inferior to models which are consistent with the assumption that an opponent can be rational.     

Two-noisy-versus-one-silent duel with equal accuracy functions

This paper deals with the two-noisy-versus-one-silent duel which is still open, as pointed out by Styszyński (Ref. 1). Player I has a noisy gun with two bullets, and player II has a silent gun with one bullet. Each player fires his bullets aiming at his opponent at any time in [0, 1]. The accuracy function (the probability that one player hits his opponent if he fires at timet) isp(t)=t for each player. If player I hits player II, without being hit himself before, the payoff of the duel is +1; if player I is hit by player II, without hitting player II before, the payoff is taken to be ?1. In this paper, we determine the optimal strategies and the value of the game. The strategy for player II depends explicitly on the firing moment of player I's first shot.