Realizability and inscribability for simplicial polytopes via nonlinear optimization

We consider the problem of screening where a seller puts up for sale an indivisible good, and a buyer with a valuation unknown to the seller wishes to acquire the good. We assume that the buyer valuations are represented as discrete types drawn from some distribution, which is also unknown to the seller. The seller is averse to possible mis-specification of types distribution, and considers the unknown type density as member of an ambiguity set and seeks an optimal pricing mechanism in a worst case sense. We specify four choices for the ambiguity set and derive the optimal mechanism in each case.     

Prevention efforts,insurance demand and price incentives under coherent risk measures

This paper studies an equilibrium model between an insurance buyer and an insurance seller, where both parties’ risk preferences are given by convex risk measures. The interaction is modeled through a Stackelberg type game, where the insurance seller plays first by offering prices, in the form of safety loadings. Then the insurance buyer chooses his optimal proportional insurance share and his optimal prevention effort in order to minimize his risk measure. The loss distribution is given by a family of stochastically ordered probability measures, indexed by the prevention effort. We give special attention to the problems of self-insurance and self-protection, and show that if the buyer’s risk measure decreases faster in effort than his expected loss, optimal effort is non-decreasing in the safety loading with a potential discontinuity when optimal coverage switches from full to zero. On the contrary, if the decrease of the buyer’s risk measure is slower than the expected loss, optimal effort may or may not be non-decreasing in the safety loading. In case of Pareto distributed losses, the seller sets the highest possible price under which the buyer still prefers full insurance over no insurance. We also analyze the case of discrete distributions: on the one hand, for self-protection, under the assumption that the marginal impact of the effort is higher on small losses than it is on catastrophic losses, the optimal effort is non-decreasing in the safety loading. On the other hand, in the case of self-protection, more conditions are needed, in particular, we obtain sufficient conditions for the optimal effort to be non-decreasing or non-monotone in the safety loading.     

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer’s utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.     

Optimal multi-period insurance contracts

In multi-period insurance contracts (such as automobile insurance contracts), unlike single-period ones, the premiums that the insured must pay increase whenever he files a claim. Hence, the buyer faces a problem that is absent in one-period models, namely: he must determine for which damages he should file a claim and for which he should not.The optimal claims policy of the buyer is presented for a large class of insurance contracts. It is shown that the buyer will file a claim only if it is larger than some critical value. Based on this it is shown that the buyer prefers a contract that provides full coverage above a deductible for damages that exceed his critical value. In this case the optimal contract is not unique since the buyer is indifferent to the form of the contract for damages below his critical value. It is shown, however, that as in one-period models (Arrow (1963, 1974)) there exists an optimal contract that provides full coverage above a deductible. In multi-period setting, however, the buyer will file a claim only if the damage is sufficiently higher than the deductible.It is also shown that the buyer prefers a strictly positive deductible. Unlike the one-period case (Mossin (1968)), this result holds true even if the premium rates equal the expected payments.     

Revenue deficiency under second-price auctions in a supply-chain setting

Consider a firm, called the buyer, that satisfies its demand over two periods by assigning both demands to a supplier via a second-price procurement auction; call this the Standard auction. In the hope of lowering its purchase cost, the firm is considering an alternative procedure in which it will also allow bids on each period individually, where there can be either one or two winners covering the two demands; call this the Multiple Winner auction. Choosing the Multiple Winner auction over the Standard auction can in fact result in a higher cost to the buyer. We provide a bound on how much greater the buyer’s cost can be in the Multiple Winner auction and show that this bound is tight. We then sharpen this bound for two scenarios that can arise when the buyer announces his demands close to the beginning of the demand horizon. Under a monotonicity condition, we achieve a further sharpening of the bound in one of the scenarios. Finally, this monotonicity condition allows us to generalize this bound to the T-period case in which bids are allowed on any subset of period demands.     

考虑相对业绩的最优投资选择博弈问题研究

研究了具有相互作用的两个竞争机构投资者之间的离散时间最优投资选择博弈问题,每个机构投资者都考虑其竞争对手的相对业绩.机构投资者可以投资于相同的无风险资产和不同的具有相关关系的风险股票,以反映投资的资产专门化.机构投资者选择投资组合策略使得期望终端绝对财富和相对财富的效用最大.首先,定义了Nash均衡投资组合选择策略.然后,在机构投资者具有指数效用函数的假设下,得到了Nash均衡投资组合选择策略和值函数的显示表达式,分析了机构投资者之间的竞争对Nash均衡投资组合选择策略的影响.最后,通过数值计算给出了各种情况下Nash均衡投资组合选择策略和值函数与模型主要参数之间的关系.结果表明:机构投资者之间的竞争会影响其对风险的承担,投资机会集对机构投资者的Nash均衡投资组合选择策略和值函数与模型主要参数之间的关系会产生很大的影响.     

Inverse stochastic dominance constraints and rank dependent expected utility theory

We consider optimization problems with second order stochastic dominance constraints formulated as a relation of Lorenz curves. We characterize the relation in terms of rank dependent utility functions, which generalize Yaari's utility functions. We develop optimality conditions and duality theory for problems with Lorenz dominance constraints. We prove that Lagrange multipliers associated with these constraints can be identified with rank dependent utility functions. The problem is numerically tractable in the case of discrete distributions with equally probable realizations. Research supported by the NSF awards DMS-0303545, DMS-0303728, DMI-0354500 and DMI-0354678.     

Heuristics for sourcing from multiple suppliers with alternative quantity discounts

In this paper, we analyze the impact of supplier pricing schemes and supplier capacity limitations on the optimal sourcing policy for a single firm. We consider the situation where the total quantity to be procured for a single period is known by the firm and communicated to the supplier set. In response to this communication, each supplier quotes a price and a capacity limit in terms of a maximum quantity that can be supplied to the buyer. Based on this information, the buyer makes a quantity allocation decision among the suppliers and corresponding to this decision is the choice of a subset of suppliers who will receive an order. Based on industry observations, a variety of supplier pricing schemes from the constituent group of suppliers are analyzed, including linear discounts, incremental units discounts, and all units discounts. Given the complexity of the optimization problem for certain types of pricing schemes, heuristic solution methodologies are developed to identify a quantity allocation decision for the firm. Through an extensive computational comparison, we find that these heuristics generate near-optimal solutions very quickly. Data from a major office products retailer is used to illustrate the resulting sourcing strategies given different pricing schemes and capacity limitations of suppliers in this industry. We find for the case of capacity constrained suppliers, the optimal quantity allocations for two complex pricing schemes (linear discount, and incremental units discount) are such that at most one selected supplier will receive an order quantity that is less than its capacity.     

Management Adopts Operational Research: Session Report

Quantitative methods are derived to assist buyers purchasing commodities in fluctuating price markets. Demand is known whilst price is a stochastic variable which may contain trends or seasonal fluctuations. The essential feature of the problem is that the buyer has many opportunities to make a purchase.Mathematical models are formulated to describe particular commodity buying problems. The optimal purchasing policy is derived by using dynamic programming. It consists of a set of discrete price breaks at each buying opportunity together with the associated stock levels the buyer should aim to achieve at each price break with his purchase at this opportunity. The price breaks are dependent on the probability density functions of future prices and the number of future buying opportunities. Recurrence relations are derived to calculate these price breaks. The case of restrictions on the purchase quantity at each price offer, either because of supply limitations or by the buyer as a policy decision, and price discounts are also considered.A case study illustrating the techniques is given and the methods are extended to purchasing for a blending problem with substitutable commodities.     

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In this paper, the insurer is allowed to buy reinsurance and allocate his money among three financial securities: a defaultable corporate zero-coupon bond, a default-free bank account, and a stock, while the instantaneous rate of the stock is described by an Ornstein-Uhlenbeck process. The objective is to maximize the exponential utility of the terminal wealth. We decompose the original optimization problem into two subproblems: a pre-default case and a post-default case. Using dynamic programming principle, and then solving the corresponding HJB equations, we derive the closed-form solutions for the optimal reinsurance and investment strategies and the corresponding value functions     

Bayesian optimal knapsack procurement

A budget-constrained buyer wants to purchase items from a shortlisted set. Items are differentiated by observable quality and sellers have private reserve prices for their items. The buyer’s problem is to select a subset of maximal quality. Money does not enter the buyer’s objective function, but only his constraints. Sellers quote prices strategically, inducing a knapsack game. We report the Bayesian optimal mechanism for the buyer’s problem. We find that simultaneous take-it-or-leave-it offers are interim optimal.     

Cooperative games with imcomplete information

A bargaining solution concept which generalizes the Nash bargaining solution and the Shapley NTU value is defined for cooperative games with incomplete information. These bargaining solutions are efficient and equitable when interpersonal comparisons are made in terms of certainvirtual utility scales. A player's virtual utility differs from his real utility by exaggerating the difference from the preferences of false types that jeopardize his true type. In any incentive-efficient mechanism, the players always maximize their total virtual utility ex post. Conditionally-transferable virtual utility is the strongest possible transferability assumption for games with incomplete information.     

Discrete maximal surfaces with singularities in Minkowski space

We describe discrete maximal surfaces with singularities in 3-dimensional Minkowski space and give a Weierstrass type representation for them. In the smooth case, maximal surfaces (spacelike surfaces with mean curvature identically 0) in Minkowski 3-space generally have certain singularities. We give a criterion that naturally describes the “singular set” for discrete maximal surfaces, including a classification of the various types of singularities that are possible in the discrete case.     

Integrated inventory model with quantity discount and price-sensitive demand

Quantity discount has been a subject of study for a long time; however, little is known about its effect on integrated inventory models when price-sensitive demand is placed. The objective of this study is to find the optimal pricing and ordering strategies for an integrated inventory system when a quantity discount policy is applied. The pricing strategy discussed here is one in which the vendor offers a quantity discount to the buyer. Then, the buyer will adjust his retail price based on the purchasing cost, which will influence the customer demand as a result. Consequently, an integrated inventory model is established to find the optimal solutions for order quantity, retail price, and the number of shipments from vendor to buyer in one production run, so that the joint total profit incurred has the maximum value. Also, numerical examples and a sensitivity analysis are given to illustrate the results of the model.     

Consistent values and the core in continuum market games with two types of players

The consistent value is an extension of the Shapley value to the class of games with non-transferable utility.? In this paper, the consistent value will be characterized for market games with a continuum of players of two types. We will show that for such games the consistent value need not belong to the core, and provide conditions under which there is equivalence between the two concepts. Received: October 1998 RID="*" ID="*"  This thesis was completed under the supervision of Professor Sergiu Hart, The Center for Rationality and Interactive Decision Theory, Department of Mathematics, Department of Economics, The Hebrew University of Jerusalem. I would like to thank Professor Hart for introducing me to this area of research, for his help and guidance, and, especially, for all his patience.? I would also like to thank Michael Borns for improving the style, and an anonymous referee for helpful comments.     

Pattern Hit-and-Run for sampling efficiently on polytopes

Pattern Hit-and-Run (PHR) is a Markov chain Monte Carlo sampler for a target distribution that was originally designed for general sets embedded in a box. A specific set of interest to many applications is a polytope intersected with discrete or mixed continuous/discrete lattices. PHR requires an acceptance/rejection mechanism along a bidirectional walk to guarantee feasibility. We remove this inefficiency by utilizing the linearity of the constraints defining the polytope, so each iteration of PHR can be efficiently implemented even though the variables are allowed to be integer valued. Moreover, PHR converges to a uniform distribution in polynomial time for a class of discrete polytopes.     

Supply chain coordination with controllable lead time and asymmetric information

This paper considers coordinated decisions in a decentralized supply chain consisting of a vendor and a buyer with controllable lead time. We analyze two supply chain inventory models. In the first model we assume the vendor has complete information about the buyer’s cost structure. By taking both the vendor and the buyer’s individual rationalities into consideration, a side payment coordination mechanism is designed to realize supply chain Pareto dominance. In the second model we consider a setting where the buyer possesses private cost information. We design the coordination mechanism by using principal-agent model to induce the buyer to report his true cost structure. The solution procedures are also developed to get the optimal solutions of these two models. The results of numerical examples show that shortening lead time to certain extent can reduce inventory cost and the coordination mechanisms designed for both symmetric and asymmetric information situations are effective.     

时变单车路径优化模型及动态规划算法

车辆路径问题由于其广泛的应用领域及经济价值而成为学术研究热点。然而,在已有的研究文献中,车辆的速度时变与服务多任务特性很少被关注。本文讨论了具有这两个特性的单车路径优化问题。建立了以送货完成时间最早为优化目标的时变单车送货路径优化模型。由于很难获得该模型的精确解,本文提出了一种贪婪补货策略压缩原问题解空间,设计动态规划算法给出了车辆行驶时间满足FIFO规则的送货顺序近似最优解。数值算例验证了该算法所得到的解仅是原问题的近似最优解这一结论。算例同时表明优化配送时间随着车辆装载能力的增大而缩短,并在车辆装载能力超过所有客户配送总需求时实现最短配送时间,即,使用较大装载能力车辆能节约更多配送时间。     

Optimal investment and consumption models with non-linear stock dynamics

We study a generalization of the Merton's original problem of optimal consumption and portfolio choice for a single investor in an intertemporal economy. The agent trades between a bond and a stock account and he may consume out of his bond holdings. The price of the bond is deterministic as opposed to the stock price which is modelled as a diffusion process. The main assumption is that the coefficients of the stock price diffusion are arbitrary nonlinear functions of the underlying process. The investor's goal is to maximize his expected utility from terminal wealth and/or his expected utility of intermediate consumption. The individual preferences are of Constant Relative Risk Aversion (CRRA) type for both the consumption stream and the terminal wealth. Employing a novel transformation, we are able to produce closed form solutions for the value function and the optimal policies. In the absence of intermediate consumption, the value function can be expressed in terms of a power of the solution of a homogeneous linear parabolic equation. When intermediate consumption is allowed, the value function is expressed via the solution of a non-homogeneous linear parabolic equation.     

Existence of value in differential games with terminal cost function

A two-person, zero-sum differential game with general type phase constraints and terminal (not fixed) cost function is investigated. Player II (possessing complete information) can choose any strategy in the Varaiya-Lin sense, while his opponent (having incomplete information) can select any lower II-strategy introduced by Friedman (Ref. 1). The existence of a value and an optimal player II's strategy is obtained under assumptions ensuring that the sets of all admissible trajectories for the two players are compact in the Banach space of all continuous functions. The present paper largely extends the results of Ref. 2.