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.     

Dynamic contracting under permanent and transitory private information

I examine a model of long-term contracting in which the buyer is privately informed about the discrete probability distribution for his future value for a divisible product, and fully characterize the optimal long term contract that will be offered by a monopolistic seller in a simple case where two types of buyers can have two types of utility in any period. In such a case, the buyer more likely to have a high utility type will receive the first-best allocations indifferent of his value report, while the lower type will receive the first best only if he makes a high utility report. The paper also supplements the current literature on infinite dynamic games with continuous buyer types, which relies on the use of a distribution of types with full support and an envelope theorem. With discrete types, the number of compatibility constraints considered can be greatly reduced by sandwiching the border of the space of solutions allowed by constraints: formulate the maximization problem in a wider space with fewer constraints and prove that the solution obeys a simpler set of stronger constraints that places it in the allowed region.     

Buyer behavior adaptation based on a fuzzy logic controller and prediction techniques

The current form of Web provides numerous product resources available to users. Users can rely on intelligent agents for purchase actions. These actions are taken in specific environments such as Electronic Markets (EMs). In this paper, we study the interaction process between buyers and sellers and focus on the buyer side. Each buyer has the opportunity to interact with a number of sellers trying to buy the most appropriate products. This interaction can be modeled as a finite horizon Bargaining Game (BG). In this game, players have opposite goals concerning the product price. We adopt a number of techniques in the buyer side trying to give the appropriate level of efficiency in the buyer decision process. The buyer uses a prediction mechanism in combination with the use of Fuzzy Logic (FL) theory in order to be able to predict the upcoming seller proposal and, thus, understand the seller pricing policy. Based on this, he/she can adapt his/her behavior when trying to purchase products. The buyer adaptation mechanism produces the belief that the buyer has about the seller pricing policy and a parameter that indicates his/her own pricing policy which yields the buyer offers in the upcoming rounds. Moreover, the buyer is based on FL system that derives the appropriate actions at every round of the BG. Our results show that the combination of Fuzzy Logic (FL) with the above-mentioned techniques provides an efficient decision mechanism in the buyer side that in specific scenarios outperforms an optimal stopping model.     

Seller—buyer System Co-operation in a Monopolistic Market

This paper deals with a situation in which the buyer is in a monopolistic position with respect to the seller, and examines the issues and advantages of co-operation in a seller–buyer inventory control system. Game theory concepts form the foundation for the analysis of these issues. Initially, the relationship between the seller and the buyer is modelled as a non-cooperative two-stage game, and it is noted that the traditional EOQ formula is one of the results. Then, interactive game theory is utilized to address the problem of system co-operation as well as to determine optimal system order quantity-pricing strategies. Mutual incentives and motivations for system co-operation are also discussed. Among several alternative methods, the combination of an equal profit sharing role implemented via quantity discounting is demonstrated as the best mechanism for achieving system co-operation. Finally, the similarities and differences between the proposed model and those in the literature are discussed.     

Bargaining order and delays in multilateral bargaining with heterogeneous sellers

We consider a complete-information multilateral bargaining game in which a single buyer negotiates with two heterogeneous sellers selling perfect complementary units. While bilateral negotiations take place through a sequence of offers and counteroffers, the bargaining order is exogenously given. We solve for the conditions under which (a) the buyer prefers to negotiate with the lower-valuation seller first and (b) efficient (inefficient) outcomes emerge for the two bargaining orders. We find that the buyer prefers to negotiate with the lower-valuation seller first whenever the players are relatively impatient or the sellers are sufficiently heterogeneous. We show that there exists a unique efficient outcome when the buyer negotiates first with the lower-valuation seller and the sellers are sufficiently heterogeneous; however, significant delay in reaching agreements may arise when they are not. In case the buyer bargains with the higher-valuation seller first, an inefficient outcome is shown to exist even when players are extremely impatient.     

Auctions with an asking price

This paper studies a sales mechanism, prevalent in housing markets, where the seller does not reveal or commit to a reserve price but instead publicly announces an asking price. We show that the seller sets an asking price such that, in equilibrium, buyers of certain types would accept it with positive probability. We also show that this sales mechanism, with an optimally chosen asking price set prior to the seller learning her value, does better than any standard auction with a reserve price equal to the seller’s reservation value. We then extend the analysis to the case where the asking price reveals information about the seller’s reservation value. We show that in this case there is a separating equilibrium with fully-revealing asking prices, which is revenue-equivalent to a standard auction with a reserve price set at the seller’s reservation value.     

The valuation of options on capacity with cost and demand uncertainty

Options contracts can provide trading partners with enhanced flexibility to respond to uncertain market conditions and allow for superior capacity planning thanks to early information on future demand. We develop an analytical framework to value options on capacity for production of non-storable goods or dated services. The market consists of a sequence of contract and spot market. Reservations are made during the contract market session in period 0, where the buyer’s future demand, the seller’s future marginal costs as well as the future spot price are uncertain, the latter being impacted neither by the buyer nor the seller. During the spot market session in period 1, the buyer may execute his options or satisfy his entire or additional demand from a competing seller in the spot market. The seller allocates reserved capacity now being called and attempts to sell remaining capacity into the spot market. Analytical expressions for the buyer’s optimal reservation quantity and the seller’s tariff are derived, making explicit the risk-sharing benefits of options contracts. The combination of an options contract and a spot market is demonstrated to be Pareto improving as compared to alternative market schemes. An analysis of the determinants of the efficiency gain characterizes industries particularly suitable to the options approach.     

The effect of deception on optimal decisions

The buyer in a seller-first offer bargaining situation use deception to cause the seller to arrive at an inaccurate subjective probability distribution of the buyer's position. The effects on the optimal behavior of the seller and on the payoffs to the two parties are examined.     

Manipulation via endowments in auctions with multiple goods

We study manipulation via endowments in a market in an auction setting with multiple goods. In the market, there are buyers whose valuations are their private information, and a seller whose set of endowments is her private information. A social planner, who wants to implement a socially desirable allocation, faces the seller’s manipulation via endowments, in addition to buyers’ manipulation of misreporting their valuations. We call a mechanism immune to the seller’s manipulation via endowments destruction-proof. In general, there exists no mechanism which is destruction-proof, together with strategy-proofness of the buyers, efficiency, and participation. Nevertheless, we find a restricted domain of the buyers’ valuation profiles satisfying a new condition called per-capita goods–buyer submodularity. We show that, in this domain, there exists a mechanism which is destruction-proof, together with the above properties. The restriction is likely to be met when each winner’s valuation is close to the next-highest valuation. We also provide a relation to monopoly theory, and argue that per-capita goods–buyer submodularity is independent of the standard elasticity argument.     

Designing win-win financial loan products for consumers and businesses

With the many possible designs that a financial company can offer to a consumer (eg terms, price, quality, features), a company can identify win-win products for both the consumer and the company. A key to identifying win-win products is to explicitly integrate the consumer's preferences for price and quality with the company's preferences for profit and market share. This paper builds a model that identifies the set of win-win products by integrating the preferences of buyer and seller. For any product not in this set, there is at least one product in the set that is better for both buyer and seller. The company's preferences are then used to select the optimal offer from the win-win set. Our development logically derives the results by focusing on financial products (eg loans, mortgages, credit cards) to consumers in the multitrillion dollar retail credit business.     

A fuzzy mathematical model for the joint economic lot size problem with multiple price breaks

This paper applies fuzzy mathematical programming to solve the joint economic lot size problem with multiple price breaks. In order to entice the buyer to increase the order quantity, it is a common practice for the seller to offer quantity discounts to the buyer. From the system viewpoint, the joint cost for the seller and buyer can be minimized only when the buyer increases his economic order quantity. The problem is how to determine the number of price breaks, as well as the quantity discount and order quantity at each price break, to achieve the optimal joint cost. Fuzzy mathematical programming provides a very efficient algorithm to solve the above problem simultaneously from the perspectives of the seller and the buyer. Another common problem in joint economic lot size model is how to split the system profit between the seller and the buyer. Whereas the traditional approach to this problem is to divide the profit based on a certain ratio determined by the bargaining power of both parties, fuzzy mathematical programming can achieve the same satisfaction level to both parties based on their respective cost functions.     

A game theory approach in seller–buyer supply chain

In this paper, several seller–buyer supply chain models are proposed which incorporate both cost factors as well as elements of competition and cooperation between seller and buyer. We assume that unit marketing expenditure and unit price charged by the buyer influence the demand of the product being sold. The relationships between seller and buyer will be modeled by non-cooperative and cooperative games, respectively. The non-cooperative game is based on the Stackelberg strategy solution concept, where we consider separately the case when the seller is the leader (Seller-Stackelberg) and also when the buyer is the leader (Buyer-Stackelberg). Pareto efficient solutions will be provided for the cooperative game model. Numerical examples presented in this paper, including sensitivity analysis of some key parameters, will compare the results between different models considered.     

The infinite time horizon dynamic pricing problem with multi-unit demand

This paper studies a dynamic pricing problem for a monopolist selling multiple identical items to potential buyers arriving over time, where the time horizon is infinite, the goods are imperishable and the buyers’ arrival follows a renewal process. Each potential buyer has some private information about his purchasing will, and this private information is unknown to the seller and therefore characterized as a random variable in this paper. Thus, the buyers may have multi-unit demand. Meanwhile, the seller needs to determine the optimal posted price such that his expected discounted revenue is maximized. This problem is formulated as a stochastic dynamic programming in this paper and then how to obtain the solution is explored. A numerical study shows that the optimal posted price performs better than that of optimal fixed price, and this advantage becomes obvious as the interest rate and/or the number of initial items increases.     

Pareto-optimal insurance contracts with premium budget and minimum charge constraints

In view of the fact that minimum charge and premium budget constraints are natural economic considerations in any risk-transfer between the insurance buyer and seller, this paper revisits the optimal insurance contract design problem in terms of Pareto optimality with imposing these practical constraints. Pareto optimal insurance contracts, with indemnity schedule and premium payment, are solved in the cases when the risk preferences of the buyer and seller are given by Value-at-Risk or Tail Value-at-Risk. The effect of our constraints and the relative bargaining powers of the buyer and seller on the Pareto optimal insurance contracts are highlighted. Numerical experiments are employed to further examine these effects for some given risk preferences.     

Default swap games driven by spectrally negative Lévy processes

This paper studies game-type credit default swaps that allow the protection buyer and seller to raise or reduce their respective positions once prior to default. This leads to the study of an optimal stopping game subject to early default termination. Under a structural credit risk model based on spectrally negative Lévy processes, we apply the principles of smooth and continuous fit to identify the equilibrium exercise strategies for the buyer and the seller. We then rigorously prove the existence of the Nash equilibrium and compute the contract value at equilibrium. Numerical examples are provided to illustrate the impacts of default risk and other contractual features on the players’ exercise timing at equilibrium.     

Assignment markets that are uniquely determined by their core

A matrix A defines an assignment market, where each row represents a buyer and each column a seller. If buyer i is matched with seller j, the market produces a_ij units of utility. Quint (1991) points out that usually many different assignment matrices exist that define markets with the same core and poses the question of when the matrix is uniquely determined by the core of the related market. We characterize these matrices in terms of a strong form of the doubly dominant diagonal property. A matching between buyers and sellers is optimal if it produces the maximum units of utility. Our characterization allows us to show that the number of optimal matchings in markets uniquely characterized by their core is a power of two.     

Equilibrium points of a class of many-person games

This paper deals with the competitive equilibrium points of a class of (n+1)-person games, the players beingn sellers and a single buyer. Each seller offers a price for his product; depending upon the prices offered, the buyer then determines the amount he wishes to buy from each seller. Two types of equilibrium points in this class of games are discussed.The authors are indebted to Professor G. Leitmann for his valuable comments on an earlier version of this paper.     

Strategic capability investments and competition for supply contracts

Suppliers often make proactive investments to strategically position themselves to win contracts with a large buyer. Such investments reduce the suppliers’ variable costs of serving the buyer’s demand. We show that an auction mechanism does not always benefit the buyer, the supply chain, or the society. We identify scenarios where the buyer can implement the supply chain and socially optimal solution by committing to a bilateral relationship with fair reimbursement, and forgoing the benefits of competition altogether. We explore the role of commitment by the buyer (to a procurement mechanism) and by the suppliers (to an investment level) by analyzing different timing games under symmetric and asymmetric information about suppliers’ types. We show that it never benefits anyone for the suppliers to commit first. Equilibrium investments and cost structures depend upon the buyer’s bargaining power (opportunity cost). However, the winning supplier’s investments are almost always below the supply chain optimal level.     

Optimal XL-insurance under Wasserstein-type ambiguity

We study the problem of optimal insurance contract design for risk management under a budget constraint. The contract holder takes into consideration that the loss distribution is not entirely known and therefore faces an ambiguity problem. For a given set of models, we formulate a minimax optimization problem of finding an optimal insurance contract that minimizes the distortion risk functional of the retained loss with premium limitation. We demonstrate that under the average value-at-risk measure, the entrance-excess of loss contracts are optimal under ambiguity, and we solve the distributionally robust optimal contract-design problem. It is assumed that the insurance premium is calculated according to a given baseline loss distribution and that the ambiguity set of possible distributions forms a neighborhood of the baseline distribution. To this end, we introduce a contorted Wasserstein distance. This distance is finer in the tails of the distributions compared to the usual Wasserstein distance.     

Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction.