**A Decision Analysis Tool for Evaluating Fundraising Tiers.**K.F. McCardle, K. Rajaram, C.S. Tang.

*Decision Analysis*. 6(1): 4-13. March 2009.

This paper presents a utility function model of donors who need to determine their donation to a charity organization that structures and publishes donations by tiers. By considering the prestige associated with each tier level, our analysis suggests that a tiered scheme generates an incentive for donors to raise their donation to the next tier when the originally intended donation is close to the minimum amount for the next tier level. In addition, we develop a decision analysis tool to illustrate how a charity can evaluate the effectiveness of different tier structures. Using publicly available federal tax return data and actual donations to a private high school, we present a method to estimate the model parameters. By taking other non-quantifiable factors into consideration, our tool can guide a charitable organization to determine an effective fundraising tier strategy.

**Bundling Retail Products: Models and Analysis.**K.F. McCardle, K. Rajaram, C.S. Tang.

*European Journal of Operational Research*. 177(2): 1197-1217. 2007.

We consider the impact of bundling products on retail merchandising. We consider two broad classes of retail products: basic and fashion. For these product classes, we develop models to calculate the optimal bundle prices, order quantities, and profits under bundling. We use this analysis to establish conditions and insights under which bundling is profitable. Our analysis confirms that bundling profitability depends on individual product demands, bundling costs, and the nature of the relationship between the demands of the products to be bundled. We also provide detailed numerical examples.

**Joint Coherence: The Infinite Case.**K.F. McCardle. April 2003.

De Finetti (1974) provides an operational definition of coherent subjective probability for an individual decision maker. In essence, a decision maker is coherent according to De Finetti if he does not represent a money pump for an outside observer, i.e., he is not subject to arbitrage. Nau and McCardle (1990) apply a similar no-arbitrage condition to noncooperative games with finite action spaces. Game outcomes that are not subject to arbitrage are called jointly coherent. In this paper we continue the development of joint coherence by considering the case wherein the decision makers have potentially infinite action spaces. We show that an outcome is jointly coherent if and only if it is supported by some finitely additive correlated equilibrium.

**Sex, Lies and the Hillblom Estate: A Decision Analysis.**S.A. Lippman, K.F. McCardle.

*Decision Analysis*. 1(3): 149-166. September 2004.

We present three approaches to evaluating a publicly detailed, high-stakes, high-risk decision made by an heir-claimant to the estate of Larry Hillblom. The first two approaches are standard: a decision tree focusing on risk aversion and a multiperiod consumption model that highlights the effects of foregone consumption. We believe the third approach is novel: we embed several Nash bargaining games into the decision tree developed in the first approach. The three approaches yield three different valuations, an indication of the inherent difficulty in modeling real world decisions.

**Advance Booking Discount Programs under Retail Competition.**K.F. McCardle, K. Rajaram, C.S. Tang.

*Management Science*. 50(5): 701-708. May 2004.

As product demand uncertainty increases and life cycles shorten, retailers respond by developing mechanisms for more accurate demand forecasting and supply planning to avoid over-stocking or under-stocking a product. We consider a situation in which two retailers consider launching one such mechanism, known as the `Advance Booking Discount' (ABD) program. In this program customers are enticed to pre-commit their orders at a discount-selling season. While the ABD program enables the retailers to lock in a portion of the customer demand and use this demand information to develop more accurate forecasts and supply plans, the advance booking discount price reduces profit margin. We analyze the four possible scenarios wherein each of the two firms offer an ABD program or not, and establish conditions under which the unique equilibrium calls for launching the ABD program at both retailers. We also provide a detailed numerical example to illustrate how these conditions are affected by the level of demand uncertainty, demand correlation, market share, and fixed costs for instituting an ABD program.

**Comparative Statics of Cell Phone Plans.**S.A. Lippman, K.F. McCardle.

*Operations Research Letters*. 31: 63-65. 2003.

Employing an order relation that is more restrictive than second-order stochastic dominance, our analysis reveals when increased variability in monthly usage of one's cell phone induces the user to increase the base amount purchased and when increased variability induces the user to decrease the base amount. We also give conditions under which increased variability leads to an increase in expected cost.

**Structural Properties of Stochastic Dynamic Programs.**J.E. Smith, K.F. McCardle.

*Operations Research*. 50(5): 796-809. September-October 2002.

In Markov models of sequential decision processes, one is often interested in showing that the value function is monotonic, convex and/or supermodular in the state variables. These kinds of results can be used to develop a qualitative understanding of the model and characterize how the results will change with changes in model parameters. In this paper we present several fundamental results for establishing these kinds of properties. The results are, in essence, “meta theorems” showing that the value functions satisfy property P if the reward functions satisfy property P and the transition probabilities satisfy a stochastic version of this property. We focus our attention on closed convex cone properties, a large class of properties that includes monotonicity, convexity, and supermodularity, as well as combinations of these and many other properties of interest.

**Options in the Real World: Lessons Learned in Evaluating Oil and Gas Investments.**J.E. Smith, K.F. McCardle.

*Operations Research*. 47(1): 1-15. January-February 1999.

Many firms in the oil and gas business have long used decision analysis techniques to evaluate exploration and development opportunities and have looked at recent development in option pricing theory as potentially offering improvements over the decision analysis approach. Unfortunately, it is difficult to discern the benefits of the options approach from the literature on the topic: most of the published examples greatly oversimplify the kinds of projects encountered in practice, and comparisons are typically made to traditional discounted cash flow analysis, which, unlike the option pricing and decision analytic approaches, does not explicitly consider the uncertainty in project cash flows. A tutorial introduction to option pricing methods is provided, with a focus on how they relate to and can be integrated with decision analysis methods. Some lessons learned in using these methods to evaluate some real oil and gas investments are described.

**Valuing Oil Properties: Integrating Option Pricing and Decision Analysis Approaches.**J.E. Smith, K.F. McCardle.

*Operations Research*. 46(2): 198-217. March-April 1998.

There are two major competing procedures for evaluating risky projects where managerial flexibility plays an important role: one is decision analytic, based on stochastic dynamic programming, and the other is option pricing theory, based on the no-arbitrage theory of financial markets. In this paper, it is shown how these two approaches can be profitably integrated to evaluate oil properties. A model of an oil property is developed and utilized. The decision maker is assumed to be risk averse and can hedge price risks by trading oil futures contracts. Extensions of this model are also described that incorporate additional uncertainties and options, and its use in exploration decisions and in evaluating a portfolio of properties rather than a single property is discussed. The other potential applications of this integrated methodology are briefly described.

**The Stochastic Learning Curve: Optimal Production in the Presence of Learning-Curve Uncertainty.**J.B. Mazzola, K.F. McCardle.

*Operations Research*. 45(3): 440-450. May-June 1997

Theoretical analyses incorporating production learning are typically deterministic: costs are posited to decrease in a known, deterministic fashion as cumulative production increases. A stochastic learning-curve model that incorporates random variation in the decreasing cost function is introduced. A discrete-time, infinite-horizon, dynamic programming formulation of monopolistic production planning when costs follow a learning curve is considered. This basic formulation is then extended to allow for random variation in the learning process. Properties of the resulting optimal policies are also explored. New insights in the deterministic setting are provided.

**The Competitive Newsboy.**S.A. Lippman, K.F. McCardle.

*Operations Research*. 45(1): 54-65. 1997.

A competitive version of the classical newsboy problem is considered, and the impact of competition upon industry inventory is investigated. A splitting rule specifies how initial industry demand is allocated among competing firms and how any excess demand is allocated among firms with remaining inventory. The relation between equilibrium inventory levels and the splitting rule is examined, and conditions are provided under which there is a unique equilibrium. The most general result is that if all excess demand is reallocated, then competition never leads to a decrease in industry inventory.

**A Bayesian Approach to Managing Learning Curve Uncertainty.**J.B. Mazzola, K.F. McCardle.

*Management Science*. 42: 680-692. 1996.

A Bayesian decision theoretic model of optimal production in the presence of learning-curve uncertainty is introduced. The well-known learning-curve model is extended to allow for random variation in the learning process with uncertainty regarding some parameter of the variation. A production run generates excess value (above its current revenue) for a Bayesian manager in 2 ways: 1. It pushes the firm further along the learning curve, increasing the likelihood of lower costs for future runs. 2. It provides information, through the observed costs, that reduces the uncertainty regarding the rate at which costs are decreasing. Conditions under which one of the classical deterministic learning-curve results - namely, that optimal production exceeds the myopic level - carries over to the extended framework are provided. It is demonstrated that another classical deterministic learning-curve result - namely, that optimal production increases with cumulative production - does not hold in the Bayesian setting.