An incentive compatible conjoint ranking mechanism.

Conjoint analysis is one of the most popular marketing research tools, used in hundreds if not thousands of academic and business research studies, e.g., see Green, Krieger, and Wind (2001) or Green and Srinivasan (1990) for reviews. Conjoint analysis typically involves people rating, ranking, or choosing among various options that differ by several attributes so as to elicit consumer preferences, estimate demand, and/or forecast market share. Until very recently, virtually all conjoint applications were hypothetical. That is, a person's ranking, rating, or choice had no immediate financial consequence. At worst, hypothetical valuation questions are open to strategic manipulation on the part of the participant, and at the least they do not provide incentives for people to put cognitive effort into their decisions. Over the past twenty years, a wealth of evidence has accumulated indicating that responses to hypothetical willingness to pay and purchase intention questions are not always consistent with decisions when real money is on the line. For example, see the Meta analyses in List and Gallet (2001) and Murphy et al. (2005).

Such findings have led researchers to investigate methods for making traditional conjoint methods incentive compatible (IC). (1) One solution, proposed by Alfnes et al. (2006), Carlsson and Martinsson (2001), Ding, Grewal, and Liechty (2005), and Lusk and Schroeder (2004) is to make use of traditional choice-based conjoint analysis methods, but to randomly select one of several repeated choices between competing product profiles as binding. The participant purchases the product indicated as most preferred in the randomly selected choice set. Although this approach represents a useful step forward, it is limited to choice-based applications. Choice-based conjoint analysis has many advantageous properties, but it is lacking on at least one front: it is informationally inefficient relative to ranking-based conjoint applications. In a choice-based conjoint application, all that is known is which one option is most preferred out of a set of options. By contrast, ranking-based conjoint applications provide information on the consumer's complete preference ordering. These facts coupled with the observation that ranking-based conjoint applications remain widespread, e.g., see Baker and Burnham (2001) or Sayadi, Roa, and Requena (2005) for recent examples, suggest it is prudent to identify methods for making ranking-based conjoint methods IC.

The objectives of this study are: (1) to introduce a mechanism for making conjoint rankings IC and (2) to empirically compare traditional, hypothetical conjoint ranking responses to those using the new IC conjoint mechanism. The new mechanism entails people ranking profiles as in traditional conjoint analysis. However, unlike traditional conjoint analysis, an individual actually purchases a product profile with a probability proportional to its assigned rank. We illustrate a simple and straightforward method to implement the IC conjoint ranking mechanism in an empirical application related to consumer preferences for beef attributes.

An Incentive Compatible Conjoint Ranking Mechanism

Consider a mechanism where individual i is asked to rank J products that differ in terms of a vector of attributes [X.sub.j]. The utility individual i derives from product j is

(1) [V.sub.ij] = [[beta].sub.i] [X.sub.j] - [[gamma].sub.i] [P.sub.j]

where [[beta].sub.i] is a vector of marginal utilities, [P.sub.j] is the price of alternative j, and [[gamma].sub.i] is the marginal utility of income. Index the products such that [V.sub.i1] > [V.sub.i2] > ... > [V.sub.i(J-1)] > [V.sub.iJ]. That is, product j = 1 is the most preferred product, j = 2 is the next most preferred product, and so on. The goal is to create a mechanism in which an individual has an economic incentive to truthfully reveal their preference ranking for the J goods. Stated differently, a mechanism is needed wherein an individual's expected utility is maximized when the products are ranked such that product j = 1 receives a rank of 1, product j = 2 receives a rank of 2, and so on.

Let [r.sub.j] represent the ranking of product j such that [r.sub.1] is the ranking of product 1, [r.sub.2] is the ranking of product 2, and so on. An IC mechanism can be constructed by requesting individuals to rank the J products where there is a J + 1 - [r.sub.j]/[[summation].sup.J.sub.j=1] x 100% chance of actually receiving product j and paying [P.sub.j]. The formula implies there is a higher chance that an individual receives a product with a lower rank than a product with a higher rank. For example, if there were five products to be considered, the mechanism would ensure that there would be a ((5 + 1 - 1)/15) x 100 = 33.3% chance of an individual actually receiving the product ranked first, a ((5 + 1 - 2)/15) x 100 = 26.7% chance the product ranked second would be received, and so on. The equation stated above also ensures that there is a 100% chance that one of the products will actually be purchased (although as will be shown in the empirical application, one of the ranked options can be a no-purchase option). (2)

An individual's expected utility from ranking J products is:

(2) E[U.sub.i] = [J.summation over (j=1)](J + 1 - [r.sub.j]/[[summation over].sup.J.sub.j=1] j) [V.sub.ij].

Equation (2) shows that the expected utility derived from ranking J goods is the sum of the probability of receiving product j times the utility received from purchasing product j. An individual's goal is to rank the products in a way so as to maximize expected utility given in equation (2). It should be clear that expected utility is maximized when an individual assigns the highest rank to product j = 1, the second highest rank to product j = 2, and so on. This result arises because product I was previously defined as the most preferred product, product 2 as the second most preferred product, and so on. Expected utility cannot be made higher by assigning a more preferred product a higher rank. This implies that the mechanism is IC; an individual's best strategy in terms of maximizing expected utility is to assign the most preferred product the lowest rank, the second most preferred product the second lowest rank, and so on. The mechanism is also IC under a variety of other nonexpected utility models. For example, under prospect theor.y (Kahneman and Tversky 1979), individuals mis-perceive probabilities such that probabilities, p, are weighted, w(p), in such a way that low probabilities are over-weighted and high probabilities events are under-weighted. However, so long as [partial derivative]w(p)/[partial derivative]p > 0, as is assumed in prospect theory, equation (2) is maximized by ranking the most preferred product first, the second most preferred product second, and so on.

Although this mechanism might, at first, seem a bit complicated to utilize in an empirical setting, it is actually quite easy to implement. In what follows, we briefly describe one such way the mechanism can be implemented. In many conjoint ranking applications, individuals are given a set of cards describing each product and are requested to sort them in terms of their preference ranking. One easy way to make such a task IC is to create a spinning wheel of the sort found in board games or seen on the Wheel of Fortune, where the wheel is divided into a number of "slices" that differ in size. To facilitate decision making, the slices can be ordered such that the largest slice appears adjacent to the second largest slice, which appears adjacent to the third largest slice, etc. Instead of simply requesting that individuals hypothetical sort the cards, as is typically done, the following steps can be taken: (1) the individual places each card on a slice on the wheel, where the number of slices exactly equals the number of cards; (2) after all cards are allocated, the wheel is spun; and (3) the individual purchases the product indicated by the fixed-pointer around which the wheel is spun. Of course, any number of other implementation methods could be used that would ensure a profile with a lower rank has a higher probability of being purchased. (3)

Empirical Application

In what follows, we describe an empirical application of the IC conjoint ranking mechanism and test whether results from such a mechanism are similar to results from traditional, hypothetical conjoint rankings. We also explore whether making the conjoint task IC interacts with two other treatment variables: information and product type.

Subjects

Participants were recruited near the meat counter in a suburban grocery store located in the southeastern United States Harrison and List (2004) discuss the advantages of carrying out value elicitation in a field setting such as a grocery store. Subjects were offered the chance to enter a drawing for $500 in free groceries in exchange for participation in the study and, as will be described in more detail, some subjects were offered free cuts of meat for participation. A total of 515 subjects took part in the study.

Procedures