Empirical evaluation of four models of buying and selling prices of gambles

Judges assigned values to gambles from viewpoints of buyers (willingness to pay) and sellers (willingness to accept). Consistent with previous results, selling prices exceed buying prices, and these two judgments are notmonotonically related to each other. There are systematic violations of consequencemonotonicity when the consequence of zero is increased to a small positive value. Models based on loss aversion combined with cumulative prospect theory (CPT) do not give accurate accounts of the data. In particular, judgments violate complementary symmetry, which is implied by third-generation prospect theory. In addition, there are violations of first order stochastic dominance in judgments of three-branch gambles. Models based on the theory of joint receipts by R. D. Luce fit better than third-generation prospect theory, but the best-fitting of six suchmodels does not give an adequate account of judgments involving a lowest consequence that might be zero or positive. Two configural weight models give better fits to the data using the same number or fewer parameters estimated from the data. © 2016 Elsevier Inc. All rights reserved. When people are asked to judge the highest price they would pay to buy something, they report a much lower value than when they are asked to judge the least they would accept to sell the same thing. This finding has been observed for both goods of uncertain or ambiguous value, such as used cars, and for gambles with defined probabilities and consequences (Birnbaum& Stegner, 1979; Coombs, Bezembinder, & Goode, 1967). The discrepancy between judged willingness to pay (WTP) and willingness to accept (WTA) is surprisingly large from the perspective of classical expected utility theory in economics (Horowitz & McConnell, 2002; Knetsch & Sinden, 1984). The same phenomenon has been discussed in different isolated segments of the scientific literature and attributed to different sources. Birnbaum and Stegner (1979) referred to it as an effect of the judge’s point of view and tested rank-affected configural weight models. Thaler (1980) used the term ‘‘endowment effect’’ and theorized that it might be due to loss aversion. It is also known as a contingent valuation effect (Irwin, Slovic, Lichtenstein, & ∗ Correspondence to: Department of Psychology, H-830M, California State University, Fullerton, P.O. Box 6846, Fullerton, CA 92834-6846, United States. Tel.: +1 657 278 2102; fax: +1 657 278 7134. E-mail address:[email protected] (M.H. Birnbaum). http://dx.doi.org/10.1016/j.jmp.2016.05.007 0022-2496/© 2016 Elsevier Inc. All rights reserved. McClelland, 1993). Luce (1991, 2000) theorized that the difference could be described by a joint receipt model in which buying involved the joint receipt of a positive gain of the item purchased and the loss of the price paid and selling involved a loss of the item sold and a gain of the purchase price. This article brings together and compares models that have arisen in different branches of the scientific literature. We report an experiment that evaluates them as empirical representations. Before presenting the models, it is useful to cite briefly three approaches to this topic, because recent reviews have focused on only part of the literature. For example, Erickson, Marzilli, and Fuster (2014), in the Annual Review of Economics, do not cite research on configural weighting and the point of view effect, and althoughMorewedge andGiblin (2015) take a broader perspective, like Erickson et al., they do not cite Duncan Luce, despite the relevance of his work. The three groups of models considered here are (1) prospect theory loss aversion models (Birnbaum & Zimmermann, 1998; Kahneman, Knetsch, & Thaler, 1990; Schmidt, Starmer, & Sugden, 2008; Tversky&Kahneman, 1991, 1992)which assume that buyers and sellers experience different patterns of losses and gains, even when buying or selling objects or lotteries that are strictly positive; (2) joint receipt models (Luce, 1991, 2000), which assume that the price paid to buy a lottery or to sell one are integrated into 184 M.H. Birnbaum et al. / Journal of Mathematical Psychology 75 (2016) 183–193 the consequences by means of a joint receipt operation; and (3) configural weight models (Birnbaum, 1982; Birnbaum, Coffey, Mellers, &Weiss, 1992; Birnbaum & Stegner, 1979), which assume that buyers assign higher weight to lower-valued consequences, estimates, or attributes of an entity than do sellers. 1. Theoretical approaches 1.1. Cumulative prospect theory and loss aversion Let g = (x, p; y) represent a binary gamble (lottery) with probability p to receive x and otherwise receive y, where x > y. The CPT model represents the utility of g as follows: U(x, p; y) = U(x)W (p) + U(y)[1 − W(p)] if (x ≥ y ≥ 0) U(x)W(p) + U(y)W(1 − p) if (x ≥ 0 ≥ y) U(x)[1 − W(1 − p)] + U(y)W(1 − p) if (0 ≥ x ≥ y). (1) In these expressions,U(x) is the utility (value) function;we assume U(0) = 0; W(p) and W(p) are the weighting functions for a gain or loss received with probability p, respectively; W(0) = W(0) = 0 andW(1) = W(1) = 1. Loss aversion has most commonly been modeled as follows: U(−x) = −λU(x), x > 0, (2) where U(−x) is the utility of a loss of x (x > 0), and λ is the factor by which losses are said to ‘‘loom larger’’ than gains, when λ > 1. The term, loss aversion, unfortunately, has been used in two different ways that may have created confusion in the literature: first, it has been used to refer to the behavioral phenomenon of risk aversion for certain mixed gambles; second, it has been used to refer to the theory that empirical findings of risk aversion for such mixed gambles are due to the shape of the utility function. A rival explanation for the behavioral phenomena is that negative consequences receive greater weight (e.g., Birnbaum & Bahra, 2007), rather than more extreme utility. It has been reported that choices satisfy reflection, which is the behavioral property that if gamble g = (x, p; y), where x > y > 0, is preferred to gamble f , then gamble −f is preferred to gamble −g , where −g = (−x, p; −y). If Eq. (2) holds and if W(p) = W(p), then reflection would be satisfied (Tversky & Kahneman, 1992). A more general form of ‘‘loss aversion’’ that need not satisfy reflection is considered in the discussion. 1.1.1. Models refuted by two types of preference reversals Within the framework of prospect theory, various models for buying and selling prices of gambles can be constructed. However, two of these have already been rejected by considerable evidence, and it is worth noting why these theories do not work and have been rejected. Let b(g) and s(g) represent the highest buying price and lowest selling price for gamble g , respectively (when gamble, g , is fixed, we use b and s, for simplicity). First, one might theorize that people are willing to buy a gamble whenever U(g) ≥ U(b) and to sell whenever U(s) ≥ U(g). However, this model implies that if a person sets a higher price on gamble g than f then U(g) > U(f ), so the person should prefer g to f . However, there are gambles for which many individuals systematically set a higher selling or buying price on g than f and then prefer f over g in direct choice (Johnson & Busemeyer, 2005; Lichtenstein & Slovic, 1971; Mellers, Chang, Birnbaum, & Ordóñez, 1992; Mellers, Ordóñez, & Birnbaum, 1992; Tversky, Sattath, & Slovic, 1988). Such results are called preference reversals because one way of comparing the utility of the gambles (direct choice) yields one preference relation, and another method (selling or buying price) yields an apparent contradiction. Furthermore, if buying and selling prices are each equal to the gamble’s utility, then they should be equal to each other. Therefore, this first model can be rejected. A second theory was proposed by Tversky and Kahneman (1991) for goods. Extended to gambles, the buying price is assumed to reflect an implicit comparison between the positive utility of receiving the gamble against the negative utility of (losing) the buying price, U(−b). Similarly, selling price reflects an implicit comparison between the positive utility of receiving the sales price versus the loss of the gamble, U(−g), where U(−g) represents the utility of −g = (−x, p; −y). In contrast to the first approach, this model involves comparisons between gains and losses, whereas the first approach involved only positive values. According to this theory, U(g) + U(−b) = U(0); (3) U(s) + U(−g) = U(0). (4) Birnbaum and Zimmermann (1998, p. 176–178) proved that if we were to assume Eqs. (3) and (4), CPT (Eq. (1)) and loss aversion (Eq. (2)), and if W(p) = W(p) (as assumed by Kahneman & Tversky, 1979 and reported as a good approximation by Tversky & Kahneman, 1992), it would follow that U(s) = λ2U(b) ; therefore, s = U−1[U(b)λ2] ; where U−1 is the inverse of U(x). Thus, this model implies that selling and buying prices should be monotonically related to each other. Johnson and Busemeyer (2005) presented a similar proof, and concur that preference reversals between buying and selling prices in Birnbaum and Beeghley (1997) refute this (extended) model of Tversky and Kahneman (1991). If in addition to Eqs. (1), (2), (3), and (4) we also assumed that U(x) = x , as in Tversky and Kahneman (1992); it would follow that s = λ2/βb, so the ratio of selling price to buying price (WTA/WTP) should be a constant. Kahneman et al. (1990) listed empirical values of the ratio of WTA/WTP = s/b from different studies, with a median slightly exceeding 4. If β = 0.88 (Tversky & Kahneman, 1992), then s/b = 4 implies λ = 1.84. Tversky and Kahneman (1991) realized that their theory implied that WTA/WTP of a $5 bill must also be 4, so they postulated exceptions for cash or goods held for exchange. More damaging than the need for such exceptions, data from several studies reported that selling prices and buying prices are not even monotonically related to each other, which contradicts this model (Birnbaum, 1982, p. 470–472; Birnbaum & Stegner, 1979). For example, Birnbaum and Sutton (1992) found that from the viewpoint of the seller, g = ($96, 0.5; $0) is judged by the majority of individuals higher than f = ($48; 0.5; $36); whereas, from the viewpoint of the buyer, g is judged lower than f by the majority of the same individuals. Such reversals between buying and selling prices represent a second type of preference reversal that has been found in several other studies (Birnbaum&Beeghley, 1997; Birnbaum et al., 1992; Birnbaum & Zimmermann, 1998). These preference reversals between WTA and WTP rule out this model of loss aversion and any other model in which the ratio, WTA/WTP, is constant. Having ruled out these two models, we next take up another approach that is not rejected by these two types of preference reversals. 1.1.2. Third-generation prospect theory This third approach, formulated by Birnbaumand Zimmermann (1998, p. 178–180) and independently by Schmidt et al. (2008), treats decisions to buy or sell as the result of an adjustment of the consequences of the gambles to reflect buying or selling prices. In the case of buying prices, the buyer is presumed to evaluate a new M.H. Birnbaum et al. / Journal of Mathematical Psychology 75 (2016) 183–193 185 gamble, which we denote g − b = (x − b, p; y − b), where x − b is the profit if the gamble wins and y − b is the loss if the gamble yields only y. Similarly, the seller considers it a ‘‘loss’’ of x − s if the gamble might win x, since the seller gave up the opportunity to win; and if the gamble pays only y, the seller considers that a ‘‘win’’ of s−y. In decisions to sell, the integrated gamble is denoted as s − g = (s − x, p; s − y); therefore, U(g − b) = U(0); (5) U(s − g) = U(0). (6) Definition. Third-generation prospect theory (TGPT) consists of the following assumptions: Eqs. (5) and (6), CPT (Eq. (1)), Eq. (2) (loss aversion), and U(x) = x . Theorem 1. For gambles g = (x, p; y) and g ′ = (x, 1 − p; y), TGPT implies b(g) + s(g ) = x + y, (7) where b(g) and s(g ) represent the buying and selling prices for gamble g and g . Expression (7) is called complementary symmetry, even though the complementary gambles, g and g′ might be played independently on different trials. Intuitively, this property follows from the symmetry between buyers and sellers: a loss to one is a gain to the other. Proof. From (5) and (2),and because U(0) = 0, highest buying prices of gamble, g = (x, p; y), x > y > 0, are given by the following: 0 = W(p)U(x − b) + W(1 − p)U(y − b) = W(p)U(x − b) − W(1 − p)λU(b − y).