The curse of hope

Violations of first-order stochastic dominance are observed in the experimental literature, but to the best of our knowledge, none directly relates to the role of expectations in the formation of reference points. For instance, Butler et al. (2014) link such violations to the complexity of the presentation format of the lotteries, and Birnbaum (2005) to cognitive processes (attention transfers). Relatedly, Leland (1998) shows the role played by the presentation format of lotteries to obtain such violations. Another instance of stochastically dominated choices is provided by Gneezy et al. (2006). But, for their findings, dominated-choices disappear in the within-subject treatment and the replication attempts have given mixed results (Simonsohn 2009; Rydval et al. 2009; Keren and Willemsen 2009). Perhaps the closest evidence for violations related to reference point is given by Loomes et al. (1992). Indeed, as stated by KR in their 2007’s paper, there exists a close link between their model and regret theory. Nevertheless, Loomes et al. (1992) find only mixed support for regret theory based on the violations observed. Generally speaking, none of the experiments referred to here concern future lotteries but present ones, none of them relies on lagged expectations. As will be clear in the second section, the type of violations predicted by KR concerns choices between simple (binary) lotteries, for which dominance is transparent, i.e., when the framing and the choice set make it evident to the decision-maker that one lottery dominates the other (Fishburn 1978).

It should be noted however that the results by Abeler et al. (2011) and Ericson and Fuster (2011) are not replicated by Camerer et al. (2016). Some other experiments do not find much, if any, effect of expectations, for instance (Heffetz and List 2014).

Some of our results can be generalized to more complex discrete lotteries, and a section of the Appendix is dedicated to this issue.

Using Eqs. 1 and 2, \(U(L_p|L_p)=pu(y|L_p)+(1-p)u(x|L_p)\) with \(u(y|L_p)=m(y)+(1-p)\mu (m(y)-m(x))\) and \(u(x|L_p)=m(x)+p\mu (m(x)-m(y))\).

Since by assumption, \(\mu '>0\), there exists some factor a large enough for \(a\mu '_+(0)>\frac{1}{\lambda -1}\). This is what Proposition 7 of KR (2007) also shows when m is linear. In the linear case, see also Masatlioglu and Raymond (2016) for conditions on loss aversion that avoid violations of stochastic dominance.

Another possibility is to assume that the reference-dependent component is weaker than the intrinsic utility. Indeed, as stated indirectly by Proposition 3, to fully suppress the curse of hope, it is enough that \(\mu '_+(0)\le \frac{1}{\lambda -1}\). Nevertheless, we see two weaknesses in this solution. First, it imposes restrictions on the strength of the reference-dependent component and the intensity of loss aversion in a rather ad hoc manner. Second, because the gain–loss component captures the standard properties of prospect theory (relevance of the reference point, loss aversion, diminishing sensitivity in the domains of loss and gains), it would also regrettably limit the ability of the model to explain other well-documented phenomena like strong risk-aversion for small stakes, the disposition effect, the endowment effect, etc. when expectations do not play a role in the formation of the reference point, i.e., for “surprise lotteries”.

For incomes or wealth, these normal levels refer, for instance, to the median levels of income or wealth in a reference group, like the family, other workers at the individual’s place of employment, people in the same neighborhood or region, etc. (Clark et al. 2008). For health, the reference level may correspond to the health level that individual considers “acceptable” given their age category (Wouters et al. 2015).

According to the authors, defensive pessimism not only helps individuals to cushion the potential blow of a bad outcome, but motivates them to work hard to prepare for the situation in which they can influence it, making their prediction potentially self-defeating. Given that we restrict our analysis to exogenous probabilities in this paper, we do not explore this second consequence.

One may wonder why the objective probabilities are not transformed. In fact, it is very likely that they are (following rank-dependent models or cumulative prospect theory and the extensive empirical evidence of an inverted-S shaped transformation). Here we focus on the specific transformation that individuals may apply to their stochastic reference point: the type of transformation is conceptually different and in the absence of empirical assessment of this particular transformation and its interaction with the usual lottery probabilities, we treat the simple case where objective probabilities of occurrences are not transformed, while reference point weights are, as a first pass.

In this paper, \(R_{\pi _{i} } =(y,x;\pi _{i} ,1-\pi _{i} )\) is said to be higher than \(R_{\pi _{j} } =(y,x;\pi _{j} ,1-\pi _{j} )\) if \(\pi _{i} >\pi _{j} \).

Note here that although time is present with respect to its influence on the reference point, intertemporal preferences and discounting do not matter because there is no trade-off between periods. The comparison is made between lotteries that are resolved at the same future period.

People suffering from chronic illnesses or disability, report better mood, happiness or Quality of Life (QoL) ratings than what healthy people predict they would feel if facing similar circumstances (see for instance Riis et al. 2005 concerning dialysis), and for this reason, they have different preferences regarding what is an acceptable level of health (Brouwer et al. 2005; Wouters et al. 2015).

People may also overattend to losses and gains because they underestimate how quickly they will adapt to these changes. On both of these accounts, the nature and scope of reference-dependent choices seems to reflect mistakes our fully rational model does not capture (KR 2006).

More in detail (see Appendix), \(A(q)=\frac{q(\Delta m +\Delta \mu _w)}{(1-p)(1-q)}\). Note that it is only defined for \(q\ne 1\) and \(p\ne 1\).

Indeed, we can easily verify that \(U (L_p | R(y,L_p) ) - U(x | R(y,x) ) > U ( L_p | L_p ) - U(x | x)\), that is the relative interest of the lottery (compared to x for sure) is higher in the model with inertia even when y is the initial situation.

Note that this is necessarily the case if \(\mu '_{-}(0)< + \infty \), given that \(\mu '\) decreases in the distance to the reference point, and that \(\mu '_{-}(0)\ge \mu '_{+}(0).\)

A counter-example is given by power functions for which \(\mu '_{-}(0)=\infty \). In this case, the curse of hope cannot disappear fully (or trivially by setting \(q=1\)), but a positive q shrinks the range of stakes (\(y-x\)) where it applies.