Table 2.

Examples of problems in the value allocation task when expected values of allocation choices are equal

I: Core problem pitting fuzzy trace theory’s prediction (Pwin) against cumulative prospect theory’s prediction (Lmin)
Outcome	Probability	Choose one of two allocations	Strategy	Categorical gist representation	Categorical gist principle
$75	0.20
$35	0.20
$0	0.20	+$15 = $15	Pwin	No gain to some gain.	Some gain is better than no gain.
−$25	0.20
−$75	0.20	+$15 = −$60	Lmin	Some loss to some loss.	No categorical improvement: not preferred.
II: Core problem pitting fuzzy trace theory’s prediction (Pwin) against expected utility theory’s prediction (Gmax)
Outcome	Probability	Choose one of two allocations	Strategy	Categorical gist representation	Categorical gist principle
$75	0.20	+$15 = $90	Gmax	Some gain to some gain.	No categorical improvement: not preferred.
$35	0.20
$0	0.20	+$15 = $15	Pwin	No gain to some gain.	Some gain is better than no gain.
−$25	0.20
−$75	0.20
III: Pwin unavailable problem indicating fuzzy trace theory’s predictions (Lmin and problem I differs from III) and cumulative prospect theory’s predictions (Lmin and problem I is virtually identical to III)
Outcome	Probability	Choose one of two allocations	Strategy	Categorical gist representation	Categorical gist principle
$75	0.20
$35	0.20
$5	0.20	+$15 = $20	Reference outcome (former Pwin)	Some gain to some gain.	No categorical improvement: not preferred.
−$25	0.20
−$75	0.20	+$15 = −$60	Lmin; Problem I differed from Problem III	Some loss to some loss: Revert to more precise representations; see Allais Problem 2 in Reyna & Brainerd, 2011; Reyna & Brust-Renck, 2020.	Losses hurt more than gains feel good (CPT).
IV: Pwin exaggerated problem indicating fuzzy trace theory’s prediction (exaggerated Pwin) and cumulative prospect theory’s prediction (Lmin)
Outcome (only total shown)	Probability	Choose one of two allocations	Strategy	Categorical gist representation	Categorical gist principle
$75 + $70 = $145	0.20
$35 + $70 = $105	0.20
$0 + $70 = $70	0.20	+$15 = $85	Reference outcome (former Pwin)	Some gain to some gain.	No categorical improvement: not preferred.
−$25 + $70 = $50	0.20
−$75 + $70 = −$5	0.20	+$15 = $10	FTT: Pwin+ CPT: Lmin	Some loss to some gain: Gamble becomes certain gain (possible loss eliminated).	Some gain is better than some loss. Categorical improvement: preferred.

Modal observed preference indicated by bolding (see text). According to fuzzy-trace theory (FTT), uncertainty at the categorical gist level of representation means that an option presents more than one qualitatively different outcome: no gain vs. gain, no loss vs. loss, and gain vs. loss. Contrary to Venkatraman et al. (2009, p. 9), the categorical gist explanation here does “apply to risky choice problems that involve only ‘pure’ gain or loss options,” introduced in Reyna and Brainerd (1991). Pwin unavailable problems also rule out expected value and expected utility theory because a small amount added or subtracted from the intermediate outcome does not appreciably change overall value or utility, and yet large differences in preference are observed (per FTT). Additionally, Cumulative Prospect Theory (CPT) predicts that preferences should not change as a function of whether adding money to the intermediate outcome changes its valence (e.g., improving from −10 to $10 or improving from −$30 to −$10) due to the rank-dependent transformations applied to the extreme outcomes (as long as intermediate outcomes are not higher in probability; Venkatraman et al., 2014, p. 75). However, when the intermediate probability is lowered (not shown), and thus the expected value of the allocation is lower compared with the other option (Gmax or Lmin), Pwin should be avoided according to expected value, expected utility, and CPT, whereas FTT predicts that Pwin remains most preferred though at a smaller margin because of the competing expected value (verbatim) representation (see Allais Problem 1 in Broniatowski & Reyna, 2018; Reyna & Brust-Renck, 2020). Eye tracking and developmental differences further differentiate FTT’s and CPT’s predictions, supporting FTT and ruling out CPT (e.g., Kwak et al., 2015; Payne, 2005; Venkatraman et al., 2014)