Probability weighting explains Independence Axiom violations of Expected Utility Theory in monkeys

Expected Utility Theory (EUT) provides axioms for maximizing utility in risky choice. The independence axiom (IA) is its most demanding axiom: preferences between two options should not change when altering both options equally by mixing them with a common gamble. We tested common consequence (CC) and common ratio (CR) violations of the IA in thousands of stochastic choice over several months using a large variety of binary option sets. Three monkeys showed few outright Preference Reversals (8%) but substantial graded Preference Changes (46%) between the initial preferred gamble and the corresponding altered gamble. Linear Discriminant Analysis (LDA) indicated that gamble probabilities predicted most Preference Changes in CC (72%) and CR (87%) tests. The Akaike Information Criterion indicated that probability weighting within Cumulative Prospect Theory (CPT) explained choices better than models using Expected Value (EV) or EUT. Fitting by utility and probability weighting functions of CPT resulted in nonlinear and non-parallel indifference curves (IC) in the Marschak-Machina triangle and suggested IA non-compliance of models using EV or EUT. Indeed, CPT models predicted Preference Changes better than EV and EUT models. Indifference points in out-of-sample tests were closer to CPT-estimated ICs than EV and EUT ICs. Finally, while the few outright Preference Reversals may reflect the long experience of our monkeys, their more graded Preference Changes corresponded to those reported for humans. In benefitting from the wide testing possibilities in monkeys, our stringent axiomatic tests contribute critical information about risky decision-making and serves as basis for investigating neuronal decision mechanisms.

S. We characterized the changes with a leave-one-out procedure using Linear Discriminant Analysis (LDA). We trained the LDA with all data except for those from the predicted leave-out 3 0 6 choice tests to build 36 models (18 CC tests and 18 CR tests) for each animal (we discarded one 3 0 7 option set in CR tests with Monkey C in which S was zero). As each of the 36 models was used to 3 0 8 predict the left-out data, we obtained 36 predictions for each animal. We compared these 3 0 9 predictions to the measured directions of Preference Change to check the accuracy of the prediction. To illustrate the test sensitivity (true positive rate / ability to predict one class) and specificity (false 3 1 1 positive rate / ability to predict the other class), we drew a confusion matrix for the CC and CR tests, 3 1 2 separately for each animal (see Fig. 4). choice probability according to Eq. 7.

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In the EV model, each option's value was its objective Expected Value: For a generic three outcome gamble in our task, it corresponded to (݉ ଵ was zero in our task and In the EUT model, each option's value was defined via the utility function (u): In our gambles' space this mapped to: index i corresponding to the outcomes ordered from worst to best (݉ ଵ and ݉ ଷ respectively, in our , Eq. 12 becomes: which, with our set of magnitudes and normalized utility, corresponds to In these three value-estimating equations, each ‫‬ represents the probability of getting the respective magnitudes may result in more complex utility functions, a power function would be sufficient to 3 5 2 account for the difference in subjective evaluation of the three reward magnitudes used in our study.

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In the CPT model, cumulative probability weighting was defined as a two-parameter Prelec 3 5 4 function (Prelec) as used before (Ferrari-Toniolo et al., 2019): where ߙ allows the function to vary from inverse-S-shaped (ߙ shifts the function vertically.

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We estimated the functions' parameters (ߠ) with the maximum likelihood estimation (MLE) 3 5 9 method, by maximizing the log-likelihood function defined (for a choice between generic options A 3 6 0 and B; using fminsearch in Matlab) as: To validate our economic models, we used an out-of-sample dataset that consisted of a set of 3 6 5 gambles that differed from the gamble set used for the IA tests. We presented monkeys with choices 3 6 6 between one fixed option (J) on the x axis (p1 between 0 and 0.8 in 0.2 increments) or on the y axis 3 6 7 (p3 between 0.2 and 0.8 in 0.2 increments) and another option (K) with variable p1 (and p3) and probability p1 in option K, we identified an indifference point (IP) as the point within the triangle   3  7  0 where a fitted softmax preference function would take the value of 0.5. All choice trials in the out- of-sample test were pseudo-randomly intermingled. IPs were estimated separately in each 3 7 2 weekdaily session. Comparison with human choices. We tested whether the observed IA violations in the 18 CC tests using data pooled from all three monkeys. We used two different comparison methods, a confusion 3 7 7 matrix using binary classes of Preference Change (either S > 0 or S < 0), and a Pearson correlation using real-number Preference Changes (S varying between -0.5 and +0.5). However, the gambles 3 7 9 used in our monkeys differed from the gambles used in the human studies (see Fig. 7A). Therefore, S that would have occurred in the human studies had they used the same gambles as we did in our 3 8 4 monkeys.

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For the confusion matrix, we predicted the S's for the unused gambles with an LDA reverse direction, for predicting the human S's for the gambles used in our monkeys, we trained the 3 9 0 LDA with the binary human S's, the probabilities for the low and the high magnitudes of the human 3 9 1 gambles (p1, p3), and the ratio of the middle and high magnitudes of the human gambles (m2 / m3).

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For the Pearson correlation, we predicted the S's for the unused gambles with two different the beta parameters for the S's measured in monkeys as follows: Eq. 18 in the 39 human studies to predict the numeric Preference Change S for these gambles in monkeys: Eq. 19  In the reverse direction, comparing measured monkey S's with predicted human S's, the with the modified regression model: with p1-human and p3-human as probabilities of lowest and highest magnitudes of gamble B, and 4 1 6 m2-human and m3-human as middle and highest magnitude used in humans (magnitudes varied 4 1 7 across the human studies but were constant in all monkey gambles). Then we applied the estimated 4 1 8 betas from Eq. 20 to all gambles used in our monkeys to predict the numeric Preference Change S 4 1 9 for these gambles in humans: with p1-monkey and p3-monkey as probabilities of lowest and highest magnitudes of gamble B, Experimental design. We used stochastic choices to test compliance with the independence axiom (higher was more), and the probability of delivering each magnitude was indicated by bar length 4 3 6 away from stimulus center (longer was higher).

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Testing the IA began with two gambles A and B that formed option set {A,B}. Gamble A 4 3 8 was a degenerate gamble with safe and fixed middle reward magnitude (m2 = 0.25 ml; p2 = 1.0), 4 3 9 whereas gambles B, C and D were two-or three-outcome gambles. The test gambles C and D 4 4 0 derived from the common addition of gamble G and constituted option set {C,D} (Eq. 1).

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Stochastic compliance with the IA requires that preferences do not change significantly between change for a CR test . Conlisk'S and tested for significance using a one-sample t-test against S = 0 in pooled sessions from 4 6 8 a given monkey (P < 0.05). In contrast to the few outright Preference Reversals, significant 4 6 9 Preference Changes using the metric S were rather frequent in all animals (N=21 for Monkey A, N=12 for Monkey B, N=17 for Monkey C; total of 46%; Table 2). preserve their preference within their respective option sets and thus produce no violation. Mean, SEM; all P < 0.05; one-sample t-test) ( Fig. 2A). The strongest negative S's were also measured absolute S's were insignificant (Fig. 3C). Fig. S2 shows the full pattern of Preference Changes in all CR tests. To summarize, all monkeys showed significant Preference Changes in both CC and CR  outcomes, irrespective of particular preferences between the initial gambles A and B.

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For the CC test, we varied the probability of the low outcome of gamble B (Bp1; i.e. the probability of receiving 0.5 ml in option B; thus, Bp2 = 1 -Bp3 -Bp1). We defined gambles C and for each monkey; 72% for Monkey A, 50% for Monkey B and 61% for Monkey C) (Fig. 4A insets).

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These results suggested a systematic relationship between Preference Changes and reward 5 1 7 probabilities in the CC test.

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For the CR test, we varied the ratio r and the high-outcome probability in gamble B (Bp3; i.e. was no less than prediction with majority class (94% for Monkey A, 67% for Monkey B, 82% for 5 2 7 Monkey C) (Fig. 4B  depended on gamble probabilities.

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The systematic nature of the observed Preference Changes in both the CC tests and the CR 5 3 0 test encouraged us to model the observed changes mathematically using economic theory.

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Therefore, we next fitted our data using different economic choice models and tested whether these 5 3 2 models might explain the observed violations. to stochastic implementations of basic constructs of three economic theories: objective Expected Value (EV), Expected Utility Theory (EUT) and Cumulative Prospect Theory (CPT). difference between the two options and includes a noise term that accounts for variability in choices.

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The difference between the choice models consisted of different value computations: in the EV were smaller for the CPT model compared to the EV and EUT models (Fig. 6B), which indicated 6 0 1 that CPT captured the out-of-sample IPs more accurately than the other models. (see Methods below Eq. 6). We thus investigated in more detail how much the EUT and EV models  As a further control, we explicitly tested the hypothesis of ICs being linear and parallel, as method has never been used to investigate EUT. In the current study, we tested separately the points in each IC that were estimated with linear least squares using out-of-sample IPs (one-sample 6 2 0 t-test against 0; Fig. S6C). We found significant non-linearity (p<0.001) and non-parallelism 6 2 1 (p<0.05) for some ICs, suggesting that EUT was not able to capture the subjective values for Taken together, these results showed that the CPT-based economic choice model predicted 6 2 7 the IA violations in both CC and CR tests, outperforming both the EV and the EUT models. These weighting of reward probabilities, in line with the explanation provided by CPT. Comparison of Preference Changes with humans. To explore the possibility of common p3 of gamble B), were mostly concentrated in the lower left area (Fig. 7A). The human studies 6 3 6 reported significant Preference Changes characterized by S > 0 or S < 0, as well as insignificant 6 3 7 changes (S ~ 0) (Fig. 7B).

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We used two methods to compare our monkey data with the published human data, a Eqs. 18 -21).

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To test the robustness of these comparisons, we reversed the direction of predicting S's for corresponded to the LDA-predicted human S's with 70% accuracy, which exceeded random (50%) 6 5 6 and majority class (61%) (Fig. 7D left). The Pearson correlation between the measured monkey S's Thus, while we saw less Preference Reversals than are generally reported in humans, the between risky and riskless choice in rhesus monkeys. Anim Cog (in press). Harless BYDW, Camerer CF (1994) The predictive utility of generalized Expected Utility Theories.