UC Merced Proceedings of the Annual Meeting of the Cognitive Science Society Title A Unified, Resource-Rational Account of the Allais and Ellsberg Paradoxes Permalink https://escholarship.org/uc/item/4p8865bz Journal Proceedings of the Annual Meeting of the Cognitive Science Society, 43(43) ISSN 1069-7977 Authors Nobandegani, Ardavan S. Shultz, Thomas Dubé, Laurette Publication Date 2021 Peer reviewed eScholarship.org Powered by the California Digital Library University of California A Unified, Resource-Rational Account of the Allais and Ellsberg Paradoxes Ardavan S. Nobandegani1;3, Thomas R. Shultz2;3, & Laurette Dube´4 [email protected] fthomas.shultz, [email protected] 1Department of Electrical & Computer Engineering, McGill University 2School of Computer Science, McGill University 3Department of Psychology, McGill University 4Desautels Faculty of Management, McGill University Abstract them can explain both paradoxes (we discuss these models in the Discussion section). Decades of empirical and theoretical research on human decision-making has broadly categorized it into two, separate As a step toward a unified treatment of these two realms: decision-making under risk and decision-making un- types of decision-making, here we investigate whether the der uncertainty, with the Allais paradox and the Ellsberg para- the broad framework of resource-rationality (Nobandegani, dox being a prominent example of each, respectively. In this work, we present the first unified, resource-rational account 2017; Lieder & Griffiths, 2020) could provide a unified ac- of these two paradoxes. Specifically, we show that Nobande- count of the Allais paradox and the Ellsberg paradox. That is, gani et al.’s (2018) sample-based expected utility model pro- we ask if these two paradoxes could be understood in terms vides a unified, process-level account of the two variants of the Allais paradox (the common-consequence effect and the of optimal use of limited cognitive resources. Answering this common-ratio effect) and the Ellsberg paradox. Our work sug- in the affirmative, we show that a resource-rational process gests that the broad framework of resource-rationality could model, sample-based expected utility (SbEU; Nobandegani permit a unified treatment of decision-making under risk and decision-making under uncertainty, thus approaching a unified et al., 2018), provides a unified, process-level account of the account of human decision-making. two variants of the Allais paradox (the common-consequence Keywords: decision-making under risk; decision-making un- effect and the common-ratio effect) and the Ellsberg paradox. der uncertainty; Allais paradox; Ellsberg paradox; resource- This paper is organized as follows. We first elaborate on rationality; sample-based expected utility model the computational underpinnings of SbEU (Sec. 2). We then introduce the Allais paradox (Sec. 3) and the Ellsberg para- 1 Introduction dox (Sec. 4), and present our simulation results. We conclude When choosing among several alternatives, either the objec- by discussing the implications of our work for a unified treat- tive probabilities associated with the possible outcomes of ment of decision-making under risk and under uncertainty, each alternative are fully known, or these objective proba- and, more broadly, for human rationality. bilities are partially or fully unknown. The former is known as decision-making under risk, while the latter is studied un- 2 Sample-based Expected Utility Model der the rubric of decision-making under uncertainty (Knight, SbEU is a resource-rational process model of risky choice 1921; Weber & Camerer, 1987; Camerer & Weber, 1992). that posits that an agent rationally adapts their strategy de- Decades of empirical and theoretical research on human pending on the amount of time available for decision-making decision-making has extensively studied decision-making un- (Nobandegani et al., 2018). Concretely, SbEU assumes that der risk and decision-making under uncertainty, predom- an agent estimates expected utility inantly treating them as two distinct modes of decision- Z making, with potentially different cognitive underpinnings E[u(o)] = p(o)u(o)do; (1) (e.g., Camerer & Weber, 1992; Bonatti et al., 2009; John- son & Busemeyer, 2010; Buckert et al., 2014; De Groot & using self-normalized importance sampling (Hammersley & Thurik, 2018). However, recent work has called for a unified Handscomb, 1964; Geweke, 1989), with its importance distri- ∗ treatment of decision-making under risk and decision-making bution q aiming to optimally minimize mean-squared error under uncertainty (e.g., Hsu et al., 2005; Deany & Ortoleva, (MSE): 2017), potentially understanding them as limiting cases of a s p(o ) ˆ 1 ∗ i broader framework (Hsu et al., 2005). E = s ∑ wiu(oi); 8i : oi ∼ q ; wi = ∗ ; (2) ∑ w j q (oi) Initially introduced as two major violations of expected j=1 i=1 utility theory, the Allais paradox (1953) and the Ellsberg para- s p dox (1961) have been a driving force for developing mod- 1 + ju(o)j s q∗(o) ∝ p(o)ju(o)j p : (3) els of decision-making under risk and decision-making un- ju(o)j s der uncertainty, respectively. However, as Deany and Ortol- eva (2017) point out, among the remarkably large number of MSE is a standard measure of estimation quality, widely used decision-making models developed in the literature, few of in decision theory and mathematical statistics (Poor, 2013). 1208 In Eqs. (1-3), o denotes an outcome of a risky gamble, p(o) 2019; Nobandegani et al., 2019a). There is also experimental the objective probability of outcome o, u(o) the subjective confirmation of a counterintuitive prediction of SbEU: delib- utility of outcome o, Eˆ the importance-sampling estimate of eration makes people move from one cognitive bias, the fram- expected utility given in Eq. (1), q∗ the importance-sampling ing effect, to another, the fourfold pattern of risk preferences ∗ distribution, oi an outcome randomly sampled from q , and s (da Silva Castanheira; Nobandegani, & Otto, 2019). Impor- the number of samples drawn from q∗. tantly, SbEU is the first, and so far the only, resource-rational SbEU assumes that, when choosing between a pair of risky process model that bridges between risky, value-based, and gambles A;B, people consider whether the expected value of game-theoretic decision-making. the utility difference Du(o) is negative or positive (w.p. stands for “with probability”): 3 The Allais Paradox Introduced as a violation of expected utility theory, the Al- o w.p. P A = A A (4) lais paradox (1953) has been a driving force for developing 0 w.p. 1 − PA models of decision-making under risk (Kahneman & Tversky, 1979; Katsikopoulos & Gigerenzer, 2008; Dean & Ortoleva, 2017). The Allais paradox has two variants: the common- o w.p. P B = B B (5) consequence effect and the common-ratio effect. We present 0 w.p. 1 − P B each of these variants and demonstrate that SbEU provides a unified, resource-rational, process model of them. 8 u(o ) − u(o ) P P > A B w.p. A B 3.1 The Common-Consequence Effect <> u(o ) − u(0) w.p. P (1 − P ) Du(o) = A A B (6) u(0) − u(oB) w.p. (1 − PA)PB As its name implies, the common-consequence effect (CCE) > : 0 w.p. (1 − PA)(1 − PB) concerns choosing between two risky gambles that share a common consequence, with known objective probability. In Eq. (6), u(·) denotes the subjective utility function of a As a canonical example of the CCE, imagine choosing be- decision-maker. Consistent with past work (Nobandegani et tween the following two risky gambles (Kahneman & Tver- al., 2018; Nobandegani et al., 2019b; Nobandegani, Destais, sky, 1979): & Shultz, 2020, Nobandegani & Shultz, 2020a), in this paper 8 we use the following utility function: < z w.p. 66% A = $2;500 w.p. 33% (8) x0:85 if x ≥ 0, : 0 w.p. 1% u(x) = (7) −2jxj0:95 if x < 0: Nobandegani et al. (2018) showed that SbEU can explain z w.p. 66% B = (9) the well-known fourfold pattern of risk preferences in out- $2;400 w.p. 34% come probability (Tversky & Kahneman, 1992) and in out- come magnitude (Markovitz, 1952; Scholten & Read, 2014). As can be seen, if you choose A or B, you will get z dollars Notably, SbEU is the first rational process model to score with probability 66% either way (hence, the term common- near-perfectly in optimality, economical use of limited cog- consequence effect). nitive resources, and robustness, all at the same time (see When choosing between A;B, according to expected utility Nobandegani et al., 2018; Nobandegani et al., 2019). theory, preference should not be affected by the value of z. Relatedly, recent work has shown that SbEU provides a The rationale behind this is as follows. Note that the gap resource-rational mechanistic account of cooperation in one- between the expected utility of gamble A, EU(A), and the shot Prisoner’s Dilemma games (Nobandegani et al., 2019b), expected utility of B, EU(B), does not depend on z: inequality aversion in the Ultimatum game (Nobandegani et al., 2020), emotions in both one-shot Prisoner’s Dilemma EU(A) − EU(B) = u(z) × 0:66 + u($2;500) × 0:33 − and the Ultimatum games (Lizotte, Nobandegani, & Shultz, u(z) × 0:66 + u($2;400) × 0:34 2021), and human coordination strategies in coordination = u($2;500) × 0:33 − u($2;400) × 0:34: games (Nobandegani & Shultz, 2020a), thus successfully bridging between game-theoretic and risky decision-making. Therefore, varying z will not change the sign of EU(A) − SbEU can also account for violation of betweenness in risky EU(B). (If EU(A)−EU(B) < 0, gamble B should be chosen, choice (Nobandegani, da Silva Castanheira, Shultz, & Otto, and if EU(A) − EU(B) > 0, gamble A should be chosen.) 2019a) and the centuries-old St.