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Wolfram Schultz - One of the best experts on this subject based on the ideXlab platform.
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Dopamine Reward Prediction Error Responses Reflect Marginal Utility
Current biology : CB, 2014Co-Authors: William R. Stauffer, Armin Lak, Wolfram SchultzAbstract:Summary Background Optimal choices require an accurate neuronal representation of economic value. In economics, Utility functions are mathematical representations of subjective value that can be constructed from choices under risk. Utility usually exhibits a nonlinear relationship to physical reward value that corresponds to risk attitudes and reflects the increasing or decreasing Marginal Utility obtained with each additional unit of reward. Accordingly, neuronal reward responses coding Utility should robustly reflect this nonlinearity. Results In two monkeys, we measured Utility as a function of physical reward value from meaningful choices under risk (that adhered to first- and second-order stochastic dominance). The resulting nonlinear Utility functions predicted the certainty equivalents for new gambles, indicating that the functions' shapes were meaningful. The monkeys were risk seeking (convex Utility function) for low reward and risk avoiding (concave Utility function) with higher amounts. Critically, the dopamine prediction error responses at the time of reward itself reflected the nonlinear Utility functions measured at the time of choices. In particular, the reward response magnitude depended on the first derivative of the Utility function and thus reflected the Marginal Utility. Furthermore, dopamine responses recorded outside of the task reflected the Marginal Utility of unpredicted reward. Accordingly, these responses were sufficient to train reinforcement learning models to predict the behaviorally defined expected Utility of gambles. Conclusions These data suggest a neuronal manifestation of Marginal Utility in dopamine neurons and indicate a common neuronal basis for fundamental explanatory constructs in animal learning theory (prediction error) and economic decision theory (Marginal Utility).
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article dopamine reward prediction error responses reflect Marginal Utility
2014Co-Authors: William R. Stauffer, Armin Lak, Wolfram SchultzAbstract:Department of Physiology, Development, and Neuroscience,University of Cambridge, Downing Street, Cambridge CB23DY, UKSummaryBackground: Optimal choices require an accurate neuronalrepresentation of economic value. In economics, Utility func-tions are mathematical representations of subjective valuethatcanbeconstructedfromchoicesunderrisk.Utilityusuallyexhibits a nonlinear relationship to physical reward value thatcorresponds to risk attitudes and reflects the increasing ordecreasing Marginal Utility obtained with each additional unitof reward. Accordingly, neuronal reward responses codingUtility should robustly reflect this nonlinearity.Results: In two monkeys, we measured Utility as a functionof physical reward value from meaningful choices under risk(that adhered to first- and second-order stochastic domi-nance). The resulting nonlinear Utility functions predictedthe certainty equivalents for new gambles, indicating thatthe functions’ shapes were meaningful. The monkeys wererisk seeking (convex Utility function) for low reward and riskavoiding (concave Utility function) with higher amounts. Criti-cally, the dopamine prediction error responses at the time ofreward itself reflected the nonlinear Utility functions measuredat the time of choices. In particular, the reward responsemagnitude depended on the first derivative of the Utilityfunction and thus reflected the Marginal Utility. Furthermore,dopamine responses recorded outside of the task reflectedthe Marginal Utility of unpredicted reward. Accordingly, theseresponses were sufficient to train reinforcement learningmodels to predict the behaviorally defined expected Utilityof gambles.Conclusions: These data suggest a neuronal manifestation ofMarginal Utility in dopamine neurons and indicate a commonneuronal basis for fundamental explanatory constructs in ani-mal learning theory (prediction error) and economic decisiontheory (Marginal Utility).IntroductionThe St. Petersburg paradox famously demonstrated that eco-nomic choices could not be predicted from physical value.Bernoulli’s enduring solution to this paradox illustrated thatdecision makers maximized the satisfaction gained fromreward, rather than physical value (wealth) [1]. In modern eco-nomictheory, theconcept ofsatisfaction wasdemystified andformalized as ‘‘Utility.’’ Utility functions are mathematical rep-resentations of subjective value, based on observable choicebehavior (rather than unobservable satisfactions) [2]. In ex-pectedUtilitytheory,thequantitativerelationshipbetweenutil-ityandphysicalvalue,U(x),canbereconstructedfromchoicesunder risk [3]. Such ‘‘von-Neumann and Morgenstern’’ (vNM)Utility functions are cardinal, in the strict sense that they aredefined up to a positive affine (shape-preserving) transforma-tion [4], in contrast to ordinal Utility relationships that are onlydefined up to a monotonic (rank-preserving) transformation[2]. Thus, the shapes of vNM Utility functions are unique, andthis formalism permits meaningful approximation of MarginalUtility—the additional Utility gained by consuming additionalunits of reward—as the first derivative, dU/dx [5]. Despiteconsiderable progress demonstrating that numerous brainstructures are involved in economic decision-making [6–17],no animal neurophysiology study has investigated how neu-rons encode the nonlinear relationship between Utility andphysical value, as defined by expected Utility theory. Mostimportantly, measurement of neuronal reward responseswhenUtilityfunctions aredefined withregardtophysicalvaluecould provide biological insight regarding the relationship be-tween the satisfaction experienced from reward and the Utilityfunction defined from choices.Midbrain dopamine neurons code reward prediction error, avalue interval important for learning [18–20]. Learning modelsthat faithfully reproduce the actions of dopamine neuronstacitlyassumethecoding ofobjectivevalue[18,21].However,the dopamine signal shows hyperbolic temporal discounting[10] and incorporates risk and different reward types onto acommon currency scale [17]. Prediction error and MarginalUtilitybothrepresentavalueinterval,andbothassumearefer-ence state (prediction and current wealth or rational expecta-tion [22, 23], respectively) and a gain or loss relative to thatstate. Therefore, the dopamine prediction error signal couldbe an ideal substrate for coding Marginal Utility.Here, we sought to define Utility as a function of physicalvalue using risky choices and investigate whether dopaminereward responses reflected the Marginal Utility calculatedfrom the Utility function. We used a classical method formeasuring vNM Utility functions (the ‘‘fractile’’ procedure)that iteratively aligns gamble outcomes with previously deter-mined points on the Utility axis [24, 25]. This procedure re-sulted in closely spaced estimates of the physical rewardamounts mapped onto predefined utilities. The data were fitwith a continuous Utility function, U(x) [24], and the MarginalUtility was computed as the first derivative, dU/dx, of the fittedfunction [5]. We then recorded dopamine responses to gam-bles and outcomes and related them to the measured Utilityfunction. The absence of common anchor points makes inter-subjective Utility comparisons generally implausible; there-fore, we did not average behavioral and neuronal data acrossthe individual animals studied.ResultsExperimental Design and BehaviorTwomonkeysmadebinarychoicesbetweengamblesandsafe(riskless)reward (Figure 1A).Therisky cue predicted agamblewithtwoequiprobable,no-zeroamountsofjuice(eachp=0.5),whereasthesafecuewasassociatedwithaspecificamountofthe same juice. The cues were bars whose vertical positionsindicated juice amount (see the Experimental Procedures).Both animals received extensive training with >10,000 trials
Marek Hudík - One of the best experts on this subject based on the ideXlab platform.
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Reference-Dependence and Marginal Utility: Alt, Samuelson, and Bernardelli
History of Political Economy, 2014Co-Authors: Marek HudíkAbstract:This article shows that the possibility of formally incorporating reference-dependence into the theory of consumer behavior was explored well before Kahneman and Tversky in the early 1990s; specifically, in separate papers by Alt, Samuelson, and Bernardelli in the late 1930s. These papers emerged within a debate on the relationship between Marginal Utility and the ordinality/cardinality of Utility. The present article identifies Alt’s, Samuelson’s, and Bernardelli’s contributions and delineates the relationship between Marginal Utility and reference-dependence. It also discusses the reception of their ideas by other economists and suggests some reasons for their neglect.
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Reference-Dependence and Marginal Utility: Alt, Samuelson, and Bernardelli
SSRN Electronic Journal, 2013Co-Authors: Marek HudíkAbstract:This paper shows that the possibility of formally incorporating reference-dependence into the theory of consumer behavior was explored well before Kahneman and Tversky (1991); specifically, in separate papers by Alt, Samuelson or Bernardelli in late 1930s. These papers emerged within a debate on the relationship between Marginal Utility and ordinality/cardinality of Utility. The present paper identifies Alt’s, Samuelson’s and Bernardelli’s contributions and delineates the relationship between Marginal Utility and reference-dependence. It also discusses the reception of their ideas by other economists and suggests some reasons for their neglect.
William R. Stauffer - One of the best experts on this subject based on the ideXlab platform.
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Dopamine Reward Prediction Error Responses Reflect Marginal Utility
Current biology : CB, 2014Co-Authors: William R. Stauffer, Armin Lak, Wolfram SchultzAbstract:Summary Background Optimal choices require an accurate neuronal representation of economic value. In economics, Utility functions are mathematical representations of subjective value that can be constructed from choices under risk. Utility usually exhibits a nonlinear relationship to physical reward value that corresponds to risk attitudes and reflects the increasing or decreasing Marginal Utility obtained with each additional unit of reward. Accordingly, neuronal reward responses coding Utility should robustly reflect this nonlinearity. Results In two monkeys, we measured Utility as a function of physical reward value from meaningful choices under risk (that adhered to first- and second-order stochastic dominance). The resulting nonlinear Utility functions predicted the certainty equivalents for new gambles, indicating that the functions' shapes were meaningful. The monkeys were risk seeking (convex Utility function) for low reward and risk avoiding (concave Utility function) with higher amounts. Critically, the dopamine prediction error responses at the time of reward itself reflected the nonlinear Utility functions measured at the time of choices. In particular, the reward response magnitude depended on the first derivative of the Utility function and thus reflected the Marginal Utility. Furthermore, dopamine responses recorded outside of the task reflected the Marginal Utility of unpredicted reward. Accordingly, these responses were sufficient to train reinforcement learning models to predict the behaviorally defined expected Utility of gambles. Conclusions These data suggest a neuronal manifestation of Marginal Utility in dopamine neurons and indicate a common neuronal basis for fundamental explanatory constructs in animal learning theory (prediction error) and economic decision theory (Marginal Utility).
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article dopamine reward prediction error responses reflect Marginal Utility
2014Co-Authors: William R. Stauffer, Armin Lak, Wolfram SchultzAbstract:Department of Physiology, Development, and Neuroscience,University of Cambridge, Downing Street, Cambridge CB23DY, UKSummaryBackground: Optimal choices require an accurate neuronalrepresentation of economic value. In economics, Utility func-tions are mathematical representations of subjective valuethatcanbeconstructedfromchoicesunderrisk.Utilityusuallyexhibits a nonlinear relationship to physical reward value thatcorresponds to risk attitudes and reflects the increasing ordecreasing Marginal Utility obtained with each additional unitof reward. Accordingly, neuronal reward responses codingUtility should robustly reflect this nonlinearity.Results: In two monkeys, we measured Utility as a functionof physical reward value from meaningful choices under risk(that adhered to first- and second-order stochastic domi-nance). The resulting nonlinear Utility functions predictedthe certainty equivalents for new gambles, indicating thatthe functions’ shapes were meaningful. The monkeys wererisk seeking (convex Utility function) for low reward and riskavoiding (concave Utility function) with higher amounts. Criti-cally, the dopamine prediction error responses at the time ofreward itself reflected the nonlinear Utility functions measuredat the time of choices. In particular, the reward responsemagnitude depended on the first derivative of the Utilityfunction and thus reflected the Marginal Utility. Furthermore,dopamine responses recorded outside of the task reflectedthe Marginal Utility of unpredicted reward. Accordingly, theseresponses were sufficient to train reinforcement learningmodels to predict the behaviorally defined expected Utilityof gambles.Conclusions: These data suggest a neuronal manifestation ofMarginal Utility in dopamine neurons and indicate a commonneuronal basis for fundamental explanatory constructs in ani-mal learning theory (prediction error) and economic decisiontheory (Marginal Utility).IntroductionThe St. Petersburg paradox famously demonstrated that eco-nomic choices could not be predicted from physical value.Bernoulli’s enduring solution to this paradox illustrated thatdecision makers maximized the satisfaction gained fromreward, rather than physical value (wealth) [1]. In modern eco-nomictheory, theconcept ofsatisfaction wasdemystified andformalized as ‘‘Utility.’’ Utility functions are mathematical rep-resentations of subjective value, based on observable choicebehavior (rather than unobservable satisfactions) [2]. In ex-pectedUtilitytheory,thequantitativerelationshipbetweenutil-ityandphysicalvalue,U(x),canbereconstructedfromchoicesunder risk [3]. Such ‘‘von-Neumann and Morgenstern’’ (vNM)Utility functions are cardinal, in the strict sense that they aredefined up to a positive affine (shape-preserving) transforma-tion [4], in contrast to ordinal Utility relationships that are onlydefined up to a monotonic (rank-preserving) transformation[2]. Thus, the shapes of vNM Utility functions are unique, andthis formalism permits meaningful approximation of MarginalUtility—the additional Utility gained by consuming additionalunits of reward—as the first derivative, dU/dx [5]. Despiteconsiderable progress demonstrating that numerous brainstructures are involved in economic decision-making [6–17],no animal neurophysiology study has investigated how neu-rons encode the nonlinear relationship between Utility andphysical value, as defined by expected Utility theory. Mostimportantly, measurement of neuronal reward responseswhenUtilityfunctions aredefined withregardtophysicalvaluecould provide biological insight regarding the relationship be-tween the satisfaction experienced from reward and the Utilityfunction defined from choices.Midbrain dopamine neurons code reward prediction error, avalue interval important for learning [18–20]. Learning modelsthat faithfully reproduce the actions of dopamine neuronstacitlyassumethecoding ofobjectivevalue[18,21].However,the dopamine signal shows hyperbolic temporal discounting[10] and incorporates risk and different reward types onto acommon currency scale [17]. Prediction error and MarginalUtilitybothrepresentavalueinterval,andbothassumearefer-ence state (prediction and current wealth or rational expecta-tion [22, 23], respectively) and a gain or loss relative to thatstate. Therefore, the dopamine prediction error signal couldbe an ideal substrate for coding Marginal Utility.Here, we sought to define Utility as a function of physicalvalue using risky choices and investigate whether dopaminereward responses reflected the Marginal Utility calculatedfrom the Utility function. We used a classical method formeasuring vNM Utility functions (the ‘‘fractile’’ procedure)that iteratively aligns gamble outcomes with previously deter-mined points on the Utility axis [24, 25]. This procedure re-sulted in closely spaced estimates of the physical rewardamounts mapped onto predefined utilities. The data were fitwith a continuous Utility function, U(x) [24], and the MarginalUtility was computed as the first derivative, dU/dx, of the fittedfunction [5]. We then recorded dopamine responses to gam-bles and outcomes and related them to the measured Utilityfunction. The absence of common anchor points makes inter-subjective Utility comparisons generally implausible; there-fore, we did not average behavioral and neuronal data acrossthe individual animals studied.ResultsExperimental Design and BehaviorTwomonkeysmadebinarychoicesbetweengamblesandsafe(riskless)reward (Figure 1A).Therisky cue predicted agamblewithtwoequiprobable,no-zeroamountsofjuice(eachp=0.5),whereasthesafecuewasassociatedwithaspecificamountofthe same juice. The cues were bars whose vertical positionsindicated juice amount (see the Experimental Procedures).Both animals received extensive training with >10,000 trials
Jeffrey T. Lafrance - One of the best experts on this subject based on the ideXlab platform.
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The law of demand versus diminishing Marginal Utility
Research Papers in Economics, 2005Co-Authors: Bruce R. Beattie, Jeffrey T. LafranceAbstract:Diminishing Marginal Utility (DMU) is neither necessary nor sufficient for downward-sloping demand. Yet, upper-division undergraduate and beginning graduate students often presumeotherwise. This paper provides two simple counter-examples that can be used to help students understand that the Law of Demand does not depend on DMU. The examples are accompaniedwith the geometry and basic mathematics of the Utility functions and the implied ordinary/Marshallian demands.
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The Law of Demand Versus Diminishing Marginal Utility
Social Science Research Network, 2003Co-Authors: Bruce R. Beattie, Jeffrey T. LafranceAbstract:Diminishing Marginal Utility is neither necessary nor sufficient for downward sloping demand, and it is not necessary for convex indifference curves. We illustrate these facts with two simple counter examples, using valid and easy to understand Utility functions. The examples are accompanied with intuition, geometry, and basic mathematics of the Utility functions, Marginal utilities, Marginal Utility slopes, indifference curves, indifference curve slopes and curvatures, and ordinary demands and slopes. The wisdom of continuing to provide the conventional diminishing Marginal Utility rationale for downward sloping demand in principles textbooks is questioned.
Sarah B. Lawsky - One of the best experts on this subject based on the ideXlab platform.
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On the edge: Declining Marginal Utility and Tax Policy
Minnesota Law Review, 2011Co-Authors: Sarah B. LawskyAbstract:The assumption of declining Marginal Utility of income— that the next dollar a person receives is “worth less”2 to a wealthy person than a poor person—has been crucial in tax scholarship over the last sixty or so years, as optimal tax theory and welfarism have become important ways that many in the legal academy evaluate tax policy.3 In spite of, or perhaps because of, the importance of this assumption, declining Marginal Utility has received little extended attention from the legal academy.4 This Article begins to fill that gap.
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On the Edge: Declining Marginal Utility and Tax Policy
Social Science Research Network, 2011Co-Authors: Sarah B. LawskyAbstract:Tax policy and scholarship generally assume that income has declining Marginal Utility (that is, that the next dollar is worth less to a wealthier person than to a poorer person). This assumption provides an easy justification for redistributive taxation. But the legal literature provides no firm grounding for the assumption of declining Marginal Utility. The Article shows that while some evidence does support declining Marginal Utility, other evidence suggests that a significant number of people actually experience increasing Marginal Utility, at least over some range of wealth. If some people’s Marginal Utility increases, a non-egalitarian welfarist analysis still supports redistribution, but not, or at least not only, from the rich to the poor: it also supports redistribution from (some) less wealthy people to (certain) wealthier people. A welfarist who finds poor-to-rich redistribution unpalatable could explicitly incorporate equality into his analysis (by, for example, adopting a social welfare function that somehow incorporates equality). Or he could continue to use a nonegalitarian approach and assume declining Marginal Utility, but also acknowledge that declining Marginal Utility is not a fact about the world, but rather a normative judgment: a rich person should value his next dollar less than a poorer person values her next dollar, whether or not he actually values it less.