Working Papers
By Year:
Paper #  Author  Title  

15026 v. 2 
Patrick DeJarnette David Dillenberger Daniel Gottlieb Pietro Ortoleva 
Time Lotteries  
We study preferences over lotteries that pay a specific prize at uncertain future dates: time lotteries. The standard model of time preferences, Expected Discounted Utility (EDU),
implies that individuals must be risk seeking in this case. As a motivation, we show in an incentivized experiment that most subjects exhibit the opposite behavior, i.e., they are risk averse over time lotteries (RATL). We then make two theoretical contributions. First, we show that RATL can be captured by a generalization of EDU that is obtained by keeping the postulates of Discounted Utility and Expected Utility. Second, we introduce a new property termed Stochastic Impatience, a risky counterpart of standard Impatience, and show that not only the model above, but also substantial generalizations that allow for nonExpected Utility and non exponential discounting, cannot jointly accommodate it and RATL, showing a fundamental tension between the two. Download Paper


17013 
Simone CerreiaVioglio David Dillenberger Pietro Ortoleva Gil Riella 
Deliberately Stochastic  
We study stochastic choice as the outcome of deliberate randomization. After first deriving a general representation of a stochastic choice function with such property, we proceed to characterize a model in which the agent has preferences over lotteries that belong to the Cautious Expected Utility class (Cerreia Vioglio et al., 2015), and the stochastic choice is the optimal mix among available options. This model links stochasticity of choice and the phenomenon of Certainty Bias, with both behaviors stemming from the same source: multiple utilities and caution. We show that this model is behaviorally distinct from models of Random Utility, as it typically violates the property of Regularity, shared by all of them. Download Paper


16015 
David Dillenberger Collin Raymond 
GroupShift and the Consensus Effect, Second Version  
Individuals often tend to conform to the choices of others in group decisions, compared to choices made in isolation, giving rise to phenomena such as group polarization and the bandwagon effect. We show that this behavior, which we term the consensus effect, is equivalent to a wellknown violation of expected utility, namely strict quasiconvexity of preferences. In contrast to the equilibrium outcome when individuals are expected utility maximizers, quasiconvexity of preferences imply that group decisions may fail to properly aggregate preferences and strictly Paretodominated equilibria may arise. Moreover, these problems become more severe as the size of the group grows. Download Paper


16011 
David Dillenberger R. Vijay Krishna Philipp Sadowski 
Subjective Dynamics Information Constraints  
We axiomatize a new class of recursive dynamic models that capture subjective constraints on the amount of information a decision maker can obtain, pay attention to, or absorb, via a Markov Decision Process for Information Choice (MIC). An MIC is a subjective decision process that specifies what type of information about the payoffrelevant state is feasible in the current period, and how the choice of what to learn now affects what can be learned in the future. The constraint imposed by the MIC is identified from choice behavior up to a recursive extension of Blackwell dominance. All the other parameters of the model, namely the anticipated evolution of the payoffrelevant state, state dependent consumption utilities, and the discount factor are also uniquely identified. Download Paper


16005 
David Dillenberger Collin Raymond 
GroupShift and the Consensus Effect  
It is well documented that individuals make different choices in the context of group decisions, such as elections, from choices made in isolation. In particular, individuals tend to conform to the decisions of others a property we call the consensus effect  which in turn implies phenomena such as group polarization and the bandwagon effect. We show that the consensus effect is equivalent to a wellknown violation of expected utility, namely strict quasiconvexity of preferences. Our results qualify and extend those of Eliaz, Ray and Razin (2006), who focus on choiceshifts in group when one option is safe (i.e., a degenerate lottery). In contrast to the equilibrium outcome when individuals are expected utility maximizers, the consensus effect implies that group decisions may fail to properly aggregate preferences in strategic contexts and strictly Paretodominated equilibria may arise. Moreover, these problems become more severe as the size of the group grows. Download Paper


15026A 
Patrick DeJarnette David Dillenberger Daniel Gottlieb Pietro Ortoleva 
"Time Lotteries: Online Appendix"  
This online appendix provides additional proofs, extensions, and all experiment instructions and questionnaire. Download Paper


15026 
Patrick DeJarnette David Dillenberger Daniel Gottlieb Pietro Ortoleva 
"Time Lotteries"  
We study preferences over lotteries that pay a specific prize at uncertain dates. Expected Utility with convex discounting implies that individuals prefer receiving $x in a random date with mean t over receiving $x in t days for sure. Our experiment rejects this prediction. It suggests a link between preferences for payments at certain dates and standard risk aversion. EpsteinZin (1989) preferences accommodate such behavior, and fit the data better than a model with probability weighting. We thus provide another justification for disentangling attitudes toward risk and time, as in EpsteinZin, and suggest new theoretical restrictions on its key parameters. Download Paper


15015 
David Dillenberger Uzi Segal 
"Skewed Noise"  
We study the attitude of decision makers to skewed noise. For a binary lottery that yields the better outcome with probability $p$, we identify noise around $p$, with a compound lottery that induces a distribution over the exact value of the probability and has an average value p. We propose and characterize a new notion of skewed distributions, and use a recursive nonexpected utility model to provide conditions under which rejection of symmetric noise implies rejection of skewed to the left noise as well. We demonstrate that rejection of these types of noises does not preclude acceptance of some skewed to the right noise, in agreement with recent experimental evidence. We apply the model to study random allocation problems (onesided matching) and show that it can predict systematic preference for one allocation mechanism over the other, even if the two agree on the overall probability distribution over assignments. The model can also be used to address the phenomenon of ambiguity seeking in the context of decision making under uncertainty. Download Paper


15009 
David Dillenberger Andrew Postlewaite Kareen Rozen 
"Optimism and Pessimism with Expected Utility, Fifth Version"  
Maximizing subjective expected utility is the classic model of decision making under uncertainty. Savage (1954) provides axioms on preference over acts that are equivalent to the existence of a subjective expected utility representation, and further establishes that such a representation is essentially unique. We show that there is a continuum of other \expected utility" representations in which the probability distributions over states used to evaluate acts depend on the set of possible outcomes of the act and suggest that these alternate representations can capture pessimism or optimism. We then extend the DM's preferences to be defined over both subjective acts and objective lotteries, allowing for sourcedependent preferences. Our result permits modeling ambiguity aversion in Ellsberg's twourn experiment using a single utility function and pessimistic probability assessments over prizes for lotteries and acts, while maintaining the axioms of Savage and von NeumannMorganstern on the appropriate domains. Download Paper


14005 
Simone CerreiaVioglio David Dillenberger Pietro Ortoleva 
"Cautious Expected Utility and the Certainty Effect"  
Many violations of the Independence axiom of Expected Utility can be traced to subjects' attraction to riskfree prospects. The key axiom in this paper, Negative Certainty Independence (Dillenberger, 2010), formalizes this tendency. Our main result is a utility representation of all preferences over monetary lotteries that satisfy Negative Certainty Independence together with basic rationality postulates. Such preferences can be represented as if the agent were unsure of how to evaluate a given lottery p; instead, she has in mind a set of possible utility functions over outcomes and displays a cautious behavior: she computes the certainty equivalent of p with respect to each possible function in the set and picks the smallest one. The set of utilities is unique in a welldefined sense. We show that our representation can also be derived from a `cautious' completion of an incomplete preference relation. Download Paper


13068 
David Dillenberger Andrew Postlewaite Kareen Rozen 
"Optimism and Pessimism with Expected Utility", Fourth Version  
Maximizing subjective expected utility is the classic model of decisionmaking under uncertainty. Savage (1954) provides axioms on preference over acts that are equivalent to the existence of a subjective expected utility representation, and further establishes that such a representation is essentially unique. We show that there is a continuum of other "expected utility" representations in which the probability distributions over states used to evaluate acts depend on the set of possible outcomes of the act and suggest that these alternate representations can capture pessimism or optimism. We then extend the DM's preferences to be defined over both subjective acts and objective lotteries, allowing for sourcedependent preferences. Our result permits modeling ambiguity aversion in Ellsberg's twourn experiment using a single utility function and pessimistic probability assessments over prizes for lotteries and acts, while maintaining the axioms of Savage and von NeumannMorganstern on the appropriate domains Download Paper


13067 
David Dillenberger Juan Sebastian Lleras Philipp Sadowski Norio Takeoka 
"A Theory of Subjective Learning", Third Version  
We study an individual who faces a dynamic decision problem in which the process of information arrival is unobserved by the analyst. We elicit subjective information directly from choice behavior by deriving two utility representations of preferences over menus of acts. One representation uniquely identifies information as a probability measure over posteriors and the other identifies information as a partition of the state space. We compare individuals who expect to learn differently in terms of their preference for flexibility. On the extended domain of datedmenus, we show how to accommodate gradual learning over time by means of a subjective filtration. Download Paper


13066 
David Dillenberger Uzi Segal 
"Skewed Noise"  
Experimental evidence suggests that individuals who face an asymmetric distribution over the likelihood of a specific event might actually prefer not to know the exact value of this probability. We address these findings by studying a decision maker who has recursive, nonexpected utility preferences over twostage lotteries. For a binary lottery that yields the better outcome with probability p, we identify noise around p with a compound lottery that induces a probability distribution over the exact value of the probability and has an average value p. We first propose and characterize a new notion of skewed distributions. We then use this result to provide conditions under which a decision maker who always rejects symmetric noise around p will always reject skewed to the left noise, but might accept skewed to the right noise. The model can be applied to the areas of investment under risk, medical decision making, and criminal law procedures, and can also be used to address the phenomenon of ambiguity seeking in the context of decision making under uncertainty. Download Paper


13037 
Simone CerreiaVioglio David Dillenberger Pietro Ortoleva 
"Cautious Expected Utility and the Certainty Effect"  
Many violations of the Independence axiom of Expected Utility can be traced to subjects' attraction to riskfree prospects. Negative Certainty Independence, the key axiom in this paper, formalizes this tendency. Our main result is a utility representation of all preferences over monetary lotteries that satisfy Negative Certainty Independence together with basic rationality postulates. Such preferences can be represented as if the agent were unsure of how risk averse to be when evaluating a lottery p; instead, she has in mind a set of possible utility functions over outcomes and displays a cautious behavior: she computes the certainty equivalent of p with respect to each possible function in the set and picks the smallest one. The set of utilities is unique in a welldefined sense. We show that our representation can also be derived from a `cautious' completion of an incomplete preference relation. Download Paper


13014 
David Dillenberger Juan Sebastian Lleras Philipp Sadowski Norio Takeoka 
“A Theory of Subjective Learning, Second Version”  
We study an individual who faces a dynamic decision problem in which the process of information arrival is unobserved by the analyst. We elicit subjective information directly from choice behavior by deriving two utility representations of preferences over menus of acts. The most general representation identifies a unique probability distribution over the set of posteriors that the decision maker might face at the time of choosing from the menu. We use this representation to characterize a notion of ”more preference for flexibility” via a subjective analogue of Blackwell’s (1951, 1953) comparisons of experiments. A more specialized representation uniquely identifies information as a partition of the state space. This result allows us to compare individuals who expect to learn differently, even if they do not agree on their prior beliefs. On the extended domain of datedmenus, we show how to accommodate an individual who expects to learn gradually over time by means of a subjective filtration. Download Paper


13001 
David Dillenberger Andrew Postlewaite Kareen Rozen 
“Optimism and Pessimism with Expected Utility”, Third Version  
Savage (1954) provides axioms on preferences over acts that are equivalent to the existence of a subjective expected utility representation. We show that there is a continuum of other \expected utility" representations in which for any act, the probability distribution over states depends on the corresponding outcomes and is firstorder stochastically dominated by (dominates) the Savage distribution. We suggest that pessimism (optimism) can be captured by the stakedependent probabilities in these alternate representations. We then extend the DM's preferences to be defined over both subjective acts and objective lotteries. Our result permits modeling ambiguity aversion in Ellsberg's twourn experiment using pessimistic probability assessments, the same utility over prizes for lotteries and acts, and without relaxing Savage's axioms. An implication of our results is that the large body of existing research based on expected utility can, with a simple reinterpretation, be understood as modeling the behavior of optimistic or pessimistic decision makers. Download Paper


12036 
David Dillenberger Philipp Sadowski 
"Generalized Partition and Subjective Filtration"  
We study an individual who faces a dynamic decision problem in which the process of information arrival is unobserved by the analyst, and hence should be identified from observed choice data. An information structure is objectively describable if signals correspond to events of the objective state space. We derive a representation of preferences over menus of acts that captures the behavior of a Bayesian decision maker who expects to receive such signals. The class of information structures that can support such a representation generalizes the notion of a partition of the state space. The representation allows us to compare individuals in terms of the preciseness of their information structures without requiring that they share the same prior beliefs. We apply the model to study an individual who anticipates gradual resolution of uncertainty over time. Both the filtration (the timing of information arrival with the sequence of partitions it induces) and prior beliefs are uniquely identified. Download Paper


12034 
David Dillenberger Philipp Sadowski Juan Sebastian Lleras Norio Takeoka 
"A Theory of Subjective Learning"  
We study an individual who faces a dynamic decision problem in which the process of information arrival is unobserved by the analyst. We derive two utility representations of preferences over menus of acts that capture the individual’s uncertainty about his future beliefs. The most general representation identifies a unique probability distribution over the set of posteriors that the decision maker might face at the time of choosing from the menu. We use this representation to characterize a notion of “more preference for flexibility” via a subjective analogue of Blackwell’s (1951, 1953) comparisons of experiments. A more specialized representation uniquely identifies information as a partition of the state space. This result allows us to compare individuals who expect to learn differently, even if they do not agree on their prior beliefs. We conclude by extending the basic model to accommodate an individual who expects to learn gradually over time by means of a subjective filtration. Download Paper


12031 
David Dillenberger Andrew Postlewaite Kareen Rozen 
“Optimism and Pessimism with Expected Utility”, Second Version  
Savage (1954) provided axioms on preferences over acts that were equivalent to the existence of an expected utility representation. We show that there is a continuum of other expected utility" representations in which for any act, the probability distribution over states depends on the corresponding outcomes. We suggest that optimism and pessimism can be captured by the stakedependent probabilities in these alternative representations. Extending the DM's preferences to be defined on both subjective acts and objective lotteries, we suggest how one may distinguish optimists from pessimists and separate attitude towards uncertainty from curvature of the utility function over monetary prizes. Download Paper


12029 
David Dillenberger Kareen Rozen 
"HistoryDependent Risk Attitude" Second Version  
We propose a model of historydependent risk attitude, allowing a decision maker’s risk attitude to be affected by his history of disappointments and elations. The decision maker recursively evaluates compound risks, classifying realizations as disappointing or elating using a threshold rule. We establish equivalence between the model and two cognitive biases: risk attitudes are reinforced by experiences (one is more risk averse after disappointment than after elation) and there is a primacy effect (early outcomes have the greatest impact on risk attitude). In dynamic asset pricing, the model yields volatile, pathdependent prices. Download Paper


12021 
David Dillenberger Uzi Segal 
"Recursive Ambiguity and Machina’s Examples"  
Machina (2009, 2012) lists a number of situations where standard models of ambiguity aversion are unable to capture plausible features of ambiguity attitudes. Most of these problems arise in choice over prospects involving three or more outcomes. We show that the recursive nonexpected utility model of Segal (1987) is rich enough to accommodate all these situations. Download Paper


12007 
David Dillenberger Philipp Sadowski 
"Subjective Learning", Second Version  
We study an individual who faces a dynamic decision problem in which the process of information arrival is unobserved by the analyst. We derive a sequence of representations of preferences over menus of acts that capture the individual's uncertainty about his future beliefs. Using the most general representation, we characterize a notion of "more preference for flexibility" via a subjective analogue of Blackwell's (1951, 1953) comparisons of experiments. A more refined representation allows us to compare individuals who expect to learn differently, even if they do not agree on their prior beliefs. The class of information structures that can support such a representation generalizes the notion of a partition of the state space. We apply the model to study an individual who anticipates gradual resolution of uncertainty over time. Both the filtration (the timing of information arrival with the sequence of partitions it induces) and prior beliefs are uniquely identified. Download Paper


11042 
David Dillenberger Philipp Sadowski 
"Models of Subjective Learning"  
We study a decision maker who faces a dynamic decision problem in which the process of information arrival is subjective. By studying preferences over menus of acts, we derive a sequence of utility representations that captures the decision maker’s uncertainty about the beliefs he will hold when choosing from a menu. In the most general model of secondorder beliefs, we characterize a notion of "more preference for flexibility" via a subjective analogue of Blackwell’s (1951, 1953) comparisons of experiments. We proceed to analyze a model in which signals are subsets of the state space. The corresponding representation enables us to compare the behavior of two decision makers who expect to learn differently, even if they do not agree on their prior beliefs. The class of information systems that can support such a representation generalizes the notion of modeling information as a partition of the state space. We apply the model to study a decision maker who anticipates subjective uncertainty to be resolved gradually over time. We derive a representation that uniquely identifies both the filtration, which is the timing of information arrival with the sequence of partitions it induces, and the decision maker’s prior beliefs. Download Paper


11036 
David Dillenberger Andrew Postlewaite Kareen Rozen 
"Optimism and Pessimism with Expected Utility"  
Savage (1954) provided a set of axioms on preferences over acts that were equivalent to the existence of an expected utility representation. We show that in addition to this representation, there is a continuum of other expected utilityrepresentations in which for any act, the probability distribution over states depends on the corresponding outcomes. We suggest that optimism and pessimism can be captured by the stakedependent probabilities in these alternative representations; e.g., for a pessimist, the probability of every outcome except the worst is distorted down from the Savage probability. Extending the DMs preferences to be de
ned on both subjective acts and objective lotteries, we show how one may distinguish optimists from pessimists and separate attitude towards uncertainty from curvature of the utility function over monetary prizes. Download Paper


11004 
David Dillenberger Kareen Rozen 
“HistoryDependent Risk Attitude”  
We propose a model of historydependent risk attitude (HDRA), allowing the attitude of a decisionmaker (DM) towards risk at each stage of a Tstage lottery to evolve as a function of his history of disappointments and elations in prior stages. We establish an equivalence between the existence of an HDRA representation and two documented cognitive biases. First, the DM’s risk attitudes are reinforced by prior experiences: he becomes more risk averse after suffering a disappointment and less risk averse after being elated. Second, the DM displays a primacy effect: early outcomes have the strongest effect on risk attitude. Furthermore, the DM lowers his threshold for elation after a disappointing outcome and raises it after an elating outcome; this makes disappointment more likely after elation and viceversa, leading to statistically reversing risk attitudes. “Gray areas” in the elationdisappointment assignment are connected to optimism and pessimism in determining endogenous reference points. Download Paper


10028 
David Dillenberger Kareen Rozen 
"Disappointment Cycles"  
We propose a model of history dependent disappointment aversion (HDDA), allowing the attitude of a decisionmaker (DM) towards disappointment at each stage of a Tstage lottery to evolve as a function of his history of disappointments and elations in prior stages. We establish an equivalence between the existence of an HDDA representation and two documented cognitive biases. First, the DM overreacts to news: after suffering a disappointment, the DM lowers his threshold for elation and becomes more risk averse; similarly, after an elating outcome, the DM raises his threshold for elation and becomes less risk averse. This makes disappointment more likely after elation and viceversa, leading to statistically cycling risk attitudes. Second, the DM displays a primacy effect: early outcomes have the strongest effect on risk attitude. “Gray areas” in the elationdisappointment assignment are connected to optimism and pessimism in determining endogenous reference points. Download Paper


10025 
Shiri ArtsteinAvidan David Dillenberger 
"Dynamic Disappointment Aversion: Don't Tell Me Anything Until You Know For Sure"  
We show that for a disappointmentaverse decision maker, splitting a lottery into several stages reduces its value. To do this, we extend Gul.s (1991) model of disappointment aversion into a dynamic setting while keeping its basic characteristics intact. The result depends solely on the sign of the coefficient of disappointment aversion. It can help explain why people often buy periodic insurance for moderately priced objects, such as electrical appliances and cellular phones, at much more than the actuarially fair rate. Download Paper


09033 
David Dillenberger 
"Preferences for OneShot Resolution of Uncertainty and AllaisType Behavior"  
Experimental evidence suggests that individuals are more risk averse when they perceive risk that is gradually resolved over time. We address these findings by studying a decision maker (DM) who has recursive, nonexpected utility preferences over compound lotteries. DM has preferences for oneshot resolution of uncertainty (PORU) if he always prefers any compound lottery to be resolved in a single stage. We establish an equivalence between dynamic PORU and static preferences that are identified with commonly observed behavior in Allaistype experiments. The implications of this equivalence on preferences over information systems are examined. We define the gradual resolution premium and demonstrate its magnifying effect when combined with the usual risk premium. In an intertemporal context, PORU captures "loss aversion with narrow framing". Download Paper


09014 
David Dillenberger Philipp Sadowski 
"Ashamed to be Selfish", Second Version  
We study a twostage choice problem. In the first stage, the decision maker (DM) chooses a set of payoffallocations between herself and a passive recipient. In the second stage, DM chooses an allocation from the set. The recipient is only aware of the second stage choice. Choosing selfishly in the second stage, in the face of a fairer available alternative, may inflict shame on DM. We axiomatize a representation of DM's preferences over sets that identifies DM's selfish ranking, her norm of fairness and shame. It has been suggested that altruism is a prominent motive for nonselfish choice. We identify a condition under which shame to be selfish can mimic altruism, when the experimenter only records the second stage choice. An additional condition implies that the norm of fairness can be characterized as the Nash solution of a bargaining game induced by the secondstage choice problem. The representation is applied to a simple strategic situation, a game of trust. Download Paper


08037 
David Dillenberger Philipp Sadowski 
"Ashamed to be Selfish"  
We study a twostage choice problem, where alternatives are allocations between the decision maker (DM) and a passive recipient. The recipient observes choice behavior in stage two, while stage one choice is unobserved. Choosing selfishly in stage two, in the face of a fairer available alternative, may inflict shame on DM. DM has preferences over sets of alternatives that represent period two choices. We axiomatize a representation that identifies DM's selfish ranking, her norm of fairness and shame. Altruism is the most prominent motive that can explain nonselfish choice. We identify a condition under which shame to be selfish can mimic altruism, when only stagetwo choice is observed by the experimenter. An additional condition implies that the norm of fairness can be characterized as the Nash solution of a bargaining game induced by the secondstage choice problem. The representation is generalized to allow for finitely many recipients and applied to a simple strategic situation, a game of trust. Download Paper


08036 
David Dillenberger 
"Preferences for OneShot Resolution of Uncertainty and AllaisType Behavior"  
Experimental evidence suggests that individuals are more risk averse when they perceive risk gradually. We address these findings by studying a decision maker (DM) who has recursive preferences over compound lotteries and who cares about the way uncertainty is resolved over time. DM has preferences for oneshot resolution of uncertainty (PORU) if he always prefers any compound lottery to be resolved in a single stage. We establish an equivalence between dynamic PORU and static preferences that are identified with the behavior observed in Allaistype experiments. The implications of this equivalence on preferences over information systems are examined. We define the gradual resolution premium and demonstrate its magnifying effect when combined with the usual risk premium. In an intertemporal context, PORU captures "loss aversion with narrow framing". Download Paper
