A theory of Bayesian decision making with action-dependent subjective probabilities

This paper presents a complete, choice-based, axiomatic Bayesian decision theory. It introduces a new choice set consisting of information-contingent plans for choosing actions and bets and subjective expected utility model with effect-dependent utility functions and action-dependent subjective probabilities which, in conjunction with the updating of the probabilities using Bayes’ rule, gives rise to a unique prior and a set of action-dependent posterior probabilities representing the decision maker’s prior and posterior beliefs.     

Half empty, half full and why we can agree to disagree forever

Aumann [Aumann R., 1976. Agreeing to disagree. Annals of Statisitics 4, 1236–1239] derives his famous we cannot agree to disagree result under the assumption that people are expected utility (=EU) decision makers. Motivated by empirical evidence against EU theory, we study the possibility of agreeing to disagree within the framework of Choquet expected utility (=CEU) theory which generalizes EU theory by allowing for ambiguous beliefs. As our first main contribution, we show that people may well agree to disagree if their Bayesian updating of ambiguous beliefs is psychologically biased in our sense. Remarkably, this finding holds regardless of whether people with identical priors apply the same psychologically biased Bayesian update rule or not. As our second main contribution, we develop a formal model of Bayesian learning under ambiguity. As a key feature of our approach the posterior subjective beliefs do, in general, not converge to “true” probabilities which is in line with psychological evidence against converging learning behavior. This finding thus formally establishes that CEU decision makers may even agree to disagree in the long-run despite the fact that they always received the same information.     

ANTICIPATED UTILITY AND RATIONAL EXPECTATIONS AS APPROXIMATIONS OF BAYESIAN DECISION MAKING*

We study a Markov decision problem with unknown transition probabilities. We compute the exact Bayesian decision rule and compare it with two approximations. The first is an infinite‐history, rational‐expectations approximation that assumes that the decision maker knows the transition probabilities. The second is a version of Kreps' anticipated‐utility model in which decision makers update using Bayes' law but optimize in a way that is myopic with respect to their updating of probabilities. For several consumption‐smoothing examples, the anticipated‐utility approximation outperforms the rational expectations approximation. The rational expectations approximation misrepresents the market price of risk.     

A note on Kuhn’s Theorem with ambiguity averse players

Kuhn’s Theorem shows that extensive games with perfect recall can equivalently be analyzed using mixed or behavioral strategies, as long as players are expected utility maximizers. This note constructs an example that illustrates the limits of Kuhn’s Theorem in an environment with ambiguity averse players who use a maxmin decision rule and full Bayesian updating.     

Weak monotonicity and Bayes–Nash incentive compatibility

An allocation rule is called Bayes–Nash incentive compatible, if there exists a payment rule, such that truthful reports of agents' types form a Bayes–Nash equilibrium in the direct revelation mechanism consisting of the allocation rule and the payment rule. This paper provides a characterization of Bayes–Nash incentive compatible allocation rules in social choice settings where agents have multi-dimensional types, quasi-linear utility functions and interdependent valuations. The characterization is derived by constructing complete directed graphs on agents' type spaces with cost of manipulation as lengths of edges. Weak monotonicity of the allocation rule corresponds to the condition that all 2-cycles in these graphs have non-negative length. For the case that type spaces are convex and the valuation for each outcome is a linear function in the agent's type, we show that weak monotonicity of the allocation rule together with an integrability condition is a necessary and sufficient condition for Bayes–Nash incentive compatibility.     

Bayesian consistent belief selection

A subjective expected utility agent is given information about the state of the world in the form of a set of possible priors. She is assumed to form her beliefs given this information. A set of priors may be updated according to Bayes' rule, prior-by-prior, upon learning that some state of the world has not obtained. In a model in which information is completely summarized by this set of priors, we show that there exists no decision maker who obeys Bayes' rule, conditions her prior only on the available information (by selecting a belief in the announced set), and who updates the information prior-by-prior using Bayes' rule.     

GROUP DECISION MAKING WITH UNCERTAIN OUTCOMES: UNPACKING CHILD–PARENT CHOICE OF THE HIGH SCHOOL TRACK

Predicting group decisions under uncertainty requires disentangling individual members' utilities over the consequences of choice, their expectations for uncertain outcomes, and their choice process as a group. I estimate simple Bayesian models of child–parent choice of high school track with subjective risk and unilateral or bilateral, nonstrategic decisions, by combining families' actual choices with novel survey information about children's and parents' subjective probabilities over choice consequences, their individually preferred choices, and their decision roles. A set of policy counterfactuals confirms the importance of introducing the beliefs and decision roles of individual members in models and policy analysis of group decisions.     

SEQUENTIAL TWO‐PLAYER GAMES WITH AMBIGUITY*

If players' beliefs are strictly nonadditive, the Dempster–Shafer updating rule can be used to define beliefs off the equilibrium path. We define an equilibrium concept in sequential two‐person games where players update their beliefs with the Dempster–Shafer updating rule. We show that in the limit as uncertainty tends to zero, our equilibrium approximates Bayesian Nash equilibrium. We argue that our equilibrium can be used to define a refinement of Bayesian Nash equilibrium by imposing context‐dependent constraints on beliefs under uncertainty.     

Bank runs as coordination failures: An experimental study

We use experimental methods to investigate what factors contribute to breakdowns in coordination among a bank’s depositors. Subjects in our experiment decide whether to leave their money deposited in a bank or withdraw it early; a bank run occurs when there are too many early withdrawals. We explore the effects of adding uncertainty about fundamental withdrawal demand and of changing the number of opportunities subjects have to withdraw. Our results show that (i) bank runs are rare when fundamental withdrawal demand is known but occur frequently when it is stochastic, and (ii) subjects are more likely to withdraw when given multiple opportunities to do so than when presented with a single decision. For the multiple-opportunity case, we evaluate individual withdrawal decisions according to a set of simple cutoff rules. We find that the cutoff rule corresponding to the payoff-dominant equilibrium of the game, which involves Bayesian updating of probabilities, explains subject behavior better than other rules.     

A mechanism for eliciting a probability distribution

This work extends Karni’s direct revelation mechanism for eliciting agents’ subjective beliefs over the distribution of a random variable.     

Asymmetric information without common priors: an indirect evolutionary analysis of quantity competition

The common prior assumption justifies private beliefs as posterior probabilities when updating a common prior based on individual information. We dispose of the common prior assumption for a homogeneous oligopoly market with uncertain costs and firms entertaining arbitrary priors about other firms’ cost-type. We show that true prior beliefs can not be evolutionarily stable when truly expected profit measures (reproductive) success.     

DOES NEAR‐RATIONALITY MATTER IN FIRST‐ORDER APPROXIMATE SOLUTIONS? A PERTURBATION APPROACH

This paper studies first‐order approximate solutions to near‐rational dynamic equilibrium models. Under near‐rationality, agents' subjective beliefs are distorted away from rational expectations via a change of measure process which fulfills some regularity conditions. In most applications, the beliefs distortion process is also directly observed by (a subset of) the decision‐makers – e.g., ambiguity‐averse households or policy‐makers with a concern for robustness – and therefore included into their optimization problems. We investigate conditions for existence and local uniqueness of solutions under endogenous distortions, as well as the relation with their rational expectations counterparts. We show that linearly perturbed solutions may well be affected by the presence of distorted beliefs, depending on the underlying model economy. In particular, while directly affecting first‐order decision rules, near‐rationality may also induce failure of the certainty equivalence principle. Moreover, the martingale representation of distorted beliefs might prove non‐unique, pointing to a subtle form of equilibrium indeterminacy.     

Recursive robust estimation and control without commitment

In a Markov decision problem with hidden state variables, a posterior distribution serves as a state variable and Bayes’ law under an approximating model gives its law of motion. A decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nearby when measured by their expected log likelihood ratios (entropies). Martingales represent alternative models. A decision maker constructs a sequence of robust decision rules by pretending that a sequence of minimizing players choose increments to martingales and distortions to the prior over the hidden state. A risk sensitivity operator induces robustness to perturbations of the approximating model conditioned on the hidden state. Another risk sensitivity operator induces robustness to the prior distribution over the hidden state. We use these operators to extend the approach of Hansen and Sargent [Discounted linear exponential quadratic Gaussian control, IEEE Trans. Automat. Control 40(5) (1995) 968-971] to problems that contain hidden states.     

Recursive multiple-priors

This paper axiomatizes an intertemporal version of multiple-priors utility. A central axiom is dynamic consistency, which leads to a recursive structure for utility, to ‘rectangular’ sets of priors and to prior-by-prior Bayesian updating as the updating rule for such sets of priors. It is argued that dynamic consistency is intuitive in a wide range of situations and that the model is consistent with a rich set of possibilities for dynamic behavior under ambiguity.     

A new approach to modeling decision-making under uncertainty

This paper presents two axiomatic models of decision making under uncertainty that avoid the use of a state space. The first is a subjective expected utility model with action-dependent subjective probabilities and effect-dependent preferences (the case of effect-independent preferences is obtained as a special instance). The second is a nonexpected utility model involving well-defined families of action-dependent subjective probabilities on effects and a utility representation that is not necessarily linear in these probabilities (a probabilistic sophistication version of this model, with action-dependent subjective probabilities is obtained as a special case). The hospitality of EUREQua, University of Paris 1, and financial support by the National Science Foundation grant SES-0314249 are gratefully acknowledged.     

Internal rationality, imperfect market knowledge and asset prices

We present a decision theoretic framework in which agents are learning about market behavior and that provides microfoundations for models of adaptive learning. Agents are ‘internally rational’, i.e., maximize discounted expected utility under uncertainty given dynamically consistent subjective beliefs about the future, but agents may not be ‘externally rational’, i.e., may not know the true stochastic process for payoff relevant variables beyond their control. This includes future market outcomes and fundamentals. We apply this approach to a simple asset pricing model and show that the equilibrium stock price is then determined by investors? expectations of the price and dividend in the next period, rather than by expectations of the discounted sum of dividends. As a result, learning about price behavior affects market outcomes, while learning about the discounted sum of dividends is irrelevant for equilibrium prices. Stock prices equal the discounted sum of dividends only after making very strong assumptions about agents? market knowledge.     

WHY DOES IT MATTER THAT BELIEFS AND VALUATIONS BE CORRECTLY REPRESENTED?*

This article contains an analysis of a simple principal–agent problem illustrating possible problems that may arise when the prinicpal ascribes to the agent subjective probabilities and utilities that are implied by the subjective expected utility model but do not represent the agent's beliefs and valuations. In particular, it is possible that an incentive contract designed by the principal induces the agent to choose an action that is not in the principal's best interest.     

Revisiting Savage in a conditional world

Summary. I present an axiomatization of subjective expected utility and Bayesian updating in a conditional decision problem. This result improves our understanding of the Bayesian standard from two perspectives: 1) it uses a set of axioms which are weak and intuitive; 2) it provides a formal proof to results on the relation between dynamic consistency, expected utility and Bayesian updating which have never been explicitly proved in a fully subjective framework. Received: December 1, 2000; revised version: February 26, 2001     

Hope springs eternal: learning and the stability of cooperation in short horizon repeated games

This paper considers learning rates in finitely repeated prisoners’ dilemmas. If players think their opponents might be relatively cooperative (e.g., tit-for-tat or grim types), they will cooperate in finitely repeated prisoners’ dilemmas (see Kreps et al., J. Econom. Theory 27 (1982) 245). However, if there are actually no cooperative types, players will eventually learn this and cooperation will break down. This paper shows that this learning is extremely slow, so it will take an extremely long time for cooperation to break down.Thus, suppose the world is either “good” or “bad.” The probability of a grim type is δ>0 if the world is good, and zero if the world is bad. Successive generations pair up to play finitely repeated prisoners’ dilemmas. Players observe play in previous generations and use Bayes’ rule to update their prior, π, that the world is good. We show that, if the world is really bad, then π falls per generation on average. Thus, if δ is small, there is less cooperation if the world is good, but cooperation may become more stable. For a representative 19 period repeated prisoners’ dilemma, beliefs fall one percentage point on average after a thousand generations.To derive these learning rates, we must refine existing results on the sensitivity of repeated games to Kreps et al. (1982) type perturbations. Specifically, we show cooperation is possible in perturbed prisoners’ dilemmas repeated O(log(1/δ)) times. This improves significantly on the O(1/δ) results in previous work. The paper thus provides two new reasons why cooperation tends to be stable, even in short horizon repeated games.     

The impossibility of compromise: some uniqueness properties of expected utility preferences

Summary. We focus on the following uniqueness property of expected utility preferences: Agreement of two preferences on one interior indifference class implies their equality. We show that, besides expected utility preferences under (objective) risk, this uniqueness property holds for subjective expected utility preferences in Anscombe-Aumann's (partially subjective) and Savage's (fully subjective) settings, while it does not hold for subjective expected utility preferences in settings without rich state spaces. Indeed, when it holds the uniqueness property is even stronger than described above, as it needs only agreement on binary acts. The extension of the uniqueness property to the subjective case is possible because beliefs in the mentioned settings are shown to satisfy an analogous property: If two decision makers agree on a likelihood indifference class, they must have identical beliefs. Received: November 15, 1999; revised version: December 29, 1999