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 illustrate the limits of Kuhn's Theorem in an environment with ambiguity averse players who use maxmin decision rule and full Bayesian updating.Comment: 7 figure

Ambiguity and the Centipede Game: Strategic Uncertainty in Multi-Stage Games

We propose a solution concept for a class of extensive form games with ambiguity. Specifically we consider multi-stage games. Players have CEU preferences. The associated ambiguous beliefs are revised by Generalized Bayesian Updating. We assume individuals take account of possible changes in their preferences by using consistent planning. We show that if there is ambiguity in the centipede game it is possible to sustain 'cooperation' for many periods as part of a consistent-planning equilibrium under ambiguity. In a non-cooperative bargaining game we show that ambiguity may be a cause of delay in bargaining

Sequential Two-Player Games with Ambiguity

If players' beliefs are strictly non-additive, 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 by imposing context-dependent constraints on beliefs under uncertainty.

Attitude polarization

Psychological evidence suggests that people’s learning behavior is often prone to a “myside bias”or “irrational belief persistence”in contrast to learning behavior exclusively based on objective data. In the context of Bayesian learning such a bias may result in diverging posterior beliefs and attitude polarization even if agents receive identical information. Such patterns cannot be explained by the standard model of rational Bayesian learning that implies convergent beliefs. As our key contribution, we therefore develop formal models of Bayesian learning with psychological bias as alternatives to rational Bayesian learning. We derive conditions under which beliefs may diverge in the learning process and thus conform with the psychological evidence. Key to our approach is the assumption of ambiguous beliefs that are formalized as non-additive probability measures arising in Choquet expected utility theory. As a speci…c feature of our approach, our models of Bayesian learning with psychological bias reduce to rational Bayesian learning in the absence of ambiguity.

Incomplete Information Games with Multiple Priors

We present a model of incomplete information games with sets of priors. Upon arrival of private information, each player "updates" by the Bayes rule each of priors in this set to construct the set of posteriors consistent with the arrived piece of information. Then the player uses a possibly proper subset of this set of posteriors to form beliefs about the opponents' strategic choices. And finally the player evaluates his actions by the most pessimistic posterior beliefs `a la Gilboa and Schmeidler (1989). So each player's preferences may exhibit non-linearity in probabilities which can be interpreted as the player's aversion to ambiguity or uncertainty. In this setup, we define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players' views on uncertainty about the environment affect the strategic outcomes.incomplete information games; multiple priors; ambiguity aversion; uncertainty aversion

CHR(PRISM)-based Probabilistic Logic Learning

PRISM is an extension of Prolog with probabilistic predicates and built-in support for expectation-maximization learning. Constraint Handling Rules (CHR) is a high-level programming language based on multi-headed multiset rewrite rules. In this paper, we introduce a new probabilistic logic formalism, called CHRiSM, based on a combination of CHR and PRISM. It can be used for high-level rapid prototyping of complex statistical models by means of "chance rules". The underlying PRISM system can then be used for several probabilistic inference tasks, including probability computation and parameter learning. We define the CHRiSM language in terms of syntax and operational semantics, and illustrate it with examples. We define the notion of ambiguous programs and define a distribution semantics for unambiguous programs. Next, we describe an implementation of CHRiSM, based on CHR(PRISM). We discuss the relation between CHRiSM and other probabilistic logic programming languages, in particular PCHR. Finally we identify potential application domains

Attitude polarization

Psychological evidence suggests that people’s learning behavior is often prone to a “myside bias” or “irrational belief persistence” in contrast to learning behavior exclusively based on objective data. In the context of Bayesian learning such a bias may result in diverging posterior beliefs and attitude polarization even if agents receive identical information. Such patterns cannot be explained by the standard model of rational Bayesian learning that implies convergent beliefs. As our key contribution, we therefore develop formal models of Bayesian learning with psychological bias as alternatives to rational Bayesian learning. We derive condi- tions under which beliefs may diverge in the learning process and thus conform with the psychological evidence. Key to our approach is the assumption of ambiguous beliefs that are formalized as non-additive probability measures arising in Choquet expected utility theory. As a specific feature of our approach, our models of Bayesian learning with psychological bias reduce to rational Bayesian learning in the absence of ambiguity.

A Limit Theorem for Equilibria under Ambiguous Beliefs Correspondences

Previous literature shows that, in many different models, limits of equilibria of perturbed games are equilibria of the unperturbed game when the sequence of perturbed games converges to the unperturbed one in an appropriate sense. The question whether such limit property extends to the equilibrium notions in ambiguous games is not yet clear as it seems; in fact, previous literature shows that the extension fails in simple examples. The contribution in this paper is to show that the limit property holds for equilibria under ambiguous beliefs correspondences (presented by the authors in a previous paper). Key for our result is the sequential convergence assumption imposed on the sequence of beliefs correspondences. Counterexamples show why this assumption cannot be removed.Ambiguous games, beliefs correspondences, limit equilibria

An overview of economic applications of David Schmeidler`s models of decision making under uncertainty

This paper surveys some economic applications of the decision theoretic framework pioneered by David Schmeidler to model effects of ambiguity. We have organized the discussion principally around three themes: financial markets, contractual arrangements and game theory. The first section discusses papers that have contributed to a better understanding of financial market outcomes based on ambiguity aversion. The second section focusses on contractual arrangements and is divided into two sub-sections. The first sub-section reports research on optimal risk sharing arrangements, while in the second sub-section, discusses research on incentive contracts. The third section concentrates on strategic interaction and reviews several papers that have extended different game theoretic solution concepts to settings with ambiguity averse players. A final section deals with several contributions which while not dealing with ambiguity per se, are linked at a formal level, in terms of the pure mathematical structures involved, with Schmeidler`s models of decision making under ambiguity. These contributions involve issues such as, inequality measurement, intertemporal decision making and multi-attribute choice.Ellsberg Paradox, Ambiguity aversion, Uncertainty aversion