Jonathan Weinstein

THE AMBIGUITY AVERSION LITERATURE: A CRITICAL ASSESSMENT

We provide a critical assessment of the ambiguity aversion literature, which we characterize in terms of the view that Ellsberg choices are rational responses to ambiguity, to be explained by relaxing Savages Sure-Thing principle and adding an ambiguity-aversion postulate. First, admitting Ellsberg choices as rational leads to behaviour, such as sensitivity to irrelevant sunk cost, or aversion to information, which most economists would consider absurd or irrational. Second, we argue that the mathematical objects referred to as “beliefs†in the ambiguity aversion literature have little to do with how an economist or game theorist understands and uses the concept. This is because of the lack of a useful notion of updating. Third, the anomaly of the Ellsberg choices can be explained simply and without tampering with the foundations of choice theory. These choices can arise when decision makers form heuristics that serve them well in real-life situations where odds are manipulable, and misapply them to experimental settings.

Two Notes on the Blotto Game

We exhibit a new equilibrium of the classic Blotto game in which players allocate one unit of resources among three coordinates and try to defeat their opponent in two out of three. It is well known that a mixed strategy will be an equilibrium strategy if the marginal distribution on each coordinate is U[0,(2/3)]. All classic examples of such distributions have two-dimensional support. Here we exhibit a distribution which has one-dimensional support and is simpler to describe than previous examples. The construction generalizes to give one-dimensional distributions with the same property in higher-dimensional simplices as well.As our second note, we give some results on the equilibrium payoffs when the game is modified so that players have unequal budgets. Our results suggest a criterion for equilibrium selection in the original symmetric game, in terms of robustness with respect to a small asymmetry in resources.

Impact of higher-order uncertainty

In some games, the impact of higher-order uncertainty is very large, implying that present economic theories may be misleading as these theories assume common knowledge of the type structure after specifying the first or the second orders of beliefs. Focusing on normal-form games in which the players strategy spaces are compact metric spaces, we show that our key condition, called global stability under uncertainty, implies a variety of results to the effect that the impact of higher-order uncertainty is small. Our central result states that, under global stability, the maximum change in equilibrium strategies due to changes in players beliefs at orders higher than k is exponentially decreasing in k. Therefore, given any need for precision, we can approximate equilibrium strategies by specifying only finitely many orders of beliefs (This abstract was borrowed from another version of this item.)

Testing Theories with Learnable and Predictive Representations

We study the problem of testing an expert whose theory has a learnable and predictive parametric representation, as do standard processes used in statistics. We design a test in which the expert is required to submit a date T by which he will have learned enough to deliver a sharp, testable prediction about future frequencies. We show that this test passes an expert who knows the data-generating process and cannot be manipulated by a uninformed one. Such a test is not possible if the theory is unrestricted.

Finite-Order Implications of any Equilibrium

Present economic theories make a common-knowledge assumption that implies that the first or the second-order beliefs determine all higher order beliefs. We analyze the role of such closing assumptions at finite orders by instead allowing higher orders to vary arbitrarily. Assuming that the space of underlying uncertainty is sufficiently rich, we show that the resulting set of possible outcomes, under an arbitrary fixed equilibrium, must include all outcomes that survive iterated elimination of strategies that are never a strict best reply. For many games, this implies that, unless the game is dominance solvable, every equilibrium will be highly sensitive to higher-order beliefs, and thus economic theories based on such equilibria may be misleading. Moreover, every equilibrium is discontinuous at each type for which two or more actions survive our elimination process.

Sensitivity of equilibrium behavior to higher-order beliefs in nice games

We analyze nice games (where action spaces are compact intervals, utilities continuous and strictly concave in own action), which are used frequently in classical economic models. Without making any richness assumption, we characterize the sensitivity of any given Bayesian Nash equilibrium to higher-order beliefs. That is, for each type, we characterize the set of actions that can be played in equilibrium by some type whose lower-order beliefs are all as in the original type. We show that this set is given by a local version of interim correlated rationalizability. This allows us to characterize the robust predictions of a given model under arbitrary common knowledge restrictions. We apply our framework to a Cournot game with many players. There we show that we can never robustly rule out any production level below the monopoly production of each firm.

Impact of Higher-Order Uncertainty

In some games, the impact of higher-order uncertainty is very large, implying that present economic theories may be misleading as these theories assume common knowledge of the type structure after specifying the first or the second orders of beliefs. Focusing on normal-form games in which the players strategy spaces are compact metric spaces, we show that our key condition, called global stability under uncertainty, implies a variety of results to the effect that the impact of higher-order uncertainty is small. Our central result states that, under global stability, the maximum change in equilibrium strategies due to changes in players beliefs at orders higher than k is exponentially decreasing in k. Therefore, given any need for precision, we can approximate equilibrium strategies by specifying only finitely many orders of beliefs.

REJOINDER: THE “AMBIGUITY AVERSION LITERATURE: A CRITICAL ASSESSMENT”

The pioneering contributions of Bewley, Gilboa and Schmeidler highlighted important weaknesses in the foundations of economics and game theory. The Bayesian methodology on which these fields are based does not answer such basic questions as what makes beliefs reasonable, or how agents should form beliefs and expectations. Providing the initial impetus for debating these issues is a contribution that will have the lasting value it deserves.

Reputation without commitment in finitely repeated games

In the reputation literature, players have emph{commitment types} which represent the possibility that they do not have standard payoffs but instead are constrained to follow a particular plan. In this paper, we show that arbitrary commitment types can emerge from incomplete information about the stage payoffs. In particular, any finitely repeated game with commitment types is strategically equivalent to a standard finitely repeated game with incomplete information about the stage payoffs. Then, classic reputation results can be achieved with uncertainty concerning only the stage payoffs.

The Effect of Changes in Risk Attitude on Strategic Behavior

We study families of normal‐form games with fixed preferences over pure action profiles but varied preferences over lotteries. That is, we subject players utilities to monotone but nonlinear transformations and examine changes in the rationalizable set and set of equilibria. Among our results: The rationalizable set always grows under concave transformations (risk aversion) and shrinks under convex transformations (risk love). The rationalizable set reaches an upper bound under extreme risk aversion, and lower bound under risk love, and both of these bounds are characterized by elimination processes. For generic two‐player games, under extreme risk love or aversion, all Nash equilibria are close to pure and the limiting set of equilibria can be described using preferences over pure action profiles.