Ehud Lehrer

A New Integral for Capacities

A new integral for capacities is introduced and characterized. It differs from the Choquet integral on non-convex capacities. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is extended to fuzzy capacities, which assign subjective expected values to random variables (e.g., portfolios) and may assign subjective probability only to a partial set of events. An equivalence between the minimum over sets of additive capacities (not necessarily probability distributions) and the integral w.r.t. fuzzy capacities is demonstrated. The extension to fuzzy capacities enables one to calculate the integral also in cases where the information available is limited to a few events.

On the Core of Dynamic Cooperative Games

We consider dynamic cooperative games, where the worth of coalitions varies over time according to the history of allocations. When defining the core of a dynamic game, we allow the possibility for coalitions to deviate at any time and thereby to give rise to a new environment. A coalition that considers a deviation needs to take the consequences into account because from the deviation point on, the game is no longer played with the original set of players. The deviating coalition becomes the new grand coalition which, in turn, induces a new dynamic game. Each stage game of the new dynamic game depends on the previous allocation. We define and characterize a new solution concept, the intertemporal core.

A general internal regret-free strategy

We study sequential decision problems where the decision maker does not observe the states of nature, but rather receives a noisy signal, whose distribution depends on the current state and on the action that she plays. We do not assume that the decision maker considers the worst-case scenario, but rather has a response correspondence, which maps distributions over signals to subjective best responses. We extend the concept of internal regret-free strategy to this setup and provide an algorithm that generates such a strategy.

Subjective Independence and Concave Expected Utility

When a potential hedge between alternatives does not reduce the exposure to uncertainty, we say that the decision maker considers these alternatives structurally similar. We offer a novel approach and suggest that structural similarity is subjective and should be different across decision makers. Structural similarity can be recovered through a property of the individuals preferences referred to as subjective codecomposable independence. This property characterizes a class of event-separable models and allows us to differentiate between perception of uncertainty and attitude towards it. In addition, our approach provides a behavioral foundation to Concave Expected Utility preferences.

Belief consistency and trade consistency

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agentsʼ beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinsteinʼs email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.

Joint Games and Compatibility

Abstract We introduce the concepts of joint games and compatibility. In a joint game, members of the grand coalition have the option to split and participate in different underlying games, thereby maximizing their total worths. In order to determine whether the grand coalition will remain intact, we introduce the notion of compatibility of these games. A set of games is compatible if the core of the joint game is non-empty. We find a necessary and sufficient condition for compatibility.

The value functions of Markov decision processes

We provide a full characterization of the set of value functions of Markov decision processes.

Approachability with delayed information

We consider a sequential decision problem where the decision maker is informed of the actual payoff with delay. We introduce a new condition, which generalizes the condition given by Blackwell and ensures that the decision maker can approach a fixed closed and convex set under delay. We show how the convergence rate to the approachable set is sensitive to changes in the information lag and apply our approachability strategy to games with one-sided incomplete information and to regret-free strategies.