Updating Non-Additive Probabilities -- A Geometric Approach

A geometric approach, analogous to the approach used in the additive case, is proposed to determine the conditional expectation with non- additive probabilities. The conditional expectation is then applied for (i) updating the probability when new information becomes available; and (ii) defining the notion of independence of non-additive probabilities and Nash equilibrium.updating, non-additive probabilities, conditional expectation

A new integral for capacities

A new integral for capacities, different from the Choquet integral, is introduced and characterized. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is then 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 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 when there is information only about a few events and not about all of them.new integral, capacity, choquet integral, fuzzy capacity, concavity

Cooperative investment games or population games

The model of a cooperative fuzzy game is interpreted as both a population game and a cooperative investment game. Three types of core- like solutions induced by these interpretations are introduced and investigated. The interpretation of a game as a population game allows us to define sub-games. We show that, unlike the well-known Shapley- Shubik theorem on market games (Shapley-Shubik) there might be a population game such that each of its sub-games has a non-empty core and, nevertheless, it is not a market game. It turns out that, in order to be a market game, a population game needs to be also homogeneous. We also discuss some special classes of population games such as convex games, exact games, homogeneousgames and additive games.investment game, population game, fuzzy game, core-like solution, market game

Categorization generated by prototypes -- an axiomatic approach

We present a model of categorization based on prototypes. A prototype is an image or template of an idealized member of the category. Once a set of prototypes is defined, entities are sorted into categories on the basis of the prototypes they are closest to. We provide a characterization of those categorizations that are generated by prototypes.categorization, prototype, prototype-orineted decision making

Subjective Games and Equilibria

Applying the concepts of Nash, Bayesian or correlated equilibrium to analysis of strategic interaction, requires that players possess objective knowledge of the game and opponents' strategies. Such knowledge is often not available. The proposed notions of subjective games, and subjective Na.sh and correlated equilibria, replace unavailable objective knowledge by subjective assessments. When playing such a game repeatedly, subjective optimizers will converge to a subjective equilibrium. We apply this approach to some well known examples including a single multi-arm bandit player, multi-person multi-arm bandit games, and repeated Cournot oligopoly games

What restrictions do Bayesian games impose on the value of information?

In a Bayesian game players play an unknown game. Before the game starts some players may receive a signal regarding the specific game actually played. Typically, information structures that determine different signals, induce different equilibrium payoffs.In zero-sum games the equilibrium payoff measures the value of the particular information structure which induces it. We pose a question as to what restrictions do Bayesian games impose on the value of information. We provide answers in two kinds of information structures: symmetric, where both players are equally informed, and one-sided where only one player is informed.value of information, zero-sum, information structure, partition, Beyesian game

Agreeing to agree

Aumann has shown that agents who have a common prior cannot have common knowledge of their posteriors for event $E$ if these posteriors do not coincide. But given an event $E$, can the agents have posteriors with a common prior such that it is common knowledge that the posteriors for $E$ \emph{do} coincide? We show that a necessary and sufficient condition for this is the existence of a nonempty \emph{finite} event $F$ with the following two properties. First, it is common knowledge at $F$ that the agents cannot tell whether or not $E$ occurred. Second, this still holds true at $F$, when $F$ itself becomes common knowledge.Agreeing theorem, common knowledge, common prior, no trade theorem

On Concavification and Convex Games

We propose a new geometric approach for the analysis of cooperative games. A cooperative game is viewed as a real valued function $u$ defined on a finite set of points in the unit simplex. We define the \emph{concavification} of $u$ on the simplex as the minimal concave function on the simplex which is greater than or equal to $u$. The concavification of $u$ induces a game which is the \emph{totally balanced cover} of the game. The concavification of $u$ is used to characterize well-known classes of games, such as balanced, totally balanced, exact and convex games. As a consequence of the analysis it turns out that a game is convex if and only if each one of its sub-games is exact.concavification, convex games, core, totally balanced, exact games