A "conditional probability system" (CPS) is a mathematical structure that determines what an agent's probabilistic beliefs would be, conditional on some possible background information. Formally, let S be a space of possible states of nature, let A be a Boolean algebra of subsets of S, and let B be an arbitrary subset of A, not containing the empty set. A CPS is a real-valued function p defined on the Cartesian product of A and B. For any event a in A and any "background knowledge" b in B, p(a|b) is interpreted as the probability of a, conditional on b.

In the simplest cases, a CPS is entirely determined by a standard probability distribution, via Bayes rule: If S itself is an element of B, and p(b|S)>0, then p(a|b)=p(a|S)/p(b|S) for all a in A. But a CPS allows us to meaningfully define p(a|b) even if p(b|S)=0. Furthermore, a CPS can be defined even when S itself is not an element of B. Both of these features are useful in many applications.

The classical theory of "belief aggregation" considers how to combine the probabilistic beliefs of two or more rational individuals into a collective probabilistic belief. I extend the classical theory to the aggregation of conditional beliefs. This generalization allows us to escape some of the antinomies which plague the classical theory, but also introduces some surprising new phenomena.

Consider an urn filled with balls, each labeled with one of several possible collective decisions. Now, draw two balls from the urn, let a random voter pick her more preferred as the collective decision, relabel the losing ball with the collective decision, put both balls back into the urn, and repeat. In order to prevent the permanent disappearance of some types of balls, once in a while, a randomly drawn ball is relabeled with a random collective decision. We prove that the empirical distribution of collective decisions converges towards a maximal lottery, a celebrated probabilistic voting rule proposed by Peter C. Fishburn (Rev. Econ. Stud., 51(4), 1984). In fact, the probability that the collective decision in round n is made according to a maximal lottery increases exponentially in n. The proposed procedure is more flexible than traditional voting rules and bears strong similarities to natural processes studied in biology, physics, and chemistry as well as algorithms proposed in machine learning.

We describe a two-stage mechanism that fully implements the set of efficient outcomes in two-agent environments with quasi-linear utilities. The mechanism asks one agent to set prices for each outcome, and the other agent to make a choice, paying the corresponding price: Price \& Choose. We extend our implementation result in three main directions: an arbitrary number of players, non-quasi linear utilities, and robustness to max-min behavior. Finally, we discuss how to reduce the payoff inequality between players while still achieving efficiency.

A set of kn indivisible items is to be allocated to n agents; each agent is to get exactly k items (one might think of managers choosing teams of equal size). We consider models with cardinal or ordinal preferences, and look for efficient and approximately fair allocations. We work with notions of approximate fairness based on exchange of single objects, rather than on the mainstream idea of adding/subtracting objects. Our notions continue to operate within the meaningful domain of k-size bundles (“teams”) and are insensitive to positive affine transformations of utilities, hence do not require a separate treatment of “goods, “bads” and “mixed objects”.
A famous Round Robin rule fares very well on all fairness accounts, but fails efficiency, while rules based on collective welfare maximisation (like Nash or Leximin) cannot guarantee fairness, except on some special sub domains (two agents, identical valuations, or binary utilities). For the model with ordinal input, using ordinal domination, we introduce notions of ordinal efficiency and approximate fairness, which prove to be compatible, and describe the sets of ordinally fair allocations. We also treat the case of lexicographic preferences.