LCD'07 : Workshop on Logics and Collective Decision making
Leornard Savage's "Foundations of Statistics" says that it would be preposterous to use Bayesian decision theory outside what he calls a small world. This paper endorses this judgement and proposes a minimal extension with an application to game theory.
Judgment aggregation aims to find collective judgments on logically interconnected propositions. According to a (contestable) democratic intuition, the collective judgment on any proposition (issue) should be determined by the individuals' judgments on this proposition; that is, the group should take an independent vote on each proposition. But propositionwise independence is known to often create collective inconsistencies, as many impossibility results show. I propose to give up propositionwise independence in favour of a new condition: independence of irrelevant information, which requires that the collective judgment on any proposition p is determined by the individuals' judgments on all propositions relevant to p, possibly including propositions other than p itself. Relevance can have many interpretations. The nature and strength of the relevance-based independence condition depend on the notion of relevance used. The more permissive this notion is, the more possibilities of aggregation arise. I will discuss several notions of relevance, and show how relevance should (not) be specified in order to obtain possibilities of aggregation.
Belief revision investigates the dynamics of the process of belief change: when an agent is faced with new information which contradicts her current beliefs, she will have to retract some of the old beliefs in order to accommodate the new information consistently. Recently, the problem of belief revision has been generalized to consider the aggregation of potentially conflicting individual belief bases into a collective one. This new area is called belief merging. On the other hand, judgment aggregation studies how to aggregate consistent individual judgments on several logically interconnected propositions into a consistent collective judgment on the same propositions. In particular, a paradox (the discursive dilemma) can arise when individual judgments are merged. To avoid the paradox, several aggregation procedures have been investigated. One of the proposals is to import the belief merging approach and apply it to judgment aggregation. Hence, the natural question that arises is how the different aggregation procedures can be evaluated. If we assume that the decision which the group is trying to reach is factually right or wrong, one possible dimension of the evaluation is to assess how well each approach is in tracking the truth. Results on the truth-tracking power of the merging operators, as well as a comparison with other procedures, will be presented.
We discuss logics for cooperation in which in which agents have control over certain aspects of the world. Moreover, this control can be delegated to other agents. We then discuss Coalition Logic: we show how adding preferences to Pauly's coalition logic CL enables one to express several game theoretical concepts, and we finally show how adding for a restricted form of quantication gives a language that is equally expressive, but exponentially more succingt than CL.