We present a modal logic for reasoning about what groups of agents can bring about by collective action. Given a set of states, we introduce game frames which associate with every state a strategic game among the agents. Game frames are essentially extensive games of perfect information with simultaneous actions, where every action profile is associated with a new state, the outcome of the game. A coalition of players is effective for a set of states in a game if the coalition can guarantee the outcome of the game to lie in . We propose a modal logic (Coalition Logic) to formalize reasoning about effectivity in game frames, where expresses that coalition is effective for . An axiomatization is presented and completeness proved. Coalition Logic provides a unifying game-theoretic view of modal logic: Since nondeterministic processes and extensive games without parallel moves emerge as particular instances of game frames, normal and non-normal modal logics correspond to 1- and 2-player versions of Coalition Logic. The satisfiability problem for Coalition Logic is shown to be PSPACE-complete.

A Modal Logic for Coalitional Power in Games

Logic for social software

Implementation, Contracts, and Renegotiation in Environments With Complete Information

https://www.cs.ox.ac.uk/people/michael.wooldridge/pubs/aij2005a.pdf

On the logic of cooperation and propositional control

Day 5: Applications and Extensions The precise topics for the course will depend in part on the interests and background of the students attending the lectures. This document contains an introduction to the course including some pointers to relevant literature. The focus is on the basics of neighborhood semantics with only pointers to literature relating to the more advanced topics. The website for the course is staff.science.uva.nl/∼epacuit/nbhd esslli.html On this website you will find the lecture slides (and possibly course notes) updated daily. Two other sources will also be useful:

Neighborhood Semantics for Modal Logic

1. Introduction.- 1. Collective decisions and power structure.- 2. Social choice.- 3. Games and game forms.- 4. The effectivity functions of a game form.- 5. Effectivity functions and cooperative games.- 6. An overview of the book.- 7. Notes.- 2. The structure of effectivity functions.- 1. Introduction.- 2. Fundamental concepts.- 3. Polarity.- 4. Selected families of effectivity functions.- 5. Topological effectivity functions.- 6. Notes.- 3. Stability.- 1. Introduction.- 2. The core of an effectivity function.- 3. Stable effectivity functions.- 4. Stability of convex effectivity functions.- 5. Acyclicity and stability.- 6. Stable veto functions.- 7. Notes.- 4. Implementation.- 1. Introduction.- 2. Implementation of social choice correspondences by game forms.- 3. Monotonic social choice functions.- 4. Notes.- 5. Implementation in strong Nash equilibrium.- 1. Strong Nash equilibrium.- 2. The effectivity functions of a strongly Nash implementable social choice correspondence.- 3. The fundamental theorem.- 4. The nucleus.- 5. The supernucleus.- 6. The subcore.- 7. A characterization of strong Nash implementable social choice correspondences.- 8. Notes.- 6. Exact and strong consistency.- 1. Self-implementation.- 2. Successive elimination and exact and.- consistency.- 3. The Holzman conditions.- 4. Notes.- 7. Implementation in coalition proof Nash equilibrium.- 1. Coalition proof Nash equilibrium.- 2. Implementation in coalition proof Nash equilibrium.- 3. Uniform domination.- 4. U-effectiveness and indirect u-domination.- 5. Notes.- 8. Other effectivity functions.- 1. Introduction.- 2. The ?-effectivity function.- 3. Cooperative equilibria of type 1.- 4. Equilibria of type 2 and 3.- 5. Notes.- 9. Value of effectivity functions.- 1. Introduction.- 2. Effectivity functions with convex feasible sets.- 3. Correlated strategies and effectivity functions.- 4. Effectivity functions of repeated games.- 5. Notes.- References.

Effectivity functions in social choice

We give an alternative proof of a theorem by Gurvich: A game form G is equivalent to a free extensive game form with perfect information if and only if G is rectangular and tight. It follows that any rectangular and tight game form is Nash solvable. We also present some basic properties of rectangular game forms.

Rectangularity and Tightness: a Normal Form Characterization of Perfect Information Extensive Game Forms

A two-player game form is Nash-consistent if and only if it is tight (Gurvich). Therefore Nash-consistency of two-player game forms depends only on the effectivity structure. This fact is no longer true for strong consistency. In this paper we introduce a new object called the joint effectivity structure and define the exact joint effectivity set. These notions are similar though more sophisticated than the usual effectivity functions. We prove that a two-player game form is strongly consistent if and only if it is tight and jointly exact. Joint exactness is a property of the exact joint effectivity set which basically requires that the joint exact effectivity set coincides with the classical effectivity function. As a corollary we have a characterization of two-player strongly implementable social choice correspondences.

Nash and strongly consistent two-player game forms

On necessary and sufficient conditions for solvability of game forms

We study the structure of unstable power mechanisms. A power mechanism is modeled by an interaction form, the solution of which is called a settlement. By stability, we mean the existence of some settlement for any preference profile. Configurations that produce instability are called cycles. We introduce a stability index that measures the difficulty of emergence of cycles. Structural properties such as exactness, superadditivity, subadditivity and maximality provide indications about the type of instability that may affect the mechanism. We apply our analysis to strategic game forms in the context of Nash-like solutions or core-like solutions. In particular, we establish an upper bound on the stability index of maximal interaction forms.

Joseph Abdou

Papers

Effectivity functions in social choice

Rectangularity and Tightness: a Normal Form Characterization of Perfect Information Extensive Game Forms

Nash and strongly consistent two-player game forms

On necessary and sufficient conditions for solvability of game forms

The structure of unstable power mechanisms