A logic of propositional control for truthful implementations Nicolas Troquard Wiebe van der Hoek University of Liverpool Department of Computer Science Liverpool L69 3BX, UK { nico, wiebe.van-der-hoek, mjw }@liv.ac.uk Michael Wooldridge Abstract We introduce a logic designed to support reason- ing about social choice functions. The logic in- cludes operators to capture strategic ability, and operators to capture agent preferences. We give a correspondence between formulae in the logic and properties of social choice functions, and show that the logic is expressively complete with respect to social choice functions, i.e., that every social choice function can be characterised as a formula of the logic. We show the decidability of the logic and give a complete axiomatization. To demonstrate the value of the logic, we show in particular how it can be applied to the problem of determining whether a social choice function is strategy-proof. 1 Introduction Social choice theory (SCT) – the theory of collective decision-making in situations where preferences over the outcomes may differ – is a topic of fundamental importance in human society [Arrow et al., 2002]. For example, the de- sign and analysis of voting procedures, such as those used in political elections across the world, has a direct effect on our lives. Our aim in this work is to develop rigorous tools to assist in the analysis and design of social choice procedures. In particular, a long-term goal is to develop techniques that will permit the automated analysis of social choice pro- cedures. To this end, we aim to develop languages that will allow us to formally express the properties of social choice procedures, such that these languages may be pro- cessed automatically and rigorously. Such languages can then be used as query languages for social choice proce- dures: given some property P of a social choice procedure (such as, e.g., the fact that the procedure is strategy-proof), we aim to be able to encode the property P as an expres- sion ρ P of our language, which we then pose as a query to an automated analysis system. Our aim in the present paper is to set out a formal language intended for the specification of social choice properties. The language is basically that of a modal logic [Chellas, 1980], partially derived from the Coalition Logic of Propo- sitional Control (CL-PC) [van der Hoek and Wooldridge, 2005]. The logic includes operators to capture strategic ability, and operators to capture agent preferences. After first recalling some key concepts from social choice and game theory, we introduce the logic. The basic idea is that we model an agent’s preferences via atomic propositions: a proposition p i x>y will be used to represent the fact that agent i has reported that he prefers outcome x at least as much as outcome y. The strategic abilities of agents are captured using a CL-PC-like operator: an agent can choose any assignment of values for its preference variables that corresponds to a preference ordering. After giving the syn- tax and semantics of the logic, we show how the logic can be used to characterise social choice functions, and show that the logic is expressively complete with respect to so- cial choice functions, i.e., that every social choice function can be characterised as a formula of the logic. We give a complete axiomatization for the logic. To demonstrate the value of the logic, we formalise some properties of social choice functions and show in particular how it can be ap- plied to the problem of determining whether a social choice function is strategy-proof. 2 Background We present the basic definitions in game theory and social choice upon which we construct our framework. As the main references to the literature we use [Dasgupta et al., 1979] and [Osborne and Rubinstein, 1994]. We assume that game forms and social choice functions (to be defined hereafter) are over the same domains of agents and consequences. We denote by N the set of agents and by K the set of consequences. Typically, the agents are the voters and the consequences will be the candidates in some