Download Extended Modal Logics for Social Software (extended abstract of

Extended Modal Logics for Social Software
Wiebe van der Hoek1 ,2
Department of Computer Science
University of Liverpool, UK
Thomas Ågotnes3
Faculty of Engineering
Bergen University College, Norway
Michael Wooldridge4
Department of Computer Science
University of Liverpool, UK
Abstract
This is a short motivational report on work that we have done in two specific areas of Social Software,
i.e. Coalitional Games and Judgement Aggregation. We argue that Extended Modal Logics prove to be
particularly successful in the modeling and axiomatisation of reasoning problems in those areas. Here, we
restrict ourselves to a description of the domains: technical details are to be found in [1] and [2], respectively.
Keywords: Coalition Logic, Coalitional Games, Judgment Aggregation, Modal Logic
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Introduction
Recently there has been a lot of attention to what is sometimes called Social Software: the systematic study of constructing and verifying social procedures. This is
an exciting arena where Social Scientists, Computer Scientists, Logicians and Game
Theorists meet. In fact the problems that are addressed are not that new, and many
of them (elections, governments, auctions, governments) have been studied for quite
some time under the term Social Engineering. What is new is the general perspective on these problems in organising social activity, and the emphasis on formal and
algorithmic aspects — after all, the notion of software is a rather novel one.
1 This abstract and my talk presents joint work with Thomas Ågotnes and Michael Wooldridge. In particular, this brief overview is based on the introductions of [1,2]
2 Email: [email protected]
3 Email: [email protected]
4 Email: [email protected]
This paper is electronically published in
Electronic Notes in Theoretical Computer Science
URL: www.elsevier.nl/locate/entcs
van der Hoek, Agotnes, & Wooldridge
In this talk, I will present two examples of formal languages designed to enable
reasoning in two areas of Social Software, i.e., Coalitional Games and Judgement
Aggregation. In the former, one is interested in what coalitions of players, or agents,
can achieve, in the latter, one tries to design a mechanisem that, given the preferences of a set of agents, combines them into an overall preference.
For technical details of the two formal systems, we refer to [1] and [2] respectively.
Or main claim is that extended modal logics appear to be very promising when it
comes to a ‘natural’ way of representing the domans of interest. In particular,
in [1] we incorporate preferences of agents in a modal logic. In [2] we show that
modal logic extended with the D-operators (see de Rijke’s Extended Modal Logic
[?] and the more recent [?]). It appears that this modal operator provides an
appropriate level of expressiveness to guarantee soundness and completeness of our
axiomatisation in a rather straightforward way.
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Reasoning about Coalitional Games
Coalitional games are games in which agents can potentially benefit by cooperating [13, pp.257–298]. Such games provide a natural and compelling model through
which to understand cooperative action, and have been widely studied in the context
of both natural and artificial multi-agent systems. In the game theory literature,
two basic questions are asked in the context of coalitional games: Which coalitions
will form? and How will the benefits of cooperation be shared within a coalition?
With respect to the first question, solution concepts such as the core have been
proposed, which try to capture the idea of rational participation in a coalition [13,
p.258]. With respect to the second question, solution concepts such as the Shapley value have been proposed, which attempt to define a “fair” distribution of the
benefits of cooperation to agents within a coalition [13, p.291]
In the context of artificial intelligence (and indeed in computer science generally), the use of coalitional game models and cooperative solution concepts raises
a number of important issues. Perhaps the most fundamental issues are those of
representing coalitional games, and reasoning with such representations. The issue
of representation – which is of course central to the field of artificial intelligence
– is of particular importance in the context of coalitional games, as the obvious
representations for them have completely unrealistic space requirements (see, e.g.,
the discussion in [17, pp.34–41]). Some effort has therefore been devoted to developing succinct representations for coalitional games. Given a specific representation
scheme, it is possible to ask concrete questions about, for example, the complexity
of computing solution concepts – see, e.g., [6,17,8,5,12,18,7] for recent examples of
such work and discussions of the associated issues.
Despite this interest, research on the representation of coalitional games has
focussed largely on the underlying mathematical models for such games, rather
than on the logical, declarative representation schemes that are commonly used in
the knowledge representation community [15]. There is good reason to suppose
that such logic based representations will be of value in reasoning about coalition
games: for example, they can be used as query languages, for expressing properties
of games to be checked via techniques such as model checking, and can also be
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van der Hoek, Agotnes, & Wooldridge
used for directly reasoning about such games via theorem proving [16]. Moreover,
logical representation schemes are frequently very succinct when compared to the
alternatives.
Our aim in [1] is thus to develop and study logic-based knowledge representation
formalisms for coalitional games (more precisely, coalitional games without transferable payoffs [13, p.268]). We develop two logical languages that are interpreted
directly as statements of such games. We study the axiomatisation and computational complexity of these logical languages, and demonstrate how they can be used
to characterise and reason about coalitional games:
•
First, we develop a Coalitional Game Logic (cgl). Syntactically, cgl contains
modal cooperation expressions of the form !C"φ with the meaning that coalition
C can achieve an outcome satisfying φ. We interpret formulae of cgl directly with
respect to coalitional games without transferable payoff, thereby establishing an
explicit link between formulae of the logic and properties of coalitional games.
In addition, cgl includes operators that make it directly possible to represent an
agent’s preferences over outcomes.
•
Second, we develop a Modal Coalitional Game Logic (mcgl), a normal modal
logic interpreted directly in coalitional games by using the preference relations in
coalitional games as modal accessibility relations.
Both logics can be used to characterise and reason about many important properties
of coalitional games, such as non-emptiness of the core. They differ, however, in
that cgl can only express such properties under the assumption that the possible
outcomes of the games are finite, while mcgl does not have this restriction. On the
other hand, if we make the finiteness assumption, cgl is more expressive than mcgl,
while the latter can often express interesting properties such as non-emptiness of
the core much more succinctly.
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Towards a Logic for Social Welfare
In the recent years there has been a great deal of interest in the logical aspects
of societies. For example, Alternating-time Temporal Logic (atl) [3] and Coalition
Logic (cl) [14] can be used to reason about the strategic abilities of individual agents
and of coalitions. There is a close connection between these logics and game theory.
A related field which, like game theory, also is concerned with social interaction,
is social choice theory. A key issue in the latter field is the construction of social
welfare functions, (SWFs), mapping individual preferences into “social preferences”.
Many of the most well known results in social choice theory are impossibility results
such as Arrow’s theorem [4]: there is no SWF that meets all of a certain number
of reasonable conditions. Formal logics related to social choice have focused mostly
on the logical representation of preferences when the set of alternatives is large and
on the computational properties of computing aggregated preferences for a given
representation [9,10,11].
In this [2], we present a formal logic which makes it possible to explicitly represent and reason about individual preferences and social preferences. The main
differences to the logics mentioned above are as follows. First, the logical language
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van der Hoek, Agotnes, & Wooldridge
is interpreted directly by social welfare functions and thus that formulae can be
read as properties of such functions; second, that preferences are represented in a
more abstract way; and, third, that the expressive power is sufficient for interesting
problems as discussed below.
Motivations for modeling social choice using logic are manyfold. In particular,
logic enables formal knowledge representation and reasoning. For example, in multiagent systems [13], agents must be able to represent and reason about propositions
involving other agents’ preferences and preference aggregation. For social choice
theory, logic can enable tools for, e.g., mechanically generating proofs, checking the
soundness of proofs, mechanically generating possibly interesting theorems, checkin
gproperties of particular social welfare functions, etc.
An example of a property of (some) social welfare functions is so-called independence of irrelevant alternatives (IIA): given two preference profiles and two
alternatives, if for each agent the two alternatives have the same order in the two
preference profiles, then the two alternatives must have the same order in the two
preference relations resulting from applying the SWF to the two preference profiles,
respectively. From this example it seems that a formal language about SWFs should
be able to express:
•
Quantification on several levels: over alternatives; over preference profiles, i.e.,
over relations over alternatives (second-order quantification); and over agents.
•
Properties of preference relations for different agents, and properties of several
different preference relations for the same agent in the same formula.
•
Comparison of different preference relations.
•
The preference relation resulting from applying a SWF to other preference relations.
From these points it seems that such a language would be complex (in particular,
they seem to rule out a “standard” propositional modal logic). However, perhaps
surprisingly, the language we present in this [2] is syntactically and semantically
rather simple; and yet the language is, nevertheless, expressive enough to give an
elegant and succinct expression of properties such as IIA.
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