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Measuring Inconsistency in Multi-Agent Systems

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Abstract

We introduce and investigate formal quantitative measures of inconsistency between the beliefs of agents in multi-agent systems. We start by recalling a well-known model of belief in multi-agent systems, and then, using this model, present two classes of inconsistency metrics. First, we consider metrics that attempt to characterise the overall degree of inconsistency of a multi-agent system in a single numeric value, where inconsistency is considered to be individuals within the system having contradictory beliefs. While this metric is useful as a high-level indicator of the degree of inconsistency between the beliefs of members of a multi-agent system, it is of limited value for understanding the structure of inconsistency in a system: it gives no indication of the sources of inconsistency. We therefore introduce metrics that quantify for a given individual the extent to which that individual is in conflict with other members of the society. These metrics are based on power indices, which were developed within the cooperative game theory community in order to understand the power that individuals wield in cooperative settings.

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Notes

  1. In the present paper, we will not be concerned with the axiom known as additivity.

  2. Consider the case where \(\Delta = \{\lnot K p \rightarrow q, \lnot K q \rightarrow p\}\). In this case there are two expansions of \(\Delta \): one containing \(p\) but not \(q\), the other containing \(q\) but not \(p\). The set \(\Delta = \{K p\}\) has no expansions.

  3. Exactly how they achieve this isn’t relevant here, but in essence they recursively construct a grounded extension [6] so that when the dialogue terminates both agents agree on the acceptability of a common set of beliefs.

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Acknowledgments

Wooldridge gratefully acknowledges the support of the ERC under Advanced Investigator Grant 291528 (“RACE”).

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Authors and Affiliations

  1. Department of Computer Science, University College London, London, UK

    A. Hunter

  2. Department of Computer Science, University of Liverpool, Liverpool, UK

    S. Parsons

  3. Department of Computer Science, University of Oxford, Oxford, UK

    M. Wooldridge

Authors
  1. A. Hunter
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  2. S. Parsons
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  3. M. Wooldridge
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Correspondence to M. Wooldridge.

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Hunter, A., Parsons, S. & Wooldridge, M. Measuring Inconsistency in Multi-Agent Systems. Künstl Intell 28, 169–178 (2014). https://doi.org/10.1007/s13218-014-0306-3

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  • DOI: https://doi.org/10.1007/s13218-014-0306-3

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