Game Relations and Metrics

Luca de Alfaro, Rupak Majumdar, Viswanath Raman, Mariëlle Ida Antoinette Stoelinga

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    We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal (e.g., “reach a target state��?), the question of winning is thus a probabilistic one: “what is the maximal probability of winning from a given state?��?. On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state “similarity��?. We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative μ-calculus. The quantitative μ- calculus can express a large set of goals, including reachability, safety, and ω-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation.
    Original languageUndefined
    Title of host publicationTwenty-Second Annual IEEE Symposium on LOGIC IN COMPUTER SCIENCE (LICS 2007)
    Place of PublicationLos Alamitos
    Number of pages10
    ISBN (Print)0-7695-2908-9
    Publication statusPublished - Jul 2007
    Event22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007 - University of Wroclaw, Wroclav, Poland
    Duration: 10 Jul 200714 Jul 2007
    Conference number: 22

    Publication series

    PublisherIEEE Computer Society Press
    ISSN (Print)1055-0143


    Conference22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007
    Abbreviated titleLICS


    • EWI-11579
    • IR-62057
    • METIS-245863

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