Citation
Krishnendu Chatterjee. "Nash Equilibrium for Upward-Closed Objectives". CSL 06, September, 2006.

Abstract
We study infinite stochastic games played by n-players on a finite graph with goals specified by sets of infinite traces. The games are concurrent (each player simultaneously and independently chooses an action at each round), stochastic (the next state is determined by a probability distribution depending on the current state and the chosen actions), infinite (the game continues for an infinite number of rounds), nonzero-sum (the players' goals are not necessarily conflicting), and undiscounted. We show that if each player has an upward-closed objective, then there exists an epsilon-Nash equilibrium in memoryless strategies, for every epsilon>0; and exact Nash equilibria need not exist. Upward-closure of an objective means that if a set Z of infinitely repeating states is winning, then all supersets of Z of infinitely repeating states are also winning. Memoryless strategies are strategies that are independent of history of plays and depend only on the current state. We also study the complexity of finding values (payoff profile) of an epsilon-Nash equilibrium. We show that the values of an epsilon-Nash equilibrium in nonzero-sum concurrent games with upward-closed objectives for all players can be computed by computing epsilon-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives for all players and a polynomial procedure. As a consequence we establish that values of an epsilon-Nash equilibrium can be computed in TFNP (total functional NP), and hence in EXPTIME.

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• HTML

  Krishnendu Chatterjee. <a
  href="http://chess.eecs.berkeley.edu/pubs/236.html"
  >Nash Equilibrium for Upward-Closed Objectives</a>,
  CSL 06, September, 2006.

• Plain text

  Krishnendu Chatterjee. "Nash Equilibrium for
  Upward-Closed Objectives". CSL 06, September, 2006.

• BibTeX

  @inproceedings{Chatterjee06_NashEquilibriumForUpwardClosedObjectives,
      author = {Krishnendu Chatterjee},
      title = {Nash Equilibrium for Upward-Closed Objectives},
      booktitle = {CSL 06},
      month = {September},
      year = {2006},
      abstract = {We study infinite stochastic games played by
                n-players on a finite graph with goals specified
                by sets of infinite traces. The games are
                concurrent (each player simultaneously and
                independently chooses an action at each round),
                stochastic (the next state is determined by a
                probability distribution depending on the current
                state and the chosen actions), infinite (the game
                continues for an infinite number of rounds),
                nonzero-sum (the players' goals are not
                necessarily conflicting), and undiscounted. We
                show that if each player has an upward-closed
                objective, then there exists an epsilon-Nash
                equilibrium in memoryless strategies, for every
                epsilon>0; and exact Nash equilibria need not
                exist. Upward-closure of an objective means that
                if a set Z of infinitely repeating states is
                winning, then all supersets of Z of infinitely
                repeating states are also winning. Memoryless
                strategies are strategies that are independent of
                history of plays and depend only on the current
                state. We also study the complexity of finding
                values (payoff profile) of an epsilon-Nash
                equilibrium. We show that the values of an
                epsilon-Nash equilibrium in nonzero-sum concurrent
                games with upward-closed objectives for all
                players can be computed by computing epsilon-Nash
                equilibrium values of nonzero-sum concurrent games
                with reachability objectives for all players and a
                polynomial procedure. As a consequence we
                establish that values of an epsilon-Nash
                equilibrium can be computed in TFNP (total
                functional NP), and hence in EXPTIME. },
      URL = {http://chess.eecs.berkeley.edu/pubs/236.html}
  }
  

Posted by Krishnendu Chatterjee, PhD on 13 May 2007.
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