# Two-player Nonzero-sum omega-regular Games

### Krishnendu Chatterjee

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-04-1364

December 2004

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/CSD-04-1364.pdf

We study infinite stochastic games played by two-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 omega-regular objective expressed as a parity objective, then there exists an epsilon-Nash equilibrium, for every epsilon > 0. However, exact Nash equilibria need not exist. We study the complexity of finding values (payoff profile) of some epsilon-Nash equilibrium. We show that the values of some epsilon-Nash equilibrium in nonzero-sum concurrent parity games can be computed by solving the following two simpler problems: computing the values of zero-sum (the goals of the players are strictly conflicting) concurrent parity games and computing epsilon-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives. As a consequence we establish that values of some epsilon-Nash equilibrium can be approximated in FNP (functional NP), and hence in EXPTIME.

BibTeX citation:

@techreport{Chatterjee:CSD-04-1364, Author = {Chatterjee, Krishnendu}, Title = {Two-player Nonzero-sum omega-regular Games}, Institution = {EECS Department, University of California, Berkeley}, Year = {2004}, Month = {Dec}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/5331.html}, Number = {UCB/CSD-04-1364}, Abstract = {We study infinite stochastic games played by two-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 omega-regular objective expressed as a parity objective, then there exists an epsilon-Nash equilibrium, for every epsilon > 0. However, exact Nash equilibria need not exist. We study the complexity of finding values (payoff profile) of some epsilon-Nash equilibrium. We show that the values of some epsilon-Nash equilibrium in nonzero-sum concurrent parity games can be computed by solving the following two simpler problems: computing the values of zero-sum (the goals of the players are strictly conflicting) concurrent parity games and computing epsilon-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives. As a consequence we establish that values of some epsilon-Nash equilibrium can be approximated in FNP (functional NP), and hence in EXPTIME.} }

EndNote citation:

%0 Report %A Chatterjee, Krishnendu %T Two-player Nonzero-sum omega-regular Games %I EECS Department, University of California, Berkeley %D 2004 %@ UCB/CSD-04-1364 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2004/5331.html %F Chatterjee:CSD-04-1364