Mean-Payoff Games Exact Solutions Approximation On the Approximation of Mean-Payoff Games Raffaella Gentilini University of Perugia Convegno Italiano Logica Computazionale (CILC2011) 1 / 20
Mean-Payoff Games Exact Solutions Approximation Contents 1. Mean-Payoff Games (MPG) Problems 2. Exact Solutions for MPG 3. Approximate Solutions for MPG: The Additive Setting 4. Approximate Solutions for MPG: The Multiplicative Setting 2 / 20
Mean-Payoff Games Exact Solutions Approximation Mean-Payoff Games MPG • 2 players: Maximazer B � b vs Minimazer △ lice 3 / 20
Mean-Payoff Games Exact Solutions Approximation Mean-Payoff Games MPG • 2 players: Maximazer B � b vs Minimazer △ lice • played on a finite graph (arena) 3 / 20
Mean-Payoff Games Exact Solutions Approximation Mean-Payoff Games MPG • 2 players: Maximazer B � b vs Minimazer △ lice • played on a finite graph (arena) • turn based 3 / 20
Mean-Payoff Games Exact Solutions Approximation Mean-Payoff Games MPG • 2 players: Maximazer B � b vs Minimazer △ lice • played on a finite graph (arena) • turn based • infinite number of turns 3 / 20
Mean-Payoff Games Exact Solutions Approximation Mean-Payoff Games MPG • 2 players: Maximazer B � b vs Minimazer △ lice • played on a finite graph (arena) • turn based • infinite number of turns • goal (for Bob): maximazing the long-run average weight 3 / 20
Mean-Payoff Games Exact Solutions Approximation MPG in Formal Term In a MPG Γ = ( V , E , w : V → Z , � V � , V △ � ): B � b wants to maximize his payoff, i.e. the long-run average weight in a play . Given a play p = { v i } i ∈ N in Γ, the payoff of B � b on p is: n − 1 � 1 MP( v 0 v 1 . . . v n . . . ) = lim inf n · w ( v i , v i +1 ) n →∞ i =0 4 / 20
Mean-Payoff Games Exact Solutions Approximation MPG in Formal Term The value secured by a strategy σ � : V ∗ · V � → V in vertex v is: MP(outcome Γ ( v , σ � , σ △ )) val σ � ( v ) = inf σ △ ∈ Σ △ sup σ � ∈ Σ � (val σ � ( v )) is the optimal value that B � b can secure in v 5 / 20
Mean-Payoff Games Exact Solutions Approximation MPG in Formal Term Example The value secured by a strategy σ � : V ∗ · V � → V in vertex v is: MP(outcome Γ ( v , σ � , σ △ )) val σ � ( v ) = inf σ △ ∈ Σ △ sup σ � ∈ Σ � (val σ � ( v )) is the optimal value that B � b can secure in v 5 / 20
Mean-Payoff Games Exact Solutions Approximation MPG in Formal Term Example val σ � ( v ) = − 4 2 The value secured by a strategy σ � : V ∗ · V � → V in vertex v is: MP(outcome Γ ( v , σ � , σ △ )) val σ � ( v ) = inf σ △ ∈ Σ △ sup σ � ∈ Σ � (val σ � ( v )) is the optimal value that B � b can secure in v 5 / 20
Mean-Payoff Games Exact Solutions Approximation MPG in Formal Term Example The value secured by a strategy σ � : V ∗ · V � → V in vertex v is: MP(outcome Γ ( v , σ � , σ △ )) val σ � ( v ) = inf σ △ ∈ Σ △ sup σ � ∈ Σ � (val σ � ( v )) is the optimal value that B � b can secure in v 5 / 20
Mean-Payoff Games Exact Solutions Approximation MPG in Formal Term Example sup σ � ∈ Σ � (val σ � ( v )) = 2 2 The value secured by a strategy σ � : V ∗ · V � → V in vertex v is: MP(outcome Γ ( v , σ � , σ △ )) val σ � ( v ) = inf σ △ ∈ Σ △ sup σ � ∈ Σ � (val σ � ( v )) is the optimal value that B � b can secure in v 5 / 20
Mean-Payoff Games Exact Solutions Approximation MPG are Memoryless Determined Theorem [Ehrenfeucht&Mycielsky’79] val Γ ( v ) = sup σ � ∈ Σ � inf σ △ ∈ Σ △ MP(outcome Γ ( v , σ � , σ △ )) = = inf σ △ ∈ Σ △ sup σ � ∈ Σ � MP(outcome Γ ( v , σ � , σ � )). There exist uniform memoryless strategies, π � : V � → V for B � b , π △ : V △ → V for △ lice such that: val Γ ( v ) = val π � ( v ) = val π △ ( v ) . 6 / 20
Mean-Payoff Games Exact Solutions Approximation MPG are Memoryless Determined Example Theorem [Ehrenfeucht&Mycielsky’79] val Γ ( v ) = sup σ � ∈ Σ � inf σ △ ∈ Σ △ MP(outcome Γ ( v , σ � , σ △ )) = = inf σ △ ∈ Σ △ sup σ � ∈ Σ � MP(outcome Γ ( v , σ � , σ � )). There exist uniform memoryless strategies, π � : V � → V for B � b , π △ : V △ → V for △ lice such that: val Γ ( v ) = val π � ( v ) = val π △ ( v ) . 6 / 20
Mean-Payoff Games Exact Solutions Approximation MPG are Memoryless Determined Example Theorem [Ehrenfeucht&Mycielsky’79] val Γ ( v ) = sup σ � ∈ Σ � inf σ △ ∈ Σ △ MP(outcome Γ ( v , σ � , σ △ )) = = inf σ △ ∈ Σ △ sup σ � ∈ Σ � MP(outcome Γ ( v , σ � , σ � )). There exist uniform memoryless strategies, π � : V � → V for B � b , π △ : V △ → V for △ lice such that: val Γ ( v ) = val π � ( v ) = val π △ ( v ) . 6 / 20
Mean-Payoff Games Exact Solutions Approximation MPG are Memoryless Determined Example d ∈ Q : 0 ≤ d ≤ | V | and | n | val Γ ( v )= n d ≤ M = max e ∈ E {| w ( e ) |} . Theorem [Ehrenfeucht&Mycielsky’79] val Γ ( v ) = sup σ � ∈ Σ � inf σ △ ∈ Σ △ MP(outcome Γ ( v , σ � , σ △ )) = = inf σ △ ∈ Σ △ sup σ � ∈ Σ � MP(outcome Γ ( v , σ � , σ � )). There exist uniform memoryless strategies, π � : V � → V for B � b , π △ : V △ → V for △ lice such that: 6 / 20 Γ π � π
Mean-Payoff Games Exact Solutions Approximation MPG Problems 7 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems 1. Decision Problem Given v ∈ V , µ ∈ Z , decide if B � b has a strategy π � to secure val π � ( v ) ≥ µ . 7 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems 1. Decision Problem Given v ∈ V , µ ∈ Z , decide if B � b has a strategy π � to secure val π � ( v ) ≥ µ . 2. Value Problem: Compute the set of (rational) values: { val Γ ( v ) | v ∈ V } 7 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems 1. Decision Problem Given v ∈ V , µ ∈ Z , decide if B � b has a strategy π � to secure val π � ( v ) ≥ µ . 2. Value Problem: Compute the set of (rational) values: { val Γ ( v ) | v ∈ V } 3. (Optimal) Strategy Synthesis Construct an (optimal) strategy for B � b. 7 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems: Why They Matter? | = Correctness Relation Quantitative Requirements : • limited resources • average performance . . . 8 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems: Why They Matter? | = Correctness Relation Quantitative Requirements : System Model? • limited resources • average performance . . . 8 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems: Why They Matter? Solved as a game: System vs Environment Solution = Winning Strategy | = Correctness Relation Quantitative Requirements : System Model? • limited resources • average performance . . . 8 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems: Why They Matter? • MPG significative for theoretical and applicative aspects PTIME PTIME • µ -calculus model checking ⇐ ⇒ parity games = ⇒ MPG PTIME • MPG = ⇒ simple stochastic games PTIME • MPG = ⇒ discounted payoff games 9 / 20
Mean-Payoff Games Exact Solutions Approximation MPG Problems: Why They Matter? • MPG significative for theoretical and applicative aspects PTIME PTIME • µ -calculus model checking ⇐ ⇒ parity games = ⇒ MPG PTIME • MPG = ⇒ simple stochastic games PTIME • MPG = ⇒ discounted payoff games • MPG problems have an interesting complexity status • MPG decision problem belongs to NP ∩ coNP (and even to UP ∩ coUP) • No polynomial algorithm known so far 9 / 20
Mean-Payoff Games Exact Solutions Approximation Solving MPG Problems Consider Γ = ( V , E , w , � V � , V △ � ), where w : V → [ − M · · · + M ]: 10 / 20
Mean-Payoff Games Exact Solutions Approximation Solving MPG Problems Consider Γ = ( V , E , w , � V � , V △ � ), where w : V → [ − M · · · + M ]: U. Zwick and M. Paterson, 1996 • Θ( EV 2 M ) algorithm for the decision problem • Θ( EV 3 M ) algorithm for the value problem • Θ( EV 4 M log( E V )) algorithm for optimal strategy synthesis 10 / 20
Mean-Payoff Games Exact Solutions Approximation Solving MPG Problems Consider Γ = ( V , E , w , � V � , V △ � ), where w : V → [ − M · · · + M ]: U. Zwick and M. Paterson, 1996 • Θ( EV 2 M ) algorithm for the decision problem H. Bjorklund and S. Vorobyov, 2004: Use a randomized framework • Θ( EV 3 M ) algorithm for the value problem • O (min( EV 2 M , 2 O ( √ V log V ) )) for the decision prob. • Θ( EV 4 M log( E V )) algorithm for optimal strategy synthesis • O (min( EV 3 M (log V + log M ) , 2 O ( √ V log V ) )) for the value prob. • Θ( EV 4 M log( E V )) algorithm for optimal strategy synthesis 10 / 20
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