Relaxing Exclusive Control in Boolean Games Arianna Novaro IRIT, University of Toulouse F. Belardinelli U. Grandi A. Herzig D. Longin E. Lorini L. Perrussel SEGA Workshop, Prague 2018
Relaxing Exclusive Control in Boolean Games SEGA 2018 Scenario 1: Friends Organize a Potluck meat wine fish “If we have steak “I hope we eat “I hate herring and I want red wine.” steak or herring.” I like white wine.” Arianna Novaro 2/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Scenario 2: Friends Organize a Visit Decide together which places to visit . Should we go check out the bridge? Should we go see the clock? Should we visit the castle? Arianna Novaro 3/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Talk Outline 1. Games of Propositional Control Boolean Games and Iterated Boolean Games 2. Strategics Abilities in Logic Concurrent Game Structures with Exclusive Control Concurrent Game Structures with Shared Control 3. Main Results Relationship between Exclusive and Shared Control Computational Complexity 4. Conclusions Arianna Novaro 4/30
Games of Propositional Control
Relaxing Exclusive Control in Boolean Games SEGA 2018 Boolean Games, Intuitively ( a ∨ d ) → g e ∧ f b ↔ ( c ∧ f ) agent 1 agent 2 agent 3 f , g a , b , c d , e Harrenstein, van der Hoek, Meyer and Witteveen. Boolean games . TARK-2001. Bonzon, Lagasquie-Schiex, Lang and Zanuttini. Boolean games revisited . ECAI-2006. Arianna Novaro 6/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Boolean Games, Formally A Boolean Game is a tuple G = ( N, Φ , π, Γ ) such that: ◮ N = { 1 , . . . , n } is a set of agents ◮ Φ is a finite set of variables ◮ π : N → 2 Φ is a control function (a partition of Φ ) ◮ Γ = { γ 1 , . . . , γ n } is a set of propositional formulas over Φ N = { 1 , 2 , 3 } Φ = { a, b, c, d, e, f, g } π (1) = { a, b, c } , π (2) = { d, e } , π (3) = { f, g } Γ = { ( a ∨ d ) → g , e ∧ f , b ↔ ( c ∧ f ) } Arianna Novaro 7/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Strategies and Utilities for Boolean Games A strategy σ i is an assignment to the variables in π ( i ) . A strategy profile is a tuple σ = ( σ 1 , . . . , σ n ) : a valuation on Φ . The (binary) utility of agent i is 1 if σ | = γ i , and 0 otherwise. π (1) = { a, b, c } , π (2) = { d, e } , π (3) = { f, g } σ 1 = { a, b } σ 1 ( a ) = σ 1 ( b ) = 1 , σ 1 ( c ) = 0 σ 2 ( d ) = 0 , σ 2 ( e ) = 1 σ 2 = { e } σ 3 = { f, g } σ 3 ( f ) = σ 3 ( g ) = 1 Which are the utilities of the agents? Γ = { ( a ∨ d ) → g , e ∧ f , b ↔ ( c ∧ f ) } Arianna Novaro 8/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Winning Strategies σ − i = ( σ 1 , . . . , σ i − 1 , σ i +1 , . . . , σ n ) is the projection of σ on N \ { i } A winning strategy σ i for i is such that ( σ − i , σ i ) | = γ i for all σ − i . ( a ∨ d ) → c e ↔ b agent 1 agent 2 a , b , c d , e A winning strategy for agent 1 ? And for agent 2 ? Arianna Novaro 9/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Iterated Boolean Games, Intuitively ( a ∨ d ) U g e ∧ f b ↔ � ( c ∧ f ) agent 1 agent 2 agent 3 a , b , c d , e f , g 0 1 2 3 4 5 . . . • • • • • • Gutierrez, Harrenstein, Wooldridge. Iterated Boolean Games . Information and Computation 242:53-79. (2015). Arianna Novaro 10/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Iterated Boolean Games, Formally An iterated Boolean Game is a tuple G = ( N, Φ , π, Γ ) such that: ◮ N = { 1 , . . . , n } is a set of agents ◮ Φ is a finite set of variables ◮ π : N → 2 Φ is a control function (a partition of Φ ) ◮ Γ = { γ 1 , . . . , γ n } is a set of LTL formulas over Φ We assume that agents have memory-less strategies = their choice of action depends on the current state only. Arianna Novaro 11/30
Strategic Abilities in Logic
Relaxing Exclusive Control in Boolean Games SEGA 2018 What Can Agents Do? ATL ∗ Syntax Alternating-time Temporal Logic ( ∗ ) allows us to talk about the strategic abilities of the agents, when time is involved. ϕ ::= p | ¬ ϕ | ϕ ∨ ϕ | � � C � � ψ ::= ϕ | ¬ ψ | ψ ∨ ψ | � ψ | ψ U ψ ψ � � C � � ψ agents in C can enforce ψ , regardless of actions of others � ψ ψ holds at the next step ψ 1 U ψ 2 ψ 2 holds in the future, and until then ψ 1 holds Interpreted over Concurrent Game Structures (CGS), such as . . . Arianna Novaro 13/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Concurrent Game Structures with Exclusive Propositional Control A CGS-EPC is a tuple G = ( N, Φ 1 , . . . , Φ n , S, d, τ ) where: ◮ N = { 1 , . . . , n } is a set of agents ◮ Φ = Φ 1 ∪ · · · ∪ Φ n is a set of variables (partition) ◮ S = 2 Φ is the set of states, i.e., all valuations over Φ ◮ d : N × S → (2 A \ ∅ ) , for A = 2 Φ , is the protocol function, such that d ( i, s ) ⊆ A i for A i = 2 Φ i ◮ τ : S × A n → S is the transition function, such that τ ( s, α 1 , . . . , α n ) = � i ∈ N α i Belardinelli, Herzig. On Logics of Strategic Ability based on Propositional Control . IJCAI-2016. Arianna Novaro 14/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Example of CGS-EPC: Friends Organize a Potluck ◮ N = { 1 , 2 , 3 } ◮ Φ = Φ 1 ∪ Φ 2 ∪ Φ 3 = { wine } ∪ { steak } ∪ { herring } ◮ S = {∅ , { wine } , { wine , steak } , { wine , steak , herring } , . . . } ◮ for any s ∈ S , d (1 , s ) = {∅ , { wine }} , d (2 , s ) = {∅ , { steak }} , d (3 , s ) = {∅ , { herring }} ◮ τ ( s, α 1 , α 2 , α 3 ) = α 1 ∪ α 2 ∪ α 3 • τ ( s, { wine } , { steak } , ∅ ) = { wine , steak } = s ′ Arianna Novaro 15/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Concurrent Game Structures with Shared Propositional Control A CGS-SPC is a tuple G = ( N, Φ 0 , . . . , Φ n , S, d, τ ) where: ◮ N, S, and d are defined as for CGS-EPC ◮ Φ = Φ 0 ∪ Φ 1 ∪ · · · ∪ Φ n is a set of variables ◮ τ : S × A n → S is the transition function Belardinelli, Grandi, Herzig, Longin, Lorini, Novaro, Perrussel. Relaxing Exclusive Control in Boolean Games . TARK-2017. Arianna Novaro 16/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Example of CGS-SPC: Friends Organize a Visit ◮ N = { 1 , 2 , 3 } ◮ Φ = Φ 1 = Φ 2 = Φ 3 = { bridge , clock , castle } ◮ S = {∅ , { bridge } , { bridge , clock } , { clock , castle } , . . . } ◮ for any s ∈ S , d (1 , s ) = d (2 , s ) = d (3 , s ) = S ◮ p ∈ τ ( s, α 1 , α 2 , α 3 ) if and only if |{ i ∈ N | p ∈ α i }| ≥ 2 • τ ( s, { bridge, castle } , { clock } , { castle } ) = { castle } = s ′ Arianna Novaro 17/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 What Can Agents Do? ATL ∗ Semantics ◮ λ = s 0 s 1 . . . is a path if, for all k ≥ 0 , τ ( s k , α ) = s k +1 such that α = ( α 1 , . . . , α n ) and α i ∈ d ( i, s k ) for i ∈ N ◮ out ( s, σ C ) = { λ | s 0 = s and, for k ≥ 0 , there is α such that σ C ( i )( s k ) = α i for all i ∈ C and τ ( s k , α ) = s k +1 } ( G , s ) | p ∈ s = p iff ( G , s ) | = � � C � � ψ iff for some σ C , for all λ ∈ out ( s, σ C ) , ( G , λ ) | = ψ ( G , λ ) | = ϕ iff ( G , λ [0]) | = ϕ ( G , λ ) | = � ϕ iff ( G , λ [1 , ∞ ]) | = ϕ there is t ′ ≥ 0 such that ( G , λ [ t ′ , ∞ ]) | � ( G , λ ) | = ϕ U ψ iff = ψ and for all 0 ≤ t ′′ < t ′ : ( G , λ [ t ′′ , ∞ ]) | � = ϕ Arianna Novaro 18/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 Iterated Boolean Games as CGS An Iterated Boolean Game is a tuple ( G , γ 1 , . . . , γ n ) such that ◮ G is a CGS-EPC where d ( i, s ) = A i for every i ∈ N and s ∈ S ◮ for every i ∈ N the goal γ i is an LTL formula An Iterated Boolean Game with shared control is a tuple ( G , γ 1 , . . . , γ n ) such that ◮ G is a CGS-SPC ◮ for every i ∈ N the goal γ i is an LTL formula We can also express influence games and aggregation games. Grandi, Lorini, Novaro, Perrussel. Strategic Disclosure of Opinions on a Social Network . AAMAS-2017. Grandi, Grossi, Turrini. Equilibrium Refinement through Negotiation in Binary Voting . IJCAI-2015. Arianna Novaro 19/30
Main Results
Relaxing Exclusive Control in Boolean Games SEGA 2018 Exclusive and Shared Control Structures A CGS-SPC can be simulated by a CGS-EPC. • ◦ ◦ Define a corresponding CGS-EPC from a given CGS-SPC • • ◦ Define a translation function tr within ATL ∗ • • • Show that the CGS-SPC satisfies ϕ if and only if the corresponding CGS-EPC satisfies tr ( ϕ ) Arianna Novaro 21/30
Relaxing Exclusive Control in Boolean Games SEGA 2018 • ◦ ◦ | The corresponding CGS-EPC G = ( N, Φ 0 , . . . , Φ n , S, d, τ ) Shared control (CGS-SPC) G ′ = ( N ′ , Φ ′ 1 , . . . , Φ ′ n , S ′ , d ′ , τ ′ ) Exclusive control (CGS-EPC) N ′ = adding a dummy agent Φ ′ = adding a turn variable and local copies of variables in Φ • agent i controls her copies; dummy controls Φ and turn S ′ = all valuations over Φ ′ d ′ = depends on the truth value of turn variable: agents act when turn false; dummy acts when turn true τ ′ = updates according to agents’ actions Arianna Novaro 22/30
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