2. Reasoning about Coalitions 2. ATL Definition 2.2 (Concurrent Game Structure) A concurrent game structure is a tuple M = � A gt , St, π, Act, d, o � , where: A gt : a finite set of all agents St : a set of states π : a valuation of propositions Act : a finite set of (atomic) actions d : A gt × St → P ( Act ) defines actions available to an agent in a state o : a deterministic transition function that assigns outcome states q ′ = o ( q, α 1 , . . . , α k ) to states and tuples of actions Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 16/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage 1 2 pos 0 1 1 2 pos 1 pos 2 2 2 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 17/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait,wait push,push 1 2 pos 0 q 0 pos 0 push,wait push,wait wait,push wait,push 1 1 2 wait,wait pos 1 wait,wait pos 2 push,push push,push q 2 q 1 2 2 wait,push 1 push,wait pos 2 pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 17/70
2. Reasoning about Coalitions 2. ATL Definition 2.3 (Strategy) A strategy is a conditional plan. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 18/70
2. Reasoning about Coalitions 2. ATL Definition 2.3 (Strategy) A strategy is a conditional plan. We represent strategies by functions s a : St → Act . Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 18/70
2. Reasoning about Coalitions 2. ATL Definition 2.3 (Strategy) A strategy is a conditional plan. We represent strategies by functions s a : St → Act . Function out ( q , S A ) returns the set of all paths that may result from agents A executing strategy S A from state q onward. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 18/70
2. Reasoning about Coalitions 2. ATL Definition 2.4 (Semantics of ATL) iff there is a collective strategy S A such M, q | = � � A � � Φ that, for every path λ ∈ out ( q, S A ) , we have M, λ | = Φ . Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70
2. Reasoning about Coalitions 2. ATL Definition 2.4 (Semantics of ATL) iff there is a collective strategy S A such M, q | = � � A � � Φ that, for every path λ ∈ out ( q, S A ) , we have M, λ | = Φ . iff M, λ [1] | = ϕ ; M, λ | = � ϕ Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70
2. Reasoning about Coalitions 2. ATL Definition 2.4 (Semantics of ATL) iff there is a collective strategy S A such M, q | = � � A � � Φ that, for every path λ ∈ out ( q, S A ) , we have M, λ | = Φ . iff M, λ [1] | = ϕ ; M, λ | = � ϕ iff M, λ [ i ] | = ϕ for some i ≥ 0 ; M, λ | = ♦ ϕ Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70
2. Reasoning about Coalitions 2. ATL Definition 2.4 (Semantics of ATL) iff there is a collective strategy S A such M, q | = � � A � � Φ that, for every path λ ∈ out ( q, S A ) , we have M, λ | = Φ . iff M, λ [1] | = ϕ ; M, λ | = � ϕ iff M, λ [ i ] | = ϕ for some i ≥ 0 ; M, λ | = ♦ ϕ iff M, λ [ i ] | = ϕ for all i ≥ 0 ; M, λ | = � ϕ Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70
2. Reasoning about Coalitions 2. ATL Definition 2.4 (Semantics of ATL) iff there is a collective strategy S A such M, q | = � � A � � Φ that, for every path λ ∈ out ( q, S A ) , we have M, λ | = Φ . iff M, λ [1] | = ϕ ; M, λ | = � ϕ iff M, λ [ i ] | = ϕ for some i ≥ 0 ; M, λ | = ♦ ϕ iff M, λ [ i ] | = ϕ for all i ≥ 0 ; M, λ | = � ϕ iff M, λ [ i ] | = ψ for some i ≥ 0 , and M, λ | = ϕ U ψ = ϕ forall 0 ≤ j ≤ i . M, λ [ j ] | Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70
2. Reasoning about Coalitions 2. ATL Definition 2.4 (Semantics of ATL) iff p is in π ( q ) ; M, q | = p iff M, q | = ϕ and M, q | = ψ ; M, q | = ϕ ∧ ψ iff there is a collective strategy S A such M, q | = � � A � � Φ that, for every path λ ∈ out ( q, S A ) , we have M, λ | = Φ . iff M, λ [1] | = ϕ ; M, λ | = � ϕ iff M, λ [ i ] | = ϕ for some i ≥ 0 ; M, λ | = ♦ ϕ iff M, λ [ i ] | = ϕ for all i ≥ 0 ; M, λ | = � ϕ iff M, λ [ i ] | = ψ for some i ≥ 0 , and M, λ | = ϕ U ψ = ϕ forall 0 ≤ j ≤ i . M, λ [ j ] | Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 19/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait,wait push,push q 0 pos 0 push,wait push,wait wait,push wait,push pos 0 → � � 1 � � � ¬ pos 1 wait,wait wait,wait push,push push,push q 2 q 1 wait,push pos 2 push,wait pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait,wait push,push q 0 pos 0 push,wait push,wait wait,push wait,push pos 0 → � � 1 � � � ¬ pos 1 wait,wait wait,wait push,push push,push q 2 q 1 wait,push pos 2 push,wait pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait,wait push,push q 0 wait wait pos 0 push,wait push,wait wait,push wait,push pos 0 → � � 1 � � � ¬ pos 1 wait,wait wait,wait push,push push,push q 2 q 1 wait,push push push wait wait pos 2 push,wait pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait ,wait wait push,push q 0 pos 0 push,wait push ,wait wait ,push wait ,push wait push pos 0 → � � 1 � � � ¬ pos 1 wait wait,wait wait wait ,wait push, push push push,push q 1 q 2 wait,push push,wait pos 2 pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait ,wait wait push,push q 0 pos 0 push,wait push ,wait wait ,push wait ,push wait push pos 0 → � � 1 � � � ¬ pos 1 wait wait,wait wait wait ,wait push, push push push,push q 1 q 2 wait,push push,wait pos 2 pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait ,wait wait ,wait push,push push,push q 0 pos 0 push,wait push,wait push ,wait push ,wait wait ,push wait ,push wait ,push wait ,push pos 0 → � � 1 � � � ¬ pos 1 wait,wait wait,wait wait ,wait wait ,wait push, push push, push push,push push,push q 1 q 2 wait,push wait,push push,wait push,wait pos 2 pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Example: Robots and Carriage wait ,wait wait ,wait push,push push,push q 0 pos 0 push,wait push,wait push ,wait push ,wait wait ,push wait ,push wait ,push wait ,push pos 0 → � � 1 � � � ¬ pos 1 wait,wait wait,wait wait ,wait wait ,wait push, push push, push push,push push,push q 1 q 1 q 2 wait,push wait,push push,wait push,wait pos 2 pos 1 pos 1 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 20/70
2. Reasoning about Coalitions 2. ATL Temporal operators allow a number of useful concepts to be formally specified Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 21/70
2. Reasoning about Coalitions 2. ATL Temporal operators allow a number of useful concepts to be formally specified safety properties liveness properties fairness properties Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 21/70
2. Reasoning about Coalitions 2. ATL Safety (maintenance goals): “something bad will not happen” “something good will always hold” Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70
2. Reasoning about Coalitions 2. ATL Safety (maintenance goals): “something bad will not happen” “something good will always hold” Typical example: � ¬ bankrupt Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70
2. Reasoning about Coalitions 2. ATL Safety (maintenance goals): “something bad will not happen” “something good will always hold” Typical example: � ¬ bankrupt Usually: � ¬ .... Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70
2. Reasoning about Coalitions 2. ATL Safety (maintenance goals): “something bad will not happen” “something good will always hold” Typical example: � ¬ bankrupt Usually: � ¬ .... In ATL: � � os � � � ¬ crash Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 22/70
2. Reasoning about Coalitions 2. ATL Liveness (achievement goals): “something good will happen” Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 23/70
2. Reasoning about Coalitions 2. ATL Liveness (achievement goals): “something good will happen” Typical example: ♦ rich Usually: ♦ .... Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 23/70
2. Reasoning about Coalitions 2. ATL Liveness (achievement goals): “something good will happen” Typical example: ♦ rich Usually: ♦ .... In ATL: � � alice, bob � � ♦ paperAccepted Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 23/70
2. Reasoning about Coalitions 2. ATL Fairness (service goals): “if something is attempted/requested, then it will be successful/allocated” Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 24/70
2. Reasoning about Coalitions 2. ATL Fairness (service goals): “if something is attempted/requested, then it will be successful/allocated” Typical examples: � ( attempt → ♦ success ) �♦ attempt → �♦ success Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 24/70
2. Reasoning about Coalitions 2. ATL Fairness (service goals): “if something is attempted/requested, then it will be successful/allocated” Typical examples: � ( attempt → ♦ success ) �♦ attempt → �♦ success In ATL* (!): � � prod, dlr � � � ( carRequested → ♦ carDelivered ) Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 24/70
2. Reasoning about Coalitions 2. ATL Connection to Games Concurrent game structure = generalized extensive game Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70
2. Reasoning about Coalitions 2. ATL Connection to Games Concurrent game structure = generalized extensive game � γ : � � splits the agents into proponents and � � A � � A � opponents γ defines the winning condition Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70
2. Reasoning about Coalitions 2. ATL Connection to Games Concurrent game structure = generalized extensive game � γ : � � splits the agents into proponents and � � A � � A � opponents γ defines the winning condition � infinite 2-player, binary, zero-sum game Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70
2. Reasoning about Coalitions 2. ATL Connection to Games Concurrent game structure = generalized extensive game � γ : � � splits the agents into proponents and � � A � � A � opponents γ defines the winning condition � infinite 2-player, binary, zero-sum game Flexible and compact specification of winning conditions Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 25/70
2. Reasoning about Coalitions 2. ATL Solving a game ≈ checking if M, q | = � � A � � γ Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 26/70
2. Reasoning about Coalitions 2. ATL Solving a game ≈ checking if M, q | = � � A � � γ But: do we really want to consider all the possible plays? Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 26/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) 2.3 Rational Play (ATLP) Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 27/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Game-theoretical analysis of games: Solution concepts define rationality of players Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Game-theoretical analysis of games: Solution concepts define rationality of players maxmin Nash equilibrium subgame-perfect Nash undominated strategies Pareto optimality Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Game-theoretical analysis of games: Solution concepts define rationality of players maxmin Nash equilibrium subgame-perfect Nash undominated strategies Pareto optimality Then: we assume that players play rationally ...and we ask about the outcome of the game under this assumption Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Game-theoretical analysis of games: Solution concepts define rationality of players maxmin Nash equilibrium subgame-perfect Nash undominated strategies Pareto optimality Then: we assume that players play rationally ...and we ask about the outcome of the game under this assumption Role of rationality criteria: constrain the possible game moves to “sensible” ones Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 28/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) start q 0 T t T h h H H money 1 t money 2 q 1 q 2 Hh H t T t H T t h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) start q 0 T t T h h H H money 1 t money 2 start → ¬� � 1 � � ♦ money 1 q 1 q 2 Hh H t T t H T t h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) start q 0 T t T h h H H money 1 t money 2 start → ¬� � 1 � � ♦ money 1 q 1 q 2 start → ¬� � 2 � � ♦ money 2 Hh H t T t H T t h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) start q 0 T t T h h H H money 1 t money 2 start → ¬� � 1 � � ♦ money 1 q 1 q 2 start → ¬� � 2 � � ♦ money 2 Hh H t T t H T t h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 29/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) ATL + Plausibility (ATLP) ATL: reasoning about all possible behaviors. � ϕ : agents A have some collective strategy to enforce ϕ � � A � against any response of their opponents. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 30/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) ATL + Plausibility (ATLP) ATL: reasoning about all possible behaviors. � ϕ : agents A have some collective strategy to enforce ϕ � � A � against any response of their opponents. ATLP: reasoning about plausible behaviors. � ϕ : agents A have a plausible collective strategy to � � A � enforce ϕ against any plausible response of their opponents. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 30/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) ATL + Plausibility (ATLP) ATL: reasoning about all possible behaviors. � ϕ : agents A have some collective strategy to enforce ϕ � � A � against any response of their opponents. ATLP: reasoning about plausible behaviors. � ϕ : agents A have a plausible collective strategy to � � A � enforce ϕ against any plausible response of their opponents. Important The possible strategies of both A and A gt \ A are restricted. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 30/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) New in ATLP: ( set-pl ω ) : the set of plausible profiles is set/reset to the strategies described by ω . Only plausible strategy profiles are considered! Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 31/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) New in ATLP: ( set-pl ω ) : the set of plausible profiles is set/reset to the strategies described by ω . Only plausible strategy profiles are considered! Example: ( set-pl greedy 1 ) � � 2 � � ♦ money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 31/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Concurrent game structures with plausibility M = ( A gt , St, Π , π, Act, d, δ, Υ , Ω , �·� ) Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Concurrent game structures with plausibility M = ( A gt , St, Π , π, Act, d, δ, Υ , Ω , �·� ) Υ ⊆ Σ : set of (plausible) strategy profiles Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Concurrent game structures with plausibility M = ( A gt , St, Π , π, Act, d, δ, Υ , Ω , �·� ) Υ ⊆ Σ : set of (plausible) strategy profiles Ω = { ω 1 , ω 2 , . . . } : set of plausibility terms Example: ω NE may stand for all Nash equilibria Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Concurrent game structures with plausibility M = ( A gt , St, Π , π, Act, d, δ, Υ , Ω , �·� ) Υ ⊆ Σ : set of (plausible) strategy profiles Ω = { ω 1 , ω 2 , . . . } : set of plausibility terms Example: ω NE may stand for all Nash equilibria �·� : St → (Ω → P (()Σ)) : plausibility mapping Example: � ω NE � q = { ( confess , confess ) } Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 32/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Outcome = Paths that may occur when agents A perform s A Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Outcome = Paths that may occur when agents A perform s A when only plausible strategy profiles from Υ are played Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Outcome = Paths that may occur when agents A perform s A when only plausible strategy profiles from Υ are played out Υ ( q, s A ) = { λ ∈ St + | ∃ t ∈ Υ( s A ) ∀ i ∈ N � � λ [ i + 1] = δ ( λ [ i ] , t ( λ [ i ])) } Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Outcome = Paths that may occur when agents A perform s A when only plausible strategy profiles from Υ are played out Υ ( q, s A ) = { λ ∈ St + | ∃ t ∈ Υ( s A ) ∀ i ∈ N � � λ [ i + 1] = δ ( λ [ i ] , t ( λ [ i ])) } q 0 start T t T h h H H money 1 t money 2 q 1 q 2 Hh H t T t H t T h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Outcome = Paths that may occur when agents A perform s A when only plausible strategy profiles from Υ are played out Υ ( q, s A ) = { λ ∈ St + | ∃ t ∈ Υ( s A ) ∀ i ∈ N � � λ [ i + 1] = δ ( λ [ i ] , t ( λ [ i ])) } q 0 start T t T h h H P : the players always show H money 1 t money 2 same sides of their coins q 1 q 2 Hh H t T t H t T h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Outcome = Paths that may occur when agents A perform s A when only plausible strategy profiles from Υ are played out Υ ( q, s A ) = { λ ∈ St + | ∃ t ∈ Υ( s A ) ∀ i ∈ N � � λ [ i + 1] = δ ( λ [ i ] , t ( λ [ i ])) } P : the players always show same sides of their coins s 1 : always show “heads” Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 33/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Semantics of ATLP � γ iff there is a strategy s A consistent with Υ M, q | = � � A � such that M, λ | = γ for all λ ∈ out Υ ( q, s A ) = ( set-pl ω ) ϕ iff M ω , q | = ϕ where the new model M, q | M ω is equal to M but the new set Υ ω of plausible strategy profiles is set to � ω � q . Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 34/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Example: Pennies Game q 0 start T t T h Hh Ht money 1 money 2 q 1 q 2 Hh Ht Ht T t T h T h h T H t q 3 money 1 q 4 q 5 money 2 money 2 M, q 0 | = ( set-pl ω NE ) � � 2 � � ♦ money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 35/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Example: Pennies Game q 0 start T t T h Hh Ht money 1 money 2 q 1 q 2 Hh Ht Ht T t T h T h h T H t q 3 money 1 q 4 q 5 money 2 money 2 M, q 0 | = ( set-pl ω NE ) � � 2 � � ♦ money 2 What is a Nash equilibrium in this game? We need some kind of winning criteria! Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 35/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Agent 1 “wins”, if γ 1 ≡ � ( ¬ start → money 1 ) is satisfied. Agent 2 “wins”, if γ 2 ≡ ♦ money 2 is satisfied. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Agent 1 “wins”, if γ 1 ≡ � ( ¬ start → money 1 ) is satisfied. Agent 2 “wins”, if γ 2 ≡ ♦ money 2 is satisfied. q 0 start T t T h h H H money 1 money 2 t q 1 q 2 Hh H t T t H t T h h T Hh T t q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Agent 1 “wins”, if γ 1 ≡ � ( ¬ start → money 1 ) is satisfied. Agent 2 “wins”, if γ 2 ≡ ♦ money 2 is satisfied. q 0 start γ 1 \ γ 2 hh ht th tt T t T h h H H money 1 t money 2 0 , 0 0 , 1 0 , 1 HH 1 , 1 q 1 q 2 HT 0 , 0 0 , 1 0 , 1 0 , 1 Hh H t T t H t T h 0 , 1 0 , 1 0 , 0 TH 1 , 1 h T Hh 0 , 1 0 , 1 0 , 0 0 , 1 T t TT q 3 money 1 q 4 q 5 money 2 money 2 Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Agent 1 “wins”, if γ 1 ≡ � ( ¬ start → money 1 ) is satisfied. Agent 2 “wins”, if γ 2 ≡ ♦ money 2 is satisfied. q 0 start γ 1 \ γ 2 hh ht th tt T t T h h H H money 1 t money 2 0 , 0 0 , 1 0 , 1 HH 1 , 1 q 1 q 2 HT 0 , 0 0 , 1 0 , 1 0 , 1 Hh H t T t H t T h 0 , 1 0 , 1 0 , 0 TH 1 , 1 h T Hh 0 , 1 0 , 1 0 , 0 0 , 1 T t TT q 3 money 1 q 4 q 5 money 2 money 2 Now we have a qualitative notion of success. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) Agent 1 “wins”, if γ 1 ≡ � ( ¬ start → money 1 ) is satisfied. Agent 2 “wins”, if γ 2 ≡ ♦ money 2 is satisfied. q 0 start γ 1 \ γ 2 hh ht th tt T t T h h H H money 1 t money 2 0 , 0 0 , 1 0 , 1 HH 1 , 1 q 1 q 2 HT 0 , 0 0 , 1 0 , 1 0 , 1 Hh H t T t H t T h 0 , 1 0 , 1 0 , 0 TH 1 , 1 h T Hh 0 , 1 0 , 1 0 , 0 0 , 1 T t TT q 3 money 1 q 4 q 5 money 2 money 2 Now we have a qualitative notion of success. M, q 0 | = ( set-pl ω NE ) � � 2 � � � ( ¬ start → money 1 ) where � ω NE � q 0 = “all profiles belonging to grey cells”. Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 36/70
2. Reasoning about Coalitions 3. Rational Play (ATLP) How to obtain plausibility terms? Stéphane Airiau and Wojtek Jamroga · Coalitional Games EASSS’09 @ Torino 37/70
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