Representing and reasoning about Bayesian coalitional games Oldooz - PowerPoint PPT Presentation

Representing and reasoning about Bayesian coalitional games Oldooz Dianat Supervisor: Prof. Mehmet Orgun Co-supervisor: Dr. Lee Flax Macquarie University Department of Computing June 2012 Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 1 / 28

Outline Research gap and aims 1 Background 2 Methodology and research progress 3 Conclusion 4 Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 2 / 28

Research gap and aims Outline Research gap and aims 1 Background 2 Methodology and research progress 3 Conclusion 4 Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 3 / 28

Research gap and aims Multi-agent systems Agents Computer systems that are capable of independent, autonomous action in order to satisfy their design objectives. In multi-agent systems with self-interested agents; this is the case in most economic applications. The optimal action for one agent to take depends on the actions that other agents take. To determine the optimal action for the agent under specific circumstance, game theory can provide useful models. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 4 / 28

Research gap and aims Game theory Tries to predict or explain the behavior of agents under a sequence of interactions. Is the mathematical study of interaction among independent, self interested agents. Quantifies an agent’s degree of preference across a set of available alternatives Describes how these preferences change when an agent faces uncertainty about which alternative he will receive. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 5 / 28

Research gap and aims Game theory and logic Game theory does not consider the reasoning abilities of agents. Logical declarative languages These languages enable reasoning about the best strategy in games by considering other players rationality and reasoning abilities. These languages are used to represent game model explicitly. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 6 / 28

Research gap and aims Research objectives The specific objectives of this project include: Develop the syntax and semantics for a logic of Bayesian coalition games. Identify a suitable model checker. Develop a method for translating the specification language to the input language of a model checker. Verify the properties of the system by the model checker. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 7 / 28

Background Outline Research gap and aims 1 Background 2 Methodology and research progress 3 Conclusion 4 Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 8 / 28

Background Classical coalitional games Three or more players. Each player is individually rational and thus seeks to maximize individual gain. Players are free to negotiate binding and enforceable agreements about the formation of coalitions. Division of payoffs, which is the result from coordinated actions. Mathematically speaking a colitional game is a pair( N , v ): N is a finite set of players,indexed by i v : 2 N �→ R associates with each coalition S ⊆ N a real-valued payoff v ( S ) that the coalition,s members can distribute among themselves. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 9 / 28

Background From classical models to Bayesian models Coalitional Game assumptions The payoff to each coalition is given by a fixed, deterministic value. These values are common knowledge among all agents. They often fail to hold for real-world problems. Bayesian coalitional games are: Study cooperation under uncertainty Generalization of coalitional games to a Bayesian framework using the information partition model Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 10 / 28

Background Uncertainty in games and epistemic logic Harsanyi analyzed uncertainty about the structure of games, specifically about the players’ payoff functions. He introduced the fundamental concept of a player’s type. It could be used to encode what the player believes the payoff functions to be, what the player believes other players believe the payoff functions to be, and so on, indefinitely. His formalism introduced the players hierarchies of beliefs over: strategies, and rationality. Interactive epistemology deals with the beliefs and the knowledge of players. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 11 / 28

Background Epistemic logic Epistemic logic is the logic of knowledge and belief Reasons about the knowledge of agents in a group. Agents may have different information and thus different epistemic alternatives at each possible state. The logical symbols used are : ¬ , ∧ , ∨ , → , and ↔ . The conjunction (disjunction) of all sentences from a finite set Σ is abbreviated by � Σ ( � Σ), assuming commutativity. The language has knowledge operator K i . Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 12 / 28

Methodology and research progress Outline Research gap and aims 1 Background 2 Methodology and research progress 3 Conclusion 4 Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 13 / 28

Methodology and research progress Epistemic logic by B.De Burin We abstract from the specific actions and reason about what specific agents can achieve. B.De Burin introduced an epistemic language for representing the solution concept in normal games and extensive games. He proposed a formula that expresses the solution concept such as the Nash equilibrium. This formula simply says, if a player knows his opponent to be rational, and knows that his opponent knows the utility structure, these beliefs form a pure strategy Nash equilibrium. We refine the notation of rationality axiom, to adapt it for pure strategy in normal form games. Then we extend it for representing and reasoning about Bayesian games. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 14 / 28

Methodology and research progress Propositions The proposition letters i m stand for the statement ’ i plays his m th strategy i m ’. The proposition u i (1 k 1 , ..., N k 1 ) = r i , 1 k 1 ,..., N k 1 denotes that the utility for player i , when the strategy profile (1 k 1 , ..., N k 1 ) is played, equals the number r . R is a set of countably many symbols such as r . The elements of R represent real numbers. The proposition rat i denotes the rationality of player i , in the sense that i is an expected utility maximizer. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 15 / 28

Methodology and research progress Axioms Any axiomatization for propositional logic K i ( ϕ → ψ ) → ( K i ϕ → K i ψ ) K i ϕ → ¬ K i ¬ ϕ K i ϕ → K i K i ϕ ( K i ϕ ∧ K i ( ϕ → ψ )) → K i ψ K i ϕ → ϕ The inference rules are: If ⊢ ϕ → ψ and ⊢ ϕ then ⊢ ψ If ⊢ ϕ then K i ϕ Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 16 / 28

Methodology and research progress Rationality axiom and Nash equilibrium Rationality axiom The proposition that captures the rationality of player i is called rat i . The axiom RAT which is the formalism of utility maximization captures the rat i as the following implication: � � � (( K i u i ( k , l ) = r i , k , l ∧ i m ) → (1) rat i ↔ r i , m , l ≥ r i , k , l m k , l k The above axiom states that player i aims at his utility maximizer, if he decides to play his m th strategy in a situation in which he has certain beliefs about utility (captured by the r i , k , l ) then the m th strategy is better than any other, given his beliefs. Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 17 / 28

Methodology and research progress Assumptions for normal form games There are some assumptions needed to use the epistemic logic for reasoning about Nash equilibrium of normal form games: All players know their own utility function, which is: � i K i � k , l u i ( k , l ) = r i , k , l All players are rational, which is: � i rat i All players know each player’s actual choice of an action, which is: K 2 1 m ∧ K 1 2 n Then the actual action profile played constitutes a Nash equilibrium. It means the solution concept for player 1 is � r 1 , m , l ≥ r 1 , k , l (2) k and for player 2 is � r 2 , k , n ≥ r 2 , k , l (3) l . Oldooz Dianat (Department of computing) Bayesian coalitional game logic June 2012 18 / 28