Agent-Based Systems Agent-Based Systems Where are we? Agent-Based Systems • Discussed procedures for making group decisions • Simple mechanisms: plurality, sequential majority • Advanced mechanisms: Borda Count, Slater Ranking Michael Rovatsos • Desirable properties, paradoxes and dictatorships mrovatso@inf.ed.ac.uk • Strategic manipulation and computational complexity Today . . . Lecture 10 – Coalition Formation • Forming Coalitions 1 / 16 2 / 16 Agent-Based Systems Agent-Based Systems Forming Coalitions Terminology • In games like the Prisoner’s Dilemma cooperation is prevented • Ag = { 1 , . . . , n } agents (typically n > 2) because: • Any subset C of Ag is called a coalition - Binding agreements are not possible • C = Ag is the grand coalition , - Utility is given directly to individuals as the result of individual action • These features do not hold in many real world situations: • A cooperative game is a pair G = � Ag , ν � • ν : 2 Ag → R is the characteristic function of the game - Contracts can form binding arrangements - Revenue that a company earns is not credited to an individual • ν ( C ) is the utility C can achieve, regardless of Ag − C ’s behaviour • When we lift these assumptions cooperation is both possible and • Singleton coalitions contain one agent (describe what agents can rational achieve alone) • Cooperative game theory asks which contracts are meaningful • Neither individual actions and utilities matter, nor the origin of ν solutions among self-interested agents 3 / 16 4 / 16
Agent-Based Systems Agent-Based Systems Three Stages of Cooperative Action Outcomes and Objections • An outcome x = � x 1 , . . . , x k � for a coalition C in game � Ag , ν � is a • Coalition structure generation distribution of C ’s utility to members of C - Asking which coalitions will form, concerned with stability • Outcomes must be feasible (don’t overspend) and efficient (don’t - For example, a productive agent has the incentive to defect from a underspend): � i ∈ C x i = ν ( C ) coalition with a lazy agent • Example: - Necessary but not sufficient condition for establishment of a coalition - Ag = { 1 , 2 } , ν ( { 1 } ) = 5, ν ( { 2 } ) = 5 and ν ( { 1 , 2 } ) = 20 • Solving the optimisation problem of each coalition - Possible outcomes for C = { 1 , 2 } are � 20 , 0 � , � 19 , 1 � , . . . , � 0 , 20 � - Decide on collective plans • C objects to an outcome for the grand coalition if there is some - Maximise the collective utility of the coalition • Dividing the value of the solution of each coaltion outcome for C in which all members of C are strictly better off - Concerned with fairness of contract • Formally, C ⊆ Ag objects to x = � x 1 , . . . , x n � for the grand coalition, iff there exists some outcome x ′ = � x ′ 1 , . . . , x ′ k � for C , - How much an agent should receive based on her contribution such that x ′ i > x i for all i ∈ C 5 / 16 6 / 16 Agent-Based Systems Agent-Based Systems The Core The Shapley Value (I) • The core of a coalitional game is the set of outcomes that no • To eliminate unfair distribution, try to divide surplus according to sub-coalition can object to contribution • If the core is non-empty, then the grand coalition is stable • Define marginal contribution of i to C : µ i ( C ) = ν ( C ∪ { i } ) − ν ( C ) • The core of the previous example contains all outcomes between • Axioms any fair distribution should satisfy: � 15 , 5 � and � 5 , 15 � inclusive - Symmetry: if two agents contribute the same they should receive • Problems: the same pay-off (they are interchangeable) - Sometimes the core is empty - Dummy player: agents that do not add value to any coalition should - Fairness: � 15 , 5 � distributes all the surplus generated by the get what they earn on their own cooperation to one agent (fairness?) - Additivity: if two games are combined, the value a player gets - The definition of the core involves quantification over all possible should be the sum of the values it gets in individual games coalitions, so all of them have to be enumerated 7 / 16 8 / 16
Agent-Based Systems Agent-Based Systems The Shapley Value (I) Representation • A naive representation of a coalition game is infeasible • The Shapley value for agent i : (exponential in the size of Ag ): 1 sh i = � o ∈ Π( Ag ) µ i ( C i ( o )) 1, 2, 3 | Ag | ! 1 = 5 - Π( Ag ) denotes the set of all possible orderings (e.g. for 2 = 5 Ag = { 1 , 2 , 3 } , Π( Ag ) = { ( 1 , 2 , 3 ) , ( 1 , 3 , 2 ) , ( 2 , 1 , 3 ) , . . . } ) 3 = 5 - C i ( o ) denotes the agents that appear before i in o 1, 2 = 10 1, 3 = 10 • Requires that 2, 3 = 20 • ν ( ∅ ) = 0 and 1, 2, 3 = 25 • ν ( C ∪ C ′ ) ≥ ν ( C ) + ν ( C ′ ) if C ∩ C ′ = ∅ • As with preference orderings, we need a succinct representations ( ν superadditive ) • Modular representations exploit Shapley’s axioms directly • Strong result: The Shapley value is the only value that satisfies the • Basic idea: divide the game into smaller games and exploit fairness axioms additivity axiom 9 / 16 10 / 16 Agent-Based Systems Agent-Based Systems Induced Subgraphs Marginal Contribution Nets • Define a characteristic function by an undirected weighted graph • Represent characteristic function as rules: pattern − → value • Value of a a coalition C ⊆ Ag : ν ( C ) = � { i , j }⊆ C w i , j - the pattern is a conjunction of agents, e.g. 1 ∧ 3 • Example: - 1 ∧ 3 would apply to { 1 , 3 } and { 1 , 3 , 5 } , but not to { 1 } or { 8 , 12 } ν ( { A , B , C } ) = 3 + 2 = 5 - C � ϕ , means the rule ϕ − → x applies to coalition C A 3 B ν ( { D } ) = 5 - rs C = { ϕ − → x ∈ rs | C � ϕ } are the rules that apply to coalition C 1 2 ν ( { B , D } ) = 1 + 5 = 6 • ν rs ( C ) = � → x ∈ rs C x ϕ − C 4 D 5 ν ( { A , C } ) = 2 • Example: • Not a complete representation (not all characteristic functions can - rs 1 = { a ∧ b − → 5 , b − → 2 } be represented) - ν rs 1 ( { a } ) = 0, ν rs 1 ( { b } ) = 2 and ν rs 1 ( { a , b } ) = 7 • But easy to compute the Shapley value for a given player in • Extension: allow negation in rules, e.g. b ∧ ¬ c − → − 2 polynomial time • Shapley value can be computed in polynomial time - sh i = 1 � j w i , j 2 • Complete representation, but not necessarily succinct • Checking emptiness of the core is NP-complete, and membership to the core is co-NP-complete 11 / 16 12 / 16
Agent-Based Systems Agent-Based Systems Representations for Simple Games Weighted Voting Games • For each agent i ∈ Ag define a weight w i and an overall quota q • A coalitional game is simple if the value of any coalition is either • A coalition is winning if the sum of their weights exceeds the quota: 0 (losing) or 1 (winning) � if � i ∈ C w i ≥ q 1 • Simple games model yes/no voting systems ν ( C ) = 0 otherwise • Y = � Ag , W � , where W ⊆ 2 Ag is the set of winning coalitions • Example: Simple majority voting , w i = 1 and q = ⌈| Ag | + 1 ⌉ • If C ∈ W , C would be able to determine the outcome, ‘yes’ or ‘no’ 2 • Succinct (but incomplete) representation: � q ; w 1 , . . . , w n � • Important conditions: • Extension: k -weighted voting games are a complete - Non-triviality: ∅ ⊂ W ⊂ 2 Ag representation - Monotonicity: if C 1 ⊆ C 2 and C 1 ∈ W then C 2 ∈ W - Zero-sum: if C ∈ W then Ag \ C �∈ W - overall game = ”conjunction” k of k different weighted voting games - Empty coalition loses: ∅ �∈ W - Winning coalition is the one that wins in all component games - Grand coalition wins: Ag ∈ W - Game dimension : k is at most exponential in the number of players - Checking whether a k -weighted voting game is minimal is • Naive representation is exponential in the number of agents NP-complete 13 / 16 14 / 16 Agent-Based Systems Agent-Based Systems Weighted Voting Games (II) Summary • Shapley-Shubic power index = Shapley value in yes/no games - Measures the power of the voter in this case • Coalition formation - Computation is NP-hard, no reasonable polynomial time • The core and the Shapley value approximation - Checking emptiness of the core can be done in polynomial time • Different representations ( veto player) • Simple games • Counter-intuitive properties: • Next time: Resource Allocation - In � 100 ; 99 , 99 , 1 � , all voters have the same power ( 1 3 ) - Dummy with non-zero power, e.g. � 10 ; 6 , 4 , 2 � , meaningful? - Adding new voters increases voter power, e.g. � 10 ; 6 , 4 , 2 , 8 � 15 / 16 16 / 16
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