Optimization and Stability in Games with Restricted Interactions Reshef Meir, Yair Zick and Jeffrey S. Rosenschein CoopMAS 2012

Lecture content Coalitional (TU) games Restricted cooperation The Cost of Stability Main result: a bound on the CoS Discussion

TU Games - Notations Agents: N = (1,…, n ) Coalition: S µ N Characteristic function: v : 2 I → R A TU game is simple, if every coalition either wins or loses, i.e. v : 2 I → {0,1} A TU game is monotone, if the value of a coalition can only increase by adding more agents to it

TU Games – notations (2) A TU game is superadditive (SA), if there is positive synergy. That is, v ( S [ T ) ≥ v ( S ) + v ( T ) for disjoint S,T. No need to consider coalition structures Results can be generalized Every game has a superadditive cover

Weighted Voting Games (WVG) A class of simple TU games Each agent has a weight w i 2 R A game has a quota q 2 R G = [ w 1, w 2,…, w n ; q ] A coalition S wins if Σ i2S w i ¸q

Payoffs Agents may freely distribute profits. An imputation is a vector x = ( x 1,…, x n ) such that Σ i2N x i = v ( N ) Individual rationality: each agent gets at least what she can make on her own: x i ≥ v ({ i }) The payoff of a coalition x ( S ) is the sum of payments to its agents.

The Core The core is the set of all stable imputations: for all S µ N we have Σ i2S x i ¸ v ( S ) May be empty in many games: No stable imputations Example: G = [2,2,3;4] Computational questions: Is the core empty? Is the vector x in the core?

Restricted cooperation Some coalitions may be impossible or unlikely due to practical reasons an underlying communication network (Myerson’77). agents are nodes. A coalition can form only if its agents are connected

Restricted cooperation - example The coalition {2,9,10,12} is allowed The coalition {3,6,7,8} is not allowed

Restricted cooperation increases stability Theorem [Demange’04] : If the underlying communication network H is a tree, then the core is non-empty. Moreover, a core imputation can be computed efficiently

What if the core is empty? A solution: subsidies Sometimes an external party is interested in the stability of a specific outcome Willing to spend money to increase stability

External Payments Originally, we divided v ( N ) between the agents. We increase the value of v ( N ), creating a “superimputation”: Division of the incremented value v ’( N )= α∙ v ( N ) Create a new game G (α) v(N)v(N) v(N)v(N)

The Cost Of Stability (CoS) Observation: With a big enough payment, every game can be stabilized α ≤ n The Cost of Stability (CoS) is the minimal subsidy α that stabilizes the grand coalition i.e. allows a non-empty core in G (α) Can also stabilize coalition structures (Bachrach et al., SAGT’09)

Back to our example G = [2,2,3;4] (core is empty) By distributing a total payoff of 1½ (rather than 1 ), the core of G (1½) is non-empty. x = (½, ½, ½) is a stable super-imputation. Thus CoS ( G ) ≤ 1½ Is this bound tight? A lower payment cannot stabilize the game Thus CoS ( G ) = 1½

Conceptual Issues How do properties of the game affect the CoS? Superadditivity, restricted cooperation, convexity… Can we stabilize other outcomes? A particular coalition, coalition structures…

Computational Issues How hard is computing the optimal coalitional structure? How hard is computing the CoS? How hard is checking whether a specific super-imputation is stable? The answer depends on game representation We assume oracle access to v ( S )

Bounds on the CoS In the general case can be as high as n For example, the WVG [1,1,1,…,1; 1] If G is superadditive, CoS ( G )≤√ n Easier to achieve cooperation If G is superadditive and symmetric, CoS ( G ) ≤ 2 Previous work Bachrach et al., SAGT’09 Meir et al., SAGT ‘10

CoS with restricted cooperation Recall that by [Demange’04] : if H is a tree, then the core is non-empty (i.e. CoS = 0 ). Sparse graphs  lower subsidies? Sparse graphs  easier computation? Theorem: If H contains a single cycle, then CoS ( G ) ≤ 2, and this is tight Previous work Meir et al., IJCAI ‘11

Graphs and tree-width Combinatorial measures to the “cyclicity” of a graph: Degree Path-width Tree-width … Many NP-hard combinatorial problems become easy when the tree-width is bounded ,2,3 2,4 2,5,9 5,9,10 5,8,10 5,6,8 6,7,8 9,11

Bounding the CoS Conjecture [MRM’11]: Let d be the maximal degree in H, then CoS ( G ) ≤ d Conjecture (fixed): Let k be the tree- width of H, then CoS ( G ) ≤ k There are games on a 3-dimensional grid (d = 6 ) with unbounded CoS

Main result Theorem: Let G be a superadditive game, then CoS ( G ) ≤ ( TW ( H ) + 1) ∙ log ( n ) Also, a stable payoff vector can be found efficiently

Proof a b c d a b e f c d i j b c k a d l m a b x y z …

Proof a b c d a b e f c d i j b c k a d l m a b x y z … ( k +1) v ( N )

Proof a b c d a b e f c d i j b c k a d l m a b x y z … ( k +1) v ( N )

Proof a b c d a b e f c d i j b c k a d l m a b x y z … f x z i j … ( k +1) v ( N ) + ( k +1)( v ( S 1 ) + v( S 2 ) + …) ≤ ( k +1) v ( N ) + ( k +1) v ( N ) + …

Proof a b c d a b e f c d i j b c k a d l m a b x y z … f x z i j … We pay at most ( k +1) v ( N ) at each iteration

Proof a b c d a b e f c d i j b c k a d l m a b x y z … f x z i j … We repeat at most log (| T |) ≤ log ( n ) times

Discussion The CoS depends on the tree-width of the underlying graph New results… Bounded tree-width does not facilitate computations (e.g. Greco et al.’11)

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