Overlapping Coalition Formation: Charting the Tractability Frontier Y. Zick, G. Chalkiadakis and E. Elkind (submitted to AAMAS 2012)

Motivation Agents have limited integer resources The benefit of interaction may be freely divided Form Bilateral Trade Contracts: coalitions

Questions What is the optimal coalition structure? How should profits be divided?

Problem Complexity Agents are nodes The problem can be modeled as a graph There is an edge between agents if they can profit from collaborating. Goal: optimal allocation

v 1 ( x ) = 5 I 5 ( x ) v 1,2 ( x, y ) = log ( x + y + 2) v 2 ( x ) = 0 w 1 = 8 w 2 = 3

v 1,2 ( x, y ) = log ( x + y + 2) v 2 ( x ) = 0 w 1 = 8 w 2 = 3 v 1 ( x ) = 5 I 5 ( x ) v 1 (5) = 5 v 1,2 (1,1) = 2

Computational complexity computing an optimal allocation is NP-hard even for a single agent (the KNAPSACK problem). One agent with large weight – find the optimal set of tasks to complete. Optimal Coalition Structure

Theorem: computing an optimal allocation is in P for constant # of agents and poly size weights. Proof: can be done by dynamic programming. Optimal Coalition Structure

Computational complexity even when weights are at most 3, complex interactions cause NP- hardness (the X3C problem). Optimal Coalition Structure

We assume that: Weights are polynomially bounded Interactions are simple. Optimal Coalition Structure

Suppose that the interaction graph is a tree Optimal Coalition Structure

Theorem: if the maximal weight is W and there are n nodes, an optimal allocation can be computed in time linear in n and polynomial in W. Optimal Coalition Structure

We set: u i ( x i ) – the most an agent can make working alone u i, j ( x i, x j ) – the most two agents can make by working together T i ( x i ) – the most the subtree rooted at i can make Optimal Coalition Structure

OPT=max{ u 1 ( x 1 ) + § u 1, j ( x 1j, y j ) + T j ( w j - y j )} T 3 ( x 3 )= max{ u 3 ( y 3 )+ § u 3, j ( y 3j, z j ) + T j ( w j - z j )}

Stability Optimal resource allocation Which profit divisions ensure group stability?

17,15 10,5 1,5 4,3 10,13 5 5,7 16,5 7 1,1 10,9 4,5 13,12 ( CS, x ) CSx Outcome Is ( CS, x ) in the core?

Deviation “Coalitional game theory [...] considers a game of n players as a set of possible 2 n – 1 coalitions, each of which, call it S, can achieve a particular value v ( S ) […] against worst case behavior of players in N \ S ” C.H. Papadimitriou, STOC 2001 Players assume they are “on their own” if they deviate.

17,15 10,5 1,5 4,3 10,13 5 5,7 16,5 7 1,1 10,9 4,5 13,

Stability Arbitration functions: agents may receive all or some of the payoff from unbroken/changed agreements. Behavior can be very general.

Arbitration Functions Others can react to deviation either locally or globally. Conservative – give nothing Refined – give all from unhurt coalitions Optimistic – deviators absorb the marginal damage of deviation; get the difference.

17,15 10,5 1,5 4,3 10,13 5 5,7 16,5 7 1,1 10,9 4,5 13,12 8,15 GlobalLocal 8,10

Stability Theorem: if there is an efficient algorithm to compute the most one can get from global arbitration functions, then P = NP.

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Stability Theorem: if the arbitration function is local, and the interaction graph is a tree, computing the most a set can get from deviating is possible in poly(n,W) time

Stability Denote the most that a set S can get by deviating by A *( S, CS, x ) Having divided payoffs, can we verify that no set wants to deviate?

Stability Theorem: if the arbitration function is local, and the interaction graph is a tree, then one can verify if an outcome is A -stable in poly(n,W) time.

Stability Given an outcome ( CS, x ), the excess of a set S is the difference between the payoff to S under ( CS, x ), denoted p S ( CS, x ) and A *( S, CS, x ) e ( S, CS, x ) = A *( S, CS, x ) - p S ( CS, x )

Stability We set: E i ( x ) – the maximal excess of a set containing i, assuming i invests x in working with that set.

E 1 ( x ) = max{ u 1 ( a 1 ) + § i2 {2,3,4} ( u 1, i ( b 1, i, y i ) + E i ( w i – y i ))} – p

Stability Corollary: Given a coalition structure CS, we can find x such that (CS, x) is A -stable in poly(n,W) time. Proof: ellipsoid method to solve an LP

Recap Optimization/Stability: Hard in general due to Weights Complex interaction

More Results Bounded hyper-treewidth: Our results can be extended to graphs with bounded hyper-treewidth. If the graph is “tree-like” we can still obtain efficient algorithms.

More Results Stable conservative core: We can find a stable outcome against worst case behavior. Each agent receives the minimum needed to make his subtree stable.

Summary Computational Issues: A major obstacle in OCF games. But: if interactions are (somewhat) local, both for values and arbitration functions, we can obtain poly-time algorithms.

Poly-time, but… Complexity is still high: Order of O(n k W 5(k+1) ) for computing optimal allocation in a graph with treewidth k Can probably do better if valuations are known.

Future Work Deterministic, Exact: randomized/ approximation algorithms? Restricted classes of games: convex, subadditive…

Thank you! Questions?