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Agent Failures in Totally Balanced Games and Convex Games Authors: Yoram Bachrach 1 Ian Kash 1

Agent Failures in Totally Balanced Games and Convex Games Authors: Yoram Bachrach 1 Ian Kash 1 Nisarg Shah 2 (speaker) 1 Microsoft Research Cambridge. 2 Carnegie Mellon University. Agenda. Agent Failures in Cooperative Games Sub-Agenda:

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Agent Failures in Totally Balanced Games and Convex Games Authors: Yoram Bachrach 1 Ian Kash 1

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  1. Agent Failures in Totally Balanced Games and Convex Games Authors: Yoram Bachrach1 Ian Kash1 Nisarg Shah2 (speaker) 1 Microsoft Research Cambridge. 2 Carnegie Mellon University.

  2. Agenda • Agent Failures in Cooperative Games • Sub-Agenda: • Effect of Agent Failures on the Existence of the Core • Initiated by [Bachrach et. al., ‘11]

  3. Cooperative Games & Core • Group of selfish agents acting together and sharing the reward. • Core: Dividing the reward in a way such that no group wants to deviate and work by itself. • Network Flow Game • Value = Flow from s to t • Total value = v({1,2,3}) = 4 • v({1,3}) = 2 • v({2,3}) = 3 • All other values are 0 • How to divide the total value between the agents? p1=0.5 p1=0 c1=2 p3=4 p3=2 c3=4 q s t c2=3 p2=1.5 p2=0

  4. Questions • Want to divide the total value among the agents such that each group gets at least its value. • So no group is better off deviating! • Existential: Does there always exist such a stable division? • Yes, NFGs are totally balanced [Kalai and Zemel, ‘82] • Computational: How to efficiently compute such a stable division? • Polynomial time algorithm for NFGs [Kalai and Zemel, ‘82]

  5. Agent Failures • Every agent “fails” independently with different probability. • Reliability = probability of not failing. • Consider the previous example, but now with failures… • Total expected value = 0.5*0 + 0.5* [ 0.2*(1-0.7)*2 + (1-0.2)*0.7*3 + 0.2*0.7*4 ]= 2.36 • Questions: • Existential: Can we divide this in a way such that no coalition is ex-ante better off deviating? • Computational: How do we compute such a stable division? r1=0.2 c1=2 r3=0.5 c3=4 q s t c2=3 r2=0.7

  6. Preliminaries • Cooperative Game: G = (N,v) where N = {1,2,…,n} is the set of agents and v : 2NR is the valuation function. • (ε-)Core: Set of all payment divisions (p1,p2,…,pn) such that • (ε-)Totally Balanced Game: the ε-core is non-empty in every sub-game.

  7. Preliminaries • Reliability Extension Model [Bachrach et. al., ’11] • Base Cooperative Game  • Reliability Game  • Reliability vector where is the reliability (probability of not failing) of agent i. • Valuation function of the reliability game = expected values that coalitions can achieve.

  8. Previous Work • Various important classes of games have been shown to be totally balanced. • Network Flow Game [Kalai & Zemel, ‘82], Linear Production Game [Owen, ’75], Assignment Game [Shapley & Shubik, ’71] etc… • [Bachrach et. al., ‘11] introduced agent failures in cooperative games through reliability extension model. • General Idea: Agent failures can only create the core (make it non-empty) but cannot make it empty. • That is, failures help stabilize the game! • Will return to this towards the end…

  9. Results I : Existential • Theorem 1: For any ε 0, if is ε-totally balanced and , then is ε-totally balanced. • Corollary 1:For any ε0, a game is ε-totally balanced iff every reliability extension of the game is ε-totally balanced. • Take . • ε = 0  every reliability extension of a totally balanced game is totally balanced, and hence has a core payment.

  10. Results 1.5  • Convex Games - Subclass of totally balanced games that capture increasing marginal returns (valuation function is supermodular). • Similar results for convex (and ε-convex) games. • A connection between ε-convexity and ε-total balancedness that generalizes a classical result by [Shapley, ’71].

  11. Results II : Computational • Every reliability extension of a totally balanced game has a non-empty core. How to compute such a core payment? • Naïve method – exponential size LP! • Using coefficients that take exponential time to be computed! • Theorem 2: For ε0, a natural linear combination of ε-core (“better than core”) payments of the sub-games of an ε-totally balanced game is an (rminε)-core (“better than core”) payment of the reliability extension, where rmin = miniri. • The linear combination is still an exponential sum! • Sampling…

  12. Results II (Continued…) • Algorithm Outline: • Approximate the linear combination through sampling. • Adjust the approximation to match the total payment. • Use enough samples so that the (rminε) cushion overcomes the inaccuracies (with high probability), and the outcome is still in the core!

  13. Agent Failures and Existence of the Core • Not totally balanced => core is non-empty and sub-game S has empty core. • Obtain sub-game S as a reliability extension by setting ri = 1 for i S and ri = 0 otherwise.

  14. Discussion Current Work • Effect of agent failures on quantitative measures of stability such as the least core value and the Cost of Stability • Effect of agent failures on other solution concepts • Power indices such as the Shapley value and the Banzhaf power index • Agent failures in other classes of games • Games with coalitional structures

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