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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.

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

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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. c 1 =2 c 2 =3 c 3 =4 s t q p 1 =0.5 p 2 =1.5 p 3 =2 p 1 =0 p 2 =0 p 3 =4 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?

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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! 1.Existential: Does there always exist such a stable division? Yes, NFGs are totally balanced [Kalai and Zemel, ‘82] 2.Computational: How to efficiently compute such a stable division? Polynomial time algorithm for NFGs [Kalai and Zemel, ‘82]

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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: 1.Existential: Can we divide this in a way such that no coalition is ex-ante better off deviating? 2.Computational: How do we compute such a stable division? c 1 =2 c 2 =3 c 3 =4 s t q r 1 =0.2 r 2 =0.7 r 3 =0.5

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Preliminaries

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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…

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Results I : Existential ε = 0 every reliability extension of a totally balanced game is totally balanced, and hence has a core payment.

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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].

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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 (r min ε)-core (“better than core”) payment of the reliability extension, where r min = min i r i. The linear combination is still an exponential sum! Sampling…

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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 (r min ε) cushion overcomes the inaccuracies (with high probability), and the outcome is still in the core!

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Agent Failures and Existence of the Core GameIntroducing FailuresIncreasing Failure Probabilities GeneralNot Preserved [Any not-totally-balanced game having non-empty core] Not Preserved [Introducing failures is a special case] Preserved [Bachrach et. al., ’11] Not Preserved [Counter-example] Totally Balanced Preserved [Special case of increasing failure probabilities] Preserved [Theorem 1] Not totally balanced => core is non-empty and sub- game S has empty core. Obtain sub-game S as a reliability extension by setting r i = 1 for i S and r i = 0 otherwise.

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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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