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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.

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Presentation on theme: "Agent Failures in Totally Balanced Games and Convex Games Authors: Yoram Bachrach 1 Ian Kash 1 Nisarg Shah 2 (speaker) 1 Microsoft Research Cambridge."— Presentation transcript:

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.

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. 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?

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! 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]

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* * [ 0.2*(1-0.7)*2 + (1-0.2)*0.7* *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

6 Preliminaries

7

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 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 (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…

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

13 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.

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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