Agent Failures in Totally Balanced Games and Convex GamesAuthors:Yoram Bachrach1

Ian Kash1

Nisarg Shah2 (speaker)1 Microsoft Research Cambridge.2 Carnegie Mellon University.

AgendaAgent Failures in

Cooperative Games Sub-Agenda:

Effect of Agent Failures on the Existence of the Core

Initiated by [Bachrach et. al., ‘11]

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.

c1=2

c2=3

c3=4s tq

p1=0.5

p2=1.5

p3=2p1=0

p2=0

p3=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?

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]

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?

c1=2

c2=3

c3=4s tq

r1=0.2

r2=0.7

r3=0.5

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.

Preliminaries• Reliability Extension Model [Bachrach et. al., ’11]

o Base Cooperative Game o Reliability Game o Reliability vector where is the reliability (probability of not failing)

of agent i.o Valuation function of the reliability game = expected values that

coalitions can achieve.

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…

Results I : Existential• Theorem 1: For any ε 0, if is ε-totally balanced and , then is

ε-totally balanced.

• ε = 0 every reliability extension of a totally balanced game is totally balanced, and hence has a core payment.

• Corollary 1: For any ε 0, a game is ε-totally balanced iff every reliability extension of the game is ε-totally balanced.

Take .

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].

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 (rminε)-core (“better than core”) payment of the reliability extension, where rmin = mini ri.

• The linear combination is still an exponential sum! Sampling…

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 (rminε) cushion

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

Agent Failures and Existence of the Core

Game Introducing Failures Increasing Failure Probabilities

General Not Preserved[Any not-totally-balanced game having non-empty core]

Not Preserved[Introducing failures is a special case]

Simple 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 ri = 1 for i S and ri = 0 otherwise.

DiscussionCurrent 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