10/14/2009Automatic Software Design Lab1 Nash Equilibria in Distributed Systems Mohamed G. Gouda & H. B. Acharya Presenter Aly Farahat Ph.D. Student Automatic Software Design Lab Computer Science Department Michigan Technological University

10/14/2009Automatic Software Design Lab2 A Nash-Equilibrium is a property of stable states in a game. It means that no player should try to perturb this state (make a move) from this point as it may decrease its gain

10/14/2009Automatic Software Design Lab3 Contents Definitions Characterization Taxonomy

10/14/2009Automatic Software Design Lab4 Definitions

10/14/2009Automatic Software Design Lab5 Nash Equilibrium Origins Concepts from Game Theory Goal Characterizing a state from which local actions might eventually lead to no gain

10/14/2009Automatic Software Design Lab6 Terminology Stabilization: All distributed system computations are finite Fixed-Point: Termination state in a distributed computation (no processes are enabled) Equilibrium Point: Fixed-Point! Local Perturbation: Transitions on a process local states while in a Fixed-Point

10/14/2009Automatic Software Design Lab7 Gain Function A set G of local functions, one per process i G={ g.i } g.i is defined only at equilibrium states and undefined elsewhere

10/14/2009Automatic Software Design Lab8 Nash Equilibrium A Fixed-Point s is a Nash Equilibrium wrt {g.i} iff For every process i, for every local perturbation, there exists a fixed-point s’ such that g.i(s’)<= g.i(s)

10/14/2009Automatic Software Design Lab9 Intuitively In a Nash-Equilibrium s, no process i has the incentive to perturb its equilibrium as it might decrease its gain function. In a non Nash-Equilibrium ns, there exists a process j that would necessarily increase its local gain g.j by perturbing ( by a specific perturbation) its equilibrium.

10/14/2009Automatic Software Design Lab10 Illustration

10/14/2009Automatic Software Design Lab11 Characterization of Nash Equilibria

10/14/2009Automatic Software Design Lab12 Sufficient Conditions Theorem 1: s is a Nash Equilibrium wrt {g.i} if any of the following is true: 1- g.i has its maximum at s, for all i. 2- For every local perturbation p i from s there exists a stable state s’ reachable by the actions of i such that g.i(s’)<=g.i(s) Why are these conditions unnecessary?

10/14/2009Automatic Software Design Lab13 Sufficient Conditions (Cont’d) Theorem 2: ns is not a Nash Equilibrium wrt { g.i } if: There exists i with a second fixed point s’ directly reachable from s by a local perturbation of i. Why this is not necessary?

10/14/2009Automatic Software Design Lab14 Absolute Nash Equilibrium (Sufficient Conditions) Theorem 3: s is a Nash Equilibrium w.r.t. any set of gain functions if: For every i, for every perturbation p i the system has a local action that returns it to state s.

10/14/2009Automatic Software Design Lab15 Construction of Gain Functions Theorem 4: For any stabilizing distributed system: a)A set of constant gain functions { g.i | g.i=c i } makes every fixed-point a Nash- Equilibrium

10/14/2009Automatic Software Design Lab16 Construction of Gain Functions (Cont’d) Theorem 4(b): For any stabilizing distributed system: If there are two fixed points, s and s’, different only in one local variable of process j. We can make s’ a non-Nash Equilibrium by forcing a local perturbation from s’ to s with g.j(s’)<g.j(s)

10/14/2009Automatic Software Design Lab17 Taxonomy based on Nash Equilibira

10/14/2009Automatic Software Design Lab18 Relatively Perturbation-Proof Systems Relatively Perturbation-Prone Systems Absolutely Perturbation-Proof Systems Absolutely Perturbation-Prone Systems (empty)

10/14/2009Automatic Software Design Lab19 A stabilizing system is relatively perturbation-proof iff: –There exists S={ g.i } such that every fixed- point is a Nash Equilibrium w.r.t S Relatively Perturbation-Proof

10/14/2009Automatic Software Design Lab20 Maximal Matching Bidirectional Ring m.i==i-1 && m.(i-1)==i-2  m.i:=i m.i==i+1 && m.(i+1)==i+2  m.i:=i m.i==i && m.(i-1)!=i-2  m.i:=i-1 m.i==i && m.(i+1)!=i+2  m.i:=i+1 g.i=0 if m.i==i g.i=1 otherwise Process i should match with one of its neighbors, otherwise it should keep its value to i.

10/14/2009Automatic Software Design Lab21 Nash Equilibrium of Matching If m.i !=i, and m.i is a fixed-point, then g.i=1. This is a maximum! From theorem 1(a), it is a Nash- Equilibrium If m.i==i, g.i=0. But no perturbation will break a match, hence, m.i == i is restablished. “Bidirectional Matching” is relatively perturbation proof

10/14/2009Automatic Software Design Lab22 Relatively-Perturbation Prone A stabilizing system is relatively perturbation-prone iff: –There exists S={ g.i } such that some fixed- point is a non-Nash Equilibrium w.r.t S –Use Theorem 4(b) to design such systems

10/14/2009Automatic Software Design Lab23 Absolutely Perturbation-Proof A stabilizing system is absolutely perturbation-proof iff: –For every S={ g.i }, every fixed-point is a Nash Equilibrium w.r.t S –Use Theorem 3 to design such systems

10/14/2009Automatic Software Design Lab24 A subclass of absolutely perturbation proof systems Theorem 5: If a stabilizing system has only one fixed- point, it is absolutely-perturbation proof Why?

10/14/2009Automatic Software Design Lab25 Absolutely Perturbation-Prone A stabilizing system is absolutely perturbation-prone iff: –For every S={ g.i }, there exists a non-Nash Equilibrium fixed-point w.r.t S –Use Theorem 4(a) to show that no such system exists: we can always construct a set of gain functions to make every fixed-point a Nash-Equilibrium

10/14/2009Automatic Software Design Lab26 Partial Order among Classes Why??

10/14/2009Automatic Software Design Lab27 Further Investigations Given a set of gain functions, automatically transforming a perturbation-prone to a perturbation-proof system –Identify the perturbations leading to other equilibria with higher gains Applicability of this concept to set of states rather than states (consider the notion of invariant) How to come up with gain-functions representing the system progress properties

10/14/2009Automatic Software Design Lab28 Further Readings -John F Nash, “Equilibrium point in n-person games,” Proceedings of the National Academy of Sciences of the United States of America, 36(1):48-49, A. Arora & M. G. Gouda, “Closure and convergence: a foundation of fault-tolerant computing.” In Proceedings of the 22 nd International Conference On Fault-Tolerant Computing Systems

10/14/2009Automatic Software Design Lab29 Thank you!