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
10/14/2009
Automatic Software Design Lab
1
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
10/14/2009
Automatic Software Design Lab
2
Contents
• Definitions
• Characterization
• Taxonomy
10/14/2009
Automatic Software Design Lab
3
Definitions
10/14/2009
Automatic Software Design Lab
4
Nash Equilibrium
• Origins
Concepts from Game Theory
• Goal
Characterizing a state from which local actions
might eventually lead to no gain
10/14/2009
Automatic Software Design Lab
5
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
10/14/2009
Automatic Software Design Lab
6
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
10/14/2009
Automatic Software Design Lab
7
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)
10/14/2009
Automatic Software Design Lab
8
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.
10/14/2009
Automatic Software Design Lab
9
Illustration
p1
g.1=0
g.2=1
g.3=2
g.1=1
g.2=2
g.3=1
p3
p2
g.1=1
g.2=3
g.3=2
g.1=2
g.2=1
g.3=0
State Space
10/14/2009
Automatic Software Design Lab
10
Characterization of Nash Equilibria
10/14/2009
Automatic Software Design Lab
11
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 pi 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?
10/14/2009
Automatic Software Design Lab
12
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?
10/14/2009
Automatic Software Design Lab
13
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 pi the system
has a local action that returns it to state s.
10/14/2009
Automatic Software Design Lab
14
Construction of Gain Functions
Theorem 4:
For any stabilizing distributed system:
a) A set of constant gain functions
{ g.i | g.i=ci } makes every fixed-point a NashEquilibrium
10/14/2009
Automatic Software Design Lab
15
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)
10/14/2009
Automatic Software Design Lab
16
Taxonomy based on Nash
Equilibira
10/14/2009
Automatic Software Design Lab
17
• Relatively Perturbation-Proof Systems
• Relatively Perturbation-Prone Systems
• Absolutely Perturbation-Proof Systems
• Absolutely Perturbation-Prone Systems (empty)
10/14/2009
Automatic Software Design Lab
18
Relatively Perturbation-Proof
• A stabilizing system is relatively
perturbation-proof iff:
– There exists S={ g.i } such that every fixedpoint is a Nash Equilibrium w.r.t S
10/14/2009
Automatic Software Design Lab
19
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.
10/14/2009
Automatic Software Design Lab
20
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 NashEquilibrium
• 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
10/14/2009
Automatic Software Design Lab
21
Relatively-Perturbation Prone
• A stabilizing system is relatively
perturbation-prone iff:
– There exists S={ g.i } such that some fixedpoint is a non-Nash Equilibrium w.r.t S
– Use Theorem 4(b) to design such systems
10/14/2009
Automatic Software Design Lab
22
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
10/14/2009
Automatic Software Design Lab
23
A subclass of absolutely
perturbation proof systems
Theorem 5:
If a stabilizing system has only one fixedpoint, it is absolutely-perturbation proof
Why?
10/14/2009
Automatic Software Design Lab
24
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
10/14/2009
Automatic Software Design Lab
25
Partial Order among Classes
Stabilizing Systems
Why??
Relatively
Perturbation-Proof
Absolutely
Perturbation-Proof
10/14/2009
Automatic Software Design Lab
Relatively
Perturbation-Prone
26
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
10/14/2009
Automatic Software Design Lab
27
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, 1950.
- A. Arora & M. G. Gouda, “Closure and convergence: a
foundation of fault-tolerant computing.” In Proceedings of
the 22nd International Conference On Fault-Tolerant
Computing Systems
10/14/2009
Automatic Software Design Lab
28
Thank you!
10/14/2009
Automatic Software Design Lab
29