Cooperation,
Competition, and
Authorization
Antoni Mazurkiewicz (ICS PAS, Warsaw)
October 29, 2003
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For whosoever hath, to him shall be given,
and he shall have more abundance;
but whosoever hath not,
from him shall be taken away even that he hath.
[Mat 13:12]
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The Problem
There is a number of agents intending to reach their
goals; the closier an agent is to its target, the better
is its situation. In any moment any agent (or their
”coalition”) is capable to improve its (their) situation;
however, improving situation of some agents may
worsen it for the others; the question is how to
organize agent actions to guarantee each agent
reaching its goal.
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System Components
R:
S:

V:
a finite set (of agents)
a set (of states)
 S  S (set of actions)
R  S  {0,1,2, ...} (evaluation)
V(r,s) is a measure how far agent r
at state s is from its goal.
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Live and Dead States
State s is live, if there is s’ s.t. (ss’);
State s is dead, if it is not live
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Winners and Losers
f = (s’  s’’) - action’
V(r,s’) > V(r,s’’) : r is a winner of f
V(r,s’) < V(r,s’’) : r is a loser of f
V(r,s’) = V(r,s’’) : r is not participating in f
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Active and Dismissed Agents
r is active at s : V(r,s)>0
r is dismissed at s : V(r,s)=0
(Agent is dismissed, if it has achieved its goal,
and active, otherwise)
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Axioms
A1. In any action there is at least one winner;
A2. Any active agent at any state can be a winner;
A3. Dismissed agents remain dismissed forever.
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Axioms (formal)
A1. (s’  s’’)   r : V(r, s’ )  V(r, s’’);
A2. V(r, s’)  0   s’’: (s’  s’’)  V(r, s’ )  V(r, s’’);
A3. V(r, s’)  0  s’’: (s’  s’’)  V(r, s’ )=V(r, s’’).
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Initial valuation
1
2
4
3
2
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Resulting valuation
1
2
8
2
2
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Independent actions:
initial valuation
1
2
3
5
2
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Independent actions:
resulting valuation
2
1
0
3
4
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Competing actions:
initial valuation
1
2
3
5
2
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Competing actions:
resulting valuation
1
2
8
2
2
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Competing actions:
another resulting valuation
1
6
2
5
2
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System Runs
System run: a finite or infinite sequence
of states (s0, s1, s2, ..., sn, ) s.t.
1. i>0: (si-1si);
2. Any live state is not the last one
(completeness condition)
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System Runs
A finite and complete run is said to terminate
Terminating run (s0, s1,s2, ..., sn) is successful, if
rR: V(r, sn) = 0
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EXAMPLE
2 1
3 0
2 2
0 0
0 3
1 2
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Sample Runs
(2,2),(1,2), (2,1),(1,2),…,(2,1),…
(infinite run)
(2,2),(2,1),(3,0),(0,0)
(2,2),(1,2),(0,3),(0,0)
(successful runs)
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EXAMPLE
2 1
3 0
2 2
0 0
0 3
1 2
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Authorization 1
Action f is authorized

The winner of f is not in worse position
than any of losers of f
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Authorization 1
(s’  s’’) is authorized

r: V(r,s’ )>V(r,s’’):p: V(p,s’ )<V(p,s’’):V(r,s’)  V(p,s’))
(one of winners is not in worse position
than any of losers)
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Forbidden actions
2 1
3 0
2 2
0 0
0 3
1 2
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Authorized System
2 1
3 0
2 2
0 0
0 3
1 2
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Older – younger relation
r’  r’’ : agent r’ is not older than agent
r’’
Relation  is a total order in
the set R of all agents
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Authorization 2
Action f is authorized

One of winners of f is not younger
than any of its losers
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Authorization 2
(s’  s’’) is authorized

r: V(r,s’ )>V(r,s’’) :p: V(p,s’ )<V(p,s’’) : (p  r)
(there is no older loser)
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Absence of deadlocks
For any system state there exists
an authorized action
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Basic Property
Any run of any authorized system
successfully terminates.
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Complexity
N – total number of agents
m – maximum initial distance
L - maximum lost
Upper bound of number of steps
needed for the completion of all tasks:
N-1
MaxSteps  m 

(L + 1)k
k=0
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Catastrophes
(relaxation of Axiom 1)
Action (s’s’’) is a catastrophe:
 r  R: V(r,s’)  V(r,s’’)
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Initial valuation
1
2
3
5
2
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Catastrophe
3
2
3
6
8
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Catastrophes

Finite number of catastrophes

Infinite number of catastrophes occurring
with frequency less than MaxSteps
does not prevent the system from successful termination
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Authorized Systems
A run is authorized, if all its actions
are authorized
System is authorized, if all its runs
are authorized
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General System Properties
• Local (changes of states concern and depend on action
participants only)
• Non-deterministic (no prescribed order of action
execution)
• Concurrent (independent actions can be executed
independently of each other).
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Specific Properties
of Authorized Systems
• Always successful
• Self-stabilizing (successful when starting with an
arbitrary initial state)
• Uniform (no particular agent is distinguished)
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Possible Extensions
• Reactivity (new obligation rather than dismissal)
• Coalition authorization
• Priority negotiations
• Dynamic position evaluation
• Partially ordered position values
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