Competitive Safety Analysis: Robust Decision-Making in MultiAgent Systems
Moshe Tennenholtz
Kyle Rokos
Introduction: Nash Equilibrium
• In game theory, a set of strategies for
each player so that neither player will
benefit from changing their strategy while
the other player keeps theirs
• Example: Prisoner’s Dilemma
Introduction: Nash Equilibrium
Don’t talk
Confess
Don’t talk
1,1
10,0
Confess
0,10
5,5
Introduction: Nash Equilibrium
Don’t talk
Confess
Don’t talk
1,1
10,0
Confess
0,10
5,5
Introduction: Nash Equilibrium
Don’t talk
Confess
Don’t talk
1,1
10,0
Confess
0,10
5,5
Competitive Safety
Analysis
• Nash equilibria assume that the other
agents also choose the best strategy
• If a competitive agent does not choose
an optimal strategy, the outcome could
potentially be very bad
• Safety strategies picks a strategy which
guarantees a certain payoff, regardless
of the competitor's strategy
Competitive Safety
Analysis
• We can think of Nash Equilibria as the
‘best’ strategy
• Goal: Find safety strategies which the
expected payoff is equal or close to the
expected payoff for the Nash Equilibrium
strategy
• Example: Decentralized Load Balancing
Competitive Safety Analysis
• 0.5<a<1
e1
e2
X/2,X/2
X,aX
• X>0
e1
e2
aX,X
aX/2,
aX/2
Competitive Safety Analysis
•Nash Equilibrium:
P(e1)X/2+P(e2)X
=
P(e1)aX+P(e2)aX/2
e1
e1
e2
X/2,X/2
X,aX
•P(e1)=(2-a)/(1+a)
•P(e2)=1-(2-a)/(1+a)
•Expected Payoff:
(3aX)/(2a+2)
e2
aX,X
aX/2,
aX/2
Competitive Safety Analysis
•Safety level strategy:
P(e1)X/2+P(e2)aX
=
P(e1)X+P(e2)aX/2
e1
e1
e2
X/2,X/2
X,aX
•P(e1)=a/(1+a)
•P(e2)=1-a/(1+a)
•Expected Payoff:
(3aX)/(2a+2)
e2
aX,X
aX/2,
aX/2
Competitive Safety
Analysis
• Not true in general that safety level and
Nash Equilibrium have the same
expected payoff
• In a non-reducible, generic, 2 person
game, with strictly mixed strategies, the
two payoffs will always coincide
• If the strategies are pure instead of
mixed, the safety level will have a lower
expected payoff than the Nash
Equilibrium
Competitive Safety Analysis
•Nash Equilibrium
q
1-q
p
a,e
b,f
1-p
c,g
d,h
hg
p
e g  f h
d b
q
a bc  d
•safety level
p
d c
a bc  d
Competitive Safety
Analysis
• This idea can be extended to n-person
games, with more than two options per
player
• As the complexity increases, it becomes
less likely that the Nash Equilibrium and
safety level strategies will produce the
same expected payoff
Competitive Safety
Analysis
• Desirable to find cases where the
expected payoff with safety level
strategies is close to the expected payoff
of the Nash Equilibrium
• A C-competitive strategy is one where
the expected payoff is 1/C of the Nash
expected payoff
Competitive Safety
Analysis
• Thus a 1-competitive strategy is ideal,
and a 2-competitive strategy is better
than a 3-competitive strategy, but not as
good as a 3/2-competitive strategy
• The load balancing problem is a 1competitive strategy
• The extended load balancing problem is
a 9/8-competitive strategy
Competitive Safety Analysis
e2
3: e1
2:
e1
e2
..
1
e1
e2
e1
e2
X/3,
X/3,
X/3
X/2,
aX,
X/2
X/2,
X/2,
aX
X,
aX/2,
aX/2
aX,
X/2,
X/2
aX/2,
aX/2,
X
aX/2,
X,
aX/2
aX/3,
aX/3,
aX/3
Conclusion
• There are many problems with Nash
Equilibria. (e.g. Prisoner’s Dilemma)
• Safety level strategies attempt to fix one
of the problems, the assumption that your
opponent will use an optimal strategy,
while retaining the expected payoff of the
Nash equilibrium
• Success depends on the specific
problem
Conclusion
• Please ask some questions now.
References
• Moshe Tennenholtz - Competitive Safety
Analysis: Robust Decision-Making in
Multi-Agent Systems.
• Lee Erlebach - An Introduction to the
Mathematics of Game Theory