Approximate Nash Equilibria
in interesting games
Constantinos Daskalakis, U.C. Berkeley
If your game is interesting, its description cannot be
astronomically long…
Game Species
normal form
games
e.g. constant number
of players
graphical
games
e.g. bounded
degree
interesting
games
what else?
Bad News…
Computing a Mixed Nash Equilibrium ?
- in normal form games
is PPAD-complete [DGP ’05]
even for 3 players [CD ’05, DP ’05]
even for 2 players [CD ’06]
- in graphical games
PPAD-complete [DGP]
even for 2 strategies per player and degree 3
So what next?
Computing approximate Equilibria
(every player plays an approximate best response)
Finding a better point in Christos’ cube
correlated
mixed Nash
[DGP06, CD06]
existence
efficiency
naturalness
pure Nash
Looking at other interesting games…
Approximate Equilibria for 2-players?
Compute a point at which each player has at most
 - regret..
for  = 2-n
PPAD-Complete [ DGP, CD]
for  = n-
PPAD-Complete for any  [CDT ’06]
( no FPTAS )
 = constant ??
LMM ’04
log n
2
- support is enough for all 
[LMM ’03] take Nash equilibrium (x, y); take
log n/ 2 independent samples from x and y
 subexponential algorithm for computing  - Nash
A simple algorithm for .5 -approximate
[DMP ’06]
Column player finds:
best response j to strategy i of row player
Row player finds:
best response k to strategy j of column player
1.0
j
0.5
i

0.5
k

G = (R, C)
0.5 approximate Nash!
[FNS ’06]: can’t do better
with small supports!
Beyond Constant Support [DMP ’07]
.38 can be achieved in polynomial time
Generalization of Previous Idea:
guess value
of the eq. u
+
sampling
(similar to LMM)
+
LP
PTAS ?
Other Interesting Games?
normal form
games
graphical
games
interesting
games
anonymous
games
“Each player is different, but sees
all other players as identical”
why interesting?
-
the succinctness argument :
n players, s strategies, all interact, ns size!
(the utility of a player depends on her strategy, and
on how many other players play each strategy)
-
ubiquity: think of your favorite large game - is it anonymous
e.g. auctions, stock market, congestion, social
phenomena, …
"How many veiled women can we expect in Cairo ?"
Characterization of equilibria in large anonymous games, [Blonski ’00]
Pure Nash Equilibria
Theorem [DP ’07]:
In any anonymous game, there exists a 2Ls2approximate pure Nash equilibrium which can
be found in polynomial time.
(L = Lipschitz constant of the utility functions)
how rapidly does the payoff change as
players change strategy?
PTAS for anonymous games
with two strategies
Big Picture:
• Discretize the space of mixed Nash equilibria.
• Discrete set achieves some approximation which
depends on the grid size.
• Reduce the problem to computing a pure Nash
equilibrium with a larger set of strategies.
Big Question:
what grid size is required to achieve approximation ?
if function of  only  PTAS
if function of n  nothing
PTAS (cont.)
[Restrict attention to 2 strategies per player]
Let p1 , p2 ,…, pn be some mixed strategy profile.
The utility of player 1 for playing pure strategy  is
n 1
 u( , t ) Pr 
t 0

X

t
j
j 1

where the Xj’s are Bernoulli random variables with
expectaion pj.
PTAS (cont.)
How is the utility affected if we replace the pi’s by
another set of probabilities {qi}?
Absolute Change in Utility
n 1
  u ( , t ) Pr   j 1 X j  t   Pr   j 1Y j  t 




t 0


j 1
X j   j 1Y j
TV
where the Yj’s are Bernoulli random variables with
expectaions qj.
PTAS(cont.)
Main Lemma: Given any constant k and any set of probabilities
{pi}i , there exists a way to round the pi’s to qi’s which are
multiples of 1/k so that
||P - Q|| = O(k-1/2),
where:
P is the distribution of the sum of the Bernoullis pi
Q is the distribution of the sum of the Bernoullis qi
no dependence on n  PTAS for anonymous games
 approximation in time n
O (1/  2 )
PTAS - complications
Two natural approaches seem to fail:
i. round to the closest multiple of 1/k
suppose pi =1/n , for all i
 qi = 0, for all i
n
1
 1
n 

 Q [0] = 1, whereas P[0]  1   
e
 n
 variation distance  1-1/e
PTAS – complications (cont.)
ii. Randomized Rounding
Let the qi be random variables taking values which
are multiples of 1/k so that
E[qi] = pi.
Then, for all t = 0,…, n,
- Q[t] is a random variable which is a function of the qi’s
e.g. Q[n]   qi
i
- Q[t] has the correct expectation!
trouble: expectations are
E[Q[t]] = P[t]
at most 1 and functions
involve products
PTAS(cont.)
Our approach: Poisson Approximations
Intuition:
If pi’s were small   X i would be close to a Poisson dist’n
i
pi
of mean 
i
q   p
 define the qi’s so that
X
i
i
i
Y
i
i
i


Poisson   pi 
 i



Poisson   qi 
 i 
i
i
PTAS(cont.)
Near the boundaries of [0,1] Poisson Approximations are sufficient
Disadvantage of Poisson distribution:
mean = variance
This is disastrous for intermediate values of the pi’s
 approximation with translated Poisson distributions
to achieve mean  and variance 2
define a Poisson(2 ) distribution; then
shift it by  - 2
Thank you for your attention!