Approximate Nash Equilibria in interesting games

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Approximate Nash Equilibriain interesting games

• Constantinos Daskalakis, U.C. Berkeley

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• If your game is interesting, its description
  cannot be astronomically long

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Game Species
e.g. bounded degree
graphical games
normal form games
interesting games
e.g. constant number of players
what else?
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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
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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
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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 ??
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LMM 04
log n
- support is enough for all ?
?2
LMM 03 take Nash equilibrium (x, y) take log
n/ ?2 independent samples from x and y
? subexponential algorithm for computing ? - Nash
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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
i
0.5
?
0.5 approximate Nash!
k
0.5
?
FNS 06 cant do better with small supports!
G (R, C)
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Beyond Constant Support DMP 07
.38 can be achieved in polynomial time
Generalization of Previous Idea
sampling (similar to LMM)
guess value of the eq. u
LP


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PTAS ?
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Other Interesting Games?
graphical games
normal form games
interesting games
anonymous games
Each player is different, but sees all other
players as identical
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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
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Pure Nash Equilibria

• Theorem DP 07
• In any anonymous game, there exists a
  2Ls2-approximate 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?
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PTAS for anonymous gameswith 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
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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
where the Xjs are Bernoulli random variables
with expectaion pj.
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PTAS (cont.)

• How is the utility affected if we replace the
  pis by another set of probabilities qi?

Absolute Change in Utility
where the Yjs are Bernoulli random variables
with expectaions qj.
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PTAS(cont.)

• Main Lemma Given any constant k and any set of
  probabilities pii , there exists a way to round
  the pis to qis 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
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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
? Q 0 1, whereas
? variation distance ?? 1-1/e
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PTAS complications (cont.)

• ii. Randomized Rounding

Let the qi be random variables taking values
which are multiples of 1/k so that Eqi pi.
Then, for all t 0,, n, - Qt is a
random variable which is a function of the qis
e.g.
- Qt has the correct expectation! EQt
Pt
trouble expectations are at most 1 and functions
involve products
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PTAS(cont.)

• Our approach Poisson Approximations

Intuition If pis were small ?
would be close to a Poisson distn of mean
? define the qis so that
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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
  pis

? approximation with translated Poisson
distributions
to achieve mean ?? and variance ?2 define a
Poisson(?2 ) distribution then shift it by ? - ?2
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Thank you for your attention!