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Approximate Nash Equilibria in interesting games

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in interesting games. Constantinos Daskalakis, U.C. Berkeley. If your game is interesting, its description cannot be astronomically long... Game Species ... – PowerPoint PPT presentation

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Title: Approximate Nash Equilibria in interesting games


1
Approximate Nash Equilibriain interesting games
  • Constantinos Daskalakis, U.C. Berkeley

2
  • If your game is interesting, its description
    cannot be astronomically long

3
Game Species
e.g. bounded degree
graphical games
normal form games
interesting games
e.g. constant number of players
what else?
4
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
5
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
6
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 ??
7
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
8
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)
9
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


10
PTAS ?
11
Other Interesting Games?
graphical games
normal form games
interesting games
anonymous games
Each player is different, but sees all other
players as identical
12
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
13
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?
14
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
15
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.
16
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.
17
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
18
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
19
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
20
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
21
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
22
Thank you for your attention!
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