6.853: Topics in Algorithmic Game Theory

Fall 2011

Constantinos Daskalakis

Lecture 13

Special Classes of Games

Special Classes of 2-player Games

small probability games ( equilibrium with non-trivial support on a linear number of strategies)

zero-sum two-player games ( ) poly-time solvable (lecture 2)

low-rank two-player games ( )

PTAS [Kannan, Theobald ’09]

sparse two-player games ( constant number of non-zero entries in each row, column)

[Daskalakis, Papadimitriou ’09]

PTAS

[Daskalakis, Papadimitriou ’09]PTAS

win-lose games (all payoff entries in {0,1})

exact is PPAD-complete [follows from Abbott, Kane, Valiant ’05]

also no FPTAS [Chen, Teng, Valiant ’07]

note : exact is PPAD-complete [Chen,Deng,Teng 06]

note : exact is PPAD-complete

no PTAS is known..

rank-1 two-player games ( R + C has rank 1 )

poly-time solvable [Adsul-Garg-Mehta-Sohoni ’10]

Special Classes of Graphical Games

line / cylic graphical games (many players, 2 strategies per player)exact algorithm [Elkind, Goldberg, Goldberg ’06]

the only class of graphs where exact equilibria can be computed

limitations on the graph structure:

trees (many players, constant #strategies)

FTPAS [Kearns, Littman, Singh ’01]

bounds on the cyclicity of the graph:

e.g. if d, s are bounded, and t = O(log n), the above algorithm is a PTAS,

since the input size is .

Theorem [Daskalakis, Papadimitriou ’08]

An -Nash equilibrium of a graphical game with n players, maximum degree d, treewidth t, and at most s strategies per player can be computed in time polynomial in n and

.2

Idea of these algorithms

dynamic programming

+discretization + TV bound

assume that the players only use mixed strategies in probabilities that are

multiples of a fixed fraction

find the best discretized collection of mixed

strategies

What is the loss in approximation due to the discretization?

[ Total Variation Distance

Def: The total variation (TV) distance between two random variables X and Y is the L1 distance of their PDFs.

]

The TV Bound

In a game, the mixed strategy of each player is a random variable independent of the random variables of the other players.

The effect of the discretization is to replace the random variable Xi corresponding to player i ’s mixed strategy with another variable Yi whose probability for every pure strategy is an integer multiple of the discretization parameter .

How much does the payoff of a player change if we replace X = (X1, X2, …, Xn) by Y = (Y1, Y2, …, Yn) ?

using independence

The TV Bound

How much does the payoff of a player change if we replace X = (X1, X2, …, Xn) by Y = (Y1, Y2, …, Yn) ?

If I’m allowed to use discretization , I can make sure that

degree

#strategies

strategy set of player i

choose for approximation of .2

Idea of these algorithms

dynamic programming

+discretization + TV bound

because of TV bound, the best discretized collection

of mixed strategies is guaranteed to be an .

-Nash equilibrium

assume that the players only use mixed strategies in probabilities that are multiples of . 2

runtime: 2

multiplayer zero-sum games (review from lecture 3)

Multiplayer Zero-Sum, wha?

Take an arbitrary two-player game, between Alice and Bob.

Add a third player, Eve, who does not affect Alice or Bob’s payoffs, but receives payoff

The game is zero-sum, but solving it is PPAD-complete.

intractability even for 3 player, if three-way interactions are allowed.

What if only pairwise interactions are allowed?

Polymatrix Games

- players are nodes of a graph G

- player’s payoff is the sum of payoffs from all adjacent edges

… … - edges are 2-player games

N.B. finding a Nash equilibrium is PPAD-complete for general games on the edges [D, Gold, Pap ’06]

What if the total sum of players’ payoffs is always 0?

Polymatrix Games

Theorem [Daskalakis-Papadimitriou ’09, Cai-Daskalakis’11]

- a Nash equilibrium can be found efficiently with linear-programming;

- if every node uses a no-regret learning algorithm (such as the MWU method from Lecture 4), the players’ behavior converges to a Nash equilibrium.

- the Nash equilibria comprise a convex set;

strong indication that Nash eq. makes sense in this setting.

i.e. payoffs approach equilibrium payoffs, and empirical strategies approach Nash equilibrium

If the global game is zero-sum:

essentially the broadest class of zero-sum games we could hope to solve

Anonymous Games

anonymous games

Every player is (potentially) different, but only cares about how many players (of each type) play each of the available strategies.

e.g. symmetry in auctions, congestion games, social phenomena, etc.

‘‘The women of Cairo: Equilibria in Large Anonymous Games.’’ Blonski, Games and Economic Behavior, 1999.

“Partially-Specified Large Games.” Ehud Kalai, WINE, 2005.

‘‘Congestion Games with Player- Specific Payoff Functions.’’ Milchtaich, Games and Economic Behavior, 1996.

- all players share the same set of strategies: S = {1,…, s}

- payoff functions: up = up (σ ; n1, n2,…,ns )

number of the other players choosing each strategy in S

choice of pDescription Size: O(min {s ns, n sn

})

PTAS

There is a PTAS for anonymous games with a constant #strategies.

Theorem [Daskalakis, Papadimitriou ’07, ’08]:

Remarks: - exact computation is not known to be PPAD-complete for multi-player anonymous games with a constant number of strategies;

- on the flip side, if n is small and s is large (few players, many strategies) then trivially PPAD-complete, since general 2-player games can be reduced to this.

sketch of algorithm for 2 strategies

0

1

0 1

p2

p1

• discretize [0,1]n into multiples of δ, and restrict search to the discrete space

• pick best point in discrete space

• since 2 strategies per player, Nash equilibrium lies in [0,1]n

sketch for 2 strategies (cont.)

Basic Question:

what grid size is required for - approximation?

if function of only PTAS

if function also of n nothing

0

1

0 1

p2

p1

First trouble:

size of search space 1

n

but will deal with this later

Theorem [Daskalakis, Papadimitriou ’07]: Given - n ind. Bernoulli’s Xi with expectations pi , i =1,…, n

there exists another set of Bernoulli’s Yi with expectations qi such that

- a constant independent of n

qi’s are integer multiples of

in fact:

N.B. argument from last lecture gives

sketch for 2 strategies (cont.)

The TV Bound

How much does player p’s payoff from pure strategy σ change if we replace X = (X1, X2, …, Xn) with Y = (Y1, Y2, …, Yn) ?

Given previous theorem, can guarantee that there exists a discretized point making the above difference at most by selecting .

Completing the algorithm

dynamic programming

+discretization + TV bound

complete this step (Exercise)

assume that the players only use mixed strategies in probabilities that are

multiples of .

enough to guarantee a discretized- Nash equilibrium

Resulting running time for 2 strategies.

Theorem [Daskalakis, Papadimitriou ’07]: Given - n ind. Bernoulli’s Xi with expectations pi , i =1,…, n

The first probabilistic approximation theorem

there exists another set of Bernoulli’s Yi with expectations qi such that

- a constant independent of n

qi’s are integer multiples of

in fact:

argument from last time gives

proof of approximation result

Law of Rare Events

+

CLT

- rounding pi’s to the closest multiple of gives total variation n- probabilistic rounding up or down quickly runs into problems

- what works:

Poisson Approximations

Berry-Esséen

(Stein’s Method)

proof of approximation result

Intuition:

If pi’s were small would be close to a Poisson with mean

define the qi’s so that

Poisson approximation is only good for small values of pi’s. (LRE)

proof of approximation result

For intermediate values of pi’s, Normals are better. (CLT)

Berry-EsséenBerry-Esséen

Anonymous Games Summary

2-strategies per player: [DP ’07]

constant #strategies per player:

bad function of s

[DP ’08]

is there a faster PTAS?

Theorem [Daskalakis ’08]: There is an oblivious PTAS with running time

- or, at most mix, and they choose mixed strategies which are integer multiples of

Theorem [D’08]: In every anonymous game there exists an ε-approximate Nash equilibrium in which

the underlying structural result…

- either all players who mix play the same mixed strategy

Lemma:

- The sum of m ≥ k3 indicators Xi with expectations in [1/k,1-1/k] is O(1/k)-close in total variation distance to a Binomial distribution with the same mean and variance

the corresponding symmetry…

… i.e. close to a sum of indicators with the same expectation

[tightness of parameters by Berry-Esséen]

proof of structural result

round some of the Xi’s falling here to 0 and some of them to ε so that the total

mean is preserved to within ε

- if more than 1/ε3 Xi’s are left here, appeal to previous slide (Binomial appx)

similarly

0 ε 1-ε 10 ε 1-εε 11-ε

- o.w. use Dask. Pap. ’07 (exists rounding into multiples of ε2)

Anonymous Games Summary

2-strategies per player: [DP ’07]

[D ’08]

constant #strategies per player:

bad function of s

[DP ’08]

Is there an even faster PTAS?

Theorem [Daskalakis, Papadimitriou ’08]: There is a non-oblivious PTAS with running time

the underlying probabilistic result [DP ’08]: If two sums of indicators have equal moments up to moment k then their

total variation distance is O(2-k).

Anonymous Games Summary

2-strategies per player: [DP ’07]

[D ’08]

[DP ’09]

constant #strategies per player:

bad function of s

is there an FPTAS?