Small clique detection and approximate Nash equilibria Danny Vilenchik UCLA Joint work with Lorenz...

Small clique detection and approximateSmall clique detection and approximateNash equilibriaNash equilibria

Danny VilenchikDanny Vilenchik

UCLAUCLA

Joint work with Lorenz MinderJoint work with Lorenz Minder

Summary

Relate three problems:

A. Approximating the best Nash equilibrium

B. Finding a planted k-clique in a random graph Gn,1/2

C. Distinguishing Gn,1/2 from Gn,1/2 with slightly larger planted clique

Executive summary:

• A is at least as hard as B (for sufficiently large constant k) [Hazan & Krauthgamer 2009]

• A is at least as hard as C [joint work with L. Minder, 2009]

Two player game

),( bapi

qj

(Mixed) Strategies: (independent)

row player: x=(p1,p2,…,pn),

column player: y=(q1,q2,…,qn)

Payoff of row player is xAyt (column playeris xByt) – expectation

Payoff for column player

Payoff for row player

Game matrix

Example: Scissors, Rock, Paper

(0,0)-(1,1)(1-,1)

(1-,1)(0,0)-(1,1)

-(1,1)(1-,1)(0,0)

3/1

3/1

3/1

3/1 3/1 3/1

• This is a zero sum game

• In this case, total payoff is 0

• No player has any incentive to deviate (payoff still 0)

Nash Equilibrium

),( bapi

qj

A strategy (x,y) is a Nash-equilibrium if

A strategy (x,y) is an ²-Nash-equilibrium if

The value of a strategy (x,y) is

xAyxAyxAyAyxyx ''','

'''',' xAyxAyxAyAyxyx

yBAx 2

1

The best equilibrium is the one with maximal value (say m)

An ²-best ²-equilibrium is:

1. An ²-equlibrium

2. Has value at least m-²

Planted k-clique (Jerrum, Kucera)

Gn,1/2

Largest clique is whp of size(2-o(1))logn

Plant a clique of size kGenerate Gn,1/2 independently

What is known for these problems?

Can find planted k-clique in O(nk)

Can find planted k-clique in poly time if k=(n1/2) [AKS’98]

Hard to distinguish between Gn,1/2 from Gn,1/2,k for k=(2-²)logn [JP’98]

Can efficiently compute a 0.34-equilinrium [TP’07]

Can compute (best) ²-equilibrium

in time [LMM’03]2/log nn

Currently no polynomial algorithm for planted O(logn)-clique

No polynomial algorithm to find a clique of size > logn in Gn,1/2

NP-Hard to compute best-Nash

Is there a PTAS for best-Nash?

Can find planted O(logn)-clique in O(nlogn)

Hardness Result for ²-best Nash

Hazan and Krauthgamer show (SODA 2009):

If there exists poly-time algorithm that finds the ²-best Nash

then

there exists a probabilistic poly-time algorithm that finds a clique of size

1000logn in Gn,1/2,1000log n

This result relates seemingly unrelated problems

How far can this technique be stretched?

Optimal would be a planted clique of size (2+½)logn for any ½ > 0

Hardness Result for ²-best Nash

Our result (with Lorenz Minder)

If there exists poly-time algorithm that finds the ²-best Nash

then

There exists a poly-time algorithm that distinguishes whp between

Gn,1/2 and Gn,1/2 with a planted clique of size > (2+²1/8)log n

Corollary of our analysis:

there exists a probabilistic poly-time algorithm that

finds a clique of size 3logn in Gn,1/2,3log n

In some sense this is the best one can expect. If k < 2logn, the two distributions may be info.

theoret. indist. ) bound too tight

Techniques

Goal: Given a graph G, incorporate it into a game so that the ²-best Nash relates to its maximum clique

First try:

0,01,10,01,1

1,11,11,10,0

0,01,11,10,0

0,01,11,11,1Game matrix is just the adjacency matrix

The value of the best Nash is 1

A)G(

1/2

1/2

1/2 1/2

1

1

Conclusion: need to “neutralize” small cliques

Techniques (Hazan and Krauthgamer)

A is the adjacency matrix of a random graph with a planted clique of size c1logn

B is an ns £ n matrix, s=s(c1)

The (i,j)-entry of B is (bi,j,-bi,j)

0B

BA T

Goal: “neutralize” small cliques

9/18

9/80, jib

Hopefully:

• Small cliques are not equilibrium

• Large planted clique is an equilirbrium

Properties of the game

Let C be the planted clique of size c1log n

0B

BT

1,11,10,00,00,0

0,01,11,11,11,1

0,01,11,11,11,1

0,01,11,11,11,1

0,01,11,11,11,11/|C|

1/|C|

1/|C|

1/|C|

1/|C| 1/|C| 1/|C| 1/|C|

Techniques (Hazan and Krauthgamer)

The value of the strategy is 1

Why is it a Nash-equilibrium?

The matrix B may interfere now

Properties of the game

0B

BT

1,11,10,00,00,0

0,01,11,11,11,1

0,01,11,11,11,1

0,01,11,11,11,1

0,01,11,11,11,1

1/|C|

1/|C|

1/|C|

1/|C|

1/|C| 1/|C| 1/|C| 1/|C|

j

1||

1

CiijbC

Techniques (Hazan and Krauthgamer)

The value of the strategy is 1

Why is it a Nash-equilibrium?

The matrix B may interfere now

The best Nash is of value at least 1

How about “neutralizing” small cliques?

Properties of the game

0B

BT

1,11,10,00,00,0

0,01,11,11,11,1

0,01,11,11,11,1

0,01,11,11,11,1

0,01,11,11,11,1

For every set of at most c2log n rows D (c2 < c1)

1/|D| 1/|D|

i

8: ijbj

Row player defects

Properties of B

The average of the c1logn columns corresponding to the clique < 1

Or else the planted clique is not an equilibrium (row player then defects)

For every set of c2logn columns there is a strike of 8’s in B

Enough to exclude small cliques as equilibria

Observation

Two contesting processes regarding B:

B shouldn’t have too many rows

Or else the average of c1logn columns > 1 (at some row)

Planted clique is not an equilibrium (row player then defects)

B shouldn’t have too few rows

Otherwise not for every set of c2logn columns there is a strike of 8’s

Small cliques not neutralized

If you choose c1 sufficiently large, c2 smaller than c1, such a B exists

Main Point of Analysis

Plant a clique of size c1log n Recover a graph of size f size c2log n

and density 0.55Such density and size do not exist in Gn,1/2 whp ) must intersect planted clique on

many vertices ) use greedy to complete to the planted clique

Main points in the analysis

If the strategy (x,y) is an ε-best Nash equilibrium then:

Fact 1: both players put most of their probability mass on A

Why?

The game outside A is 0-sum. So if one player has 2δ-probability outside A, the value of the game cannot exceed (2-2δ)/2=1- δ (maximal value on A is 1)

But, we know that the best Nash has value 1, so δ< ε

Here we use the fact that we are given a best Nash equilibrium.

OPEN PROBLEM: can you let go of the “best” assumption ?!

0B

BA T

Main points in the analysis

If the strategy (x,y) is an ε’-best Nash equilibrium played on A then:

Fact 2: Small sets of indices cannot be assigned with probability > 1/8

Why?

By the second property of B, a strike of 8’s will cause a player to defect

Fact 3: Sets of large probability correspond to high payoff, and in turn to dense subgraphs.

Again, here we use the fact that the equilibrium has value 1 (since it is the best one)

Our work

Optimal result means c1=(2+½)log n

This means that 2 < c2 < c1

Because the subgraph is small (c2logn), it has to be very dense: 1-½

Otherwise, again, such sub graphs exist in Gn,1/2

Need to preserve the separation properties of the game

The planted clique is a Nash equilibrium of value 1

Probability is placed on sets of size at least c2log n

What did we do?

Use tightest possible version of probabilistic bounds (Chernoff in our case)

Optimize over values of Bernoulli variables (in the matrix B)

Two contesting processes in B

Tighter analysis of other game properties

However, we only get detection of small cliques

To find a planted clique we need to plant a clique of size 3logn

(we don’t know an algorithm that finds a planted clique when given a piece of it of size < logn)

Limitation of the technique

Can we hope to have a reduction from finding the maximal clique in Gn,1/2?

Probably not

The main reason: the technique relates value of equilibrium to density ) value cannot exceed 1-², and there are plenty of such dense subgaphs in Gn,1/2 not connected to the cliqe

Open Questions

Remove the “best” assumption

Reduction in the other direction