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The Computational Complexityof Finding a Nash

Equilibrium

- Edith Elkind, U. of Warwick

Based On

- Reducibility Among Equilibrium Problems

(Goldberg, Papadimitriou) Aug 2005 - The Complexity of Computing a Nash Equilibrium

(Goldberg, Daskalakis, Papadimitriou) Sep 2005 - 3-NASH is PPAD-Complete

(Chen, Deng) Nov 2005 - Three-Player Games Are Hard

(Daskalakis, Papadimitriou) Nov 2005 - Settling the Complexity of 2-Player

Nash-Equilibrium (Chen, Deng) Dec 2005

Normal Form Games

- finite set of players 1, , n
- each player has k actions
- (pure strategies) 1, , k
- payoffs of the ith player Pi 1, , kn ? R

Row player

Column player

Nash Equilibrium

- Nash equilibrium a strategy profile such that
- noone wants to deviate given other players

strategies, i.e., each players strategy is a

best response to others strategies - (0, 0) and (1, 1) are both NE

Row player

Column player

Pure vs. Mixed Strategies

- NE in pure strategies may not exist!
- matching pennies
- Mixed strategy a probability distribution over

actions - 50 tail, 50 head

Row player

Column player

Existence of NE

- Theorem (Nash 1951)

any game in normal form has an equilibrium in

mixed strategies - 1 000 000 question
- how to find one?

Finding mixed NE in 2 x 2 Games

Suppose R plays 1 w.p. r EP(C) from playing

0 (1-r)1 EP(C) from playing 1 r3 1-r

3r iff r ¼

Suppose C plays 1 w.p. c EP(R) from playing 0

(1-c)2 EP(R) from playing 1 c1 (1-c)2 c

iff c 2/3

NE r1/4, c2/3

Row player

Column player

2 players, k actions

- Representation two k x k matrices
- Checking for pure NE easy
- at most k2 of them
- Checking for mixed NE
- all straightforward methods are exptime
- Lemke-Howson algorithm is exptime, too (previous

talk) - For 2 players all NE are rational
- but not for 3 and more players

n players, 2 actions

- Representation payoffs to each player for every

action profile (vector in 0, 1n) n2n numbers - graphical games
- players are associated with the vertices of a

graph - each players payoff depends on his own action

and actions of his neighbors - n players, max degree d gt n2d1 numbers

W

t0, u0, v0, w0 12 t1, u0, v0, w0 31

. t1, u1, v1, w1 -6

Ws payoffs (16 cases)

T

V

U

Algorithms for NE in Graphical

Games

- Bounded-degree trees
- Exp-time algorithm/poly-time approximation

algorithm to find all NE (Kearns, Littmann,

Singh, UAI 2001) - ??? poly-time algorithm to find a single NE

(Kearns, Littmann, Singh, NIPS2001) - shown to be incorrect in E., Goldberg, Goldberg,

ACM EC06 - Graphs of max degree 2
- poly-time algorithm (EGG06)

Is Finding NE NP-hard?

- Reminder a problem P is NP-hard if you can

reduce 3-SAT to it - yes-instance 3-SAT ? yes-instance of P
- no-instance 3-SAT ? no-instance of P
- Problem each instance of NASH is

a yes-instance! - every game has a NE
- need complexity theory for search problems
- Side note pure Nash for n players, NE of total

value gt K are NP-hard

Reducibility Among Search Problems

S X Y

T X Y

- S associates x in X with a solution set S(x)
- Total search problem for any x, S(x) is not empty

If T is easy, so is S

Equivalences GP05

deg d graphical game G NE of G

d2-player game G NE of G

d-Graphical Game GG ? d2-Player Game G

- Color the graph of GG

d(u,v) 2 ? color(u) ? color(v) - Each color is a player of G
- RED chooses a red vertex in GG

and an action for that vertex in GG - payoffpayoff1payoff2
- payoff1 BLUE tries to guess which vertex RED

chose RED pays a penalty if BLUE guesses

correctly - payoff2 if all neighbors of a chosen vertex are

also chosen, it gets same payoff as in GG, else 0

r-Player Game G ?

3-Graphical Game GG

- Si space of pure strategies of player i
- S- i S1 Si-1Si1 .. Sr
- xij the probability that ith player uses jth

strategy - xs x1s1 x2s2 xrsr (for s in S-i)
- uijs utility of the ith player when he plays j

and others play according to s

NE 0 xij 1 Sj xij 1 Ss in

S uijsxs gt Ss in S uijsxs implies xij 0

-p

-p

r-Player Game G ?

3-Graphical Game GG

- Vertex Vij for any pair (playeri, actionj)
- Want PrVij plays 1 Pr i plays j in Gxij
- Idea graphical games can do math!
- Enforce constraints from the previous slide

v1

v2

v3

Need gadgets for , , c, , min, max,

u

Set payoffs to u, v3 so that pv3pv1 pv2

Equivalences GP05

deg d graphical game G NE of G

d2-player game G NE of G

Combining Reductions GP05

Completeness Results?

- Can we prove that any total search problem is

reducible to r-NASH? - Not really the class T of all total search

problems is a semantic class - not known how to find complete problems for these
- Want to pick a large subclass S of T s.t.
- S includes some natural problems
- there are problems that are complete for S
- in particular, r-NASH is complete for S

END OF THE LINE

- Input Boolean circuits

S (Successor), P

(Predecessor) - n inputs, n outputs
- S(0n) ? 0n, P(0n) 0n
- Output x ? 0n s.t.
- S(P(x)) ? x or P(S(x)) ? x
- Intuition G(V, E)
- V Sn
- E (x,y) yS(x), xP(y)

PPAD

- PPAD Polynomial Parity Argument, Directed

version - PPAD is the class of all search problems that are

reducible to END OF THE LINE

search problem solution

g

f

circuits S, T end of the

line

r-NASH is in PPAD

- Proof on Nashs theorem
- existence of NE reduces to Brouwers fixpoint

theorem - Brouwers fixpoint theorem reduces to Sperners

lemma - Sperners lemma is proven by a parity argument

(similar to END OF THE LINE) - Reduction of r-NASH to END OF THE LINE can be

extracted from these proofs (Papadimitriou 94)

Brouwers Fixpoint Theorem

- Brouwers Theorem Any continuous mapping from

the simplex to itself has a fixpoint. - Nash ? Brouwer proof sketch
- set of all strategy profiles ? simplex
- mapping (s1, , sn) ? (s1d1, , sndn), where

di is a shift in the direction of best response

to (s1, , si-1, si1, , sn) - NE is a point where noone wants to deviate, i.e.,

a fixpoint

Sperners Lemma

- Proper coloring
- vertices on BC are not blue
- vertices on AC are not green
- vertices on AB are not yellow
- Sperners Lemma

there exists a trichromatic

triangle - Brouwers theorem ? Sperners Lemma
- x is blue if the grad(F) at x points away from A,

etc. - trichromatic triangle has no direction
- repeat at increased resolution

Opposite Direction 3D-BROUWER

- Input
- 3D unit cube divided into 23n cubelets
- cijk is the center of Kijk
- f(cijk)cijkdijk, dijk is in d0, d1, d2, d3,

where - d1(a, 0, 0), d2(0, a, 0)
- d3(0, 0, a), d0(-a, -a, -a)
- circuit C 0, 1 3n ? 0, 1, 2, 3 selects dijk
- Output
- a panchromatic cubelet, i.e., one that has all of

d0, d1, d2, d3 among its 8 neighbors

3D-BROUWER is PPAD-complete

- Papadimitriou (1994) shows that a more

complicated version of 3D-BROUWER is

PPAD-complete - This version was proven hard in DGP05
- Reduction from END OF THE LINE
- embed the line L into 3d cube
- protect L from color 0 using three other colors
- color the rest of inner cubelets with 0

r-NASH vs 3D BROUWER

- Existence of NE follows from Brouwers fixpoint

theorem - NE are special cases of Brouwers fixpoints
- just how special?
- Can any fixpoint be represented as a NE of a

game? - DGP05 YES! ? 4-NASH is PPAD complete
- Proof
- 4-NASH ? deg 3 Graphical Nash
- graphical games can compute fixpoints

4-NASH to 3-NASH

- Daskalakis, Papadimitriou modify arithmetic

gadgets so that the graph is 3-colorable - Chen, Deng same gadgets, but allow for small

error

2-NASH

- Chen, Deng
- avoid graphical games
- reduce directly from 3D-BROUWER to

2-NASH using arithmetic gadgets similar to

graphical game gadgets - Game over?

Graphical Games Open Problems

- Degree
- deg 3 PPAD-complete (DGP05b)
- deg 2 polynomial time solvable (EGG06)
- Pathwidth
- paths poly-time
- pathwidth 1 maybe algorithm from EGG06 still

works - pathwidth 2 any KLS-style algo is exptime

(EGG06) - pathwidth gt r, r constant PPAD-complete (EGG06)
- Finding NE on trees?

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