The Computational Complexity of Finding a Nash Equilibrium

1
The Computational Complexityof Finding a Nash
Equilibrium

• Edith Elkind, U. of Warwick

2
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

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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
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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
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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
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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?

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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
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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

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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
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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)

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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

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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
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Equivalences GP05
deg d graphical game G NE of G
d2-player game G NE of G
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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

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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
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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
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Equivalences GP05
deg d graphical game G NE of G
d2-player game G NE of G
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Combining Reductions GP05
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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

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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)

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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
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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)

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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

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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

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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

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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

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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

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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

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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?

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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?