纳什均衡复杂度

Complexity of Nash

Lecturer: Constantinos Daskalakis

Function NP(FNP)

The complexity class FNP is the function problem(output is more complex) extension of the decision problem (output Yes or No) class NP.

Search problem L:

​ Def. Relation RL⊆{0,1}* × {0,1}* : (x,y)∈RL iff y is a solution to x

A search problem is a total iff:

​ ∀x,∃y : (x,y)∈R

L∈FNP:

​ ∃AL(.,.) which is a p-time algorithm, and polynnomial function pL(.):

​ (1) ∀x,y: AL(x,y)=1⇔(x,y)∈RL

​ (2) ∀x: ∃y s.t. (x,y)∈RL⇒∃z with |z|≤pL(|x|) s.t. (x,z)∈R

TFNP = {L∈FNP|L is total}

FNP-completeness

Poly-time (karp) reducible

​ A search problem L∈FNP is poly-time (karp) reducible to another L′∈FNP, associated with AL and pL iff exist efficiently computable function f,g :

​ (1) f: {0,1}* → {0,1}* maps inputs x to L into inputs f(x) to L’

​ (2) ∀x,y:AL(f(x),y)=1⇒AL(x,g(y))=1

​ ∀x:AL(f(x),y)=0⇒AL(x,g(y))=0,∀y

FNP-complete

L∈FNPL′is p−time reducible to L,∀L′∈FNP

Proof of Sperner’s Lemma

• Introduce an outer boundary, that does not create new tri-chromatic triangles.
• Introduce an artificial tri-chromatic triangle (left-bottom corner).
• Define a directed walk starting from the artificial tri-chromatic triangle:

​ If exist red - yellow door, cross it with red on your left hand.

• Then it must stop somewhere inside. This can only happen at tri-chromatic triangle.

完成以上步骤，我们构造一幅图(Generic PPAD)，图的顶点由三角形组成，所有的顶点的出入度都小于等于1，图的边由以上路径组成。在这样的情况下，度数为0的点表示都是其他三角形，为1的是三色三角形，为2的是没有蓝色顶点的三角形。

下证：至少存在一个人工（起点）三色三角形则必存在另一个三色三角形。

Lemma: A directed graph with an unbalanced node (in≠out) must have another.

End of The Line: If 0n is an unbalanced node, find another unbalanced node. Otherwise output 0n.

PPAD: (“Polynomial Parity Arguments on Directed graphs”) = { Search problems in FNP reducible to END OF THE LINE }

PPAD-Completeness of NASH

• Generic PPAD → Embed PPAD→ Sperner (帮助上色的三维坐标) → Brouwer → ArithmcircuitSAT → PolymatrixNash→ Nash

Graphical Game: player 是图中的节点，player的payoff只和它们自己的侧露额以及指向它的节点决定。

Polymatrix Games [Janovskaya ‘68’]

​ Graphical games with edge-wise separable utility functions:

uv(x1,...,xn)=∑(w,v)∈Euw,v(xw,xv)=∑(w,v)∈ExTvA(v,w)xw

ArthmCircultSAT → PolyMatrixNash

​ 我们可以构造addition gadget, subtraction gadget 等

• copy: z=x
• addition: z=min{1,x+y}
• substraction: z=max{0,x-y}
• set equal to a constant: z=max{0,min{1,α}}
• multiply by a constant: z = max{0,mnin{1,αx}}
• comparison: z = 1 if (x>y) 0 if x

Escape 1: Approximation

[Daskalakis’11, Rubinstein’15]:

​ For some ϵ>0, in 2-player games, computing a pair of mixed strategies s.t. no player can improve his current payoff by more than ϵ-fraction is PPAD-complete.

• Absolute error

• • Is unlikely PPAD-hard nlog(n/ϵ2)
  • P algorithm is missing despite a long line of research
  • LMM’03 cannot be improved unless PPAD⊆Time(2n√)

Escape 2: Games w/ Special Structure

• Arbitrary normal form are hard but 2-player zero-sum aren’t
• Identify even broader families of games are tractable

Multiplayer zero-sum game

Take anyarbitrary two-player game, between Alice and Bob

Add Samwho does not affect Alice or Bob’s payoffs but receives:

PAlice(σ)+PBob(σ),∀outcome σ

In zero-sum polymatrix games:

• Found with LP
• Convex set
• If every node uses no-regret learning algorithm (will be defined soon),

Anonymous Games

Anonymous Games : Every player might have a different payoff function, which only depends symmetrically on the otherplayers’ actions

E.g.Auction, traffic, social phenomena

Arbitrarily good approximations are intractable if # strategies does not scale to infinity (Exact eq. is intractable)

Interesting relation to limit theorems in probability:

​ ∀ϵ,n, X1+…Xn of any independent Bernolli 0/1 random variables is ϵ-close in L1-distance to

​ the sum of i.i.d. Bernoullis, or

​ c+∑1/ϵ3i=1Yi, for some constant x & independent Berno Yi

Implies:
In every n-players 2-strategy anonymous game, there exists ϵ-Nash equilibrum in which at most 1/ϵ3 players randomize all players

Escape 3:Alternative solution concepts

Correlated vs Nash

Correlatedequilibrium

• Similar to Nash, except players’ randomization can be correlated
• No player has incentive to deviate given own sampled pure action from the joint distribution

Equilibrium condition expressive as linear constraints on the joint action distribution

• Solvable by LP

In normalform games, LP has polynomial size in game description:

• LP maintains a variable for every pure strategy profile
• Same #variables as title # payoff…

Correlatedeq in P while Nash is PPAD-completed