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Computing Nash Equilibria: Hardness andApproximation Algorithms

Anthi Orfanou

May 05, 2015

Anthi Orfanou Computing Nash Equilibria May 05, 2015 0 / 43

Games

set of playersset of pure strategies (actions) available to each playerpayoffs of each player for every strategy profile


(Approximate) Nash Equilibria

Game of n players, ξ actions,Assumption: payoffs in [0,1]

xi : Mixed strategy of a player i , probability distribution over ξ(mixed) strategy profile X = (x1, . . . ,xn)

Expected Payoff (/Utility) from action s: ui (s ,X −i )

(Approximate) NE, for ε≥ 0ε-well-supported NE:

xi ,s > 0 ⇒ ui (s ,X −i )≥ ui (s′,X −i )−ε

ε-approximate NE:

ui (X ) = (∑s

xi ,s ∗ui (s ,X −i )) ≥ ui (s′,X −i )−ε


Categories of Games

n players, ξ actions

Normal form games: payoffs for all strategy profilesÏ O(nξn) payoffsÏ special case: Bimatrix games ( 2 players )

Anonymous games:Ï Payoff of player:

1. action she plays2. Partition of the other players into actions

Ï Succinctly representable for constant ξ: O(ξ ·nξ−1) valuesGraphical games:

Ï players: vertices of a graph, payoffs affected by the neighbours onlyÏ Succinctly representable for constant degreeÏ special case: Polymatrix games ( each edge plays Bimatrix )

+ Property: Bimatrix, Polymatrix have a rational equilibrium


OverviewAnonymous games:

PTAS 2 strategies, constant # strategies [DP07,Das08,DP08,DP08,DP14]

query-efficient approximation [GT14]

PPAD-hard for ε= 1/exp(n), ≥ 7 strategies [CDO14]

Normal form games: r players, n strategiesReductions: r →3,2 player games [DGP09, FT10,Bub79] , normal form ↔graphical [GP05]3-Player PPAD-hardness for 1/exp(n)-NE (via Polymatrix) [DGP09]

Bimatrix gamesÏ 1/poly(n)-NE PPAD-hardness [CDT09]Ï Approximation algorithms: [TS07], [LMM03]Ï Special case: Constant Rank Bimatrix

F rank 3 PPAD-hard [Meh14], FPTAS [KT07], Polynomial for rank 1[AGM+11]

Polymatrix gamesÏ constant-NE PPAD-hardness [Rub14]


Table of Contents

1 Anonymous Games

2 Reducibility among NE problems

3 Hardness of normal form games

4 Bimatrix Games

5 Constant Rank Bimatrix


Anonymous Games - Approximation Algorithms [DP14]

“Oblivious" algorithms:Enumerate over candidate equilibriaUse the game only for verification

Mixed NE:2 actions PTAS:

Ï [DP07]: nO(1/ε2),Ï [Das08]: faster poly(n) · (1/ε)O(1/ε2)

Ï [DP09]: "weakly"-oblivious poly(n) · (1/ε)O(log2(1/ε))

[DP08]: PTAS for ξ actions nO(g(ξ,1/ε)), g exponential in ξPure NE:

[DP07]: λ-Lipschitz games have O(λξ)-approximate pure NE


Useful properties of Anonymous Games

In polynomial time we can do:1 Expected payoff computation - Dynamic programming O(ξnξ+1)2 Given an unordered mixed profile we can decide if there is an

assignment of strategies to players that yields an ε-NE


Oblivious Algorithms

Discrete search space:Ï If we are promised ∃ ε-NE with xi ,s integer multiples of fixed quantity

1z (z independent of n, related to ε)

Ï ⇒ K =O(zξ−1) different mixed strategies (polynomial of z)

Enumerate over partitions of players to mixed strategies(polynomially many: O(nK ))For the chosen partition, check if exists an assignment that is ε-NE


"Weakly" Oblivious Algorithms (2 actions)

Oblivious: enumerate all unordered mixed profiles ⟨q⟩“weakly" oblivious: enumerate over the first O(log 1

ε ) moments ofpartition probabilities Prq[π]


Anonymous Games - Query Complexity [GT14]

Payoffs not known - ask queries

2-action query efficient algorithm [GT14]Randomized algorithm that finds (with high prob.)O(ε+ 1

εpn)-approximate NE with Ő(n11/8) payoff queries

finds c-NE for c ≥ 14pn

running time poly(n,1/ε)

Lipschitz games have approx-PNE [DP07]

find O(λ+ε)-PNE:Ï ε-accurate queries

G∗: "Smooth" the payoffs to becomeO( 1

εpn)-Lipschitz

simulate ε-queries on G∗ using G


Table of Contents

1 Anonymous Games



4 Bimatrix Games



Reducibility among NE problems

Normal ↔ Graphical [DGP09,GP05]Ï d-Graphical Nash ∝ r -Nash

F r : associated with chromatic number of a graphÏ r -Nash (normal form) ∝ 3-Graphical (binary)

F using graphical gadget gamesÏ + combined: r -Nash ∝ 3-Nash

r -Nash ∝ 2-Nash (via Polymatrix) [FT10]Ï for ε-NE only

exact NE mapping r -Nash → 3-Nash [Bub79]Ï (Irrational NE)


Generalized Matching Pennies (GMP)

Block diagonal M, k blocks(M = f (k))Peturb payoffs by +[0,1]ε-NE (ε≤ 1): uniform overblocks ± O(1/M)


d -Graphical ∝ r -Nash [DGP09,DG05]

d-Graphical Nash (GG ) ∝p r -Nash (G ):r : (even) # of colors required to color the vertices of G :

Colors of G −→ Players of G

action a[v ] of node v −→ strategy (v ,a[v ])


d -Graphical ∝ r -Nash [DGP09]

Enforce fairness: pairs play GMPRecover NE of GG : normalizeover block for node v


Graphical Gadget Games [GP05]

Graphical games that perform arithmetic/logical computations

Polymatrix Gadgets:Ï +, −, =ζ, ×ζ, copy =Ï comparison < : brittleÏ boolean ∨,∧,¬

Non polymatrix gadgets: ∗,max,min

E.g Addition gadget G+:

4 players, 2 strategies

ε-NE:p[v3]=maxp[v1]+p[v2],1±ε


r -Nash ∝ 3-Graphical [DGP05]

(binary) nodes are used to encode:Ï mixed strategies of players (probability nodes)Ï expected payoffs of players (utility nodes)

Gadgets are used to enforce the required relations between them!


Exact r -Nash → 3-Nash [Bub79]

PA represents all original players: strategies (player,action) pairsPC represents pure original action profiles

Payoffs of G given to PA

+ Need to force player PC play products of probabilities of PA!Ï Role of player PB

PA,PC also play a side GMP


r -Nash ∝ 2-Nash [FT10]

r -Nash vs Bimatrix utilities: multiplicative vs linear[FT10]: Multiplication simulated by a Polymatrix gadget (± error)

Ï ε-NE only

r -Nash ∝ Polymatrixr -Nash utility:∑

s−i payoffi (aj ,s−i ) ·∏

j 6=i pjs[j]

+ simulate∏

j 6=i pjs[j]

by G∗

Polymatrix ∝ BimatrixRow player simulates polymatrixplayersColumn player forces fairrepresentation


Table of Contents

1 Anonymous Games



4 Bimatrix Games



Hardness of normal form games

PPAD-hardness:1/exp(n)-NE for up to 4 players [DP06,DGP05,DGP09]

Ï + hardness of degree ≥ 3 graphicalÏ + hardness of polymatrixÏ improvement to 3 players [CD06b, DP06b]

1/poly(n)-NE for Bimatrix [CDT06b,CDT09]Ï (No FPTAS)

r -Nash ∈PPAD, r ≥ 2 [GDP09,Pap94]


The class PPAD

PPAD ⊆TFNPÏ total: ∃ solutionÏ FNP: poly-time verifiability

(def) PPAD: class of TFNP problems ∝ EOL

End of the Line (EOL):(exponentially large) graph given bypoly-time circuits Successor/Predecessorin/out-degree ≤ 1Given a source, find a sink / differentsource


PPAD-Hardness of normal form games [DP09]

Hardness proof:3D-Brouwer (PPAD-hard)3D-Brouwer ∝p 3-Graphical Nash (∝p 3-colorable Graphical Nash ∝p 3-Nash)

Ï 3D-Brouwer ∝ (3-Graphical) Polymatrix

Hardness for ε-approximate NEÏ w.s. ∝ approx: given ε-approx recover O(

pε · r)-NE


3D Brouwer [DP09]

3D-unit cube, subdivision to subcubes of side 2−n

C : "coloring"/displacement circuit of the sub-cubesÏ 4 colors/displacements (must satisfy a boundary condition)

Goal: find a panchromatic vertex

α displacementδ1 = (α,0,0) - (color 1 yellow)δ2 = (0,α,0) - (color 2 blue)δ3 = (0,0,α) - (color 3 red)δ4 = (−α,−α,−α) - (color 4 green)


3D-Brouwer ∝ Graphical Nash (Polymatrix)

Binary (2 actions) graphical gamepoint (x ,y ,z) ⇔ probabilities that nodes vx ,vy ,vz play strategy s1

vx

3 players representcoordinates ofpoint (x ,y ,z)

vy

vz

Bit ExtractorsExtract n MSBitsof vx ,vy ,vz[<,×ζ,−,= ζ]

auxiliary players

Circuit Simulator[∨,∧,¬]

auxiliary players

∆x ,∆y ,∆z displacements

bits

Bit extractors work well only for points not too close to facetsÏ due to brittle comparators


Deal with brittle comparators

Sampling + Averaging:Do the previous computation to 413

points around p = (x ,y ,z)

(x + iα,y + jα,z +kα)

Poorly vs Well-positioned Points

Claim: Most of the points will be well-positionedε=α2 « α= 2−2n « edge = 2−n


Simulate 3D-Brouwer with Graphical Games (Polymatrix)

1 point p + sampled points2 Extract bis + Simulate circuit3 compute average displacement r4 "Update" p=p+ r

Claim:In ε-NE the cubelet containing pointp has a panchromatic vertex.

vx

vy

vz

v ′x

v ′y

v ′z

vx ,i

vy ,j

vz ,k

Bit Extractors

Bit Extractors

Bit Extractors

Circuit Simulator

Circuit Simulator

Circuit Simulator

∆x ,∆y ,∆z

∆x ,∆y ,∆z

∆x ,∆y ,∆z

Average

+

bits

=

=

=


Table of Contents

1 Anonymous Games



4 Bimatrix Games



Hardness of Bimatrix [CDT06,CDT09]

+ d-D Brouwer ∝ Generalized Circuit ∝ Bimatrix ( 1poly(n) - NE)

d-Dimensional Brouwerd-dimensional hyper-grid - subdivision in unit hypercubesColoring/Displacement circuit C over vertices of grid

Ï d+1 colors,

Goal: Find panchromatic simplex contained in unit hypercube


Exponential vs Polynomial NE hardness

Need exp # cubelets (embed EOL graph)3D space ⇒ 2−n edge size, 1/exp(n)-NEFor 1/poly(n) need constant edge size ⇒ Increase dimensionCan’t ask for vertex, Ask for simplex


Generalized Circuit [CDT09]

Generalized Circuitset of K nodesset of gates +,−,<,=,×ζ,∨,∧,¬,= ζ (poly(K ))

Ï gate (constraint): 2 inputs, 1 unique outputÏ Cycles allowed!

ε-approximate solution:Ï node v ↔ variable x [v ] ∈ [0, 1

K +ε]Ï values assignment x [v ] s.t. ε-satisfy the gates

e.g. addition (+,v1,v2,v3) : x [v3]=minx [v1]+x [v2],1K ±ε

PPAD-hard for ε= 1/K 3


PPAD hardness for Bimatrix

GCircuit ∝ Bimatrix:Ï The GCircuit operations can be simulated by Bimatrix gadgets!Ï Embed gadgets over a GMP game

n-D Brouwer ∝ GCircuit:Ï [DGP09] framework: GCircuit encodes points +simulate coloring circuit Cbr

Ï difference with [DGP09]: Equiangle samplingF sampling around p: 41nF n3 points sufficient, parallel to (1,1, . . . ,1) vector


Hardness of constant-approx GCircuit [Rub14]

GCircuit PPAD-hard for constant-approximate solutionsÏ different choice of Brouwer function f over hyper-gridÏ f : M-Lipschitz-continous function (arithmetic circuit C )Ï f from [HPV89]: ε-fixed point computation with black box access to frequires Ω(1εMd−2)

(implicit) [DGP09]: c-approx GCircuit ∝ c-NE 3-Graphical PolymatrixÏ no PTAS for Polymatrix


Bimatrix ε-Approximation Algorithm [LMM03]

logarithmic support NE [LMM03]Bimatrix (n×n)

∃ ε-approximate NE of support size k =O( lognε2

)

mixed strategy: uniform over a multiset of pure actions

+ Oblivious algorithm:enumerate over all multisets of size k

Running time: nO(logn/ε2)


Local Search for Bimatrix [TS07,TSK08]

(0.3393+δ)-approximate NErunning time O(1/δ2) ·poly(n) for Bimatrix (A,B)

regret functions:

fA(x ,y)=maxrow i

(Ay)i −x>Ay , fB(x ,y)= . . .

maximum regret f (x ,y)=maxfA(x ,y), fB(x ,y)

Ï f (x ,y)≤ ε ↔ ε-approximate NE

Gradient decent on f ()Ï Stationary points: 0.3393-NE (either themselves or can exctract one)


Table of Contents

1 Anonymous Games



4 Bimatrix Games



Constant Rank Bimatrix

Bimatrix A,B , Rank of Game: rank(A+B)

(rank 0 are 0-sum games)[AGM+11]: rank 1 poly-time[Mehta14]: rank ≥ 3 PPAD-hard[KT07]: FPTAS


Constant Rank Bimatrix - PPAD hard [Mehta14]

+ 2D-Brouwer ∝p 2D-Linear FIXP ∝p Rank 3 Bimatrix

2D-Linear FIXP [EY07]Ï F : [0,1]2→ [0,1]2Ï Circuit CF +,×ζ,maxÏ F : piece-wise linearÏ Find fixed point of FÏ (rational)

2D-Brouwer ∝ 2D-Linear FIXP:similar to [CTD09]:extract bits + simulate Coloringcircuit (using max,×ζ,+)


2D-Linear FIXP ∝ 3-Rank Bimatrix

Circuit CF (λ1,λ2):Ï linear + quadratic contraints among λ1,λ2 and xi (output of i-th maxgate)

Can be simulated by a parametrized LP(λ), for appropriate objectiveÏ variables xi

LP + DLP → LCPÏ solutions of LCP ↔ fixed points of CF

LCP constraints embedded into NE of rank 3 Bimatrix game


Bimatrix as QP

Bimatrix game (A,B)

NE as Quadratic Program:Objective: max x>(A+B)y −πA−πBx ,y mixed strategiesπA,πB scalars for optimal utility (free variables)


Rank 1 Bimatrix in P [Meh]

Rank 1: A+B = a ·b> - Rank 1 Games described by (A,a,b)

Games (A,u,b) for u ∈Rm share same QP, except for (x> ·u)Ï maximize (x> ·a) (b> ·y)−πA−πB

for fixed value λ= x> ·u: QP transformed to parametrized LP(λ)

LP(λ) used to define f : [0,1]→ [0,1]fixed points = NE of (A,a,b) - use Binary search


FPTAS for constant Rank Bimatrix [KT07]

NE of (A,B) as QP of fixed rank[Vav91]: Fixed rank QP minz>Qz : B ·z ≤ b with bounded feasibleregion can be solved ε-approximately in poly(L,1/ε) (L: bit length ofQP)modifiy Bimatrix QP to fit the above - FPTAS


Open Problems

Anonymous Games:Ï FPTAS? (2 strategies / constant strategies)

F current best algorithm for 2 strategies Moments Search [DP09]:poly(n) · (1/ε)O(log2(1/ε))

Ï Hardness for ≤ 6 strategies? [CDO14]Bimatrix games:

Ï PTAS?F current best algorithm [LMM03]: nO( logn

ε2)

Ï Hardness for Rank 2? [Meh14]


Thank you!


(Approximate) Nash Equilibria

Game of n players, ξ actions, payoffs in [0,1]

xi : Mixed strategy of a player i , probability distribution over ξ(mixed) strategy profile X = (x1, . . . ,xn)

Expected Payoff (Utility) from action s: ui (s ,X −i )

(Approximate) NE, for ε≥ 0ε-well-supported NE:

xi ,s > 0 ⇒ ui (s ,X −i )≥ ui (s′,X −i )−ε

ε-approximate NE:

ui (X ) = (∑s

xi ,s ∗ui (s ,X −i )) ≥ ui (s′,X −i )−ε


Categories of Games

n players, ξ actions

Normal form games: payoffs for all strategy profilesÏ O(nξn) payoffsÏ special case: Bimatrix games ( 2 players )

Anonymous games:Ï Payoff of player:

1. action she plays2. Partition of the other players into actions

Ï Succinctly representable for constant ξ: O(ξ ·nξ−1) valuesGraphical games:

Ï players: vertices of a graph, payoffs affected by the neighbours onlyÏ special case: Polymatrix games ( each edge plays Bimatrix )

+ Property: Bimatrix, Polymatrix have a rational equilibrium


Overview of main results

Anonymous gamesÏ n playersÏ approximation algorithms 2 strategies, constant # strategies[PD07,Das08,DP08,DP08,DP14]

Ï query-efficient approximation algorithm [GT14]Normal form games

Ï k player games ∝ 3 player games (via graphical) [PD09]F 3-Player PPAD-hardness for 1/exp(n)-NE, n=# strategies [PD09]

Ï Bimatrix games (2 players)F 1/poly(n) - NE PPAD-hard [CD09]F Approximation algorithms: local search [TS07], oblivious

quasi-polynomial [LMM03]F Constant Rank Bimatrix: PPAD-hard [Meh14], FPTAS [KT07],

Polynomial for rank=1 [AGM+11]


Computing NE in Anonymous Games

(trivially) PPAD-hard to compute ε-NE for n players and f (n)strategies

Ï 3-player (k-strategy) normal form ≡ anonymous with 3k startegies

PPAD-hard to compute 1/2n-NE for n+2 players, constant ≥ 7strategies [CDO14]∃ 3-player game with 2 strategies with irrational NE only [GT14]


Anonymous Games - Approximation Algorithms [DP14]

“Oblivious" algorithms:Enumerate over a set of predefined candidate equilibriaUse the game only to test if a candidate is a NE

+ Require knowledge of the “structure" of candidate equilibria

Mixed NE:2 actions PTAS:

Ï [DP07]: nO(1/ε2),Ï [Das08]: faster poly(n) · (1/ε)O(1/ε2)

Ï [DP09]: “weakly"-oblivious poly(n) · (1/ε)O(log2(1/ε))

[DP08]: PTAS for ξ actions nO(g(ξ,1/ε)), g exponential in ξPure NE:

[DP07]: λ-Lipschitz games have O(λξ)-approximate pure NE (poly(n))Ï λ-Lipschitz: other player’s choice affects payoffs by ≤λ


Nice properties of Anonymous GamesIn polynomial time we can do:

1 Expected payoff computation - Dynamic programming O(ξnξ+1)

ui (sj ,X )= ∑partitions π

payoffi (sj ,π) ·PrX [π occurs]

Ï Pr [π] complex expression of xj ,s

2 Given an unordered mixed profile we can decide if there is anassignment of strategies to players that yields an ε-NE:

Ï Find the set of ε-best responses per player (expected payoffcomputation)

Ï max flow (/perfect matching) on bipartite graph

source

s

st

t

sink

n players

P1

P2

Pi

Pn

K mixed strategies

σ1

σ2

σj

σl

σK

t1t2

tK

11

1

ε-best responces

1

11


Oblivious Algorithms

Discrete search space:Ï If we are promised ∃ ε-NE with xi ,s integer multiples of fixed quantity

1z (z independent of n)

Ï ⇒ K =O(zξ−1) different mixed strategies (constant)

Enumerate over partitions of players to mixed strategies(polynomially many: O(nK ))For the chosen partition, can we assign players to mixed strategiessatisfying ε-NE conditions?


Anonymous Games - Candidate Equilibria

Promised discretized ε-NE:

2-action games s0,s1: a partition is # of players playing s1

mixed strategy of player i : pi

mixed profile ⟨p⟩ defines prob. distr. Prp[π] over partitions π

ε-TVD ⇒ ε-NEÏ Given mixed profiles ⟨p⟩, ⟨q⟩Ï If TVD ( Probp[π], Probq[π] ) ≤ εÏ then if ⟨p⟩ NE ⇒ ⟨q⟩ ε-NE


Anonymous Games - Candidate Equilibria

Discretized ε-NE: [Das08, DP14b]∀ mixed profile ⟨pi ⟩∀ integer k

∃ mixed profile ⟨qi ⟩ s.t.1 TVD ( Probp[π], Probq[π] ) is O(1/k)2 either ∃ S ⊂ [n], |S | < k3 for which qi= multiple of 1/k2

OR ∃ S ⊆ [n]: for which all qi = same multiple of 1/n, (the rest pure)

[DP07] only multiples of 1/k2, nO(1/ε2)

[Das08]: PTAS, poly(n) · (1/ε)O(1/ε2)

[DP09] "Weakly" oblivious algorithms: (1/ε)O(log2(1/ε))

+ ε-cover of size n2+n · (1/ε)O(1/ε2) for PBD


Obliviousness and efficiency

Oblivious lower bound [DP09]

Any oblivious algorithm for 6-stategy games ε-NE runs in 2Ω((1ε)1/3)

n main players, 2 strategies + 2 special players, 4 special strategies

Game Gp: Force p= (p1, . . . ,pn) be the unique NE strategy for themain players

Ï ε-NE: pi ±error

n = (1/ε)13

Exist ≥ 2Ω((1/ε)13 ) different tuples p s.t. ε-NE of games Gp,Gp′ have

different unordered mixed profiles+ Running time: For some Gp the oblivious algorithm tries all of them


"Weakly" Oblivious Algorithms

Oblivious: enumerate all unordered mixed profilesBottleneck: Search over |S | < k3, qi = multilples of 1

k2

“weakly" oblivious: enumerate over the first O(log 1ε ) moments of∑

i Yi

TV Boundmixed profiles ⟨pi ⟩, ⟨qi ⟩If Probp[π], Probq[π] have equal first d moments ⇒ TVD = 2−Θ(d)

Ï equal moments ≡ ∑i (pi )

j =∑i (qi )

j , j = 1, . . . ,d

Guess correctly the moments of ε-NE ⟨q⟩Compute O(ε)-BR mixed profile ⟨q′⟩ w.r.t. moments (DynamicProgramming)⟨q⟩ 6= ⟨q′⟩ but q′ will also be O(ε)-NE


Weakly Oblivious Algorithms - "Moments Search"

"Moments search" algorithm:

Guess moments: the first O(log 1ε ) moments of Pr [

∑i Yi ]

Ï knowledge: qi multiples 1k2

Ï (1/ε)O(log2 1ε ) choices

Given guessed moments: Find set BR(playeri ) of O(ε)-best responsesÏ Find a compatible partition + utility computation (poly(n))Ï Dynamic programming (1/ε)O(log2 1

ε )

Given moments and sets BR(): Can we assign players to mixedstrategies from BR() w.r.t. guessed moments?

Ï Dynamic programming, poly(n) · (1/ε)O(log2 1ε )


Anonymous Games - Query Complexity [GT14]

Payoffs not known, Query access to themSingle player query: player i , action s, partition π: payoffi (s ,π)

2-action query efficient algorithm [GT14]

Randomized algorithm that finds (with high prob) O(ε+ 1εpn)-approximate

NE with Ő(n11/8) payoff queries

compared to 2n2 payoffs in totalfinds c-NE for c ≥ 1

n1/4

running time poly(n,1/ε)non-oblivious


2 action Anonymous Games - Query Complexity

λ-Lipschitz games:|payoffi (s ,π)−payoffi (s ,π′)| ≤λ · ||π−π′||1[DP07] for ξ actions ∃ O(λξ)-Pure NE[GT14]: Assume we answer payoff queries with ε-accuracy∃ O(λ+ε)-PNE, can be found by O(n logn) queries

Ï binary search to find a fixed point of a function

Idea for general game G :Game G∗: "Smooth" the payoffs to make them O( 1

εpn)-Lipschitz

find O(ε+ 1εpn)-PNE

Ï need to simulate (ε-accurate) queries on G∗, using queries on GÏ pure s0/s1 → mixed (1−ε,ε) / (ε,1−ε)


2 action Anonymous Games - Query Efficient Algorithm

Input: 2 actions s0,s1, n-player game G

Create Smoothed game G∗:Partition k ↔ random variable X (PBD)

Ï X =∑Xi

Ï k of them have E [Xi ]= εÏ rest (n−1−k) have E [Xi ]= 1−ε

Set: payoffG∗

i (s ,k)=∑n−1k=0payoff

Gi (s ,k) ·Pr [X = k] (utility in profile X )

+ G∗: is O( 1εpn)-Lipschitz


2 action Anonymous Games - Query Efficient AlgorithmSimulate Payoffs - payoffG

∗i (s ,k)=∑n−1

k=0payoffGi (s ,k) ·Pr [X = k]

exact payoff G∗ involves n payoffs of Gneed a ε-accurate only:

Ï X follows a PBD ∆ (E [Xi ] ∈ ε,1−ε)Ï For ε≥ 1

n2 :F Decrease support size to D ∼

√n logn:

F cut tails: prob mass ≤ ε (causes ε error)Ï ∃ H (piecewise constant) that approximates ∆

s.t. ||H;∆||TVD ≤ εF Using H in place of ∆ causes error 2εF + By querying (uniformly at random) K payoffs in each interval of H

we can compute with high probability a value p′: |p′−payoffG | ≤ 3εF H has ∼ 1

ε2

pD intervals, from each we need K= Ő( 4p

D) queriesF Total # queries for a payoff: Ő( 1

ε2n3/8)


Reductions among NE problems

r -Nash:find ε-NE in a r -player game

ε-NE due to irrational NE

Normal ↔ Graphical [DGP09,GP05]Ï d-Graphical Nash ∝ r -Nash (Normal form)

F r : associated with chromatic number of a graphÏ r -Nash (normal form) ∝ 3-Graphical (binary)

F using graphical gadget gamesÏ + reduction r -Nash ∝ 3-Nash - (via graphical)

r -Nash ∝ 2-Nash (via Polymatrix) [FT10]Ï for ε-NE only

exact NE mapping r -Nash → 3-Nash [Bubelis79]Ï Irrational NE: For any polynomial f with rational coefficients ∃ 3 player game s.t. in

NE one player utility=α iff f (α)= 0 (irrational root)


d -Graphical ∝ r -Nash [DGP09,DG05]

d-Graphical Nash ∝p r -Nash:r : (even) # of colors required to color the vertices of G :

If G degree d then ≤ d2+1 colors sufficient



d-Graphical Nash (GG ) ∝p r -Nash (G ):Colors of G −→ Players of G

node v , action a[v ] −→ strategy (v ,a[v ]) of player color [v ]

even # players in G

pairs playingGeneralized Matching Pennies:

Ï (M > 2#blocks)embed the payoffs of GG overGMP

Ï additive perturbations in [0,1]Ï ε′-NE: blocks-uniform ± 1

MÏ (ε′ ≤ ε( 1

#block − 1M )d )

Ï Recover NE of GG : normalizeover block for node v



Embedding payoffs of GG to G :

GGG (graphical):node v , neighbours v1,v2, . . . ,vk

Payoffv (a[v ] | a[v1],a[v2] . . .)

G (normal form):Player “Color [v ]"PayoffRed(⟨v ,a[v ]⟩ | ⟨v1,a[v1]⟩,⟨v2,a[v2]⟩, . . .)


Graphical Gadget Games [GP05]

Graphical games that perform arithmetic/logical computations

Polymatrix Gadgets:Ï +, −, =ζ, ×ζ, copy =Ï comparison < (brittle)Ï boolean ∨,∧,¬

Non polymatrix gadgets: ∗,max,min

uv3 (s0)= p[w ],uv3 (s1)=1−p[w ]

uw (s0)= p[v1]+p[v2],uw (s1)= p[v3]

E.g Addition gadget:

4 players, 2 strategiesInputs v1,v2, Output v3

Directed edges: "affects"

ε-NE:p[v3]=maxp[v1]+p[v2],1±ε


r -Nash ∝ 3-Graphical

Normal form G :r players, n actions, payoffs ∈ [0,1] (input size rnr )

Ï xpi: prob of player p playing action i

Graphical (binary) GG :2 strategiesFor node v : P[v ] = prob v plays s1For every player p:

Ï Probability nodes: v(xpi) for each action i

F force:∑i P[v(x

pi)] = 1 : using gadgets −,+,∗

Ï Utility nodes: v(Upi), v(U

p≤i ) for each action i

F force: P[v(Upi)]= ∑

s−p payoffpi ,s−p

· ∏p≤q P[v(x

qsq )]

F P[v(Up≤i )] =max

P[v(U

p≤i−1)], P[v(U

pi)]

F gadgets +,∗,= ζ,max

Ï Total O(rnr ) nodes (probability + utility + auxiliary)



Utility comparator nodes: w(Upi ):

Ï compare Upiand U

p≤i−1

Probability allocator nodes: vpi :

Ï Depending on comparison of w(Upi) they allocate probability among:

F P[v(xpi)] VS

F all previous P[v(xpi−1)] . . . P[v(x

p1 )]

Degree 3:Ï all gadgets used have degree 3Ï v(x

pi) nodes involved in many computations - multiple copies (gadget

=) on a binary tree


probability nodesutility nodesutility comparison nodesprobability allocator nodes

Not pictured:Binary tree of v(xp

i )

subgraph performing utility Upi

computation over:Ï for action profiles s:Ï nodes v(x

qsq ) of players q 6= p

Ï payoff nodes:p[node]= payoff

pi ,s

* (using gadget "= ζ")



previous construction maps exact NEFor ε′-NE can’t directly use p[v(xp

i )] as NE of G

Ï∑i p[v(x

pi)] 6= 1

Ï p[Upj]> p[U

pj ′ ]+ε′ but p[v(x

pj ′ )]> 0



previous construction maps exact NEFor ε′-NE can’t directly use p[v(xp

i )] as NE of G

Ï∑i p[v(x

pi)] 6= 1

Ï |∑i p[v(xpi)]−1| ≤Θ(nε′)

Ï p[Upj]> p[U

pj ′ ]+ε′ but p[v(x

pj ′ )]> 0

Ï p[Upj]> p[U

pj ′ ]+Θ(nε′) ⇒ p[v(x

pj ′ )] ∈ [0,Θ(nε′)]

What works:Set probabilities in [0,Θ(nε′)] to 0 and normalize the restFor appropriate choice of ε′ = ε/poly(|G |):

Ï errors from substitutionÏ + errors from transformation from p[U

pj] to utilities of G

Ï absorbed into ε ⇒ we recover ε-NE of G


Exact r -Nash → 3-Nash [Bub79]

r -player game G mapped to 3 player game G ′:Ï PlayerA represents all original playersÏ PlayerC represents pure original action profilesÏ PlayerB : auxiliary player

PlayerA:Ï Pure strategies: (pi ,aj ) original player-action pairs

Partial action profiles:Ï K = 1,2, . . . ,k \ j k ∈ [r ], j ∈ [k]Ï partial profile sK : actions played by the players in K

PlayerC , PlayerB :Ï Pure strategies: all partial profiles sKÏ player PC has r extra strategies corresponding to players pi

F on which he playes GMP with PA


Exact r -Nash → 3-Nash

PA interacts only with PC :Embedding payoffs of G :

Ï The strategy profile of G ′ where:Ï PA plays strategy (pi ,aj ) - an action of player iÏ PC plays strategy sK for K = [r ]\ i - an action profile s−iÏ gets payoffi (aj ,sK ) of G

+ Need to force player PC play products of probabilities of PA!F Role of player PB

PC interacts with both PA,PB :Ï has linearly separable payoffs

PB interacts with both PA,PC :Ï Non separable payoffs


Properties of G ′ on NE (x,y,z):PC plays:

Ï zsK = ζ ·∏i∈K x(pi ,as[i ])Ï ζ> 0 probability on the player block (GMP with PA)

PA plays uniform over all his blocksRecover NE of G : Normalize x over each block


r -Nash → 2-Nash [FT10]

r -Nash ∝p Polymatrix ∝p Bimatrix

r -Nash vs Bimatrix expected payoffs:Ï r -Nash: multiplicative in other players’ strategiesÏ 2-Nash: linear in opponent’s strategy

[DGP09] r -Nash ∝ 3-Nash uses multiplicative gadget "∗"[FT10]: Multiplication simulated by a Polymatrix gadget (± error)

Ï works for ε-NE onlyÏ Gadget G∗: on inputs p1, . . . ,pm: output p = p1 . . .pm±Θ(mεc)

F 2 constructions of sizes O(mε2), O(m log 1

ε )


r -Nash ∝ Polymatrix

r -Nash utility:∑s−i payoffi (aj ,s−i ) ·

∏j 6=i p

js[j]

+ approximate∏j 6=i p

js[j]

by G∗

Polymatrix G ′: for all i ∈ [r ]:Original player Pi

O(nr−1) Mediator playersÏ simulates the probability

of each action profile s−iauxiliary players

G payoffi (aj ) on profile s−i ↔ G ′ payoffi (aj ) when Mediator Qs−i plays action 1

Recover ε-NE of G : probabilities p of original players (ε′ = poly(ε/|G |))


Polymatrix ∝ Bimatrix

Polymatrix G :Ï m players, player pi has ni actionsÏ players i , j play bimatrix M i ,j

Ï N= total # of actions

N ×N Bimatrix (A,B)strategies (pi ,aj): player-action pairs

Ï Row player matrix A:

Ï α= 8m2/εÏ Column player B: identity matrix IN

ε′-NE (ε′ ≤ 1/N) (x ,y) of (A,B):

support(y)⊆ support(x)Ï Column plays the ε-bestresponses of Row

x : each row Block gets > 0probability

Ï polymatrix playersrepresented

y is almost uniform overcolumn BlocksRecover ε-NE of Bimatrix:normalize y over each block

Ï ε′ = poly(ε/|G |)


Hardness of normal form games

PPAD-hardness:1/exp(n)-NE for up to 4 players [DP06,DGP05,DGP09]

Ï + hardness of graphical games with degree ≥ 3Ï + hardness for graphical polymatrix gamesÏ improvement to 3 players [CD06b, DP06b]

1/poly(n)-NE for Bimatrix [CDT06b,CDT09]Ï No FPTAS for Bimatrix (unless PPAD ⊆P)

Membership: r -Nash ∈PPAD, r ≥ 2 [GDP09,Pap94]


The class PPAD

PPAD ⊆TFNPÏ total: a solution always existsÏ FNP: search problems with solutions verifiable in poly-time

(def) PPAD: class of TFNP problems ∝ EOL

End of the Line (EOL):(exponentially large) graph given bypoly-time circuits Successor/PredecessorGiven a source, find a sink / differentsource


PPAD-Hardness of normal form games [DGP09]

Hardness proof:3D-Brouwer (PPAD-hard)3D-Brouwer ∝p 3-Graphical Nash (∝p 3-colorable Graphical Nash ∝p 3-Nash)

Ï 3D-Brouwer ∝ 3-Graphical (Polymatrix)

Hardness for ε-approximate NE (ws-NE ∝ approximate NE)Ï Given ε-approximate NE of r -player game, we can recover O(

pε · r)-NE


3D Brouwer [DGP09]

3D-unit cube, subdivision to cubelets of side 2−n

23n cubelets needed for hardness - embedding EOL

C : displacement / "coloring" circuit of cubeletsÏ 4 colors (must satisfy a boundary condition)

Goal: find a panchromatic vertex (∃ from Sperner’ Lemma)Ï adjacent to 4 cubes with all 4 colors

Displacement Circuit C :

Input: Grid vertex p = (i , j ,k) (3n bits)

Output: Displacement vector ofcublet Kp

Ï displacement αÏ δ1 = (α,0,0) - (color 1yellow)Ï δ2 = (0,α,0) - (color 2 blue)Ï δ3 = (0,0,α) - (color 3 red)Ï δ4 = (−α,−α,−α) - (color 4 green)


EOL ∝ 3D-BrouwerPPAD-hardness of 3D-Brouwer:

The graph of EOL problem can be embedded in 3D unit cubevertex ↔ path on the cube, edges: connect pathsnon-crossing pathsColoring:

Ï boundary colorsÏ around the line: red, blue, yellowÏ rest of the cube 4-rth color (green)


3D-Brouwer ∝ Graphical Nash (Polymatrix)Binary (2 actions) graphical gamepoint (x ,y ,z) ⇔ probabilities that nodes vx ,vy ,vz play strategy s1

# of players: poly(n,size[C ])

vx

3 players representcoordinates ofpoint (x ,y ,z)

vy

vz

Bit ExtractorsExtract n MSBitsof vx ,vy ,vzusing gadgets<,×ζ,−,= ζ

auxiliary players

Circuit Simulatorusing gadgets∨,∧,¬

auxiliary players

∆x ,∆y ,∆z displacements

bits

Bit extractors work well only for points not too close to facets(3nε-distance)

Ï due to brittle comparators



Sampling + Averaging:Do the previous computation to a setof points around p = (x ,y ,z)

413 points (constant)Fron p = (x ,y ,z) generate points(x + iα,y + jα,z +kα), i , j ,k =−20, . . . ,20

Ï can be done using gadgetsÏ α: displacement value

Poorly vs Well-positioned PointsÏ Poorly: too close to facets (∼ εn)

Claim: Most of the points will be well-positionedMust have: ε « α (exponentially approximate NE)edge = 2−n » α= 2−2n » ε=α2



1 Sampling: from p = (x ,y ,x) generate points (x + iα,y + jα,z +kα)

2 Extract bis + Simulate circuit3 compute average displacement δx ,δy ,δz : using gadgets +,×ζ4 update p = p + avg

vx

vy

vz

v ′x

v ′y

v ′z

vx ,i

vy ,j

vz ,k

Bit Extractors

Bit Extractors

Bit Extractors

Circuit Simulator

Circuit Simulator

Circuit Simulator

∆x ,∆y ,∆z

∆x ,∆y ,∆z

∆x ,∆y ,∆z

Average

+

bits

=

=

=



Claim:In ε-NE the cublet containing point p= (p[vx ],p[vy ],p[vz ]) has apanchromatic vertex.

M: sampled points, W : well-positioned points

Claim: in ε-NE (p[vx ],p[vy ],p[vz ]) is farfrom the boundary (∼ 20α distance)ε-NE ⇒ total displacement (from poorly+ well positioned) “small” (∼ |M |ε)⇒ No matter what the poorly-positioneddo, valid displacement fromwell-positioned points “small” (∼α · |W |)⇒ all valid displacements present around p(∃ panchromatic vertex)


Hardness of Bimatrix [CDT06,CDT09]

+ d-D Brouwer ∝ Generalized Circuit ∝ Bimatrix ( 1poly(n) - NE)

d-Dimensional Brouwerd-dimensional hyper-grid - subdivision in unit hypercubesColoring/Displacement circuit C over vertices of grid

Ï d+1 colors, satisfies boundary condition

Goal: Find panchromatic simplex contained in unit hypercube

PPAD-hardness: 2D-Brouwer [CD06] ∝ n-D BrouwerÏ 2-D over 0,1, . . . ,2n−12

Ï n-D: 0,1, . . . ,7n


Exponential vs Polynomial NE hardness

Need exp # cubelets for hardness (embedding EOL in Brouwer)3D space ⇒ 2−n edge size, 1/exp(n)-NEFor 1/poly(n) need constant edge size ⇒ Increase dimensionCan’t ask for vertex - panchromatic verification not poly-time!⇒ Ask for simplex


d-Dimensional Brouwer PPAD-hardness


Generalized Circuit [CDT09]

Generalized Circuitset of K nodes, set of gates +,−,<,=,×ζ,∨,∧,¬,= ζ (poly(K ))

Ï gate (constraint): 2 inputs, 1 unique outputÏ Cycles allowed!

Goal: ε-approximate solution:Ï each node v associated with variable x [v ] ∈ [0, 1

K +ε]Ï values assignment x [v ] s.t. ε-satisfy the gates constraints

e.g. addition (+,v1,v2,v3) : x [v3]=minx [v1]+x [v2],1K ±ε

Boolean values: "1" = 1K ±ε, "0"= [0,ε]

GCircuit hardnessPPAD-hard to find solution for ε= 1/K 3


GCircuit ∝ Bimatrix

GCircuit with K nodes, ε= 1/K 3 (PPAD-hard)Bimatrix with 2K strategies:

Ï node v ↔ actions (v : 0), (v : 1)

Base game: Generalized MatchingPennies (A∗,−A∗)

(M = 2K3)

scaling payoffs to [0,1] maintains1/poly(K) -NE

Gates can be simulated byBimatrix!

Perturb the GMP payoffs bythe gadgets

+ If GMP perturbed by valuesin [0,1]:(≤1)-NE: uniform (±ε= 1

K3 )over nodes v


Bimatrix Gadgets

Simulate the gates with Bimatrix games (Gadgets)E.g. Addition gadget (+,v1,v2,v3)

Player 1 (Row):

Rows v3 affected only by this gate

Ï NE (x ,y):Ï u1(v3 : 0)= y [v3 : 0]Ï u1(v3 : 1)= y [v3 : 1]

Player 2 (Column):

Columns v3 affected only by this gate

Ï u2(v3 : 0)= x[v3 : 1]Ï u2(v3 : 1)= x[v1 : 1]+v [v2 : 1]


Bimatrix Gadgets

Simulate the gates with Bimatrix games (Gadgets)E.g. Addition gadget (+,v1,v2,v3)

Add the Gadgets to GMP([0,1]-perturbation)

ε= 1/K3 NE (x ,y)

+ probabilities x [v : 1] are ε-solution toGCircuit

x [v : 1] are in valid range [0, 1K ±ε]x [v : 1] satisfy the gate constrains

Ï e.g. x[v3 :1]=minx[v1 :1]+x[v2 :1], 1K

±ε

Player 2 (Column):

Columns v3 affected only by this gate

Ï u2(v3 : 0)= x[v3 : 1]Ï u2(v3 : 1)= x[v1 : 1]+v [v2 : 1]


Hardness of Generalized Circuit(PPAD-hard) n-dimensional Brouwer:

Grid 0,1, . . . ,7n, Coloring C (vertices), K = poly(n,size[C ])n+1 colors/ displacements:

Ï “color" i : α ·~ei , “color" n+1: (−α, . . . ,−α), α= 1/K2

Finding a panchromatic Simplex:1 Given point p= (p1 . . . ,pn) of Grid2 (equiangle) Sampling: Generate n3 points pi = p+ ( i−1K , . . . , i−1K )

3 Bit extractor: 3 MSBits of pi (finds lowest vertex in cube) (brittle)4 On extracted bits (vertex) run Circuit C

5 Compute average displacement r of points pi

6 “Update" p= p+ r

+ All the above computation can be simulated with a GCircuitwith K nodes


Hardness of Generalized Circuit (ε= 1/K 3)

(Again) poorly positioned points (coordinate ∼ 1K2 close to integer)

Sampling: ≥ n3−n will be well-positionedIf total displacement (well+poor points) is small (∼ ε)

Ï + Set of extracted vertices from well-positioned points is apanchromatic simplex

Claim: In ε= 1/K 3 approximatesolution of GCircuit, the totaldisplacement is small (∼ ε)


Hardness of constant-NE in Polymatrix [Rub14]

Embeds EOL 0,1n into 2n+1 dimensional unit cubeEOL on unit cube ∝ Succinct Brouwer

Ï Succinct Brouwer: find approximate fixed point of a Lipschitzcontinuous function

Ï f whose fixed points are EOL solutionsÏ f : computed by arithmetic circuitÏ + Need hardness for Succinct Brouwer for constant ε

Succinct Brouwer (constant εb) ∝ GCircuit (constant εc)Ï similar to [CDT09]: simulation of function f over grid using operationsof GCircuit (+,−,×ζ,= ζ,=,<,∨,∧,¬)

(implicit) [DGP09]: εc -approx GCircuit ∝ εc -NE 3-GraphicalPolymatrix

Ï [DGP09] (adapted for constant degree) From ε-approximate NE we canrecover Θ(

pε)-NE


Picking a Brouwer function [Rub14]

[HPV89]: construct f : [0,1]d → [0,1]d (M-Lipschitz), s.t. given blackbox access to f , ε-approx fixed point computation needs timeΩ(1εMd−2)division in sub cubes (constant edge)f (x)= x +g(x), where g(x): displacement function (displacementparameter α constant)g(x) is defined in terms of a "tube": sequence of adjacent-sub cubes[0,1]d−1 (path, the fast dimension fixed)the path enters/exits a cube from the center of a facet


Picking a Brouwer function [Rub14]

[HPV89]:g(x): 79-Lipschitz continuous (⇒ f (x) is 80-Lipschitz continuous)||g(x)||∞ ≥ 1/88 for every x that is not endpoint of the pathg(x) depends on wether:

Ï path passes through cube(x), entering/exiting facets (locallycomputable)

Computing g(x) involves operations max,/ and × (for interpolation)


The displacement function g(x)

1. g(x) on a facet F crossed by tubez the center of facet (entrance point)local coordinates of x = ⟨r ,p⟩zr = ||x−z||∞ (distance), p= x−z

r (unitvector)the path advances through direction i

g(x)= α·ei if r = 0

-α· p if r =edge/8-α·ei if r =edge/4α·ed if r =edge/2

interpolation on the intermediate2. this defines g(x) on two facets,

interpolation on the rest3. cubes not in the tube get default ed

(last dimension - fixed in the tube)

View of facet of the tubein 3D cube:


Hardness of constant-NE in Polymatrix [Rub14]

EOL ∝ Succinct Brouwer:Ï Succinct Brouwer for dimension 2n+2, M=80, εbr = 1/88 is PPAD hard

Succinct Brouwer ∝ GCircuit:∃ constant ε s.t. ε-approximate GCircuit is PPAD-hard

Ï Computation of f ,g simulated with GCircuitF ε-approx solution of GCircuit gives O(ε1/4) fixed point of f

Ï g requires max,×,/ non-GCircuit operationsF GCircuit simulates them with error O(ε1/2)


Computing f with GCircuit

On point x extract bits (brittle <) - find cube it belongsGoal: compute g(x) - depends on cube:Simulate the circuits that give previous/next cube on the tube -Compute entrance/exit facets

Valid computation of g within error O(ε1/4) when:Ï well-positioned: ε-far from facetsÏ NEW! corner-points: ε1/4-close to 2 facets (may be poorly-positionedbut for corners we know by default their g(x))

Deal with poorly positioned: Equiangle sampling + averagingconstant number of points sufficient

Ï points x` = x + (6ε`, . . . ,6ε`) for `= 0, . . . ,1/ε1/2Ï at most 1 will be both poorly-positioned and non-corner (invalidcomputation)

rest of steps as in [CDT09]


Bimatrix Approximation Algorithms

ε-approximate NE: oblivious, quasi-polynomial [LMM03]∼ (1/3+δ)-approximate NE: Local search [TS07]


ε- Approximation Algorithm [LMM03]

logarithmic support NE [LMM03](wlog) n×n Bimatrix

∃ ε-approximate NE of support size O( lognε2

)

mixed strategy: uniform over a multiset of pure actions

+ Oblivious algorithm:k =O( logn

ε2)

enumerate over all multisets of size k (+ expected payoff computation)

Ï(n+k−1

k

)2multiset combinations for both

Running time: nO(logn/ε2)


logarithmic support NE [LMM03]

∃ k-uniform ε-NE:let (x∗,y∗) NE

draw k = 16 lognε2

samples from x∗ (y∗)set x (y) uniform over samples

+ (x ,y) ε-approximate NE w.p. ≥ 1− 4n (Chernoff Bounds)

+ Property: expected payoffs in (x ,y) ε-close to NE payoffs in (x∗,y∗)


Local Search [TS07,TSK08]

(0.3393+δ)- approximate NErunning time O(1/δ2) ·poly(n,m) for Bimatrix (A,B) m×n

(Non Oblivious)

regret functions:

fA(x ,y)=maxrow i

(Ay)i −x>Ay , fB(x ,y)=maxcol j

(x>B)j −x>By

maximum regret function: f (x ,y)=maxfA(x ,y), fB(x ,y)

Ï c-approximate NE: f (x ,y)≤ c

Gradient decent of f ()


Gradient of f

Given points (profiles) (x ,y), (x ′,y ′)New point (xε,yε)= (1−ε)(x ,y)+ε(x ′,y ′):Change of f : Df (x ,y ,x ′,y ′,ε)= f (xε,yε)− f (x ,y)

Df : piece-wise quadratic on εÏ Given x ,y ,x ′,y ′ we can find ε minimizing DfÏ Gradient of f at direction (x ′,y ′)− (x ,y):

Df (x ,y ,x ′,y ′)= limε→∞ 1εDf (x ,y ,x ′,y ′,ε)


Gradient descent on f

From point (x ,y) find feasible descent direction - towards (x ′,y ′)Explicit form of gradient of f towards (x ′,y ′):if fA(x ,y)= fB(x ,y):

Ï Df (x ,y ,x ′,y ′)=max

maxrow i∈BR(y)(Ay ′)i −x>Ay ′− (x ′)>Ay +x>Ay

maxcol j∈BR(x)(x′>B)j −x>By ′− (x ′)>By +x>Ay

− f (x ,y)

Ï linear x ′,y ′

Linear Program LP(x ,y): Find (x ′,y ′) that minimizes Df (the max term)If Df (x ,y ,x ′,y ′)< 0: steepest descent direction (x ′,y ′)If Df (x ,y ,x ′,y ′)≥ 0: stationary point - no descent direction

Given stationary point (x ,y) one of the following is a 0.3393-approx NE:(x ,y) itselfa point computed from (x ,y) and the solution of Dual(LP(x ,y))


Gradient Descent Algorithm

Start from arbitrary point (x ,y), precision parameter δ> 01 Equalize regrets: fA(x ,y)= fB(x ,y) (can be done by an LP)2 From (x ,y) find steepest descent direction (x ′,y ′) ( LP(x ,y)+Dual )

Ï If δ-Stationary point: (Df (x ,y ,x ′,y ′)≥−δ )Ï ( OR f (x ,y)≤ δ+0.339 )return 0.339+δ approximate NE

3 Perform Descent: Move to new point (1−ε)(x ,y)+ε(x ′,y ′)Ï Compute optimal ε that decreases f the most, (ε ∈ [0, δ

1+δ ])4 Repeat Ê

+ Terminates after O( 1δ2 ) iterations

in every iteration f new ≤ (1− ( δ1+δ)

2)f


Constant Rank Bimatrix

Bimatrix A,B , Rank of Game: rank(A+B)

(rank 0 are 0-sum games)[AGM+11]: rank 1 polytime[Mehta14]: rank ≥ 3 PPAD-hard[KT07]: FPTAS


Constant Rank Bimatrix - PPAD hard [Mehta14]

2D-Brouwer ∝p 2D-Linear FIXP ∝p Rank 3 Bimatrix

2D-Brouwer PPAD-hard [CD06]

2D-Linear FIXP [EY07]Ï F : [0,1]2→ [0,1]2Ï Circuit CF +,×ζ,maxÏ F : piece-wise linearÏ Find fixed point of FÏ rational fixed points


2D-Brouwer ∝ 2D-Linear FIXP

Define CF :Input: p= (p1,p2)

1 Sampling: Generate points pi = p+ ( i−1L , i−1L ), i = 1, . . . ,16(L= poly(size[Cbr ]),> 16)

2 Bit extractor: n MSBits of pi using max,×ζ,+3 On extracted bits simulate Cbr :

Ï ∨=max, ∧=min, ¬= 1−b

4 Compute average discplacement r of points pi

5 Output: p+ r+ Issue: poorly-positioned points - too close (L2) to edges of grid (∃≤2)PPAD hardnessFixed points of F (r=~0) ⇒ Panchromatic Square


2D-Brouwer ∝ 2D-Linear FIXP



Circuit CF (λ1,λ2) implemented as LPλ1,λ2

CF (λ): constraints over input λ and outputs of max gatesÏ +, ×ζ: linear constraintsÏ max: quadratic constraints

F CF DAG −→ ordering of max gates

F xi =maxi Li ,0 ≡

xi ≥ 0, xi ≥ Lixi · (xi −Li )= 0

F Li : linear λ and previous max gates

Goal: remove quadratic constraintsÏ ∃ poly-time computable cost vector c> 0 s.t. minc> ·x enforces

xi · (xi −Li )= 0Ï linear constraints: Ax≥λ1 ·u1+λ2 ·u2+b

unique optimal solution of LPλ ⇔ output of CF (λ)Ï output of CF : max gates xn−1,xn



LP(λ) ∝ LPC

LP(λ), Dual(LPλ) + complementary slackness

Ax ≥λ1u1+λ2u2+b A>y≤ cx ≥ 0 y ≥ 0

yi (Ax−λ1u1−λ2u2−b)i = 0 xi (A>y−c)i = 0

Ï OPT solution (x ,y) ⇔ (xn−1,xn) output of CF (λ)Ï Fixed points of CF : λ1 = xn−1 = e>n−1 ·x, (λ2 = . . . )



LP(λ) ∝ LPC


Ax ≥λ1u1+λ2u2+b A>y≤ cx ≥ 0 y ≥ 0

yi (Ax−u1(e>n−1 ·x)−u2(e>n ·x)−b)i = 0 xi (A>y−c)i = 0




LP(λ) ∝ LPC


Ax ≥λ1u1+λ2u2+b A>y≤ cx ≥ 0 y ≥ 0

yi ((A−e>n−1 ·u1−e>

n ·u2)x−b)i = 0 xi (A>y−c)i = 0




LP(λ) ∝ LPC

LP(λ), Dual(LPλ) + complementary slackness + fixed points: LPC(CF )

A′x ≥ b A>y≤ cx ≥ 0 y ≥ 0

yi (A′x−b)i = 0 xi (A

>y−c)i = 0Ï OPT solution (x ,y) ⇔ (xn−1,xn) output of CF (λ)Ï Fixed points of CF : λ1 = xn−1 = e>n−1 ·x, (λ2 = . . . )Ï Claim (x ,y) sol of LCP ⇔ (xn−1,xn) fixed points of CF (λ)Ï A′ =A−e>

n−1 ·u1−e>n ·u2

Ï (A−A′: rank 2)



LPC ∝Bimatrix :

R =[

A> 1−c0> 1

]C =

[ −A′> 01>+b> 1

]If (x ′,s),(y ′,t) NE of (R ,C )

Ï s ,t > 0, ⇒ scaling: ( xs , yt ) solution of LCP

Rank(R+C)=4 - Can be improved to 3 by adapting c= 1 (c> 0)


Bimatrix as QP

m×n Bimatrix game (A,B)

NE as Quadratic Program:maximize x>(A+B)y −πA−πB(x ,πB) ∈P , (y ,πA) ∈Q

Constraints Q ,P :

QA : (y ,πA)Ï y ∈∆nÏ i ∈ [m]: (A ·y)i ≤πA

PB : (x ,πB)Ï x ∈∆mÏ j ∈ [n]: (x> ·B)j ≤πB

Objective = 0 iff NE Conditions hold ∀ i : xi ((Ay)i −πA)= 0

∀ j : yj((x>B)j −πB)= 0


Rank 1 Bimatrix

Rank 1 Bimatrix is in P . [Mehta14b]

Rank 1: A+B = a ·b>Ï B =−A+a ·b>Ï Rank 1 game (A,B) : (A,a,b)

Rank-1 NE QP:

maximize x> ·:(a·b>)

(A+B) ·y −πA−πB(y ,πA) ∈QA

(x ,πB): :−(x>A)j+(x>·a)bj

(x>B)j ≤πB , x ∈∆m


Rank 1 Bimatrix

NE QP:maximize (x> ·a)(b> ·y)−πA−πB(y ,πA) ∈QA

(x ,πB): −(x>A)j + (x> ·a)bj ≤πB , x ∈∆m

Consider all games (A,u,b) for u ∈Rm

they have the same QP, except (x> ·u)


Rank 1 Bimatrix

NE QP:

maximize : (x>·u)

(x> ·a) (b> ·y)−πA−πB(y ,πA) ∈QA

(x ,πB): −(x>A)j +:(x>·u)

(x> ·a) bj ≤πB , x ∈∆m




Rank 1 Bimatrix

NE QP:maximize (x> ·u)(b> ·y)−πA−πB(y ,πA) ∈QA

(x ,πB): −(x>A)j + (x> ·u)bj ≤πB , x ∈∆m




Rank 1 Bimatrix

NE QP:

maximize :λ

(x> ·u) (b> ·y)−πA−πB(y ,πA) ∈QA

(x ,πB ,λ): −(x>A)j +:λ(x> ·u) bj ≤πB , x ∈∆m


they have the same QP, except (x> ·u)variable λ= (x> ·u)

+ Let (x ,y ,πA,πB ,λ) OPT sol of QP. For any u s.t. x> ·u=λ⇔ (x ,y) NE of (A,u,b) (with utilities πA,πB)

Ï Goal: If x> ·a=λ then NE for (A,a,b)


Rank 1 Bimatrix

λ(b> ·y) the only non-linear termReplace λ with value c ∈R ⇒ LP

Ï (λ no longer a variable)(x ,y ,πA,πB ,λ) OPT sol of QP ⇔ (x ,y ,πA,πB) OPT sol LP(λ)

Ï (previous slide): For u: λ= x> ·u, (x ,y) NE of (A,u,b)

LP(λ):

maximize λ(b> ·y) −πA−πB(y ,πA) ∈QA

(x ,πB): −(x>A)j +λbj ≤πB , x ∈∆m


Rank 1 Bimatrix

Goal: NE of G = (A,a,b)Define function fG : A →A

Ï A = [mini ai ,maxi ai ]

fG :On input c ∈A

Solve LP(c) : (x ,y ,πA,πB) (opt sol)output: x> ·a

u Claim: ∀ c ∈R the solution(s) of LP(c) have unique x (valid fG )u Fixed points of fG ⇔ (x ,y) NE of G

Ï c = x> ·a + OPT sol of LP(c) ⇔ (x ,y) NE of (A,a,b)


Rank 1 Bimatrix is in P

A fixed point of fG can be found in poly(m,n,L):L: bit length of game A,a,bBinary search

Images from [Mehta14b]


FPTAS for constant Rank Bimatrix [KT07]

Stronger approximate NE:Ï total regret function r(x ,y)= x>(A+B)y −maxi (Ay)i −maxj (x>B)jÏ strong ε-NE: r(x ,y)≤ εmax(A+B)Ï strong ε-NE ⇒ ε-NE

Again NE of Bimatrix (A,B) as QP

symmetric matrix: Q =[

0m×n (A+B)(A>+B>) 0n×m

]x>(A+B)y = 1

2z>Qz , z = (x |y)Rank(Q)=2 Rank(A+B)


FPTAS for constant Rank Bimatrix

NE as QP:minimize: π - 1

2z>Qz

π≥ (B>[∗,j] |A[i ,∗])z

z ∈∆m×∆n

+ optimal value 0 ⇔ NE

Approximate solutions for QP of fixed rank [Vav91]

QP: min f (x)= 12x>Qx +q>x s.t. Ax ≤ b

Q constant rankxmax ,xmin max/min points of f () in feasible regionε-approximate solution x∗: in poly(L,1/ε) (L: bit length of QP)f (x∗)− f (xmin)≤ ε(f (xmax)− f (xmin)

)Anthi Orfanou Computing Nash Equilibria May 05, 2015 77 / 80

FPTAS for constant Rank Bimatrix

NE as QP’:minimize: π - 1

2z>Qz

π≥ (B>[∗,j] |A[i ,∗])z , ∀i , j

z ∈∆m×∆n

π≤max(A+B)

Need bounded feasible region: π≤max(A+B)

Using [Vav91] on QP’ FPTAS:Ï f (zmin,πmin)= 0Ï f (zmax ,πmax )≤max(A+B)Ï compute ε-approximate solution z∗,π∗

z∗ = (x∗ | y∗) is ε-NEÏ (B>x∗)j + (Ay∗)i −x∗>(A+B)y∗ ≤ f (z∗,π∗)≤ εmax(A+B)


Open Problems

Anonymous Games:Ï FPTAS? (2 strategies / constant strategies)

F current best algorithm for 2 strategies Moments Search [DP09]:poly(n) · (1/ε)O(log2(1/ε))

Ï Hardness for ≤ 6 strategies? [CDO14]Bimatrix games:

Ï PTAS?F current best algorithm [LMM03]: nO( logn

ε2)

Ï Hardness for Rank 2? [Meh14]


Thank you!


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