Settling the Complexity of 2-Player Nash-Equilibrium

Transcription

1 Electronic Colloquium on Computational Complexity, Report No. 140 (2005) Settling the Complexity of 2-Player Nah-Equilibrium Xi Chen Department of Computer Science Tinghua Univerity Beijing, P.R.China Xiaotie Deng Department of Computer Science City Univerity of Hong Kong Hong Kong SAR, P.R.China Abtract We prove that finding the olution of two player Nah Equilibrium i PPAD-complete. 1 Introduction Almot ixty year ago Morgentern and von Neumann [13] initiated the tudy of game theory with their application to Economic behavior. A particularly intereting mathematical reult i their proof of the exitence of equilibrium in the 2-player zero-um game model where one player gain i the lo of the other. It exploit duality propertie of polytope, which alo lead to Dantzig linear programming method [5] for optimization problem, a well a Yao principle [16] for finding algorithmic lower bound. Nah propoed in the middle of the lat century to tudy the more general multiple peron game model, and proved that there exit a et of (mixed) trategie, now called Nah-equilibrium point, one for each player, uch that no player can benefit if it change it own trategy. While 2-player zero-um game ha a polynomial-time algorithm ince linear programming ha one, a by Khachian ellipoid algorithm [11], the exitence proof of Nah equilibrium relied on the Kakutani fixed point theorem (a generalization of the Brouwer fixed point theorem [1]) which doe not admit any polynomial-time algorithm [3, 9]. Depite much effort on the important problem, no ignificant progre ha been made on algorithm for the original Nah-equilibrium problem in the lat half century, though both hardne reult and polynomial-time algorithm have been derived for variou modified verion. An exciting breakthrough wa announced a few week ago that tated that finding Nah equilibrium i indeed hard, by Dakalaki, Goldberg and Papadimitriou [6], for game with four player or more. An ɛ approximation verion wa proven to be complete in the PPAD ( polynomial parity argument, directed verion) cla, introduced by Papadimitriou in hi eminal work about fifteen year ago [14]. The work wa improved to the 3-player cae by Chen and 1 ISSN

2 Deng [4], Dakalaki and Papadimitriou [7], independently, and with different proof. Thoe reult leave the two player Nah-equilibrium the lat opening problem in the long equel of earch for an efficient olution. Finding a Nah-equilibrium in a game between two player could be eaier for everal reaon. Firt, the zero-um verion can be olved in polynomial time by linear programming. Secondly, it admit a polynomial ize rational number olution while game between three or more player may only have olution all in irrational number. Finally, an important technique employed in the hardne proof, that color vertice of a graphical game, doe not eem poible to work down to the cae of two player. In thi work, we ettle the problem with a PPAD-complete proof for the 2-player Nahequilibrium problem. Our proof get rid of the graphical game model and derived a direct reduction from 3-Dimenional Brouwer to 2-Nah. We need to deign new gadget for variou arithmetic and logic operation [6] but they all work. The paper i arranged a follow: We review the neceary definition in Section 2. In ection 3, we ummarize the reduction from 3-Dimenional Brouwer to 3-Graphical Nah in [6], in particular the type of gadget required by the reduction. In Section 4, we preent our new gadget and prove the correctne of the reduction. We conclude in Section 5 with remark and dicuion. 2 Preliminarie 2.1 Game, Graphical Game and Nah Equilibrium A game G between r 2 player i compoed of two part. Firt, every player p [r] where [r] = {0, 1,... r } ha a et S p of pure trategie. Second, for each p [r] and S where S = S 1 S 2... S r, we have u p a the payoff or utility of player p. Here i called a pure trategy profile of the game. For any p, we ue S p to denote the et of all trategy profile of player other than p. For any j S p and S p, we ue j to denote the pure trategy profile in S, which i combined by j and. A mixed trategy x p of player p [r] i a probability ditribution over S p, that i, real number x p j 0 for any j S p and j S p x p j = 1. A profile of mixed trategie p of game G conit of r mixed trategie x p, p = 1, 2,... r. For any p [r], x p i a mixed trategy of player p. For any p [r] and S p, we define x a x = p [r], p p Now we give the definition of both accurate and approximate Nah equilibrium of a game. Intuitively, a Nah equilibrium i a profile of mixed trategie p uch that no player can gain by unilaterally chooing a different mixed trategy, where the other trategie in the profile are kept fixed. The concept of approximate Nah-equilibrium here wa firt propoed by [15]. 2 x p p

3 Definition 1. A Nah equilibrium of G i a profile of mixed trategie p = {x p } uch that for any p [r] and i, j S p. S p u p i x > S p u p j x = x p j = 0 Definition 2. An ɛ-nah equilibrium of G i a profile of mixed trategie p = {x p } uch that u p i x > u p j x + ɛ = x p j = 0 S p S p for any p [r] and i, j S p. A ueful cla of game are graphical game, which wa firt defined in [10] and then generalized by [8]. Player in a graphical game are vertice of an underlying directed graph G. A player u can affect the payoff to player v only if uv G. While general game require exponential data for their decription, graphical game have uccinct repreentation. More exactly, when the in-degree of the underlying graph G i bounded, the repreentation of the graphical game i polynomial in the number of player and trategie. 2.2 TFNP, PPAD and r-nah Let R Σ Σ be a polynomial-time computable, polynomially balanced relation (that i, there exit a polynomial p uch that for any x and y atify (x, y) R, y p( x )). The NP earch problem Q R pecified by R i thi : given input x Σ, return a y Σ uch that (x, y) R, if uch a y exit, and return the tring no otherwie. An NP earch problem i aid to be total if for every x, there exit a y uch that (x, y) R. We ue TFNP [12] to denote the cla of total NP earch problem. Definition 3. Given two problem Q R1, Q R2 TFNP, we ay that Q R1 i reducible to Q R2 if there exit a pair of polynomial-time computable function (f, g) uch that, for every input x of R 1, if y atifie (f(x), y) R 2, then (x, g(y)) R 1. One of the mot intereting ub-clae of TFNP i PPAD which i the directed verion of cla PPA. The totality of problem in PPAD i guaranteed by the following trivial fact: in a directed graph, where the in-degree and out-degree of every vertex are no more than one, if there exit a ource, there mut be another ource or ink. Many important problem were identified to be in PPAD [15], e.g. the earch verion of Brouwer fixed point theorem, Kakutani fixed point theorem, Smith theorem and Boruk-Ulam theorem. r-nah, that i, the problem of finding an approximate Nah equilibrium in a game between r player, alo belong to PPAD [15]. Definition 4. The input of problem r-nah i a pair (G, 0 k ) where G i an r-player game in normal form, and the output i a (1/2 k )-Nah equilibrium of G. 3

4 3 Review of the Reduction in [6] In thi ection, we briefly review the reduction from problem 3-Dimenional Brouwer to 3-Graphical Nah in [6]. Firt, we define the earch problem 3-Dimenional Brouwer. Definition 5 (3-Dimenional Brouwer). The input of the problem i a pair (C, 0 n ) where C i a circuit with 3n input bit and 6 output bit x +, x, y +, y, z + and z. It pecifie a Brouwer function φ of a very pecial form. For any 0 i, j, k 2 n 1, we define a cubelet K ijk in the unit cube [0, 1] 3 a K ijk = { (x, y, z) i2 n x (i + 1)2 n, j2 n y (j + 1)2 n, k2 n z (k + 1)2 n } and ue c ijk to denote it center. Brouwer function φ i a function on the et of center. For any c ijk, φ(c ijk ) = c ijk + δ where δ i one of the four increment vector δ 1, δ 2, δ 3, δ 4 below, and i pecified by the 6 output bit of C(i, j, k) a follow: cae 1 : x + = 1 and other five bit are 0 δ = δ 1 = (α, 0, 0); cae 2 : y + = 1 and other five bit are 0 δ = δ 2 = (0, α, 0); cae 3 : z + = 1 and other five bit are 0 δ = δ 3 = (0, 0, α); cae 4 : x = y = z = 1 and other three bit are 0 δ = δ 4 = ( α, α, α), where α = 2 2n i much maller than the cubelet ide. For any 0 i, j, k 2 n 1, the output bit of C(i, j, k) are guaranteed to be one of the four cae above, and C atifie the following condition on the boundary: φ(c 0jk ) = c 0jk + δ 1 φ(c i0k ) = c i0k + δ 2 φ(c ij0 ) = c ij0 + δ 3 φ(c (2 n 1)jk) = c (2 n 1)jk + δ 4 φ(c i(2 n 1)k) = c i(2 n 1)k + δ 4 φ(c ij(2 n 1)) = c ij(2 n 1) + δ 4 with conflict reolved arbitrarily. A vertex of a cubelet i aid to be panchromatic if, among the eight cubelet adjacent to it, there are four that have all four increment δ 1, δ 2, δ 3 and δ 4. The output of the problem i a panchromatic vertex of φ which i pecified by (C, 0 n ). Theorem 1 ([6]). Search problem 3-Dimenional Brouwer i PPAD-complete. In [6], a binary graphical game GG with degree 3 i contructed from (C, 0 n ). Given any 2 4n -Nah equilibrium of GG, a panchromatic vertex of (C, 0 n ) can be identified efficiently. There are two kind of vertice in GG, arithmetic vertice and interior vertice. For any arithmetic vertex v, p[v] i a meaningful real number in any Nah equilibrium p, where p[v] i the probability of v chooing trategy 1. Gadget are deigned to implement arithmetic and logic operation among arithmetic vertice, and interior vertice are ued to mediate between arithmetic vertice, o that the latter one obey the intended arithmetic relationhip. 4

5 Totally 9 gadget are neceary, i.e. G ζ, G ζ, G =, G +, G, G <, G, G and G. Every gadget contain both arithmetic vertice and interior vertice. Furthermore, arithmetic vertice in a gadget are claified a input vertice and output vertice. For example, a G + gadget contain 4 vertice v 1, v 2, v 3 and w where w i an interior vertex and other are arithmetic vertice. v 3 i the output vertex of G + and v 1, v 2 are both input vertice. A gadget only decide payoff of it interior vertex and output vertex. For example, by aying adding a G + gadget, we actually etup the payoff of v 3 and w, o that in any ɛ-nah equilibrium of GG, we have p[v 3 ] = max(p[v 1 ] + p[v 2 ], 1) ± ɛ. For any arithmetic vertex v, there exit exactly one gadget of which v i the output vertex, while it can be an input vertex of arbitrary many gadget. The main idea in the contruction of GG come from the following obervation: Let p = (x, y, z) be a point in the unit cube. If the increment of function φ at p ( interpolated from all the adjacent cubelet ) i cloe enough to zero, then there mut exit a panchromatic vertex of Brouwer function φ near point p. There are three ditinguihed vertice v x, v y and v z which encode a point p in the unit cube. After extracting the 3n bit of p[v x ], p[v y ] and p[v z ], we imulate circuit C with logic gadget G, G, G and calculate the increment vector of φ. The above computation i repeated for 41 3 point around (p[v x ], p[v y ], p[v z ]), and all the vector are averaged a the diplacement of φ at p. Finally, we add it to p[v x ], p[v y ], p[v z ], and ue G = to make ure that, in any ɛ-nah equilibrium, the diplacement of φ at p i very cloe to zero. The averaging maneuver ued in the interpolation here alo reolve the problem caued by the brittle comparator G <. Let k 0 be an integer uch that, for any input pair (C, 0 n ) of 3-Dimenional Brouwer, the number of arithmetic vertice in the graphical game GG (C, 0 n ) k0. The hardne proof of 4-Nah in [6] i baed on a combined reduction from 3-Dimenional Brouwer to 3-Graphical Nah to 4-Nah. In thi work, we developed new tructure ( which are called node here) to perform the tak of vertice in the reduction above. Gadget are deigned in the new etting, which allow u to directly reduce 3-Dimenional Brouwer to 2-Nah, and prove the latter i alo PPAD-complete. 4 Reduction from 3-Dimenional Brouwer to 2-Nah In thi ection, we give a reduction from problem 3-Dimenional Brouwer to 2-Nah and prove that the latter i alo PPAD-complete. Let (C, 0 n ) be any input of 3-Dimenional Brouwer, then a 2-player game G will be contructed. Given any ɛ-nah equilibrium of the game where ɛ = 2 (m+4n) and m i the mallet integer uch that 2 m (C, 0 n ) k 0 (contant k 0 i defined at the end of ection 3), a panchromatic vertex of φ can be identified eaily. Let call the two player P 1 and P 2. For any i {1, 2}, P i ha a et of node N i where N i = K = 2 m. Each node v contain two trategie (v, 0) and (v, 1). Thu the trategy et 5

6 S i of player P i conit of totally 2K trategie where S i = { (v, j) v N i, j {0, 1} }. To clarify the preentation, we alway ue v to denote node in N 1 and w to denote node in N 2. Given a mixed trategy profile p of G, we ue p[v] (p[w]) to denote the probability of P 1 chooing trategy (v, 1) (P 2 chooing trategy (w, 1)) and p C [v] (p C [w]) to denote the probability of P 1 chooing (v, 1) and (v, 0) (P 2 chooing (w, 1) and (w, 0)). It alo called the capacity of node v (w ) in the profile p. The idea of the contruction i decribed informally a follow: The function of node in N 1 N 2 i imilar to the vertice in ection 3. Node in N 2 are called interior node, while node in N 1 are called arithmetic node, a for any v N 1, p[v] i a meaningful real number in any ɛ-nah equilibrium p. Gadget are deigned to implement all the nine arithmetic and logic operation in the new etting. Every gadget contain exactly one interior node in N 2, which i ued to mediate between arithmetic node in the gadget, o that the latter one obey the intended arithmetic relationhip. Game G i built upon G which i a variation of the 2-player Matching Pennie with an exponentially large contant M = 2 4(m+n)+1. G ha the ame number of player and ame trategy et a G, and we ue u to denote it payoff. To get G, we add a number of gadget into G, which form a network and perform a tak imilar to the graphical game in ection 3. Every gadget contain exactly one interior node in N 2 and 3 arithmetic node in N 1. One of the arithmetic node i called the output node of the gadget, and other are called input node. Let w N 2 be the interior node and v N 1 be the output node of a gadget G. By aying adding G into G, we actually modifie the following payoff of G related to v and w : the payoff u 1 the payoff u 2 to player P 1 where the pure trategy profile contain node v to player P 2 where the pure trategy profile contain node w More exactly, contant in [0, 1] are added to thee payoff, while mot of them tay the ame. For any arithmetic node v N 1, there i exactly one gadget of which v i the output node, while it can be an input node of arbitrary many gadget. In the left part of thi ection, we firt give the payoff of game G and define a cla L of game baed on it. For any G L, player can only chooe node uniformly in a 1-Nah equilibrium. Then, we deign all the neceary gadget in the new etting. Finally, we build game G by inerting gadget into G, and prove the correctne of the reduction. 4.1 Payoff of Game G Payoff u of game G are decribed in figure 1 with contant M = 2 4(m+n)+1 = 2K 4 2 4n. Definition 6. A 2-player game G (with ame trategy et a G ) belong to L if it payoff u atify that u i [u i u i + 1] for any profile S 1 S 2 and i {1, 2}. 6

7 Payoff u of Game G 1: pick an arbitrary one-to-one correpondence C from N 1 to N 2 2: for any pure trategy profile = ((v, i 1 ), (w, i 2 )), v N 1, w N 2, i 1, i 2 {0, 1} do 3: if C(v) = w then 4: et u 1 = M and u 2 5: ele 6: et u 1 = 0 = M Figure 1: Payoff u of Game G The following property of game in L i eay to prove. Lemma 1. Let p be any 1-Nah equilibrium of game G L, then for any node v N 1, w N 2, the capacitie of v and w in profile p atify 1 K ɛ < p C[v], p C [w] < ɛ. ( recall that ɛ = K 2 m+4n = 1 K2 4n ) 4.2 Deign of Arithmetic and Logic Gadget In thi part, we deign all the nine neceary gadget, i.e. G ζ, G ζ, G =, G +, G, G <, G, G and G in the new etting. Function of them are imilar to thoe in [6]. One difference hould be noticed here i the repreentation of bit. Let v be any node in N 1, we ay v tore 1 if p[v] = p C [v] and v tore 0 if p[v] = 0. We only prove the property of gadget G +, while other can be verified imilarly. Definition 7. By x = y ± ɛ where ɛ > 0, we mean that y ɛ x y + ɛ. Propoition 1 (Gadget G + ). Let G (with payoff u ) be a 2-player game in L and node v 1, v 2, v 3 N 1, w N 2. Let pure trategy profile 1 = ((v 1, 1), (w, 1)), = ((v 2, 1), (w, 1)), = ((v 3, 1), (w, 0)), 4 = ((v 3, 1), (w, 1)) and 5 = ((v 3, 0), (w, 0)). If game G atifie 1 + 1, u and for any other which contain (w, 1), u 2 2). u and for any other which contain (w, 0), u and for any other which contain (v 3, 1), u and for any other which contain (v 3, 0), u 1 then in any ɛ-nah equilibrium p of G, we have p[v 3 ] = min(p[v 1 ] + p[v 2 ], p C [v 3 ]) ± ɛ. Proof. Propertie 1) 4) how that, in any mixed trategy profile p of game G, we have payoff to P 2 if it chooe (w, 1) payoff to P 2 if it chooe (w, 0) = p[v 1 ] + p[v 2 ] p[v 3 ] payoff to P 1 if it chooe (v 3, 1) payoff to P 1 if it chooe (v 3, 0) = p[w] (p C [w] p[w]) 7

8 If p[v 3 ] (p[v 1 ] + p[v 2 ]) > ɛ, then the firt equation how that p[w] = 0 and the econd one how p[v 3 ] = 0 which contradict with our aumption that p[v 3 ] > p[v 1 ] + p[v 2 ] + ɛ > 0. If p[v 3 ] (p[v 1 ] + p[v 2 ]) < ɛ, then the firt equation how p[w] = p C [w] and the econd one how that p[v 3 ] = p C [v 3 ]. A p C [v 3 ] = p[v 3 ] < p[v 1 ] + p[v 2 ], we have p[v 3 ] = p C [v 3 ] > p C [v 3 ] ɛ = min(p[v 1 ] + p[v 2 ], p C [v 3 ]) ɛ, and the propoition i proven. Propoition 2 (Gadget G ζ where ζ 1/K ɛ). Let G (with payoff u ) be a game in L and node v N 1, w N 2. Let pure trategy profile 1 = ((v, 1), (w, 1)), = ((v, 1), (w, 0)) and = ((v, 0), (w, 1)). If the following condition are atified and for any other which contain (w, 1), u 2 2). for any which contain (w, 0), u 2 + ζ; + 1 and for any other which contain (v, 1), u and for any other which contain (v, 0), u 1 then in any ɛ-nah equilibrium p of game G, we have p[v] = ζ ± ɛ. Propoition 3 (Gadget G ζ where 0 ζ 1/2). Let G (with payoff u ) be a 2-player game in L and node v 1, v 2 N 1, w N 2. Let pure trategy profile 1 = ((v 1, 1), (w, 1)), = ((v 2, 1), (w, 0)), = ((v 2, 1), (w, 1)) and 4 = ((v 2, 0), (w, 0)). If u atifie 1 + ζ and for any other which contain (w, 1), u 2 2). u and for any other which contain (w, 0), u and for any other which contain (v 2, 1), u and for any other which contain (v 2, 0), u 1 then in any ɛ-nah equilibrium p of game G, we have p[v 2 ] = ζp[v 1 ] ± ɛ. Propoition 4 (Gadget G = ). Gadget G = i imilar a G ζ. We jut et the contant ζ to be 1, then in any ɛ-nah equilibrium p of game G, we have p[v 2 ] = min(p[v 1 ], p C [v 2 ]) ± ɛ. Propoition 5 (Gadget G ). Let G (with payoff u ) be a 2-player game in L and node v 1, v 2, v 3 N 1, w N 2. Let pure trategy profile 1 = ((v 1, 1), (w, 1)), = ((v 2, 1), (w, 0)), = ((v 3, 1), (w, 0)), 4 = ((v 3, 1), (w, 1)) and 5 = ((v 3, 0), (w, 0)). If game G atifie and for any other which contain (w, 1), u 2 2). u 2 + 1, u and for any other which contain (w, 0), u and for any other which contain (v 3, 1), u and for any other which contain (v 3, 0), u 1 8

9 then in any ɛ-nah equilibrium p of game G, we have min(p[v 1 ] p[v 2 ], p C [v 3 ]) ɛ p[v 3 ] max(p[v 1 ] p[v 2 ], 0) + ɛ. Propoition 6 (Gadget G < ). Let G (with payoff u ) be a 2-player game in L and node v 1, v 2, v 3 N 1, w N 2. Let pure trategy profile 1 = ((v 1, 1), (w, 1)), = ((v 2, 1), (w, 0)), = ((v 3, 1), (w, 0)) and 4 = ((v 3, 0), (w, 1)). If game G atifie and for any other which contain (w, 1), u 2 2). u and for any other which contain (w, 0), u and for any other which contain (v 3, 1), u and for any other which contain (v 3, 0), u 1 then in any ɛ-nah equilibrium p of game G, we have p[v 3 ] = p C [v 3 ] if p[v 1 ] < p[v 2 ] ɛ and p[v 3 ] = 0 if p[v 1 ] > p[v 2 ] + ɛ. Propoition 7 (Gadget G ). Let G (with payoff u ) be a 2-player game in L and node v 1, v 2, v 3 N 1, w N 2. Let pure trategy profile 1 = ((v 1, 1), (w, 1)), = ((v 2, 1), (w, 1)), = ((v 3, 1), (w, 1)) and 4 = ((v 3, 0), (w, 0)). If payoff u of game G atify 1 + 1, u and for any other which contain (w, 1), u 2 2). for any which contain (w, 0), u 2 + 1/(2K); + 1 and for any other which contain (v 3, 1), u and for any other which contain (v 3, 0), u 1 then in any ɛ-nah equilibrium p, we have p[v 3 ] = p C [v 3 ] if p[v 1 ] = p C [v 1 ] or p[v 2 ] = p C [v 2 ] and p[v 3 ] = 0 if p[v 1 ] = p[v 2 ] = 0. Propoition 8 (Gadget G ). Gadget G i imilar a G. We only change the contant in 2) of Propoition 7 from 1/(2K) to 3/(2K). Propoition 9 (Gadget G ). Let G (with payoff u ) be a 2-player game in L and node v 1, v 2 N 1, w N 2. Let pure trategy profile 1 = ((v 1, 1), (w, 1)), = ((v 2, 0), (w, 0)), = ((v 2, 1), (w, 0)) and 4 = ((v 2, 0), (w, 1)). If the payoff of game G atifie and for any other which contain (w, 1), u 2 2). u and for any other which contain (w, 0), u and for any other which contain (v 2, 1), u and for any other which contain (v 2, 0), u 1 then in any ɛ-nah equilibrium p, p[v 2 ] = 0 if p[v 1 ] = p C [v 1 ] and p[v 2 ] = p C [v 2 ] if p[v 1 ] = 0. 9

10 4.3 Contruction of Game G Now we are ready to ue the gadget deigned o far to build the game G. We ue G ζ (v, w) to denote the inertion of a G ζ gadget into game G with v a it output node and w a it interior node. For gadget with one input node (G ζ, G and G = ), we ue G(v 1, v 2, w) to denote the inertion of uch a gadget into game G with v 1, v 2, w a it input node, output node and interior node repectively. For gadget with two input node, we ue G(v 1, v 2, v 3, w) to denote the inertion of uch a gadget into game G with v 1 and v 2 a it firt and econd input node repectively, v 3 a it output node and w a it interior node. The tructure of the gadget network in G i imilar to the one in [6]. There are 3 ditinguihed node v x, v y, v z in N 1, and real number p[v x ], p[v y ], p[v z ] encode a point t = (x, y, z) in the unit cube [0, 1] 3 where x = Kp[v x ], y = Kp[v y ], z = Kp[v z ]. (Strictly peaking, it may happen that Kp[v x ] > 1 according to Lemma 1, but we will prove that thi i impoible in any ɛ-nah equilibrium later.). Let K ijk be the cubelet that contain point t. Starting from node v x,v y and v z, we extract the 3n bit which encode integer i, j, k (from the (m + 1)th bit to the (m + n)th bit of p[v x ], p[v y ] and p[v z ]), and ue logic gadget to imulate C. But only getting the increment of φ at c ijk i not enough, we need to repeat the above computation for 41 3 point of the form (x + p α, y + q α, z + r α) for 20 p, q, r 20, and finally calculate the average of all thee increment. After adding the diplacement to p[v x ], p[v y ] and p[v z ], we inert gadget G = to make ure that, in any ɛ-nah equilibrium, the average increment at t i very cloe to zero, which guarantee the exitence of a panchromatic vertex near t. The averaging maneuver ued in the interpolation here reolve the problem caued by the brittle comparator G < at the ame time. The contruction of game G i divided into 5 part: Part 1. Starting from the three ditinguihed node v x, v y, v z N 1, for any 20 i 20, there are three node v xi, v yi and v zi in N 1. By adding gadget G ζ, G and G +, we make ure that in any ɛ-nah equilibrium p of G, p[v xi ] = min(p[v x ] + iα, p C [v xi ]) ± 4ɛ if i 0 and p[v xi ] = max(p[v x ] + iα, 0) ± 4ɛ if i < 0 where α = α2 m. Similar reult alo tand for node v yi and v zi. Part 2. For any 20 p 20, we extract 3n bit (from the (m + 1)th to the (m + n)th) of p[v xp ], p[v yp ] and p[v zp ] and tore them in node v i x p, v i y p and v i z p where 1 i n. Figure 2 how how to extract the thee bit from v xp. Although we hope p[v i x p ] = 0 if the (m + i)th bit of p[v xp ] i 0 and p[v i x p ] = p C [v i x p ] if it i 1, thi may not be true a the comparator G < i brittle. The following lemma i eay to check. Similar reult alo tand for v i y p and v i z p. Lemma 2. If p[v xp ] 1/K 61α, then p[vx i p ] = p C [vx i p ] for any 1 i n. If p[v xp ] 61α, then p[vx i p ] = 0 for any 1 i n. Otherwie, if p[v xp ] atifie 2 n+m p[v xp ] p[v xp ] > n 2 2 n+m ɛ, 10

11 Implementation of Part 2 1: pick unued node v 1, v 2... v n+1 N 1 and w N 2 2: G = (v xp, v 1, w), 3: for any 1 i n do 3: pick unued node v 1, v 2, v 3 N 1 and w 1, w 2, w 3, w 4 N 2 4: G 2 (m+i)(v 1, w 1 ), G < (v 1, v i, v 2, w 2 ), G 2 (m+i)(v 2, v 3, w 3 ), G (v i, v 3, v i+1, w 4 ) Figure 2: Implementation of Part 2 then p[v i x p ] = 0 if the (m + i)th bit of real number p[v xp ] i 0 and p[v i x p ] = p C [v i x p ] if it i 1, for any integer 1 i n. Part 3. For any 20 p, q, r 20, we recognize the 3n node vx i p vy i q vz i r where 1 i n a the input bit of circuit C and ue logic gadget G, G, G to imulate it. The output (6 bit) are tored in 6 node, x + pqr, x pqr, y pqr, + ypqr, z pqr + and zpqr in N 1. Part 4. Pick 6 unued node x +, x, y +, y, z + and z in N 1. By uing gadget G ζ and G +, we make ure that in any ɛ-nah equilibrium p of game G, ( p[ x + ] = p,q,r α ) ( 41 3 p[ x+ pqr ] ± ɛ p[ x ] = p,q,r α ) 41 3 p[ x pqr ] ± ɛ where α = α2 m. Similar reult alo tand for node y +, y, z + and z. Part 5. Pick unued node v 1, v 2, v 3, v x, v y, v z N 1 and w 1, w 2, w 3, w 4, w 5, w 6 N 2. Add the following nine gadget into game G : G + (v x, x +, v 1, w 1 ) G (v 1, x, v x, w 2) G + (v y, y +, v 2, w 3 ) G (v 2, y, v y, w 4) G + (v z, z +, v 3, w 5 ) G (v 3, z, v z, w 6) G = (v x, v x ) G = (v y, v y ) G = (v z, v z ) 4.4 Correctne of the Reduction Obviouly, game G belong to cla L and all the gadget inerted work well in it. The ize of game G i polynomial of (C, 0 n ), a both the number of trategie and the number of bit required to repreent a payoff u i are polynomial of (C, 0 n ), and game G can be computed from (C, 0 n ) in polynomial time. Furthermore, the following theorem how that, given any ɛ-nah equilibrium of game G where ɛ = 2 (m+4n), a panchromatic vertex of (C, 0 n ) can be identified very efficiently. 11

12 Theorem 2. Let p be any ɛ-nah equilibrium of the game G contructed above. x = Kp[v x ], y = Kp[v y ] and z = Kp[v z ] where K = 2 m. Let p, q, r be three integer atifying (p 1)2 n < x 30α < x + 30α < (p + 1)2 n ; (q 1)2 n < y 30α < y + 30α < (q + 1)2 n ; (r 1)2 n < z 30α < z + 30α < (r + 1)2 n, then vertex (p2 n, q2 n, r2 n ) mut be a panchromatic vertex of (C, 0 n ). Similarly a the proof in [6], we need the following property of the four increment. Lemma 3. Suppoe that for nonnegative integer k 1... k 4, all three coordinate of 4 i=1 k iδ i are maller in abolute value than αk/5 where k = 4 i=1 k i. Then all four k i are poitive. Proof of Theorem 2. Firt, we prove that t = (x, y, z) cannot be cloe to the boundary of the unit cube. If p[v x ] 40α, then Lemma 2 how for any 20 i, j, k 20, we have p[ + ijk ] = p C[ x + ijk ]. Thu p[ x+ ] i very cloe to α, while p[ x ] i cloe to zero. A α i much larger than ɛ, we get a contradiction in the following gadget: G = (v x, v x ). Similarly, we can prove that p[v x ] < 1/K 40α, which can be eaily generalized to p[v y ] and p[v z ]. Now we ee the exitence of integer p, q, r which atify the three condition above. Let T be the et of eight center around (p2 n, q2 n, r2 n ), V = { (i, j, k), 20 i, j, k 20 } and V 1 be the ubet of V uch that, triple (i, j, k) V 1 if p[vxi ] p2 (n+m) n 2 ɛ or p[vyj ] q2 (n+m) n 2 ɛ or p[vzk ] r2 (n+m) n 2 ɛ. A α i much larger than ɛ, we have V For any triple (i, j, k) V V 1, Lemma 2 how that all the 3n bit of p[v xi ], p[v yj ] and p[v zk ] are extracted uccefully, and x + ijk etc. value hould imply an increment which i ame a one of thoe at center in T. Let k t, where 1 t 4, be the number of triple in V V 1 whoe x + ijk etc. value imply the increment vector δ t, then all four k i mut be poitive according to Lemma 3 (otherwie, we could find a contradiction in one of the three G = gadget in Part 5), which how that vertex (p2 n, q2 n, r2 n ) i a panchromatic vertex of (C, 2 n ), and the theorem i proven. 5 Concluding Remark Even though many thought the problem of finding a Nah-equilibrium i hard in general, and ha been proven o for three or more player recently, it i not clear whether the 2-player cae can be hown in the ame cla of PPAD-complete problem. Our work ettle thi iue and a long tanding open problem that ha attracted Mathematician, Economit, Operation Reearcher, and mot recently Computer Scientit. The reult how the richne of the PPAD-complete cla introduced by Papadimitriou fifteen year ago [14]. The new proof technique which made incluion of r-nah into thi cla poible, tarted in Goldberg and Papadimitriou [8], have hown a variety of tructure, a exhibited in the hardne proof of problem 4-Nah, 2D-SPERNER [2], 3-Nah, and finally 2-Nah, may find their ue in other related problem and complexity clae. 12

13 Reference [1] L.E.J. Brouwer. 115, Über Abbildung von Mannigfaltigkeiten. Mathematiche Annalen, 71:97 [2] X. Chen and X. Deng. 2D-SPERNER i PPAD-complete. ubmitted to STOC [3] X. Chen and X. Deng. On Algorithm for Dicrete and Approximate Brouwer Fixed Point. In STOC 2005, page [4] X. Chen and X. Deng. 3-Nah i PPAD-complete. ECCC, TR05-134, [5] G.B. Danzig. Linear Programming and Extenion. Princeton Univerity Pre, [6] C. Dakalaki, P.W. Goldberg, and C.H. Papadimitriou. The Complexity of Computing a Nah Equilibrium. ECCC, TR05-115, [7] C. Dakalaki and C.H. Papadimitriou. Three-player game are hard. ECCC, TR [8] P.W. Goldberg and C.H. Papadimitriou. Reducibility Among Equilibrium Problem. ECCC, TR05-90, [9] M.D. Hirch, C.H. Papadimitriou, and S. Vavai. Exponential lower bound for finding Brouwer fixed point. J.Complexity, 5: , [10] M. Kearn, M. Littman, and S. Singh. Graphical Model for Game Theory. In Proceeding of UAI, [11] L.G. Khachian. A Polynomial Algorithm in Linear Programming. Dokl. Akad. Nauk, SSSR 244: , Englih tranlation in Soviet Math. Dokl. 20, , [12] N. Megiddo and C. Papadimitriou. On total function, exitence theorem and computational complexity. Theoret. Comput. Sci., 81: , [13] O. Morgentern and J. von Neumann. The Theory of Game and Economic Behavior. Princeton Univerity Pre, [14] C.H. Papadimitriou. On inefficient proof of exitence and complexity clae. In Proceeding of the 4th Czecholovakian Sympoium on Combinatoric, [15] C.H. Papadimitriou. On the complexity of the parity argument and other inefficient proof of exitence. Journal of Computer and Sytem Science, page , [16] A.C-C. Yao. Probabilitic computation: Toward a unified meaure of complexity. In Proceeding of FOCS 1997, page ECCC ISSN ftp://ftp.eccc.uni-trier.de/pub/eccc ftpmail@ftp.eccc.uni-trier.de, ubject help eccc