Rank-1 Bimatrix Games: A Homeomorphism and a Polynomial Time Algorithm
†
Bharat Adsul, Jugal Garg, Ruta Mehta and Milind Sohoni
Indian Institute of Technology, Bombay
H
• Question: Is EΓ homeomorphic to Γ?
Abstract
Analysis and computation of Nash equilibrium for finite two player (bimatrix) games
is a central problem in non-cooperative game theory. Homotopy of Nash equilibrium
turns out to be useful to address these problems. Given a rank-1 bimatrix game
(A, B), i.e., where rank(A + B) = 1, we construct a suitable linear subspace
of the rank-1 game space and show that this subspace is homeomorphic to its Nash
equilibrium correspondence. Using this homeomorphism, we give the first polynomial
time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game.
This settles an open question posed by Kannan and Theobald (SODA’07) . In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and
show that a similar technique may also be applied for finding a Nash equilibrium of any
bimatrix game. This technique also proves the existence, oddness and the index theorem
of Nash equilibria in a bimatrix game. Further, we extend the rank-1 homeomorphism
result to a fixed rank game space, and give a fixed point formulation on [0, 1]k for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise
linear and considerably simpler than the classical constructions.
N
Q0 = {(x, λ, π2) | xi ≥ 0, ∀i ∈ S1; xT C j + βj λ − π2 ≤ 0, ∀j ∈ S2; Σi∈S1 xi = 1}
H−
• For G(α), P (α) = P and Q(α) = Projection of {(x, λ, π2) ∈ Q0 | λ − Σi∈S1 xiαi = 0}
on (x, π2)-space.
λ↑
• The set of fully-labeled points is N = {(v, w) ∈ P × Q | (v, w) is fully-labeled} .
0
• The set N covers all the points in EΓ and vice-versa.
H
N
H
P × Q′
−
Results
• N is a single path parametrized by λ (i.e., does not contain cycles).
• Let g : N → R be such that g((y, π1), (x, λ, π2)) = β T · y + λ.
Function g strictly increases on the directed path N .
• Let f : EΓ → Γ be s.t. f (α, x, y) = (β T · y + αT · x, α2 − α1, . . . , αm − α1)T
Function f establishes homeomorphism between EΓ and Γ.
Algorithms for Rank-1 Games
• Bimatrix game (A, B) with strategy sets |S1| = m and |S2| = n.
• ∆i is the probability distribution over Si.
P × Q′
• Nash equilibrium: (x, y) ∈ ∆1 × ∆2 is a Nash equilibrium (NE) of (A, B) iff
no player gains by unilateral deviation.
• Best response polytopes:
• For a v ∈ P , L(v) denotes the set of numbers of tight
• A pair (v, w) ∈ P × Q is fully-labeled if L(v) ∪
• (x, y) is a NE of (A, B) iff ((y, π1), (x, π2)) ∈ P × Q is fully-labeled
for some π1 and π2.
Known Results
• Nash equilibrium computation in Bimatrix games
• One of the most important open question on the boundary of P. (P, STOC’01)
1
• Is PPAD-complete (CD, FOCS’06), even to nθ(1)
-approximate.
• The Lemke-Howson (LH) algorithm may take exponential steps. (SS, Econometrica’06)
• Rank of a game (A, B) is defined as rank(A + B). (KT, SODA’07)
• Polynomial time algorithm for finding an -approximate Nash equilibrium (KT).
• Rank-0 (zero-sum) games ≡ Linear Programming. (Dantzig, 1963)
• Rank-1
games - Smallest extension of rank-0 games. No polynomial
time algorithm was known.
• Shapley’s index theory (1974) assigns a sign to a Nash equilibrium.
• The
two endpoints of LH paths have opposite indices. This provides a direction and in
turn parity argument that puts Nash equilibrium in PPAD. (P, JCSS’94)
• Bimatrix game space R
is homeomorphic to it’s Nash equilibrium correspondence.
(KM, Econometrica’86)
• Useful to understand index, degree and stability of Nash equilibria. (GW, GEB’97)
• It validates the homotopy based Nash equilibrium computation methods. (HP, ET’10)
2mn
• No
NE
such result is known for special subspaces.
Subspace and it’s Nash Equilibrium Correspondence
• A ∈ Rmn, C ∈ Rmn, β ∈ Rn and G(α) = (A, C + α · β T ) for an α ∈ Rm.
• Γ = {G(α) | α ∈ Rm} is an m-dimensional affine subspace of R2mn.
• EΓ = {(α, x, y) | (x, y) is a NE of G(α)} is the NE correspondence of Γ.
• Let (A, B) = G(γ) be a rank-1 game, where B = −A + γ · β T .
• Let the hyper-plane H be λ − Σi∈S1 γixi = 0.
• u ∈ N ∩ H iff u correspond to a NE of (A, B).
−
+
• γmin = min
γ
,
γ
=
max
γ
.
OP
T
(γ
)
∈
H
,
OP
T
(γ
)
∈
H
.
i
max
i
min
max
i∈S
i∈S
Results
equations at v.
L(w) = {1, . . . , m + n}.
NE
H+
Bimatrix games and Nash equilibrium
P = {(y, π1) | Aiy − π1 ≤ 0, ∀i ∈ S1;
yj ≥ 0,
∀j ∈ S2; Σj∈S2 yj = 1}
Q = {(x, π2) |
xi ≥ 0,
∀i ∈ S1; xT B j − π2 ≤ 0, ∀j ∈ S2; Σi∈S1 xi = 1}
H+
1
1
• Let F : R → R be such that F (a) = H(N (a)).
• EΓ is connected iff N is a single connected component.
• The set of fully-labeled points N admits the following decomposition into
BinSearch Algorithm: Finding a zero of F .
F
∪ki=1
mutually disjoint connected components. N = P
Ci, k ≥ 0, where
P and Cis form a path and cycles on 1-skeleton of P × Q0.
−−→0 ←−−0
0
• Let E be the set of edges of N , and E = {u, u , u, u | u, u0 ∈ N } be the
set of directed edges. There exists a (efficiently computable) function →: E → E 0
such that it maps a cycle of N to a directed cycle and P to the directed path.
• Consider a game α ∈ Γ, and hyper-plane H : λ − Σi∈S1 αixi = 0. Every point
in N ∩ H corresponds to a NE of the game G(α) and vice-versa.
• The path P of N covers at least one NE of G(α), i.e., P ∩ H 6= ∅.
• This
shows the existence of a Nash equilibrium for a bimatrix game. It is also easy to deduce that
the number of Nash equilibria of a non-degenerate bimatrix game is odd from the fact that N
contains a set of cycles and a path, simply because a cycle must intersect H an even number of
times, and the path an odd number of times.
−−→0
0
0
• Let u, u ∈ N intersect H at NE (x, y) of G(α), and → (u, u ) = u, u .
γmin
γmax
• The index of the Nash equilibrium obtained by BinSearch is +1.
• BinSearch terminates in time poly(L, m, n), where L is the bit length of i/p.
Enumeration Algorithm: Traverses the path between N (γmin) and N (γmax).
• Appropriate call to Enumeration outputs at least one NE of any bimatrix game.
If u ∈ H − and u0 ∈ H + then index of (x, y) is +1, otherwise it is −1.
Rank-k Space and Homeomorphism
• In a component, the index of the Nash equilibria alternates, and the first and the last Nash equilibria
on the path P have index +1. This proves that the number of Nash equilibria with index +1 is
one more than the number of Nash equilibria with index −1.
Rank-1 Game Space and Homeomorphism
• Let C = −A in Γ, i.e., Γ consists of rank-1 games only.
• LP (δ) : max δ(β T · y) − π1 − π2
(y, π1) ∈ P ; (x, λ, π2) ∈ Q0; λ = δ
• OP T (δ) : Set of optimal points of LP (δ).
• N (a) = {((y, π1), (x, λ, π2)) ∈ N | λ = a}.
• ∀a ∈ R, OP T (a) = N (a) and OP T (a) 6= ∅.
λ
lT
· β ) ∈ R2mn | ∀l ≤ k, αl ∈ Rm} is a kmdimensional affine subspace of the bimatrix game space (R2mn), where {β l}kl=1 are
linearly independent.
T
1
k
km
k
l
l
• Let α = (α , . . . , α ) ∈ R , and G(α) be (A, −A + Σl=1α · β ).
• EΓk = {(α, x, y) | (x, y) is a NE of G(α) ∈ Γk }.
k
• Γ = {(A, −A +
Σkl=1αl
Results
• The NE correspondence EΓk is homeomorphic to the game space Γk .
• Finding a NE of a game G(γ) ∈ Γk reduces to finding a fixed point of a
polynomially computable piece-wise linear function f : [0, 1]k → [0, 1]k.
†arXiv:1010.3083 (STOC’11)