Discrete fixed-point theorem

In discrete mathematics, a discrete fixed-point is a fixed-point for functions defined on finite sets, typically subsets of the integer grid ${\displaystyle \mathbb {Z} ^{n}}$.

Discrete fixed-point theorems were developed by Iimura,^[1] Murota and Tamura,^[2] Chen and Deng^[3] and others. Yang^[4] provides a survey.

Basic concepts

Continuous fixed-point theorems often require a continuous function. Since continuity is not meaningful for functions on discrete sets, it is replaced by conditions such as a direction-preserving function. Such conditions imply that the function does not change too drastically when moving between neighboring points of the integer grid. There are various direction-preservation conditions, depending on whether neighboring points are considered points of a hypercube (HGDP), of a simplex (SGDP) etc. See the page on direction-preserving function for definitions.

Continuous fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set.

A fixed point of a discrete function f is defined exactly as for continuous functions: it is a point x for which f(x)=x.

For functions on discrete sets

We focus on functions ${\displaystyle f:X\to \mathbb {R} ^{n}}$, where the domain X is a nonempty subset of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$. ch(X) denotes the convex hull of X.

Iimura-Murota-Tamura theorem:^[2] If X is a finite integrally-convex subset of ${\displaystyle \mathbb {Z} ^{n}}$, and ${\displaystyle f:X\to {\text{ch}}(X)}$ is a hypercubic direction-preserving (HDP) function, then f has a fixed-point.

Chen-Deng theorem:^[3] If X is a finite subset of ${\displaystyle \mathbb {R} ^{n}}$, and ${\displaystyle f:X\to {\text{ch}}(X)}$ is simplicially direction-preserving (SDP), then f has a fixed-point.

Yang's theorems:^[4]

• [3.6] If X is a finite integrally-convex subset of ${\displaystyle \mathbb {Z} ^{n}}$, ${\displaystyle f:X\to \mathbb {R} ^{n}}$ is simplicially gross direction preserving (SGDP), and for all x in X there exists some g(x)>0 such that ${\displaystyle x+g(x)f(x)\in {\text{ch}}(X)}$, then f has a zero point.
• [3.7] If X is a finite hypercubic subset of ${\displaystyle \mathbb {Z} ^{n}}$, with minimum point a and maximum point b, ${\displaystyle f:X\to \mathbb {R} ^{n}}$ is SGDP, and for any x in X: ${\displaystyle f_{i}(x_{1},\ldots ,x_{i-1},a_{i},x_{i+1},\ldots ,x_{n})\leq 0}$ and ${\displaystyle f_{i}(x_{1},\ldots ,x_{i-1},b_{i},x_{i+1},\ldots ,x_{n})\geq 0}$, then f has a zero point. This is a discrete analogue of the Poincaré–Miranda theorem. It is a consequence of the previous theorem.
• [3.8] If X is a finite integrally-convex subset of ${\displaystyle \mathbb {Z} ^{n}}$, and ${\displaystyle f:X\to {\text{ch}}(X)}$ is such that ${\displaystyle f(x)-x}$ is SGDP, then f has a fixed-point.^[5] This is a discrete analogue of the Brouwer fixed-point theorem.
• [3.9] If X = ${\displaystyle \mathbb {Z} ^{n}}$, ${\displaystyle f:X\to \mathbb {R} ^{n}}$ is bounded and ${\displaystyle f(x)-x}$ is SGDP, then f has a fixed-point (this follows easily from the previous theorem by taking X to be a subset of ${\displaystyle \mathbb {Z} ^{n}}$ that bounds f).
• [3.10] If X is a finite integrally-convex subset of ${\displaystyle \mathbb {Z} ^{n}}$, ${\displaystyle F:X\to 2^{X}}$ a point-to-set mapping, and for all x in X: ${\displaystyle F(x)={\text{ch}}(F(x))\cap \mathbb {Z} ^{n}}$ , and there is a function f such that ${\displaystyle f(x)\in {\text{ch}}(F(x))}$ and ${\displaystyle f(x)-x}$ is SGDP, then there is a point y in X such that ${\displaystyle y\in F(y)}$. This is a discrete analogue of the Kakutani fixed-point theorem, and the function f is an analogue of a continuous selection function.
• [3.12] Suppose X is a finite integrally-convex subset of ${\displaystyle \mathbb {Z} ^{n}}$, and it is also symmetric in the sense that x is in X iff -x is in X. If ${\displaystyle f:X\to \mathbb {R} ^{n}}$ is SGDP w.r.t. a weakly-symmetric triangulation of ch(X) (in the sense that if s is a simplex on the boundary of the triangulation iff -s is), and ${\displaystyle f(x)\cdot f(-y)\leq 0}$ for every pair of simplicially-connected points x, y in the boundary of ch(X), then f has a zero point.
• See the survey^[4] for more theorems.

For discontinuous functions on continuous sets

Discrete fixed-point theorems are closely related to fixed-point theorems on discontinuous functions. These, too, use the direction-preservation condition instead of continuity.

Herings-Laan-Talman-Yang fixed-point theorem:^[6]

Let X be a non-empty convex compact subset of ${\displaystyle \mathbb {R} ^{n}}$. Let f: X → X be a locally gross direction preserving (LGDP) function: at any point x that is not a fixed point of f, the direction of ${\displaystyle f(x)-x}$ is grossly preserved in some neighborhood of x, in the sense that for any two points y, z in this neighborhood, its inner product is non-negative, i.e.: ${\displaystyle (f(y)-y)\cdot (f(z)-z)\geq 0}$. Then f has a fixed point in X.

The theorem is originally stated for polytopes, but Philippe Bich extends it to convex compact sets.^{[7]: Thm.3.7}Note that every continuous function is LGDP, but an LGDP function may be discontinuous. An LGDP function may even be neither upper nor lower semi-continuous. Moreover, there is a constructive algorithm for approximating this fixed point.

Applications

Discrete fixed-point theorems have been used to prove the existence of a Nash equilibrium in a discrete game, and the existence of a Walrasian equilibrium in a discrete market.^[8]