Knaster–Tarski theorem

In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following:
It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed-point theorem. Some time earlier, Knaster and Tarski established the result for the special case where L is the lattice of subsets of a set, the power set lattice.
The theorem has important applications in formal semantics of programming languages and abstract interpretation, as well as in game theory. It is the logical bedrock for defining the meaning of recursive or repetitive processes in computer science and for proving the existence of equilibrium states in fields like game theory. It essentially proves that when a system follows simple, non-decreasing rules, a stable, self-consistent outcome is always guaranteed to exist.
A kind of converse of this theorem was proved by Anne C. Davis: If every order-preserving function f : L → L on a lattice L has a fixed point, then L is a complete lattice.

Consequences: least and greatest fixed points

Since complete lattices cannot be empty, the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point. In many practical cases, this is the most important implication of the theorem.
The least fixpoint of f is the least element x such that f = x, or, equivalently, such that f ≤ x; the dual holds for the greatest fixpoint, the greatest element x such that f = x.
If f = lim f for all ascending sequences x_n, then the least fixpoint of f is lim fⁿ where 0 is the least element of L, thus giving a more "constructive" version of the theorem. More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of f^α, taking α over the ordinals, where f^α is defined by transfinite induction: f^α+1 = f and f^γ for a limit ordinal γ is the least upper bound of the f^β for all β ordinals less than γ. The dual theorem holds for the greatest fixpoint.
For example, in theoretical computer science, least fixed points of monotonic functions are used to define program semantics, see for an example. Often a more specialized version of the theorem is used, where L is assumed to be the lattice of all subsets of a certain set ordered by subset inclusion. This reflects the fact that in many applications only such lattices are considered. One then usually is looking for the smallest set that has the property of being a fixed point of the function f. Abstract interpretation makes ample use of the Knaster–Tarski theorem and the formulas giving the least and greatest fixpoints.
The Knaster–Tarski theorem can be used to give a simple proof of the Cantor–Bernstein–Schroeder theorem and it is also used in establishing the Banach–Tarski paradox.

Weaker versions of the theorem

Weaker versions of the Knaster–Tarski theorem can be formulated for ordered sets, but involve more complicated assumptions. For example:
This can be applied to obtain various theorems on invariant sets, e.g. Ok's theorem:
In particular, using the Knaster–Tarski principle one can develop the theory of global attractors for noncontractive discontinuous iterated function systems. For weakly contractive iterated function systems the Kantorovich theorem suffices.
Other applications of fixed-point principles for ordered sets come from the theory of differential, integral and operator equations.

Proof

Let us restate the theorem.
For a complete lattice and a monotone function on L, the set of all fixpoints of f is also a complete lattice, with:

as the greatest fixpoint of f
as the least fixpoint of f.

Proof. We begin by showing that P has both a least element and a greatest element. Let and . Then because f is monotone we have, that is.
Now let . Then for all it is true that and, so. Therefore, f is an upper bound of D, but u is the least upper bound, so, i.e.. Then = u. Because every fixpoint is in D we have that u is the greatest fixpoint of f.
The function f is monotone on the dual lattice. As we have just proved, its greatest fixpoint exists. It is the least fixpoint of L, so P has least and greatest elements, that is more generally, every monotone function on a complete lattice has a least fixpoint and a greatest fixpoint.
For a, b in L we write for the closed interval with bounds a and. If a ≤ b, then is a complete lattice.
It remains to be proven that P is a complete lattice. Let, and. We show that. Indeed, for every we have x = f and since w is the least upper bound of W,. In particular. Then from follows that, giving or simply. This allows us to look at f as a function on the complete lattice . Then it has a least fixpoint there, giving us the least upper bound of W. We've shown that an arbitrary subset of P has a supremum, that is, P is a complete lattice.

Computing a Tarski fixed-point

Chang, Lyuu and Ti present an algorithm for finding a Tarski fixed-point in a totally-ordered lattice, when the order-preserving function is given by a value oracle. Their algorithm requires queries, where L is the number of elements in the lattice. In contrast, for a general lattice, they prove a lower bound of queries.
Deng, Qi and Ye present several algorithms for finding a Tarski fixed-point. They consider two kinds of lattices: componentwise ordering and lexicographic ordering. They consider two kinds of input for the function f: value oracle, or a polynomial function. Their algorithms have the following runtime complexity :

	Polynomial function	Value oracle
Componentwise
Lexicographic

The algorithms are based on binary search. On the other hand, determining whether a given fixed point is unique is computationally hard:

	Polynomial function	Value oracle
Componentwise	coNP-complete
Lexicographic	coNP-complete

For d=2, for componentwise lattice and a value-oracle, the complexity of is optimal. But for d>2, there are faster algorithms:

Fearnley, Palvolgyi and Savani presented an algorithm using only queries. In particular, for d=3, only queries are needed.
Chen and Li presented an algorithm using only queries.

Application in game theory

Tarski's fixed-point theorem has applications to supermodular games. A supermodular game is a game in which the utility function of each player has increasing differences, so the best response of a player is a weakly-increasing function of other players' strategies. For example, consider a game of competition between two firms. Each firm has to decide how much money to spend on research. In general, if one firm spends more on research, the other firm's best response is to spend more on research too. Some common games can be modeled as supermodular games, for example Cournot competition, Bertrand competition and Investment Games.
Because the best-response functions are monotone, Tarski's fixed-point theorem can be used to prove the existence of a pure-strategy Nash equilibrium in a supermodular game. Moreover, Topkis showed that the set of PNE of a supermodular game is a complete lattice, so the game has a "smallest" PNE and a "largest" PNE.
Echenique presents an algorithm for finding all PNE in a supermodular game. His algorithm first uses best-response sequences to find the smallest and largest PNE; then, he removes some strategies and repeats, until all PNE are found. His algorithm is exponential in the worst case, but runs fast in practice. Deng, Qi and Ye show that a PNE can be computed efficiently by finding a Tarski fixed-point of an order-preserving mapping associated with the game.