Blackwell's contraction mapping theorem

In mathematics, Blackwell's contraction mapping theorem provides a set of sufficient conditions for an operator to be a contraction mapping. It is widely used in areas that rely on dynamic programming as it facilitates the proof of existence of fixed points. The result is due to David Blackwell who published it ^[1] in 1965 in the Annals of Mathematical Statistics.

Statement of the Theorem

Let ${\displaystyle T}$ be an operator defined over an ordered normed vector space ${\displaystyle X}$. ${\displaystyle T:X\rightarrow X}$ is a contraction mapping with modulus ${\displaystyle \beta }$ if it satisfies

1. ${\displaystyle (monotonicity)\quad u\leq v\implies Tu\leq Tv}$
2. ${\displaystyle (discounting)\quad T(u+c)\leq Tu+\beta c.}$

Proof of the Theorem

For all ${\displaystyle u}$ and ${\displaystyle v\in X}$, ${\displaystyle u\leq v+||v-u||}$. Properties 1. and 2. imply that ${\displaystyle T(u)\leq T(v+||v-u||)\leq T(v)+\beta ||v-u||}$, hence, ${\displaystyle T(u)-T(v)\leq \beta ||v-u||}$. The symmetric follows from a similar argument and we prove that ${\displaystyle T}$ is a contraction mapping.

Applications

The cake eating problem

An agent has access to only one cake for its entire, infinite, life. It has to decide the optimal way to consume it. It evaluates a consumption plan, ${\displaystyle c_{t}}$, by using a separable utility function, ${\displaystyle \sum _{t=0}^{\infty }\beta ^{t}{\frac {c_{t}^{1-\sigma }}{1-\sigma }}}$, with discounting factor ${\displaystyle \beta \in (0,1)}$. Its problem can be summarized as

${\displaystyle \max _{c_{t}}\sum _{t=0}^{\infty }\beta ^{t}{\frac {c_{t}^{1-\sigma }}{1-\sigma }}{\text{ subject to }}x_{t}=x_{t-1}-c_{t}{\text{, }}x_{-1}=0{\text{, }}c_{t}\geq 0{\text{ and }}x_{t}\geq 0\,\forall \,t\,\in \,\mathbb {Z} _{+}}$. (1)

Applying Bellman's principle of optimality we find ( 1 )'s corresponding Bellman equation

${\displaystyle V(c)=\max _{c'}{\frac {c^{1-\sigma }}{1-\sigma }}+\beta V(c')}$. (2)

It can be proven that the solution to this functional equation, if it exists, is equivalent to the solution of ( 1 ) ^[2] . To prove its existence we can resort to Blackwell's sufficient conditions.

Define the operator ${\displaystyle T(V(c))=\max _{c'}{\frac {c^{1-\sigma }}{1-\sigma }}+\beta V(c')}$. A solution to ( 2 ) is equivalent to finding a fixed-point for our operator. If we prove that this operator is a contraction mapping then we can use Banach's fixed-point theorem, and conclude that there is indeed a solution to ( 1 ).

First note that ${\displaystyle T}$ is defined over the space of bounded functions since for all feasible consumption plans, ${\displaystyle \sum _{t=0}^{\infty }\beta ^{t}{\frac {c_{t}^{1-\sigma }}{1-\sigma }}\leq \sum _{t=0}^{\infty }\beta ^{t}{\frac {1^{1-\sigma }}{1-\sigma }}={\frac {1}{(1-\beta )(1-\sigma )}}<\infty }$. Endowing it with the sup-norm we conclude that the domain and co-domain are ordered normed vector spaces. We are just left with verifying that the conditions for Blackwell's theorem are respected:

1. (monotonicity) ${\displaystyle {\text{if }}V(c)\geq U(c)\,\forall \,c\,\in \,[0,1]}$ then ${\displaystyle T(V(c))=\max _{c'}{\frac {c^{1-\sigma }}{1-\sigma }}+\beta V(c')\geq \max _{c'}{\frac {c^{1-\sigma }}{1-\sigma }}+\beta U(c')=T(U(c))}$
2. (discounting) ${\displaystyle T(V(c)+a)=\max _{c'}{\frac {c^{1-\sigma }}{1-\sigma }}+\beta (V(c')+a)\leq \max _{c'}{\frac {c^{1-\sigma }}{1-\sigma }}+\beta V(c')+\beta a=T(V(c))+\beta a}$ where ${\displaystyle a}$ is a constant function.

References

1. Blackwell, David. “Discounted Dynamic Programming.” The Annals of Mathematical Statistics 36, no. 1 (1965): 226–35. R, http://www.jstor.org/stable/2238089. Accessed 15 Aug. 2022.
2. Stokey, Nancy L., Robert E. Lucas, and Edward C. Prescott. Recursive Methods in Economic Dynamics. Harvard University Press, 1989. https://doi.org/10.2307/j.ctvjnrt76.