<p>In SLP, <span class="math-container">$T(v)=\sup \{f(x,y)+ \beta v(y)\}$</span>, where <span class="math-container">$v(y)$</span> is the value function in the next period and <span class="math-container">$\beta\in(0,1)$</span> is the discounting factor. Check Blackwell's discounting condition in order to get the intuition.</p>

<p><span class="math-container">\begin{align}
T(v+a) = Tv + \beta a
\end{align}</span>
That is, if there is an increase by <span class="math-container">$a$</span> in the next-period value function, the increase should be discounted to get the present value.</p>

<p>But the above intuition may be specific. Generally, the discounting condition should be termed as the shortening condition. That is, <span class="math-container">$a$</span> should be shortened after mapping.</p>

CC BY-SA 4.0
2019-10-29T04:22:26.220Z
2019-11-20T18:44:58.030Z
2019-11-20T18:44:58.030Z
SLP
<p>Without discounting, you cannot show either</p>

<p><span class="math-container">$$
T(g + || f - g||) \leq Tg + \beta || f - g|| 
$$</span></p>

<p>or </p>

<p><span class="math-container">$$
T(f + || g - f||) \leq Tf + \beta || g - f|| 
$$</span></p>

<p>and thus you cannot demonstrate that <span class="math-container">$T$</span> is a contraction mapping in the traditional proof.</p>

<hr>

<p>For intuition, note that Blackwell's sufficiency conditions for a contraction mapping is a <em>fixed-point theorem</em> used to establish the convergence of value function iterations. In the simplest case possible of a collapsed state space with the linear operator <span class="math-container">$T(W) = \sigma + \beta W$</span>, it is clear that <span class="math-container">$\beta \in (0,1)$</span> will <strong>always</strong> lead to an intersection of <span class="math-container">$T(W)$</span> (from above) with the 45-degree line <span class="math-container">$T(W) = W$</span> and thus converge at a fixed point <em>independent</em> of the initial guess. The discounting condition that pins down such a <span class="math-container">$\beta$</span> assures this.</p>

<p>In systems with fully specified state spaces, however, discounting is no longer sufficient to establish uniqueness. Monotonicity is additionally required in these cases to establish the sufficiency of Blackwell's conditions because of additional concerns over dynamics.</p>

CC BY-SA 4.0
2018-10-24T23:35:36.413Z
2018-10-25T04:00:55.880Z
2018-10-25T04:00:55.880Z
Kenneth Rios
<p>The discounting condition is as follow:</p>

<p>There exists some <span class="math-container">$\beta \in (0, 1)$</span> such that <span class="math-container">$[T(f + a)](x) ≤ (T f)(x) + βa$</span>, for all <span class="math-container">$f ∈ B(X), a ≥ 0, x ∈ X$</span>.</p>

<p>While the monotonicity condition makes sense, I can't give a nice meaning to this property. </p>

<dynamic-programming>
In Blackwell's condition for T to be a contraction mapping, we require that satisfies discounting. What is the intuition of discounting?
Tecon
# In Blackwell's condition for T to be a contraction mapping, we require that satisfies discounting. What is the intuition of discounting?

(Asked by: username_0 on 2018-10-23)

The discounting condition is as follow:

There exists some $\beta \in (0, 1)$ such that $[T(f + a)](x) ≤ (T f)(x) + βa$, for all $f ∈ B(X), a ≥ 0, x ∈ X$.

While the monotonicity condition makes sense, I can't give a nice meaning to this property.

**Tags:** dynamic-programming

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## 2 answers

### Answer by username_2 on 2018-10-24. Score: 8 | Accepted answer

Without discounting, you cannot show either

$$
T(g + || f - g||) \leq Tg + \beta || f - g||
$$

or

$$
T(f + || g - f||) \leq Tf + \beta || g - f||
$$

and thus you cannot demonstrate that $T$ is a contraction mapping in the traditional proof.

---

For intuition, note that Blackwell's sufficiency conditions for a contraction mapping is a *fixed-point theorem* used to establish the convergence of value function iterations. In the simplest case possible of a collapsed state space with the linear operator $T(W) = \sigma + \beta W$, it is clear that $\beta \in (0,1)$ will **always** lead to an intersection of $T(W)$ (from above) with the 45-degree line $T(W) = W$ and thus converge at a fixed point *independent* of the initial guess. The discounting condition that pins down such a $\beta$ assures this.

In systems with fully specified state spaces, however, discounting is no longer sufficient to establish uniqueness. Monotonicity is additionally required in these cases to establish the sufficiency of Blackwell's conditions because of additional concerns over dynamics.


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### Answer by username_1 on 2019-10-29. Score: 1

In SLP, $T(v)=\sup \{f(x,y)+ \beta v(y)\}$, where $v(y)$ is the value function in the next period and $\beta\in(0,1)$ is the discounting factor. Check Blackwell's discounting condition in order to get the intuition.

\begin{align}
T(v+a) = Tv + \beta a
\end{align}
That is, if there is an increase by $a$ in the next-period value function, the increase should be discounted to get the present value.

But the above intuition may be specific. Generally, the discounting condition should be termed as the shortening condition. That is, $a$ should be shortened after mapping.


---

