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Consider a function $\phi : \mathbb{R}^+ \rightarrow \mathbb{R}^+$. $\phi$ is convex and increasing.

Can we come up with functions $\phi_i : \mathbb{R}^+ \rightarrow \mathbb{R}^+, i \in \{1,2,\cdots,n \}$ such that $$ \phi(\sum_{i=1}^{n} \theta_i) = \sum_{i=1}^{n} \phi_i( \theta_i), $$ where $n$ is finite and $\theta_i \in \mathbb{R}^+$ for all $i$?

If yes, what properties must $\phi_i$'s satisfy?

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    Are you asking that there be functions $\phi_i: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any map $\theta: \{1;...;n\} \rightarrow \mathbb{R}^+$, $\phi(\sum \limits_{i=1}^n \theta_i) = \sum \limits_{i=1}^n \phi_i(\theta_i)$, or such that for any such map, there is a permutation $\sigma_{\theta}$ of $\{1;...;n\}$ such that $\phi(\sum \limits_{i=1}^n \theta_i) = \sum \limits_{i=1}^n \phi_{\sigma_{\theta}(i)}(\theta_i)$? – 2017-02-10
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    Either case works for me, permutation or no permutation; given a $\phi$ can we construct such $\phi_i$'s or $\phi_{\sigma_{\theta}(i)}$'s. So then what will be your response? Thanks in advance! – 2017-02-10

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Suppose we have $\phi(\sum_{i=1}^{n} \theta_i) = \sum_{i=1}^{n} \phi_i( \theta_i)$ for all $\theta_i$. Let $s=\sum_{i=1}^{n} \theta_i$. Replacing $\theta_j$ by $\theta_j + d$ for just one value of j and eliminating the unchanged terms we see

$\phi(s + d) - \phi(s) = \phi_j(\theta_j + d) - \phi_j(\theta_j)$.

This is for any j and any s and $\theta_j$ ($\theta_j < s$), so

$\phi_j(s+d)-\phi_j(s) = \phi_j(t+d)-\phi_j(t)$

for any positive s, t, d. Similarly for $\phi$.

A function $\mathbb{R} \rightarrow \mathbb{R}$ with this property is not necessarily linear (though I don't know about $\mathbb{R}^+ \rightarrow \mathbb{R}^+$) but an increasing one must be.

Note also that if we put $\theta_i = a/(n-1)$ for some a and let $b_1=\sum_{i=2}^{n} \phi_i( \theta_i)$ then, for all x,

$\phi(x+a) = \phi_1(x) + b_1$.

$\phi(x+a)-\phi(x)$ is independent of x so, letting $c_1 = \phi(x+a) - \phi(x) +b_i$ we get

$\phi_1(x) = \phi(x) + c_1$

Similarly we get c_i such that $\phi_i(x) = \phi(x) + c_i$ for the other values of i.

So $\phi$ and the $\phi_i$'s are all linear functions with the same gradient, differing only by constants.

PS. "Linear" can be ambiguous. I mean functions of the form f(x) = mx + c, whose graph is a straight line, but not necessarily passing through the origin.