A(\textbf{X})\leq\left\lceil\frac{T}{r+1}\right\rceil(B+r+1)$, where the
second inequality holds because the assignment is over paths that observe
$(r+1)$ good days and thus the maximum cost suffered by any algorithm on this
path is at most $(B+r+1)$. With this assignment, we have that

$u(A,p)=\frac{1}{T}\mathbf{E}_{\textbf{X}^{\prime}\sim\Delta(\\{0,1\\}^{T-1})}[A^{\prime}(\textbf{X}^{\prime})]$

where the distribution is generated by a binomial process with probability $p$
and length $(T-1)$. Applying 4.11, we have that if
$|p_{\alpha}-p_{\beta}|\leq\frac{\epsilon}{2(B+1)^{2}(T+1)^{2}T}$, when
$p_{\alpha},p_{\beta}\in\left[\delta,\frac{B-1}{T-1}\right]$,

$|u(A,p_{\alpha})-u(A,p_{\beta})|\leq\frac{\epsilon}{2T(B+1)(T+1)}\leq\epsilon$

where the second inequality follows from that when we apply 4.11, the worst
case cost along a path is upper bounded as
$\max_{r}\left\lceil\frac{T}{r+1}\right\rceil(B+r+1)$ which is maximized at
$r=0$. Now consider the case $p_{\alpha},p_{\beta}$ in the interval
$\left(\frac{B-1}{T-k},\min\left\\{\frac{B-1}{T-k-1},1\right\\}\right]$ such
that

$|p_{\alpha}-p_{\beta}|\leq\frac{\epsilon}{2(B+1)^{2}(T+1)^{2}T}$

We have that

$\begin{split}|u(A,p_{\alpha})-u(A,p_{\beta})|&=\left|\frac{A(p_{\alpha})}{\text{OPT}_{p_{\alpha}}(p_{\alpha})}-\frac{A(p_{\beta})}{\text{OPT}_{p_{\beta}}(p_{\beta})}\right|\\\
&=\left|\frac{A^{\prime}(p_{\alpha})}{g_{k}(p_{\alpha})}-\frac{A^{\prime}(p_{\beta})}{g_{k}(p_{\beta})}\right|\\\
&\leq\epsilon\end{split}$

where the second equality follows from D.1. Note that $g_{k}$ can also be
decomposed in an expected form and thus the Lipschitz bound derived above
holds for $g_{k}(p)$ as well. Thus the inequality follows from noting that
$g_{k}(.)\geq 1$ for all $p\in[\delta,1]$ (D.1), and an upper bound on both
$A^{\prime}(.)$ and $g_{k}(.)$ is
$\max_{r}\left\lceil\frac{T}{r+1}\right\rceil(B+r+1)$ which is upper bounded
by $(B+1)(T+1)$. ∎

###### Proof of Theorem 4.13.

The idea is to discretize each interval where the Bayesian optimal algorithm
is the same (Theorem 4.6), with a discretization of size

$\bar{\epsilon}=\frac{\epsilon}{8(B+1)^{2}(T+1)^{2}T}.$

First consider the interval $\left[\delta,\frac{B-1}{T-1}\right]$, where the
Bayesian optimal algorithm always rents. Consider the following grid points on
this interval

$G_{0}=\left\\{\delta+k\bar{\epsilon}\right\\}_{k=1}^{\left\lceil\frac{1}{\bar{\epsilon}}\left[\frac{B-1}{T-1}-\delta\right]-1\right\rceil}.$

Such a discretization covers the interval
$\left[\delta,\frac{B-1}{T-1}\right]$. We have that for any grid point in
$G_{0}$ i.e. $(p_{\epsilon})_{k}\in G_{0}$,

$|u(A,(p_{\epsilon})_{k})-u(A,p)|\leq\frac{\epsilon}{4}\quad\forall
p\in\left[(p_{\epsilon})_{k}-\bar{\epsilon},(p_{\epsilon})_{k}+\bar{\epsilon}\right]$

which follows from Lemma 4.12. We do this process for every interval
$\left[\frac{B-1}{T-l},\min\left\\{\frac{B-1}{T-l-1},1\right\\}\right]$, where
$l\in[\min\\{\left\lceil T-B\right\rceil,T-1\\}]$ to obtain a grid for that
interval $G_{l}$.

$G_{l}=\left\\{\frac{B-1}{T-l}+k\bar{\epsilon}\right\\}_{k=1}^{\left\lceil\frac{1}{\bar{\epsilon}}\left[\min\left\\{\frac{B-1}{T-l-1},1\right\\}-\frac{B-1}{T-l}\right]-1\right\rceil}$

.

By the same argument for $G_{0}$, these satisfy the same properties as well.
Hence

$\mathcal{P}_{\epsilon}=\cup_{l=0}^{\min\\{\left\lceil
T-B\right\rceil,T-1\\}}G_{l}$

satisfies Small-Cover. ∎

###### Proof of Theorem 4.14.

To compute the run-time, first note that the number of actions to the
distribution player is bounded as
$|\mathcal{P}_{\epsilon}|\leq\frac{2(B+1)^{2}(T+1)^{2}T}{\epsilon}$ (look at
proof of Theorem 4.13). Also note that the maximum possible utility
encountered by the learning algorithm is $(B+1)(T+1)$ (look at the proof of
Lemma 4.12 in Appendix D). The number of updates the algorithm takes is
$\frac{(B+1)^{2}(T+1)^{2}\log|\mathcal{P}_{\epsilon}|}{\epsilon^{2}}$. Each
update takes $O(|\mathcal{P}_{\epsilon}|T^{3})$ time to execute (Theorem 4.9).
Thus the overall running time is $\tilde{O}(T^{12}\epsilon^{-3})$ where we
have used the inequality $B\leq T$ as the problem is trivial when $B>T$ in
which case the agent always continues. ∎

##  Appendix E Proofs For Section 5

###### Proof of Lemma 5.1.

By definition, the sub-game optimal algorithm takes an optimal decision at
every stage of the game. Since the worst-case optimal algorithm takes an
optimal decision on some input sequence, it follows that the sets have a non-
empty intersection. ∎

##  Appendix F Properties of the Optimal Prior-Independent Ski-Rental
Algorithm

In this section, we describe certain properties of an optimal prior-
independent ski-rental algorithm. We first define what we mean by an algorithm
to satisfy the thresholding property. Intuitively, for any sub-sequence of
days of a fixed length, if the algorithm stops on seeing a certain count of
good days, then it will also stop on seeing a greater count of good days. We
then show a property of the worst-case distribution that the adversary plays.

######  Definition 14.

(Thresholding Algorithm) We say that an algorithm $A=(A_{1},\dots,A_{T})$ is a
thresholding algorithm if $\forall i\in[T]$, $\forall X_{i}\in\\{0,1\\}$, if
$\exists k\in\\{0\\}\cup[i-1]$ such that $A_{i}(k,X_{i})=S$, then
$A_{i}(k+1,X_{i})=S$

######  Lemma F.1.

For any distribution, $P\in\Delta([\delta,1])$ that the adversary plays, an
optimal algorithm for that distribution satisfies the thresholding property.

###### Proof.

We show the above Lemma for any discrete distribution, but the same arguments
go through for a continuous distribution as well. We use the same notation as
that used in Theorem 4.9. We do an induction over the length of sub-sequences
$n\in[T]$, while maintaining the following invariant at each stage

  * •

The function $u_{t}(k)$, where $u_{t}(k)$ is given as,

$u_{t}(k)=\frac{\sum_{p\in
supp(P)}\frac{Pr[(k,t,1)|p]Pr[P=p]}{OPT(p)}c^{p}(k,t,1)}{\sum_{p\in
supp(P)}\frac{Pr[(k,t,1)|p]Pr[P=p]}{OPT(p)}}$

is monotone non-decreasing in $k$.

  * •

The function $l_{t}(k)$, where $l_{t}(k)$ is given as,

$l_{t}(k)=\frac{\sum_{p\in
supp(P)}\frac{Pr[(k,t,0)|p]Pr[P=p]}{OPT(p)}c^{p}(k,t,0)}{\sum_{p\in
supp(P)}\frac{Pr[(k,t,0)|p]Pr[P=p]}{OPT(p)}}$

is monotone non-decreasing in $k$.

  * •

$u_{t}(k)\geq l_{t}(k)$ for all $k\in\\{0\\}\cup[t-1]$.

Let us first check the base case when $t=T$. We always continue on the last
day no matter what and thus we have that $c^{p}(k,T,1)=1$ and $c^{p}(k,T,0)=0$
for all $k\in\\{0\\}\cup[T-1]$ and $p\in supp(P)$. Thus the base case is
satisfied. Also observe that thresholding is vacuously satisfied at stage
$t=T$. We now prove that when the induction assumptions are satisfied at stage
$(n+1)$, then we get the thresholding property of the algorithm at stage $n$.
It is sufficient to prove that on observing a good day, the normalized
continuation cost of continueing is monotone non-decreasing in $k$. This is
because the algorithm takes a decision according to

$A^{P}_{t}(k,1)=arg\,min_{\\{B,R\\}}\\{\sum_{p\in
supp(P)}Pr[(k,t,1)|p]Pr[P=p]\frac{B}{OPT(p)},\\\ \sum_{p\in
supp(P)}Pr[(k,t,1)|p]Pr[P=p]\left[\frac{1+pc^{p}(k+1,t+1,1)+(1-p)c^{p}(k+1,t+1,0)}{OPT(p)}\right]\\}$

or effectively compares the stop cost $B$ against

$u^{R}_{t}(k)=\frac{\sum_{p\in
supp(P)}\frac{Pr[(k,t,1)|p]Pr[P=p]}{OPT(p)}\left[1+pc^{p}(k+1,t+1,1)+(1-p)c^{p}(k+1,t+1,0)\right]}{\sum_{p\in
supp(P)}\frac{Pr[(k,t,1)|p]Pr[P=p]}{OPT(p)}}$

Thus if the latter is monotone non-decreasing, if at any $k$ it becomes
greater than $B$, it will still be greater than $B$ for larger $k$. We ignore
writing the limits of the summation as it is clear it is over probabilites in
the support. To make the calculations more compact, we re-write the weighted
probability as

$f^{p}(k,t,X_{T})=\frac{Pr[(k,t,X_{T})|p]Pr[P=p]}{OPT(p)},\quad
X_{T}\in\\{0,1\\}$

Our aim is to prove $u^{R}_{t}(k+1)-u^{R}_{t}(k)\geq 0$. We have

$u^{R}_{t}(k+1)-u^{R}_{t}(k)=\overbrace{\sum p\left[\frac{f^{p}(k+1,t,1)}{\sum
f^{p}(k+1,t,1)}c^{p}(k+2,t+1,1)-\frac{f^{p}(k,t,1)}{\sum
f^{p}(k,t,1)}c^{p}(k+1,t+1,1)\right]}^{\mathchoice{\leavevmode\hbox
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\underbrace{\sum(1-p)\left[\frac{f^{p}(k+1,t,1)}{\sum
f^{p}(k+1,t,1)}c^{p}(k+2,t+1,0)-\frac{f^{p}(k,t,1)}{\sum
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.94pt{\vbox to4.94pt{\pgfpicture\makeatletter\hbox{\hskip
2.47173pt\lower-1.34395pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.27173pt}{1.12778pt}\pgfsys@curveto{2.27173pt}{2.38243pt}{1.25465pt}{3.3995pt}{0.0pt}{3.3995pt}\pgfsys@curveto{-1.25465pt}{3.3995pt}{-2.27173pt}{2.38243pt}{-2.27173pt}{1.12778pt}\pgfsys@curveto{-2.27173pt}{-0.12688pt}{-1.25465pt}{-1.14395pt}{0.0pt}{-1.14395pt}\pgfsys@curveto{1.25465pt}{-1.14395pt}{2.27173pt}{-0.12688pt}{2.27173pt}{1.12778pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.12778pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$

Considering the first summation, it simplifies to the following equation which
follows from the i.i.d assumption on the class of distributions we consider

$\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
1$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sum
f^{p}(k+2,t+1,1)}{\sum
f^{p}(k+1,t,1)}\left[\sum\frac{f^{p}(k+2,t+1,1)c^{p}(k+2,t+1,1)}{\sum
f^{p}(k+2,t+1,1)}\right]-\\\ \frac{\sum f^{p}(k+1,t+1,1)}{\sum
f^{p}(k,t,1)}\left[\sum\frac{f^{p}(k+1,t+1,1)c^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t+1,1)}\right]$ (8)

Utilizing the inductive assumption corresponding to the non-decreasing
property of $u_{t+1}(k)$, we have that

$\begin{split}\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
1$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\geq\left[\sum\frac{f^{p}(k+1,t+1,1)c^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t+1,1)}\right]\overbrace{\left[\frac{\sum f^{p}(k+2,t+1,1)}{\sum
f^{p}(k+1,t,1)}-\frac{\sum f^{p}(k+1,t+1,1)}{\sum
f^{p}(k,t,1)}\right]}^{\mathchoice{\leavevmode\hbox to10.22pt{\vbox
to10.22pt{\pgfpicture\makeatletter\hbox{\hskip 5.10966pt\lower-2.67912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.22pt{\vbox to10.22pt{\pgfpicture\makeatletter\hbox{\hskip
5.10966pt\lower-2.67912pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.77pt{\vbox to7.77pt{\pgfpicture\makeatletter\hbox{\hskip
3.8853pt\lower-2.18391pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.6853pt}{1.70139pt}\pgfsys@curveto{3.6853pt}{3.73674pt}{2.03535pt}{5.38669pt}{0.0pt}{5.38669pt}\pgfsys@curveto{-2.03535pt}{5.38669pt}{-3.6853pt}{3.73674pt}{-3.6853pt}{1.70139pt}\pgfsys@curveto{-3.6853pt}{-0.33397pt}{-2.03535pt}{-1.98392pt}{0.0pt}{-1.98392pt}\pgfsys@curveto{2.03535pt}{-1.98392pt}{3.6853pt}{-0.33397pt}{3.6853pt}{1.70139pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.70139pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.27644pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.13pt{\vbox to6.13pt{\pgfpicture\makeatletter\hbox{\hskip
3.06718pt\lower-1.85191pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.86719pt}{1.21527pt}\pgfsys@curveto{2.86719pt}{2.79878pt}{1.58351pt}{4.08246pt}{0.0pt}{4.08246pt}\pgfsys@curveto{-1.58351pt}{4.08246pt}{-2.86719pt}{2.79878pt}{-2.86719pt}{1.21527pt}\pgfsys@curveto{-2.86719pt}{-0.36824pt}{-1.58351pt}{-1.65192pt}{0.0pt}{-1.65192pt}\pgfsys@curveto{1.58351pt}{-1.65192pt}{2.86719pt}{-0.36824pt}{2.86719pt}{1.21527pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.21527pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.62604pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
1b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}$

Analyzing the term $\textstyle 1b$, we have

$\begin{split}\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
1b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&=\frac{\left(\sum
f^{p}(k+2,t+1,1)\right)\left(\sum f^{p}(k,t,1)\right)-\left(\sum
f^{p}(k+1,t+1,1)\right)\left(\sum f^{p}(k+1,t,1)\right)}{\left(\sum
f^{p}(k+1,t,1)\right)\left(\sum f^{p}(k,t,1)\right)}\\\
&=\frac{\sum_{\\{(p_{1},p_{2}):p_{1}<p_{2}\\}}\frac{Pr[P=p_{1}]Pr[P=p_{2}]}{OPT(p_{1})OPT(p_{2})}(p_{1}p_{2})^{k+1}\left[(1-p_{1})(1-p_{2})\right]^{t-k-2}(p_{1}-p_{2})^{2}}{\left(\sum
f^{p}(k+1,t,1)\right)\left(\sum f^{p}(k,t,1)\right)}\end{split}$

Analyzing term $\textstyle 2$, we obtain the following simplified equation
again following from the i.i.d assumptions on the class of distributions we
consider.

$\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sum
f^{p}(k+2,t+1,0)}{\sum
f^{p}(k+1,t,1)}\left[\sum\frac{f^{p}(k+2,t+1,0)c^{p}(k+2,t+1,0)}{\sum
f^{p}(k+2,t+1,0)}\right]-\\\ \frac{\sum f^{p}(k+1,t+1,0)}{\sum
f^{p}(k,t,1)}\left[\sum\frac{f^{p}(k+1,t+1,0)c^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t+1,0)}\right]$ (9)

Utilizing the inductive assumption corresponding to the non-decreasing
property of $l_{t+1}(k)$, we have that

$\begin{split}\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\geq\left[\sum\frac{f^{p}(k+1,t+1,0)c^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t+1,0)}\right]\overbrace{\left[\frac{\sum f^{p}(k+2,t+1,0)}{\sum
f^{p}(k+1,t,1)}-\frac{\sum f^{p}(k+1,t+1,0)}{\sum
f^{p}(k,t,1)}\right]}^{\mathchoice{\leavevmode\hbox to10.22pt{\vbox
to10.22pt{\pgfpicture\makeatletter\hbox{\hskip 5.10966pt\lower-2.67912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 2b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.22pt{\vbox to10.22pt{\pgfpicture\makeatletter\hbox{\hskip
5.10966pt\lower-2.67912pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 2b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.77pt{\vbox to7.77pt{\pgfpicture\makeatletter\hbox{\hskip
3.8853pt\lower-2.18391pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.6853pt}{1.70139pt}\pgfsys@curveto{3.6853pt}{3.73674pt}{2.03535pt}{5.38669pt}{0.0pt}{5.38669pt}\pgfsys@curveto{-2.03535pt}{5.38669pt}{-3.6853pt}{3.73674pt}{-3.6853pt}{1.70139pt}\pgfsys@curveto{-3.6853pt}{-0.33397pt}{-2.03535pt}{-1.98392pt}{0.0pt}{-1.98392pt}\pgfsys@curveto{2.03535pt}{-1.98392pt}{3.6853pt}{-0.33397pt}{3.6853pt}{1.70139pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.70139pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.27644pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 2b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.13pt{\vbox to6.13pt{\pgfpicture\makeatletter\hbox{\hskip
3.06718pt\lower-1.85191pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.86719pt}{1.21527pt}\pgfsys@curveto{2.86719pt}{2.79878pt}{1.58351pt}{4.08246pt}{0.0pt}{4.08246pt}\pgfsys@curveto{-1.58351pt}{4.08246pt}{-2.86719pt}{2.79878pt}{-2.86719pt}{1.21527pt}\pgfsys@curveto{-2.86719pt}{-0.36824pt}{-1.58351pt}{-1.65192pt}{0.0pt}{-1.65192pt}\pgfsys@curveto{1.58351pt}{-1.65192pt}{2.86719pt}{-0.36824pt}{2.86719pt}{1.21527pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.21527pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.62604pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
2b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}$

Analyzing the term $\textstyle 2b$, we have

$\begin{split}\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 2b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 2b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 2b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
2b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&=\frac{\left(\sum
f^{p}(k+2,t+1,0)\right)\left(\sum f^{p}(k,t,1)\right)-\left(\sum
f^{p}(k+1,t+1,0)\right)\left(\sum f^{p}(k+1,t,1)\right)}{\left(\sum
f^{p}(k+1,t,1)\right)\left(\sum f^{p}(k,t,1)\right)}\\\
&=-\frac{\sum_{\\{(p_{1},p_{2}):p_{1}<p_{2}\\}}\frac{Pr[P=p_{1}]Pr[P=p_{2}]}{OPT(p_{1})OPT(p_{2})}(p_{1}p_{2})^{k+1}\left[(1-p_{1})(1-p_{2})\right]^{t-k-2}(p_{1}-p_{2})^{2}}{\left(\sum
f^{p}(k+1,t,1)\right)\left(\sum f^{p}(k,t,1)\right)}\\\
&=-\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
1b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\end{split}$

Thus combining $\textstyle 1$ and $\textstyle 2$ to give a bound on
$u^{R}_{t}(k+1)-u^{R}_{t}(k)$, we have

$\begin{split}u^{R}_{t}(k+1)-u^{R}_{t}(k)&\geq\mathchoice{\leavevmode\hbox
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 1b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
1b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\left[\sum\frac{f^{p}(k+1,t+1,1)c^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t+1,1)}-\sum\frac{f^{p}(k+1,t+1,0)c^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t+1,0)}\right]\\\ &\geq 0\end{split}$

which follows from $\textstyle 1b$ being non-negative and the inductive
assumption that $u_{t+1}(k)\geq l_{t+1}(k)$.

All that is left to verify is that the inductive invariants are maintained at
stage $t$. Since the algorithm is thresholding at stage $t$, let the algorithm
continue till a count of $k^{*}$ and then stop for strictly greater than
$k^{*}$ good days. If $k^{*}=(t-1)$, then the inductive invariant holds as
$u_{t}(k)=u^{R}_{t}(k)$ for all $k\in\\{0\\}\cup[t-1]$, which we have proved
to be monotonic non-decreasing in $k$. If $k^{*}<t$, then $u_{t}(k)$ is
monotone non-decreasing for $k\in\\{0\\}\cup[k^{*}]$. We also have that
$u_{t}(k)$ is monotone non-decreasing for $k\in\\{k^{*}+1,t-1\\}$ as
$c^{p}(k,t,1)=B$ for all $p\in supp(P)$. All that is remaining to show is that
$u_{t}(k^{*}+1)\geq u_{t}(k^{*})$.

$u_{t}(k^{*}+1)=\frac{\sum_{p\in
supp(P)}\frac{Pr[(k^{*}+1,t,1)|p]Pr[P=p]}{OPT(p)}B}{\sum_{p\in
supp(P)}\frac{Pr[(k^{*}+1,t,1)|p]Pr[P=p]}{OPT(p)}}=B>u_{t}^{R}(k^{*})=u_{t}(k^{*})$

where the inequality follows from the fact that the algorithm chose to
continue on seeing $k^{*}$ good days in the first $(t-1)$ days. Now to verify
the monotone non-decreasing nature of $l_{t}(k)$, we follows the same
procedure as that done to show the monotonicity of $u_{t}^{R}(k)$. Note that
the algorithm always continues on seeing a bad day of weather. Thus,

$l_{t}(k+1)-l_{t}(k)=\overbrace{\sum p\left[\frac{f^{p}(k+1,t,0)}{\sum
f^{p}(k+1,t,0)}c^{p}(k+1,t+1,1)-\frac{f^{p}(k,t,0)}{\sum
f^{p}(k,t,0)}c^{p}(k,t+1,1)\right]}^{\mathchoice{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.07pt{\vbox to6.07pt{\pgfpicture\makeatletter\hbox{\hskip
3.03496pt\lower-1.45607pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.22499pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.94pt{\vbox to4.94pt{\pgfpicture\makeatletter\hbox{\hskip
2.47173pt\lower-1.34395pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.27173pt}{1.12778pt}\pgfsys@curveto{2.27173pt}{2.38243pt}{1.25465pt}{3.3995pt}{0.0pt}{3.3995pt}\pgfsys@curveto{-1.25465pt}{3.3995pt}{-2.27173pt}{2.38243pt}{-2.27173pt}{1.12778pt}\pgfsys@curveto{-2.27173pt}{-0.12688pt}{-1.25465pt}{-1.14395pt}{0.0pt}{-1.14395pt}\pgfsys@curveto{1.25465pt}{-1.14395pt}{2.27173pt}{-0.12688pt}{2.27173pt}{1.12778pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.12778pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\\\
\underbrace{\sum(1-p)\left[\frac{f^{p}(k+1,t,0)}{\sum
f^{p}(k+1,t,0)}c^{p}(k+1,t+1,0)-\frac{f^{p}(k,t,0)}{\sum
f^{p}(k,t,0)}c^{p}(k,t+1,0)\right]}_{\mathchoice{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.79pt{\vbox to7.79pt{\pgfpicture\makeatletter\hbox{\hskip
3.89555pt\lower-1.64pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69556pt}{2.25555pt}\pgfsys@curveto{3.69556pt}{4.29657pt}{2.04102pt}{5.95111pt}{0.0pt}{5.95111pt}\pgfsys@curveto{-2.04102pt}{5.95111pt}{-3.69556pt}{4.29657pt}{-3.69556pt}{2.25555pt}\pgfsys@curveto{-3.69556pt}{0.21454pt}{-2.04102pt}{-1.44pt}{0.0pt}{-1.44pt}\pgfsys@curveto{2.04102pt}{-1.44pt}{3.69556pt}{0.21454pt}{3.69556pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.07pt{\vbox to6.07pt{\pgfpicture\makeatletter\hbox{\hskip
3.03496pt\lower-1.45607pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.83496pt}{1.57889pt}\pgfsys@curveto{2.83496pt}{3.1446pt}{1.56572pt}{4.41385pt}{0.0pt}{4.41385pt}\pgfsys@curveto{-1.56572pt}{4.41385pt}{-2.83496pt}{3.1446pt}{-2.83496pt}{1.57889pt}\pgfsys@curveto{-2.83496pt}{0.01317pt}{-1.56572pt}{-1.25607pt}{0.0pt}{-1.25607pt}\pgfsys@curveto{1.56572pt}{-1.25607pt}{2.83496pt}{0.01317pt}{2.83496pt}{1.57889pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.57889pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.22499pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to4.94pt{\vbox to4.94pt{\pgfpicture\makeatletter\hbox{\hskip
2.47173pt\lower-1.34395pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.27173pt}{1.12778pt}\pgfsys@curveto{2.27173pt}{2.38243pt}{1.25465pt}{3.3995pt}{0.0pt}{3.3995pt}\pgfsys@curveto{-1.25465pt}{3.3995pt}{-2.27173pt}{2.38243pt}{-2.27173pt}{1.12778pt}\pgfsys@curveto{-2.27173pt}{-0.12688pt}{-1.25465pt}{-1.14395pt}{0.0pt}{-1.14395pt}\pgfsys@curveto{1.25465pt}{-1.14395pt}{2.27173pt}{-0.12688pt}{2.27173pt}{1.12778pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.12778pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.875pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$

Considering the first summation, it simplifies to the following equation which
follows from the i.i.d assumption on the class of distributions we consider

$\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sum
f^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t,0)}\left[\sum\frac{f^{p}(k+1,t+1,1)c^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t+1,1)}\right]-\\\ \frac{\sum f^{p}(k,t+1,1)}{\sum
f^{p}(k,t,0)}\left[\sum\frac{f^{p}(k,t+1,1)c^{p}(k,t+1,1)}{\sum
f^{p}(k,t+1,1)}\right]$ (10)

Utilizing the inductive assumption at stage $(t+1)$ on the monotonicity of
$u_{t+1}(k)$, we have that

$\begin{split}\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&\geq\left[\sum\frac{f^{p}(k,t+1,1)c^{p}(k,t+1,1)}{\sum
f^{p}(k,t+1,1)}\right]\overbrace{\left[\frac{\sum f^{p}(k+1,t+1,1)}{\sum
f^{p}(k+1,t,0)}-\frac{\sum f^{p}(k,t+1,1)}{\sum
f^{p}(k,t,0)}\right]}^{\mathchoice{\leavevmode\hbox to10.22pt{\vbox
to10.22pt{\pgfpicture\makeatletter\hbox{\hskip 5.10966pt\lower-2.67912pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.22pt{\vbox to10.22pt{\pgfpicture\makeatletter\hbox{\hskip
5.10966pt\lower-2.67912pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.90967pt}{2.43054pt}\pgfsys@curveto{4.90967pt}{5.1421pt}{2.71156pt}{7.34021pt}{0.0pt}{7.34021pt}\pgfsys@curveto{-2.71156pt}{7.34021pt}{-4.90967pt}{5.1421pt}{-4.90967pt}{2.43054pt}\pgfsys@curveto{-4.90967pt}{-0.28102pt}{-2.71156pt}{-2.47913pt}{0.0pt}{-2.47913pt}\pgfsys@curveto{2.71156pt}{-2.47913pt}{4.90967pt}{-0.28102pt}{4.90967pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to7.77pt{\vbox to7.77pt{\pgfpicture\makeatletter\hbox{\hskip
3.8853pt\lower-2.18391pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.6853pt}{1.70139pt}\pgfsys@curveto{3.6853pt}{3.73674pt}{2.03535pt}{5.38669pt}{0.0pt}{5.38669pt}\pgfsys@curveto{-2.03535pt}{5.38669pt}{-3.6853pt}{3.73674pt}{-3.6853pt}{1.70139pt}\pgfsys@curveto{-3.6853pt}{-0.33397pt}{-2.03535pt}{-1.98392pt}{0.0pt}{-1.98392pt}\pgfsys@curveto{2.03535pt}{-1.98392pt}{3.6853pt}{-0.33397pt}{3.6853pt}{1.70139pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.70139pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.27644pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.13pt{\vbox to6.13pt{\pgfpicture\makeatletter\hbox{\hskip
3.06718pt\lower-1.85191pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{2.86719pt}{1.21527pt}\pgfsys@curveto{2.86719pt}{2.79878pt}{1.58351pt}{4.08246pt}{0.0pt}{4.08246pt}\pgfsys@curveto{-1.58351pt}{4.08246pt}{-2.86719pt}{2.79878pt}{-2.86719pt}{1.21527pt}\pgfsys@curveto{-2.86719pt}{-0.36824pt}{-1.58351pt}{-1.65192pt}{0.0pt}{-1.65192pt}\pgfsys@curveto{1.58351pt}{-1.65192pt}{2.86719pt}{-0.36824pt}{2.86719pt}{1.21527pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.21527pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.62604pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{split}$

Analyzing the term $\textstyle 3b$, we have

$\begin{split}\mathchoice{\leavevmode\hbox to14.43pt{\vbox
to14.43pt{\pgfpicture\makeatletter\hbox{\hskip 7.21367pt\lower-3.74146pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to14.43pt{\vbox to14.43pt{\pgfpicture\makeatletter\hbox{\hskip
7.21367pt\lower-3.74146pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{7.01367pt}{3.47221pt}\pgfsys@curveto{7.01367pt}{7.3458pt}{3.87358pt}{10.48589pt}{0.0pt}{10.48589pt}\pgfsys@curveto{-3.87358pt}{10.48589pt}{-7.01367pt}{7.3458pt}{-7.01367pt}{3.47221pt}\pgfsys@curveto{-7.01367pt}{-0.40137pt}{-3.87358pt}{-3.54146pt}{0.0pt}{-3.54146pt}\pgfsys@curveto{3.87358pt}{-3.54146pt}{7.01367pt}{-0.40137pt}{7.01367pt}{3.47221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.47221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.64583pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.93pt{\vbox to10.93pt{\pgfpicture\makeatletter\hbox{\hskip
5.46465pt\lower-3.0341pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.26465pt}{2.43054pt}\pgfsys@curveto{5.26465pt}{5.33815pt}{2.90761pt}{7.69519pt}{0.0pt}{7.69519pt}\pgfsys@curveto{-2.90761pt}{7.69519pt}{-5.26465pt}{5.33815pt}{-5.26465pt}{2.43054pt}\pgfsys@curveto{-5.26465pt}{-0.47707pt}{-2.90761pt}{-2.8341pt}{0.0pt}{-2.8341pt}\pgfsys@curveto{2.90761pt}{-2.8341pt}{5.26465pt}{-0.47707pt}{5.26465pt}{2.43054pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.43054pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.25208pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 3b$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.59pt{\vbox to8.59pt{\pgfpicture\makeatletter\hbox{\hskip
4.29594pt\lower-2.55984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.09595pt}{1.7361pt}\pgfsys@curveto{4.09595pt}{3.99825pt}{2.26215pt}{5.83205pt}{0.0pt}{5.83205pt}\pgfsys@curveto{-2.26215pt}{5.83205pt}{-4.09595pt}{3.99825pt}{-4.09595pt}{1.7361pt}\pgfsys@curveto{-4.09595pt}{-0.52605pt}{-2.26215pt}{-2.35985pt}{0.0pt}{-2.35985pt}\pgfsys@curveto{2.26215pt}{-2.35985pt}{4.09595pt}{-0.52605pt}{4.09595pt}{1.7361pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.7361pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.3229pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
3b$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}&=\frac{\left(\sum
f^{p}(k+1,t+1,1)\right)\left(\sum f^{p}(k,t,0)\right)-\left(\sum
f^{p}(k,t+1,1)\right)\left(\sum f^{p}(k+1,t,0)\right)}{\left(\sum
f^{p}(k+1,t,0)\right)\left(\sum f^{p}(k,t,0)\right)}\\\
&\geq\frac{\sum_{\\{(p_{1},p_{2}):p_{1}<p_{2},p_{2}\neq
1\\}}\frac{Pr[P=p_{1}]Pr[P=p_{2}]}{OPT(p_{1})OPT(p_{2})}(p_{1}p_{2})^{k}\left[(1-p_{1})(1-p_{2})\right]^{t-k-1}(p_{1}-p_{2})^{2}}{\left(\sum
f^{p}(k+1,t,0)\right)\left(\sum f^{p}(k,t,0)\right)}\end{split}$

Analyzing term $\textstyle 4$, we obtain the following simplified equation
again following from the i.i.d assumptions on the class of distributions we
consider.

$\mathchoice{\leavevmode\hbox to10.96pt{\vbox
to10.96pt{\pgfpicture\makeatletter\hbox{\hskip 5.4793pt\lower-2.25708pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\displaystyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to10.96pt{\vbox to10.96pt{\pgfpicture\makeatletter\hbox{\hskip
5.4793pt\lower-2.25708pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{5.2793pt}{3.22221pt}\pgfsys@curveto{5.2793pt}{6.13792pt}{2.91571pt}{8.50151pt}{0.0pt}{8.50151pt}\pgfsys@curveto{-2.91571pt}{8.50151pt}{-5.2793pt}{6.13792pt}{-5.2793pt}{3.22221pt}\pgfsys@curveto{-5.2793pt}{0.3065pt}{-2.91571pt}{-2.05708pt}{0.0pt}{-2.05708pt}\pgfsys@curveto{2.91571pt}{-2.05708pt}{5.2793pt}{0.3065pt}{5.2793pt}{3.22221pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{3.22221pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textstyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to8.5pt{\vbox to8.5pt{\pgfpicture\makeatletter\hbox{\hskip
4.25005pt\lower-1.99449pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.05005pt}{2.25555pt}\pgfsys@curveto{4.05005pt}{4.49236pt}{2.2368pt}{6.3056pt}{0.0pt}{6.3056pt}\pgfsys@curveto{-2.2368pt}{6.3056pt}{-4.05005pt}{4.49236pt}{-4.05005pt}{2.25555pt}\pgfsys@curveto{-4.05005pt}{0.01875pt}{-2.2368pt}{-1.7945pt}{0.0pt}{-1.7945pt}\pgfsys@curveto{2.2368pt}{-1.7945pt}{4.05005pt}{0.01875pt}{4.05005pt}{2.25555pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.25555pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.75pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptstyle 4$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{\leavevmode\hbox
to6.89pt{\vbox to6.89pt{\pgfpicture\makeatletter\hbox{\hskip
3.44536pt\lower-1.83426pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.24536pt}{1.6111pt}\pgfsys@curveto{3.24536pt}{3.40347pt}{1.79237pt}{4.85646pt}{0.0pt}{4.85646pt}\pgfsys@curveto{-1.79237pt}{4.85646pt}{-3.24536pt}{3.40347pt}{-3.24536pt}{1.6111pt}\pgfsys@curveto{-3.24536pt}{-0.18127pt}{-1.79237pt}{-1.63426pt}{0.0pt}{-1.63426pt}\pgfsys@curveto{1.79237pt}{-1.63426pt}{3.24536pt}{-0.18127pt}{3.24536pt}{1.6111pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{1.6111pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\scriptscriptstyle
4$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}=\frac{\sum
f^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t,0)}\left[\sum\frac{f^{p}(k+1,t+1,0)c^{p}(k+1,t+1,0)}{\sum
f^{p}(k+1,t+1,0)}\right]-\\\ \frac{\sum f^{p}(k,t+1,0)}{\sum
f^{p}(k,t,0)}\left[\sum\frac{f^{p}(k,t+1,0)c^{p}(k,t+1,0)}{\sum
f^{p}(k,t+1,0)}\right]$ (11)

Utilizing the inductive assumption corresponding to the non-decreasing
property of $l_{t+1}(k)$, we have that

$\begin{split}\mathchoice{\leavevmode\hbox to10.96pt{\vbox