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\begin{document}

\title{Dynamic Relocation in Ridesharing via Fixpoint Construction (Supplementary material)}


\author[1]{Ian A. Kash}
\author[1]{Zhongkai Wen}
\author[1]{Lenore D. Zuck}
% Add affiliations after the authors
\affil[1]{%
    %Computer Science \\
    University of Illinois Chicago\\
    Chicago, IL, USA
}


\maketitle

\appendix
%\input{dynamic_experiment}
\section{Additional Simulation Details} \Appl{didi}
We use the dataset from Braverman et al. [2019],
%~\citet{braverman2019empty}, 
whose raw data is from the Di-Tech Challenge and covers 5-6 pm (evening rush hour) interval in each day between Jan 1-2016 until Jan 21-2016. 

The transition matrix $V$ induced from the raw data is in Table \ref{tab:transitionMatrix_braverman4}. The average travel times normalized for 10-min slots are in Table\ref{tab:traveltimeMatrix_braverman4}. The demand rates (normalized for a fleet size of 1) are in Table \ref{tab:demandrate_braverman4}.

For our unit-time setup, we adapt the non-uniform travel time by adding intermediary dummy regions for routes with longer travel time. So the travel times between real regions and dummy regions and between dummy regions are unit time. We accordingly adapt transition matrix $V$ for dummy regions. For the 9-region Didi data, we use 1.5 as the normalized travel time between, which corresponds to 15 mins, thus divide the travel times in Table \ref{tab:transitionMatrix_braverman4} by 1.5 and round to the nearest integer. This results in an extension of the 9-region problem to 48-regions when the dummy regions are included. Then we construct OA based on this 48 region problem.

\iffalse
\begin{table*}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Region& 10& 11& 18& 13& 19& 27& 45& 47& 50\\
\hline
10&  0.230& 0.297& 0.372& 0.004& 0.026& 0.029& 0.009& 0.018& 0.015\\
\hline
11&  0.044& 0.655& 0.146& 0.005& 0.079& 0.038& 0.018& 0.005& 0.011\\
\hline
18&  0.165& 0.291& 0.288& 0.007& 0.054& 0.126& 0.017& 0.025& 0.027\\
\hline
13&  0.0013& 0.010& 0.006& 0.139& 0.031& 0.185& 0.101& 0.117& 0.409\\
\hline
19&  0.005& 0.096& 0.026& 0.037& 0.25& 0.333& 0.218& 0.012& 0.027\\ 
\hline
27&	0.004& 0.031& 0.032& 0.088& 0.121& 0.426& 0.148& 0.059& 0.092\\ 
\hline
45&	0.002& 0.023& 0.011& 0.066& 0.142& 0.269& 0.399& 0.020& 0.069\\ 
\hline
47&	0.004& 0.008& 0.023& 0.067& 0.011& 0.095& 0.019& 0.400&  0.374\\ 
\hline
50&	0.001& 0.004& 0.005& 0.095& 0.010& 0.059& 0.030& 0.185& 0.610\\ 
\hline
\end{tabular}
\end{center}
\caption{Transition matrix V for 9 region DiDi Data \label{tab:transitionMatrix_braverman4}}
\end{table*}

\begin{table*}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Region& 10& 11& 18& 13& 19& 27& 45& 47& 50\\
\hline
10&  0.83& 1.87& 1.07& 3.89& 3.25& 2.79& 4.25& 2.94& 4.37\\
\hline
11&  1.78& 0.89& 1.18& 3.24& 1.24& 1.99& 2.89& 3.46& 4.18\\
\hline
18&  1.02& 1.31& 0.78& 2.82& 1.45& 1.36& 3.26& 2.17& 3.04\\
\hline
13&  3.52& 3.13& 2.76& 0.93& 1.5& 1.26& 1.49& 1.75& 1.6\\
\hline
19&  2.86& 1.42& 1.64& 1.55& 0.84& 1.04& 1.45& 2.88& 2.89\\ 
\hline
27&	2.61& 2.17& 1.54& 1.31& 1.15& 0.81& 1.86& 1.78& 2.2\\ 
\hline
45&	4.38& 3.02& 2.79& 1.36& 1.35& 1.65& 0.94& 3.1& 3\\ 
\hline
47&	2.93& 3.06& 2.26& 1.75& 2.69& 1.62& 3.23& 0.9& 1.48\\ 
\hline
50&	3.58& 4.18& 2.8& 1.49& 2.46& 2.02& 2.72& 1.43& 1.01\\ 
\hline
\end{tabular}
\end{center}
\caption{Normalized travel time matrix for 9 region DiDi Data \label{tab:traveltimeMatrix_braverman4}}
\end{table*}


\begin{table*}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Region& 10& 11& 18& 13& 19& 27& 45& 47& 50\\
\hline
rate &0.0131& 0.0624& 0.0381& 0.0652& 0.0870& 0.1178& 0.0762& 0.1438& 0.2751\\ 
\hline
\end{tabular}
\end{center}
\caption{Demand rate for 9 region DiDi Data \label{tab:demandrate_braverman4}}
\end{table*}
\fi

\begin{table*}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Region& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
\hline
1&  0.230& 0.297& 0.372& 0.004& 0.026& 0.029& 0.009& 0.018& 0.015\\
\hline
2&  0.044& 0.655& 0.146& 0.005& 0.079& 0.038& 0.018& 0.005& 0.011\\
\hline
3&  0.165& 0.291& 0.288& 0.007& 0.054& 0.126& 0.017& 0.025& 0.027\\
\hline
4&  0.0013& 0.010& 0.006& 0.139& 0.031& 0.185& 0.101& 0.117& 0.409\\
\hline
5&  0.005& 0.096& 0.026& 0.037& 0.25& 0.333& 0.218& 0.012& 0.027\\ 
\hline
6&	0.004& 0.031& 0.032& 0.088& 0.121& 0.426& 0.148& 0.059& 0.092\\ 
\hline
7&	0.002& 0.023& 0.011& 0.066& 0.142& 0.269& 0.399& 0.020& 0.069\\ 
\hline
8&	0.004& 0.008& 0.023& 0.067& 0.011& 0.095& 0.019& 0.400&  0.374\\ 
\hline
9&	0.001& 0.004& 0.005& 0.095& 0.010& 0.059& 0.030& 0.185& 0.610\\ 
\hline
\end{tabular}
\end{center}
\caption{Transition matrix V for 9 region Didi Data \label{tab:transitionMatrix_braverman4}}
\end{table*}

\begin{table*}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Region& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
\hline
1&  0.83& 1.87& 1.07& 3.89& 3.25& 2.79& 4.25& 2.94& 4.37\\
\hline
2&  1.78& 0.89& 1.18& 3.24& 1.24& 1.99& 2.89& 3.46& 4.18\\
\hline
3&  1.02& 1.31& 0.78& 2.82& 1.45& 1.36& 3.26& 2.17& 3.04\\
\hline
4&  3.52& 3.13& 2.76& 0.93& 1.5& 1.26& 1.49& 1.75& 1.6\\
\hline
5&  2.86& 1.42& 1.64& 1.55& 0.84& 1.04& 1.45& 2.88& 2.89\\ 
\hline
6&	2.61& 2.17& 1.54& 1.31& 1.15& 0.81& 1.86& 1.78& 2.2\\ 
\hline
7&	4.38& 3.02& 2.79& 1.36& 1.35& 1.65& 0.94& 3.1& 3\\ 
\hline
8&	2.93& 3.06& 2.26& 1.75& 2.69& 1.62& 3.23& 0.9& 1.48\\ 
\hline
9&	3.58& 4.18& 2.8& 1.49& 2.46& 2.02& 2.72& 1.43& 1.01\\ 
\hline
\end{tabular}
\end{center}
\caption{Normalized travel times for 9 region Didi Data \label{tab:traveltimeMatrix_braverman4}}
\end{table*}


\begin{table*}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
Region& 1& 2& 3& 4& 5& 6& 7& 8& 9\\
\hline
Rate &0.0131& 0.0624& 0.0381& 0.0652& 0.0870& 0.1178& 0.0762& 0.1438& 0.2751\\ 
\hline
\end{tabular}
\end{center}
\caption{Demand rate for 9 region Didi Data \label{tab:demandrate_braverman4}}
\end{table*}











\section{Additional Simulations} \Appl{simulations}

We examine the performance of our dynamic relocation policy, referred to as \dynamic here,  relative to several baselines.
In contrast to \Secr{didi}, these simulations use synthetic data with unit travel times to more closely match our theoretical results.
We simulate a range of parameters, initial conditions, and policies and show that \dynamic converges substantially faster, coming close to a lower bound.  We also define two performance metrics, efficiency and availability, and show that our approach matches the baseline performance in terms of efficiency while being more flexibly able to target availability.


\subsection{What we compare}
 Consider the dynamics $\widehat{M}_{t+1} = \sigc{F(\widehat{M}_t)}$.  These non-mass-conserving dynamics follow the drivers carrying passengers, discarding any who would relocate.  It follows, from the relocation constraint of dynamic relocation, that $\widehat{M}_t \leq M_t$ for any dynamic relocation policy $\{M_t\}_{t=0}$.  Thus, the lowest $t$ for which   $\widehat{M}_t\leq M^*$, if such a time exists, is a lower bound on the convergence time of any dynamic policy.  We refer to this lower bound as \lb.
 
 We then compare our \dynamic with \lb and three policies.  These are essentially the same as those from \Secr{didi}, with minor variations due to the uniform distances.

\begin{description}
    \item[\static.]  This is a static policy that  sets $\pi_t = \pi$ for all $t$.  From \Crlr{limit} it follows that \static  guarantees convergence to the fixpoint, yet, as we pointed out, it may do so slowly.  Thus \static represents a baseline in the absence of a more sophisticated dynamic policy.
    \item[\greedy.] This is a greedy policy that  distributes the relocating mass proportional to the unmet demand in each region with a one-step look ahead. That is, it takes $\pi_t[i] \propto \maxz{\sigr{W} - \sigc{F(M_t)}}$, which   guarantees as many relocating drivers as possible will have a passenger at time $t+1$ while spreading them among the regions where they can be useful.  As \greedy does not depend on $\pi$, it may not converge to the fixpoint, but it does provide a meaningful baseline for other metrics based purely on the provision of service.
    \item[\hmr.] We adapt the dynamic policy of %\citet{hosseini2021dynamic}, 
    [Hosseini et al., 2021],
    which dispatches a single car at a time, to our setting.
    In particular, their algorithm computes a measure of which region will generate the most long-run service and sends the car there.  Since the results of this computation do not change until a region is saturated, we adapt their policy by assigning relocation drivers to this region until (a) it becomes saturated or (b) the mass reaches the fixpoint mass of the region.  While their policy is heuristic, with the inclusion of (b) it can often achieve convergence in our simulations.
    %0relocating the mass to the region with greatest gradient in regards to the current location.  In details,  Matrix $B$ can be obtained in the following way.  First, Matrix $T_W^{sat}$ is achieved by removing the rows and columns in $T_W$ from saturated regions; second, we calculate the inverse of $I-T_W^{sat}$. Then $B$ is achieved by adding back zeros to where rows and columns are removed into the inverse. For every saturated region $i$, the region $j^*$ where mass drivers in region $i$ relocate to should satisfy the following.
    %\begin{align*}\nonumber
    %j^* = \argmax_{\ell \neq i} %\sum_k(B[\ell,k] - B[i,k])
    %\end{align*}
    %In each duration, the passenger-less drivers relocate to region $j^*$ until either no more passenger-less  drivers or fixed point is reached in region $j^*$. We regard region $j^*$ as saturated. If there are passenger-less drivers left, we repeat above procedure and calculate new $j^*$. If all the regions are regarded as saturated and there are still passenger-less drivers, we uniformly distribute these drivers proportional to difference of current mass distribution and the fixed point.
    %\item Constructive policy, based on the policy we can prove its guarantee. In each step, following constructive policy is provable to be closer to the optimal fixed point.
 
\end{description}

\subsection{Simulation Design}

For each datapoint in each of our experiments we generate $40$ different demand matrices $W$ according the specified distribution and solve the resulting optimal allocation problem (\Figr{OA}) to determine $\pi$ and $M^*$ for each $W$.
We then choose $20$ different $M_0$ uniformly at random.  We run each dynamic policy for $50$ steps starting from each $M_0$ and report the results averaged across the choices of $W$ and $M_0$. Thus each point is an average of 800 runs.
We compare the performance of the various policies according to the following metrics, all of which have been considered in prior work on dynamic relocation
%~\citep{braverman2019empty,hosseini2021dynamic}
[Braverman et al., 2019, Hosseini et al., 2021].
\begin{itemize}
    \item Convergence Time. The number of steps of the dynamics until the ratio of current mass in a region to the fixpoint mass in that region is at most $1+10^{-6}$ for every region.
    \item Efficiency. The total number of full rides as a fraction of the total demand.  In our stylized model this captures both the total value created for passengers and the revenue of the platform.
    \item Availability. The average efficiency on a per-region basis.  That is, the ratio of full rides in a region to the demand in that region averaged across regions.  High availability ensures some degree of fairness between regions 
\end{itemize}

\subsection{Results}
%*****************************
%10 region
\iffalse
\begin{figure*}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/m_uniform/L4_optimalm_uniform_size10_allsteps.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/m_uniform/L4_optimalm_uniform_size10_alleff.png}
    \end{subfigure}     
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/m_uniform/L4_optimalm_uniform_size10_allavail.png}
    \end{subfigure}
    \caption{Performance with uniform demand; Efficiency objective \Figl{10-m-u}}
\end{figure*}
\begin{figure*}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_ uniform_PA (pow_10random)/L1_optimalf_uniformPA10power_size10_allsteps.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_ uniform_PA (pow_10random)/L1_optimalf_uniformPA10power_size10_alleff.png}
    \end{subfigure}     
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_ uniform_PA (pow_10random)/L1_optimalf_uniformPA10power_size10_allavail.png}
    \end{subfigure}
    \caption{Performance with correlated demand; Half efficiency and half fairness objective \Figl{10-f-p}}
\end{figure*}
\begin{figure*}
    \centering
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/m_uniform/L4_optimalm_uniform_size10_welfarelarge.png}
        \caption{}
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_ uniform_PA (pow_10random)/L1_optimalf_uniformPA10power_size10_welfarelarge.png}
        \caption{}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/m_uniform+PA (pow 10random)/L3_optimalm_uniformPA10power_size10_welfarelarge.png}
        \caption{}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_uniform/L2_optimalf_uniform_size10_welfarelarge.png}
        \caption{}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_uniform_PA _old_version/L0_optimalf_10region_PAoldpower_welfarelarge.png}
        \caption{}        
    \end{subfigure}    
    \caption{Relative availability of \dynamic to \hmr  \Figl{ai}}
\end{figure*}
\fi
%40 regions
\begin{figure*}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\linewidth]{Figures/40region_optimalm_uniform/40Region_uniform_optimalm_allsteps.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\linewidth]{Figures/40region_optimalm_uniform/40Region_uniform_optimalm_alleff.png}
    \end{subfigure}     
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\linewidth]{Figures/40region_optimalm_uniform/40Region_uniform_optimalm_allavail.png}
    \end{subfigure}
    \caption{Performance with uniform demand; Efficiency objective \Figl{40-m-u}}
\end{figure*}
\begin{figure*}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalf_PA20power/40Region_preferentialAttachment20_optimalf_allsteps.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalf_PA20power/40Region_preferentialAttachment20_optimalf_alleff.png}
    \end{subfigure}     
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalf_PA20power/40Region_preferentialAttachment20_optimalf_allavail.png}
    \end{subfigure}
    \caption{Performance with correlated demand; Half efficiency and half fairness objective \Figl{40-f-p}}
\end{figure*}
\begin{figure*}
    \centering
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalm_uniform/40Region_uniform_optimalm_welfarelarge.png}
        \caption{}
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalf_PA10power/40Region_preferentialAttachment10_optimalf_welfarelarge.png}
        \caption{}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalm_PA10power/40region_optimalm_PA10power_welfarelarge.png}
        \caption{}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalf_uniform/40region_L2_optimalf_uniform_welfarelarge.png}
        \caption{}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/40region_optimalf_PA40power/40Region_preferentialAttachment40_fairness_welfarelarge.png}
        \caption{}        
    \end{subfigure}    
    \caption{Relative availability of \dynamic to \hmr \Figl{ai}}
\end{figure*}
\Figr{40-m-u} shows the performance with all three of our metrics with 40 regions when the demand between each pair of regions is i.i.d. uniform, the supply of drivers is determined as a multiple of the total demand, and the policy is chosen by solving the optimization from \Figr{OA}.
At high levels of supply of drivers relative to demand (above a ratio of about 1.1) \dynamic, \hmr, and \greedy  are all able to control the relocation of enough drivers to satisfy essentially all of the demand.  Because so many drivers relocate, \lb is essentially 1 and both \dynamic and \hmr  achieve it.  \static converges slowly and has worse efficiency and availability, showing the value of dynamic policies.  As previously remarked,  \greedy does not in general converge to the fixpoint, and typically times out by reaching the 50 step limit.
At very low ratios of supply to demand, the dynamics are dominated by the full rides.  Thus \lb  is larger and even \greedy converges.
At intermediate ratios, \dynamic converges substantially faster than the other approaches.  Its performance is  close to that of \lb, but fails to match it because it only looks ahead a single step. 
The effects of \dynamic on efficiency and availability relative to \hmr and \greedy are minimal.

In \Figr{40-f-p}, we change the setup of the experiment: (1)  We introduce correlation between the demands from various regions by first choosing the total demand in each region i.i.d.~uniform and then assigning that demand to each destination proportional to the total demand in the destination raised to a random exponent determined for each source region independently and uniformly from $[0,10]$.  (2) We change the objective of the optimization from maximizing $\mb{F}$ to maximizing $0.5 \mb{F} + 0.5 \min_{i} \sigr{F}[i]$.  This shifts the objective from purely efficiency into a hybrid of efficiency and a (somewhat minimal) fairness criterion of maximizing welfare of the least-served region. 
 The overall shape of the results is similar to \Figr{40-m-u}, 
 with  a small improvement in availability relative to \hmr and \greedy.  Because the policy aims at fairness and not just efficiency, \dynamic has a tendency to do so as well.  In contrast, \hmr and \greedy are inherently efficiency-focused, yet we still match their performance on that metric which achieving the improved availability.

\Figr{ai} focuses specifically on the percentage change in availability of \dynamic  relative to \hmr. Subfigures (a) and (b) correspond to \Figr{40-m-u} and \Figr{40-f-p} respectively.  In (a) \dynamic typically performs better than \hmr, though the effect is very small, substantially less than $1\%$. In contrast (b) shows a much larger effect, a $5\%$ improvement in performance in a meaningful range where supply is somewhat less than demand.\footnote{\dynamic performs a bit worse than \hmr when the supply of drivers is sharply limited and the efficiency-focused approach appears beneficial.  However, this is not the regime ridesharing platforms strive to operate in.}  Correlated demand alone (c) or the fairness objective alone (d) do not show this large benefit.  Making the demand perfectly correlated, by making it exactly proportional to the total demand in the destination, shows benefits of nearly $10\%$.  Overall, these results show that in the more plausible ranges of supply and demand \dynamic can more effectively achieve a non-efficiency objective and are suggestive of the size of the benefit being driven by the correlation of the demand pattern.

\begin{figure*}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_uniform_PA10power_size2to50_supplydemand_1/L6_optimalf_uniformPA10power_2to50_supply1_20rep_allsteps.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_uniform_PA10power_size2to50_supplydemand_1/L6_optimalf_uniformPA10power_2to50_supply1_20rep_alleff.png}
    \end{subfigure}     
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/f_uniform_PA10power_size2to50_supplydemand_1/L6_optimalf_uniformPA10power_2to50_supply1_20rep_allavail.png}
    \end{subfigure}
    \caption{Performance varying number of regions \Figl{regions}}
\end{figure*}
\begin{figure*}
    \centering
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/5region_demand_correlation_investigation/optimalf_5region_meancoefpoint05_welfarelarge.png}
        \caption{correlation 0.05}
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/5region_demand_correlation_investigation/optimalf_5region_meancoefpoint25_welfarelarge.png}
        \caption{correlation 0.25}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/5region_demand_correlation_investigation/optimalf_5region_meancoefpoint45_welfarelarge.png}
        \caption{correlation 0.45}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/5region_demand_correlation_investigation/optimalf_5region_meancoefpoint75_welfarelarge.png}
        \caption{correlation 0.75}        
    \end{subfigure}
    \begin{subfigure}{0.19\textwidth}
        \includegraphics[width=\textwidth]{Figures/5region_demand_correlation_investigation/optimalf_5region_meancoefpoint95_welfarelarge.png}
        \caption{correlation 0.95}        
    \end{subfigure}    
    \caption{Relative availability of \dynamic~to \hmr~for demand patterns of different correlation levels \Figl{correlation}}
\end{figure*}
%10 regions
\begin{figure*}
    \centering
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/a_uniform/L9_optimala_uniform_size10_allsteps.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/a_uniform/L9_optimala_uniform_size10_alleff.png}
    \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
        \includegraphics[width=\textwidth]{Figures/a_uniform/L9_optimala_uniform_size10_allavail.png}
    \end{subfigure}
    \caption{Performance with correlated demand; availability objective \Figl{10-a-u}}
\end{figure*}
\begin{figure*}
    \centering
  \begin{subfigure}{0.3\textwidth}
        \includegraphics[width=\textwidth]{Figures/2-TO_50_100steps/L7_optimalf_uniformPA10power_2to50_supply1_20rep_allavail.png}
   \end{subfigure}
    \begin{subfigure}{0.3\textwidth}
      \includegraphics[width=\textwidth]{Figures/2-TO_50_100steps/L7_optimalf_uniformPA10power_2to50_supply1_20rep_alleff.png}
   \end{subfigure}
    \caption{Performance varying demand \Figl{demand}}
\end{figure*}



Finally, we provide the results of several additional experiments.  These are:
\begin{itemize}
    \item An experiment that examines how our results for the correlated demand pattern and hybrid objective depend on the number of regions (using a ratio of 1 between supply and demand) and shows that the benefits do not significantly depend on the number of regions except for very small values of $r$ (\Figr{regions}).
    \item An experiment that examines how correlation level of demand patterns affects the relative availability of \dynamic compared to \hmr, and shows that the relative availability of \dynamic is positively related with the correlation level of demand patterns. The higher the correlation level of demand patterns, the better relative performance of \dynamic over \hmr (\Figr{correlation}).
    \item An experiment with a third objective, directly optimizing for availability, which leads to similar results as our fairness objective (\Figr{10-a-u}).
     \item An experiment in the spirit of our second experiment on the DiDi data that runs each policy for 100 steps with demand pattern changing every 5 steps, to represent changes in demand over the course of a day (\Figr{demand}).
\end{itemize}

To summarize the results of our experiments, we have seen that \dynamic consistently converges substantially faster than other policies and has a performance that is often close to or matching \lb.  The effect of this on efficiency relative to the other policies is, however, quite small.  When targeting a policy that puts weight on fairness rather than just efficiency, \dynamic leads to economically meaningful improvements in availability, showings its ability to target a wider range of objectives than previous approaches.
%############################################################
%\input{experiment_appendix}
%\input{proofs}
\section{Omitted Proofs}\Appl{proofs}

\subsection*{Proof of \Lemr{delJ}}
If suffices to show that for $M' = M + q\pi$, $M'=\next{M'}$.  A region $i\in U_M$ is in $Z_\pi$ by
assumption, hence $M'[i]=M[i]$ and,
since  $i\in Z_{\strategy}$, $\mathit{out}(M')[j,i] = \mathit{out}(M)[j,i]$ for every $j\not\in U_M$.
Thus $\next{M}[i] = \next{M'}[i]$. 

For a region $i$ not in $U_M$, $M'=M + q\pi[i]$.  While non-$U_M$ regions contribute no additional flow into $i$, 
each other region $j$ contributes  $q\pi[j] \pi[i]$ additional flow into $i$. 
Since $\sum_{j\not\in U_M} \pi[j]\pi[i] = \pi[i]$, 
hence the additional flow into $i$ is $q\sigma[i]$.  \QED


\subsection*{Proof of \Thr{alg_correct}}
The proof is by showing that the following is an inductive invariant\footnote{An \emph{inductive invariant} is an assertion that holds upon entry to the loop, and if it holds at the beginning of the loop then it holds when the loop is next re-entered.} of the while loop: 
%$$\varphi\colon   \curM = \next{\curM} \band \bm{\curM}
%= q - \curm \band \vert U_M\vert \le i\band U_M\not\subseteq %Z_{\strategy} \to \er{T_M} $$
\begin{align*} 
\varphi\colon   \curM = \next{\curM} &\band \bm{\curM}= q - \curm\\
&\band \vert U_M\vert \le i\band U_M\not\subseteq Z_{\strategy} \to \er{T_M}
\end{align*}

Termination of the loop trivially follows since $i$, being an upper bound of the size of $U_M$, cannot go below 0.

Upon entering the loop for the first time, $\curM$ is the all-0 vector and $\curm = q$ hence $M=\next{M}$,  $\bm{\curM} = q - \curm$, $\vert U_M \vert = r = i$, and $\er{T_M}$, which establishes the base case. 

Assume now a new iteration that starts when $\varphi$ holds.  Obviously, $\curm>0$.  We now distinguish between two cases, the first where $U_M\subseteq Z_\pi$, which in fact can be true only before the last iteration, and the case where $U_M\not\subseteq Z_\pi$ where the loop may terminate or enter a new iteration with a lower $i$.

In the first case, since $\sigma$ is set to $\pi$ and $\tmin$ is set to $\curm$,  according to \Lemr{delJ}, after line \ref{Mupdate} is executed, 
$M =  \next{M}$, $\bm{M}= q$ and $\curm=0$. Hence the new $M$ is the desired one and the loop terminates. 

In the second case where there are unsaturated regions not in $Z_\pi$, the induction hypothesis implies that  $\er{T_M}$. 
Line \ref{fix} calls an \emph{external} procedure to compute, in $\sigma$, the unique fixpoint of $T_M$,
and Line \ref{dist} assigns to 
$\tmin(>0)$ the maximum available mass that can be distributed among the regions so to avoid an unsaturated
becoming oversaturated.  Hence, after $\tmin$ is distributed according to $\sigma$, at least one additional region becomes saturated. 
From \Lemr{inc_q}  it follows that $M+\tmin \sigma $, whose total mass is $\bm{M}+\tmin$, is fixpoint mass distribution. 
Line \ref{Mupdate} updates $\curM$ and, implicitly, $U_M$.  Thus $\varphi$, with the updated variables, holds also at the end of the iteration.
If the new $\curm=0$, then $\varphi$ (over the new variables) imply that $\curM = \next{\curM}$
and $\bm{\curM} = q$.  Otherwise, another iteration is executed. 
\QED

\subsection*{Proof of \Lemr{we_need}}
Consider a region $i$. From the definition of $\mathit{next}$ and the monotonicity of $\mathit{out}$ in $M$ (for every $i$ and $j$) it follows that $\next{M_0}[i] =\next{M_1}[i]$ iff  $\mathit{out}(M_0)[j,i] = \mathit{out}(M_1)[j,i]$ for every region $j$.
It thus suffices to prove that  for every region $j$ such that $M_0[j] < M_1[j]$, 
$$\mathit{out}(M_0)[j,i] = \mathit{out}(M_1)[j,i] \quad \mbox{iff} \quad i \in Z_{\strategy}~\mbox{and}~ j\not\in J$$

%In one direction assume, by way of contradiction, that $j \in J$. 
%Hence some (exactly $V[j,i](\min(M_1[j],\bm{W}[j,])-M_0[j])$) of $j$'s  additional outflow is directed towards $i$, which contradicts that assumption that $\mathit{out}(M_0)[j,i] = \mathit{out}(M_1)[j,i]$.
%It thus follows that $j \not\in J$. But then, $j$'s additional outflow is distributed according to $\strategy$, and since $i$ receives none we can conclude that $\strategy[i]=0$. 
%
%In the other direction, since $J=\emptyset$ and $M_0 < M_1$, there are some saturated regions $j$ for which $M_0[j]<M_1[j]$
%Since these regions contribute no additional drivers to $i$ it follows that $i \in Z_{\strategy[i]}$.   \QED

For $\Rightarrow$, assume for contraposition that 
$j \in J$. Since $M_0[j] < M_1[j]$, $V[j,i](\min(M_1[j],\sigr{W}[j])-M_0[j])$) of $j$'s  additional outflow is directed towards $i$ implying that $\mathit{out}(M_0)[j,i] \neq \mathit{out}(M_1)[j,i]$. If, however, $j\not\in J$,
$j$'s additional outflow is distributed according to $\strategy$, and since $i \not \in Z_\pi$,   $\pi[i]$ fraction of it towards $i$ implying $\mathit{out}(M_0)[j,i] \neq \mathit{out}(M_1)[j,i]$.

For $\Leftarrow$, since $j \not\in J$ the additional outflow is directing according to $\pi$.  Since $i \in Z_\pi$, $\mathit{out}(M_0)[j,i] = \mathit{out}(M_1)[j,i]$. \QED

\subsection*{Proof of \Thr{unique}}
%\noindent{\bf Proof of \Thr{unique}: }%The function %$\mathit{next}$ is continuous (\Obr{next_cont}) 
%and $S_q$ is compact and convex. From Brouwer's fixpoint theorem, 
%$\mathit{next}$ has a least one fixpoint in $S_q$.  
From \Thr{alg_correct}, such a fixpoint exists.
It suffices to show that the fixpoint is unique. 

Assume, by way of contradiction, that there are two distinct such fixpoints, say $\widehat{M_0}$ and $\widehat{M_1}$. 
Let $\widehat{M} = \min(\widehat{M_0}, \widehat{M_1})$. Since $\widehat{M_0}$ and $\widehat{M_1}$ are distinct fixpoints in $S_q$,
$\widehat{M}$ is not in $S_q$. 

Let 
$$ \lattice ~=~ \{M \in (\mathbb{R}_{\ge 0} \cup \{\infty\})^r:  \widehat{M} \le M\}  $$
With the vector (point-wise) $\le$ as a preorder, and vector (point-wise) $\min$ (resp.~$\max$) as meet (resp.~join), 
$\lattice$ is a complete lattice. 

While $\lattice$ is complete lattice, it may not be closed under $\mathit{next}$. 
We thus define $\aux=\max(M, \next{M})$ as an auxiliary function under which $\lattice$ is closed. 
%ian01*:added
%ldz01*:moved to main body; we need a raison d'etre for g
%\footnote{We use $g()$ here because $\next$ is not necessarily monotonic, as e.g. $\next{\hat{M}}$ need not be in $\lattice$.}
From the Knaster-Tarski theorem it follows that $\aux$ has a set of fixpoints in $\lattice$, and that the set
of $\aux$'s fixpoints in $\lattice$ is a complete lattice. 

Both $\widehat{M_0}$ and $\widehat{M_1}$ are fixpoints of $\mathit{next}$ in $S_q \subseteq \lattice$, 
as well as fixpoints of $\aux$ in $\lattice$.
Since $\widehat{M} = \widehat{M_0} \sqcap \widehat{M_1}$ is  the minimum element of $\lattice$, 
it is the least fixpoint of $\aux $ in $\lattice$.


%Recall the ``willingness" matrix $W$ of the number or riders willing to take a ride in between each two regions given the demand $D$ and the pricing $p$. 
Define $I_0 = \{ i : \widehat{M_0}[i] <  \widehat{M_1}[i]\}$ and $I_1 =  \{ i : \widehat{M_0}[i] >  \widehat{M_0}[i]\}$,
that is, for every $i \in I_0$ , $\widehat{M}[i] =  \widehat{M_0}[i]$, for every $i \in I_1$ , $\widehat{M}[i] =  \widehat{M_1}[i]$,
and for every $i \not\in I_0 \cup I_1$,  $\widehat{M}[i] =  \widehat{M_0}[i] =  \widehat{M_1}[i]$. 
Recall that $Z_\pi$ is the set $\{ j : \pi[j]=0\}$.  Obviously, $I_0$ and $I_1$ are disjoint and neither is empty. 
%ian02*:
%By construction, $\widehat{M_0}, \widehat{M_1} \lneq \widehat{M}$, and 
By construction, $\widehat{M_0}, \widehat{M_1} \gneq \widehat{M}$, and all three are fixpoints of 
$\mathit{next}$. It then follows from \Lemr{we_need} that 
for every $i \in I_0 \cup I_1$, $\widehat{M}[i]$ is saturated in $\widehat{M}$ and therefore in both $\widehat{M_0}$
and $\widehat{M_1}$. It then follows from  \Lemr{we_need} that $Z_\pi$ is pairwise disjoint from both $I_0$ and $I_1$. 

If $Z_\pi$ is empty, it follows from  \Lemr{we_need} that
%$\widehat{M}[j]  ~=~ \next{\widehat{M}}[j]$ is strictly less than both  $\widehat{M_0}[j]$ and $\widehat{M_1}[j]$, 
$\widehat{M}$ is strictly less than both  $\widehat{M_0}$ and $\widehat{M_1}$, 
which is obviously a contradiction. Assume therefore that $Z_\pi$ is nonempty. 

%We next show that for every $i \in I_k$, $k\in \{0,1\}$, for every $j \in Z_\pi$,  and in particular of $j=i$, 
%$\widehat{M}[j] < \widehat{M_k}[j]$ which implies a contradiction. 
%
%Without loss of generality, let $k=0$. Consider $i\in I_0$. Since region $i$ is saturated in both $\widehat{M_0}$, and therefore in  $\widehat{M}$,
%it follows from \Lemr{we_need} that for every $j \not \in Z_\pi$, $\next{\widehat{M}}[j] < \next{\widehat{M_0}}[j] $.
%Since $I_0$ is disjoint from $J$, it follows that 
%$\widehat{M}[i]  < \widehat{M_0}[i]$, contradicting the assumption that $i \in I_0$.  \QED

Now consider some $i \in I_0$.  Since region $i$ is saturated in both $\widehat{M_0}$ and $\widehat{M}$,
it follows from \Lemr{we_need} that for every $j \not \in Z_\pi$, $\next{\widehat{M}}[j] < \next{\widehat{M_0}}[j] $.
Since $I_0$ is disjoint from $Z_\pi$, it follows that 
$\widehat{M}[i]  < \widehat{M_0}[i]$, contradicting the assumption that $i \in I_0$.  \QED

\subsection*{Proof of \Lemr{property1}}
Assume $\langle F_0,E_0 \rangle$, $i_0$, and $j_0$ as in the lemma statement.
We construct $\langle F,E \rangle$ that is also feasible for $q$, but with
$\bm{F} > \bm{F_0}$, thus showing that $\langle F_0,E_0 \rangle$ is nonoptimal. 

Let $x$ be the vector induced by $\min (E_0[i_0,j_0], (\sigr{W-F_0)}[i_0] )$ given
$V[i_0]$. 

Define $F_x$ be such that $F_x[i_0]=x$  and 0 elsewhere, and 
$E_x$ such that:
$$ E_x[i,j] ~=~ \left\{ 
       \begin{array}{ll}
		x[i] & \mbox{$i \ne i_0, j= j_0 $} \\
		\bm{x} - x[i_0]  & \mbox{$i= j_0 \ne  i_0 = j$} \\ 
		0 & \mbox{otherwise}
	\end{array}\right.$$ 
We claim that $\langle F_x, E_x\rangle$ feasible for $2\bm{x}-x[i_0]$ if $i_0 = j_0$ and  $3\bm{x}-2x[i_0]$ otherwise:
The D- and P- constraints are satisfied by construction. As for the F-constraint, 
every region $i$, $i \ne i_0,j_0$, receives $x[i]$ full cars and sends $x[i]$ empty cars. 
Region $i_0$ sends $\bm{x}$ full cars and receives $x[i_0]$ cars full from itself,
and (whether or not $i_0 = j_0$),  $\bm{x}-x[i_0]$ empty cars from other regions. Finally, 
when $j_0 \ne i_0$, region $j_0$
receives $x[j_0]$ full cars from $i_0$ and $x[i]$ empty cars from every region $i$, $i\ne i_0$, 
thus it receives $\bm{x}-x[i_0]+x[j_0]$ cars in total, which is exactly the number of cars it sends ($\bm{x}-x[i_0]$ empty, to $i_0$ and $x[j_0]$ empty to $j_0$). 
 

Let $F_1 = F_0 + F_x$, and $E_1=E_0+E_x$. 
The choice of $x$ guarantees that $F_1 \le W$, hence it follows from \Obr{feas2} 
that $\langle F_1, E_1\rangle$ is  feasible for $q + 2\bm{x}-x[i_0]$ (or $q + 3\bm{x}-2[i_0]$ if $i_0 \neq j_0$).
Let $y = \bm{x} - x[i_0]$ ($y = 2\bm{x} - 2x[i_0]$). Obviously, $y > 0$. 
If $i_0=j_0$, then $E_0[i_0,i_0]\ge \bm{x} > y$.
If $i_0\ne j_0$,  then $E_0[i_0,j_0]\ge \bm{x} > y/2$ and $E_x[j_0,i_0] = \bm{x} - x[i_0] \geq y/2$.
It then follows from \Obr{feas2} that $\langle F_2, E_2\rangle$ where
$F_2=F_1$ and $E_2$ is just like $E$ with $y$ subtracted from $E[i_0,i_0]$ (in the case of $i_0=j_0$)
or $y/2$ subtracted from both $E_1[i_0,j_0]$ and $E_1[j_0,i_0]$ (in the case of $i_0\ne i_j$)
is feasible for $q+\bm{x}$. 

Let $c = q/(q+\bm{x})$, $F=c\cdot F_2$ and $E=c\cdot E_2$. 
From \Obr{feas1} it follows that $\langle F, E\rangle$ is feasible for $q$. 
Moreover, 
$$\bm{F} = c(\bm{F_2}) = c(\bm{F_0}+\bm{x}) = \frac{q\bm{F_0}+q\bm{x}}{q+\bm{x}} > \bm{F_0}$$
where the last inequality follows from $q > \bm{F_0}$ (which holds since $\bm{E_0}>0$.)

Consequently, $\langle F,E\rangle$ is feasible for $q$ and $\bm{F} > \bm{F_0}$ so that
$\langle F_0, E_0\rangle$ is not optimal.
\QED

\subsection*{Proof of \Lemr{psi>0}}

Let $c = \min_{i,j}(V[i,j])$. 
If $\mb{E(M_t)} > 0$, then $\maxz{\nu(q) - \sigc{F(M_t)}}$ is monotone increasing in $\nu(q)$ on an open interval including $q_{t+1}$.
%In \Appr{proofs} 
Subsequently
we show that 
%$$\sum_i\maxz{\nu(q_t+c\Delta(t))-\sigc{F(M_t)}}[i]
%\leq \sum_i\maxz{[\nu(q_{t+1}) - \sigc{F(M_t)}]}[i]$$
\begin{multline*}
\sum_i\maxz{\nu(q_t+c\Delta(t))-\sigc{F(M_t)}}[i]\\
\leq \sum_i\maxz{[\nu(q_{t+1}) - \sigc{F(M_t)}]}[i]
\end{multline*}

Monotonicity then implies that 
$\nu(q_{t+1}) \geq \nu(q_t+ c \Delta(t)$, and therefore
$q_{t+1} - q_t > c \Delta(t)$.

If $\mb{E(M_t)} = 0$, there is no relocation and we cannot apply the previous argument because we only know that $\maxz{\nu(q) - \sigc{F(M_t)})}$ is monotone non-decreasing. % In \Appr{proofs}
Subsequently
we show that then  $M_{t+1}[i] - \nu(q_t)[i] \geq  c\Delta(t)$.
%
Thus $M_{t+1} = \nu(q_t) + (M_{t+1} - \nu(q_t)) \geq \nu(q_t) + c\Delta(t)\vecc{1} \geq \nu(q_t + c\Delta(t))$ and $q_{t+1} \geq q_t + c\Delta(t)$. 
\QED

\subsection*{Proof of \Lemr{psi>0} for the case that $\mb{E(M_t)} > 0$}
%\small{
\[ \begin{array}{l}
\sum_i\maxz{\nu(q_t+c\Delta(t))-\sigc{F(M_t)}}[i] \\
\;\begin{array}{l}
\stackrel{(1)}{\leq} \sum_i\maxz{(\nu(q_t)+c\Delta(t)\ones-\sigc{F(M_t)})}[i]\\
\stackrel{(2)}{=} \sum_i\maxz{\sigc{F(\nu(q_t))}+\sigc{E(\nu(q_t))}+c\Delta(t)\ones-\sigc{F(M_t)}}[i] \\
\stackrel{(3)}{=}  \sum_i\lsq\sigc{F(\nu(q_t))}+\sigc{E(\nu(q_t))}+c(\mathbf{F(M_t)}-\mathbf{F(\nu(q_t))})\vecc{1} ~ + \\ 
\multicolumn{1}{r} {c(\mathbf{E(M_t)}-\mathbf{E(\nu(q_t))})\ones-\sigc{F(M_t)}\rsq_{_+}[i]}\\ 
\stackrel{(4)}{\le}  \sum_i\lsq\sigc{F(\nu(q_t))}+\sigc{E(\nu(q_t))}+\sigc{F(M_t)}-\sigc{F(\nu(q_t))}~+ \\
\multicolumn{1}{r}{c(\mathbf{E(M_t)}-\mathbf{E(\nu(q_t))})\ones-\sigc{F(M_t)})\rsq_{_+}[i]}\\
\stackrel{(5)}{=} \sum_i\maxz{\sigc{E(\nu(q_t))}+c(\mathbf{E(M_t)}-\mathbf{E(\nu(q_t))})\ones}[i]\\
\stackrel{(6)}{=} \mathbf{E(\nu(q_t))}+c(\mathbf{E(M_t)}-\mathbf{E(\nu(q_t))})\\
\stackrel{(7)}{<} \mathbf{E(M_t)}\\
\stackrel{(8)}{=} \sum_i\maxz{\nu(q_{t+1}) - \sigc{F(M_t)}}[i]

\end{array}
\end{array}
\] 
%}

(1) follows from the monotonicity of $\nu$; 
(2) follows  since $\nu(q_t)$ is a  fixpoint; 
(3) follows from the expansion $\Delta(t) =\mathbf{M_t}- q_t$, 
followed  mass conservation; 
(4) follows by the definition of $c$, $\sigc{F(M_t)}$ and $\sigc{F(\nu(q_t))}$;
In particular, $(\sigc{F(M_t)}-\sigc{F(\nu(q_t))})[i] = \sum_j (\sigr{F(M_t)}[j]-\sigr{F(\nu(Q_t))}[j])V[j,i] \ge  \sum_j (\sigr{F(M_t)}[j]-\\ \sigr{F(\nu(Q_t))}[j]) \min_{k,\ell} V[k,l]  = c (\mathbf{F(M_t)}-\mathbf{F(\nu(q_t))})$;
(5)  follows by cancellation;
(6) follows because $\sigc{E(\nu(q_t))}\geq \vecc{0}$, and $M_t\ge \nu(q_t)$ and so $E(M_t)\ge E(\nu(q_t))$; 
(7) follows because $c < 1$ and $E(M_t) \ge E(\nu(q_t))$; (8) is by definition of $q_{t+1}$.

\subsection*{Proof of \Lemr{psi>0} for the case that $\mb{E(M_t)} = 0$.}
\[
\begin{array}{l}
M_{t+1}[i] - \nu(q_t)[i] ~~=\\
\qquad \begin{array}{cl}
\stackrel{(1)}{=} & \sum_j (\sigr{F(M_t)}[j]-\sigr{F(\nu(q_t))}[j])V[j,i]\\
\stackrel{(2)}{\ge} & \sum_j (\sigr{F(M_t)}[j]-\sigr{F(\nu(q_t))}[j])c\\
\stackrel{(3)}{=} & c\Delta(t)
\end{array}
\end{array}
\]
where (1) and (3) hold because of lack of relocation.
Thus $M_{t+1} = \nu(q_t) + (M_{t+1} - \nu(q_t)) \geq \nu(q_t) + c\Delta(t)\vecc{1} \geq \nu(q_t + c\Delta(t))$ and $q_{t+1} \geq q_t + c\Delta(t)$. 


\subsection*{Proof of \Crlr{limit}} 
Let $c = \min(\min_{\{i \mid \pi[i] > 0\}} (\pi[i]),\min_{i,j}(V[i,j]))$.  As we are now following $\pi$ at each step, let $M_{t+1} = \mathit{next}_\pi(M_t)$ and $q_t = \nu^{-1}(M_t)$.

If $\pi[i] > 0$ then:
\[
\begin{array}{l}
M_{t+1}[i] - \nu(q_t)[i]\\
\qquad\begin{array}{cl}
\stackrel{(1)}{=} &  \sum_j (\sigr{F(M_t)}[j]-\sigr{F(\nu(q_t))}[j])V[j,i] ~+\\
& \multicolumn{1}{r}{\sum_j (\sigr{E(M_t)}[j]-\sigr{E(\nu(q_t))}[j])\pi[i]}\\
\stackrel{(2)}{\ge}& \sum_j (\sigr{F(M_t)}[j]-\sigr{F(\nu(q_t))}[j])c ~+\\
& \multicolumn{1}{r}{\sum_j (\sigr{E(M_t)}[j]-\sigr{E(\nu(q_t))}[j])c}\\
\stackrel{(3)}{=}& c\Delta(t)
\end{array}
\end{array}
\] 
If $\pi[i] = 0$ the same conclusion holds by omitting the terms involving $\mb{E}$.
Thus $M_{t+1} = \nu(q_t) + (M_{t+1} - \nu(q_t)) \geq \nu(q_t) + c\Delta(t)\vecc{1} \geq \nu(q_t + c\Delta(t))$ and 
$M_t$ converges at least linearly to $M^*$. \QED

\subsection*{Proof of \Lemr{nondecreasingq}}
\[
\begin{array}{l}
\Delta^E(t) ~\stackrel{(1)}{=}~ \mathbf{E(M_t)} - \mb{E}(\nu(q_t))\\
\qquad\begin{array}{cl}
 \stackrel{(2)}{=} &\sum_i \maxz{(\nu(q_{t+1})-\sigc{F(M_t)}}[i] -  \mb{E}(\nu(q_t))\\
\stackrel{(3)}{=}&\sum_i \maxz{\nu(q_{t+1})-\nu(q_t)+\nu(q_t)-\sigc{F(M_t)}}[i] \\  
&\multicolumn{1}{r}{-~\mb{E}(\nu(q_t)}\\
 \stackrel{(4)}{=}&\sum_i \lsq (\nu(q_{t+1})-\nu(q_t)+\sigc{F(\nu(q_t))}+\sigc{E(\nu(q_t))} \\
&\multicolumn{1}{r}{-~\sigc{F(M_t)}\rsq_{_+}[i] -  \mb{E}(\nu(q_t))}\\
 \stackrel{(5)}{\le}& \mathbf{\nuu{q_{t+1}}}-\nuu{q_t)} + \mathbf{E(\nu(q_t))} -  \mathbf{E(\nu(q_t))}\\
%  \stackrel{(6)}{=}& \mathbf{\nuu{q_{t+1}}}-\mathbf{\nuu{q_t}}\\
  \stackrel{(6)}{=}& q_{t+1}-q_t
  \end{array}
\end{array}
\] 


(1) follows from the definition of $\Delta^E(t)$;
(2) follows from definition of $q_{t+1}$; (3)  adds and subtracts $\nu(q_t)$;
(4)  follows because  $\nu(q_t)$ is a fixpoint;
(5) follows because $\nu(q_{t+1}) \ge \nu(q_t)$ and $\sigc{F(\nu(q_t))} \le \sigc{F(M_t)}$;
%The sixth line follows because $\mathbf{R_{\nu(q_t)}} = \mathbf{E(\nu(q_t))}$ by $\nu(q_t)$ being a fixpoint.
(6) follows because $q_t$ is by definition the total mass of $\nu(q_t)$.
\QED

\subsection*{Proof of \Obr{convergence}}
\[
\begin{array}{l}
\sum_t(\mathbf{E(M_t)} - \mathbf{E(M^*)} ) \\
\qquad\begin{array}{cl} 
\stackrel{(1)}{=} & \sum_t(\mathbf{E(\nu(q_t))}+\Delta^E(t) - \mathbf{E(M^*)})\\
\stackrel{(2)}{\le} & \sum_t \Delta^E(t) ~~
\stackrel{(3)}{\le} ~~ \sum_t(q_{t+1}-q_t)~~
\stackrel{(4)}{\le} ~~ \mathbf{M^*}
  \end{array}
\end{array}
\] 

(1) follows by the definition of $\Delta^E(t)$; (2)  follows since $\nu(q_t)\le M^*$,
hence $\mathbf{E(\nu(q_t))}\le \mathbf{E(M^*)}$; (3) follows from \Lemr{nondecreasingq} and(4) line follows since the $q_t$s are the masses of the $\nu(q_t)$s.
\QED
%######################################################

\end{document}




