deletions | additions       diff --git a/Theorem_maximum_soci.tex b/Theorem_maximum_soci.tex  index 314fee8..1661901 100644  --- a/Theorem_maximum_soci.tex  +++ b/Theorem_maximum_soci.tex 
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\subsection{Optimal MAPF}  A line of previous work was devoted to the optimal variant of MAPF. In the optimal variant a solution which minimizes the summation of single agent paths is required.   In CA optimizing the summed utility over all agents is known as maximum social welfare. Many CA solvers guarntee guarantee  maximum social welfare as long as agents use the myopic best response bidding strategy (Parkes 1999). According to this strategy, agents allways always  bid on the best bundle (according to thier their  evaluation) given the current prices. A myopic best response MAPF agent will aspire to minimize its own path cost.   The utility of agiven a given  path $p$, for a given agent is $-c$ where $c=cost(p)$. Since the maximal price an agent is willing to pay on a bundle (path) can not cannot  be negative, we suggest setting $price(p)=max_{p' \in P}[cost(p')]-cost(p)$ where $P$ is the set of all possibale possible  paths in the optimal solution and $p' \in P$ is the path with the maximal cost. Though finding $max_{p' \in P}[cost(p')]$ is a hared hard  problem, we can bound this value by using the $|V|^3$ upper bound defined above./footnote{The $|V|^3$ bound is defined for the number of time steps and not cost. A cost bound can be calculated by multipling multiplying  the time steps bound by the maximal cost of a single time step.} -setting Setting  the cost above price function and a myopic best response strategy  for each bundle all agents is not sufficient to guarantee an optimal MAPF solution using a CA solver. Figure xxx presents a MAPF problem  Theorem: maximum social welfare = optimal MAPF.