deletions | additions       diff --git a/subsectionBidding_Al.tex b/subsectionBidding_Al.tex  index 80fef68..2d40ce3 100644  --- a/subsectionBidding_Al.tex  +++ b/subsectionBidding_Al.tex 
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A reasonable strategy for the agents is the \emph{myopic best response} strategy~\cite{parkes1999bundle}.According to this strategy, agents bid on their best bundles at each iteration given the current prices and always place bids with the current price (i.e., they are not considering raising the price before it has been raised by the auctioneer). It has been shown that myopic best response is the best strategy for an agent given that other agents are acting based on this strategy~\cite{parkes2001iterative}. In MAPF, an agent's best response is to bid on all paths that minimize $c+p$ at their current ask price ($p$).  For example, in the first iteration of an auction for the problem shown in Figure~\ref{fig:mapf}(a), $a_1$'s best response bid in the first round is to bid on its shortest path, $[\langle A,t_1\rangle,\langle C,t_2\rangle]$. $a_2$'s $a_2replace_content#x27;s  best response is to place a $XOR$ bid on its three shortest paths:$[\langle B,t_1\rangle,\langle A,t_2,\langle C,t_3,\langle E,t_4\rangle]$; $[\langle B,t_1\rangle,\langle D,t_2,\langle C,t_3,\langle E,t_4\rangle]$; $[\langle B,t_1\rangle,\langle D,t_2,\langle G,t_3,\langle E,t_4\rangle]$. Similarly, $a_3$'s $a_3replace_content#x27;s  best response is to bid on all of its shortest paths (paths with $c=4$). Note that if agents bid on paths that end at an earlier time than some of the other agents' paths, we extend their paths by adding their last location to the next time steps. As prices increase, agents begin to consider longer paths. Specifically, their best response should include all paths that reach the goal with a minimum value of $c+p$. For example, given that all paths with $c=3$ that $a_2$ can traverse to reach its goal are priced at 1 ($p=1$) and paths with $c=4$ are priced at 0, $a_2$ will bid on all paths with $c=3$ at price 1 and all paths with $c=4$ at price 0, as they have the same total cost ($c+p=4$).  
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To address these problems, agents bid on MDDs that compactly represent their desired paths. Note that the cost used in the MDDs refers to the travel cost ($c$) and does not consider the prices of the MDDs in the auction. Each bid can therefore be represented by the pair $\langle mdd,p_{mdd} \rangle$.  Figure~\ref{fig:mapf}(b) shows the MDD for agent $a_2$ with $c=3$. This MDD represents all of the paths that $a_2$ can take from its location to its goal, at a cost of 3. Each level of the MDD includes all of the locations that $a_2$ might be at in a specific time step when traversing a path of cost 3 to the goal. An edge connects an MDD node with locations in the next level that can be arrived at from the current location. For example, at level 1, $a_2$ could be at either location $(1,2)$ or $(0,1)$. From $(1,2)$ it can move to either $(1,1)$ or $(2,2)$, while from $(0,1)$ it can only move to $(1,1)$ in order to reach the goal in cost 3. As explained above, we align the MDDs to have the same cost by replicating the goal node in the next time steps as show for agent $a_1$ in Figure~\ref{fig:mapf}(c) (dashed nodes and edges were added to match the length of $a_2$'s $a_2replace_content#x27;s  MDD).