deletions | additions       diff --git a/Some_algorithms_for_.tex b/Some_algorithms_for_.tex  index cb5fe3d..f0156f6 100644  --- a/Some_algorithms_for_.tex  +++ b/Some_algorithms_for_.tex 
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Importantly, it has been proven that when using this mechanism agents do best by revealing their true preferences and values and cannot manipulate the outcome in a useful way by not being truthful***cite Vickrey***. To illustrate, assume that $a_1$ would have bid only 5 on the path $s_{1} \rightarrow $X$ \rightarrow g_1$, and 8 for the path $s_{1} \rightarrow $X$ \rightarrow g_1$, and that $a_2$ bids truthfully as before. In this scenario the auctioneer will allocate the path $s_{1} \rightarrow $Y$ \rightarrow g_1$ to $a_1$ and $s_{2} \rightarrow $Z$ \rightarrow g_2$ to $a_2$. So $a_1$ was indeed able to manipulate the outcome. However, such a manipulation does not benefit $a_1$ due to the Vickrey payment scheme: $a_1$ will now need to pay for the harm it caused $a_2$, which is $8-7.3=0.7$. So its final utility will be $10-0.7=9.3$. Importantly this utility is lower than its utility when reporting truthfully, which was $10$.   To the best of our knowledge, the only mechanism proposed for non-cooperative pathfinding is the Iterative Taxation Framework (ITF) by Bnaya et al.~\cite{bnaya2013multi}. This mechanism assigns ``taxes'' to pairs of location and time, driving agents to follow paths with higher-social welfare. In contrast, the auction mechanism sells paths to agents. This mechanism has several drawbacks compared to optimal, strategyproof auction mechanisms: (1) it does not necessarily maximize the social welfare; (2) it requires that agents have equal costs over edges and that the mechanism knows these costs; (3) even if the costs on edges are equal, the mechanism is not strategyproof: ITF requires that agents report their start and goal locations and agents could potentially manipulate the outcome by reporting these locations untruthfully in a way that would result in lower taxes on their paths.