in Twenty Lectures on Algorithmic Game Theory problem $4\mathrm{.}2$ d I can't find an example to show that the algorithm with the parameter m set as in $($a$)$ and $($b$)$ need not yield a monotone allocation rule $*$ the vi are rounded up can i get a numbered example? Problem $4\mathrm{.}2$ Section $4\mathrm{.}2\mathrm{.}2$ gives an allocation rule for knapsack auctions that is monotone, guarantees at least $50\%$ of the maximum social welfare, and runs in polynomial time. Can we do better? We first describe a classical fully polynomial$-$time approximation scheme $($FPTAS$)$ for the knapsack problem. The input to the problem is item values v$\_1,...,$ v$\_$n$,$ item sizes w$\_1,...,$ w$\_$n$,$ and a knapsack capacity W$.$ For a user$-$supplied parameter $>0,$ we consider the following algorithm $\_$ ; m is a parameter that will be chosen shortly. $-$ Round each v$\_$i up to the nearest multiple of m$,$ call it v$\_$i$^\text{'}.-$ Divide the v$\_$i$^\text{'}\text{'}$s through by m to obtain integers $\_1,...,\_$n$.-$ For item values $\_1,...,\_$n$,$ compute the optimal solution using a pseudopolynomial$-$time algorithm. $[$You can assume that there exists such an algorithm with running time polynomial in n and max $\_$i$=1^$n$\_$i$.]($a$)$ Prove that if we run algorithm $\_$ with the parameter m set to $($max $\_$i$=1^$n v$\_$i$)/$ n$,$ then the running time of the algorithm is polynomial in n and $1/($independent of the v$\_$i $\text{'}$s$).($b$)($H$)$ Prove that if we run algorithm $\_$ with the parameter m set to $($max $\_$i$=1^$n v$\_$i$)/$ n$,$ then the algorithm outputs a solution with total value at least $1-$ times the maximum possible. $($c$)$ Prove that if we run algorithm $\_$ with the parameter m set to a fixed constant, independent of the v$\_$i $\text{'}$s$,$ then the algorithm yields a monotone allocation rule. $($d$)$ Prove that if we run algorithm $\_$ with the parameter m set as in $($a$)$ and $($b$),$ then the algorithm need not yield a monotone allocation rule. $($e$)($H$)$ Give a DSIC mechanism for knapsack auctions that, for a user$-$specified parameter $$ and assuming truthful bids, outputs an outcome with social welfare at least $1-$ times the maximum possible, in time polynomial in n and $1/.$