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| -rw-r--r-- | ps1/main.tex | 2 |
1 files changed, 1 insertions, 1 deletions
diff --git a/ps1/main.tex b/ps1/main.tex index ac1a2b9..d42f2ec 100644 --- a/ps1/main.tex +++ b/ps1/main.tex @@ -302,7 +302,7 @@ $$\begin{cases} \text {allocate to all agents if } &\sum_{i=1}^n \phi_i (v_i) \g If the virtual functions are not monotone, we can apply the same mechanism by replacing the virtual functions by the ironed virtual functions. Finally, the payments of the mechanism are obtained by applying the inverse virtual function to the threshold virtual payments: \begin{equation}\label{eq:pay} - p_i = \phi^{-1}\big(-\sum_{j\neq i}\phi_j(0)\big) + p_i = \phi^{-1}\big(-\sum_{j\neq i}\phi_j(v_j)\big) \end{equation} \item If all of the agents' values are i.i.d. from $U[0,1]$, then the agents' virtual functions are given by $$\phi_i(v_i) = (2v_i - 1),$$ and so we will allocate to all agents only if \begin{align*} &\sum_{i=1}^n (2v_i - 1) \geq 0 \\ &\implies \left(2\sum_{i=1}^n v_i \right)- n \geq 0 \\ &\implies \sum_{i=1}^n v_i \geq \frac{n}{2}. \end{align*} So to summarize, our mechanism is: |