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@@ -169,4 +169,19 @@ utility in this case will be $u' = w_1(v-b)$ which is less than $u$.
 We have now established the uniqueness of a symmetric Bayes-Nash equilibrium.
 We can now conclude by applying Theorem 2.10 which states that there are no
 asymmetric strategy profiles.
+
+\section{Exercise 3.1}
+
+We recall that the virtual value function for agent $i$ is defined as $$\phi_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$$ for a cumulative density function $F_i(v_i)$ and density function $f_i(v_i)$, with $F_i'(v_i) \stackrel{\text{def}}{=} f_i(v_i)$. The virtual function satisfies the following relationship: $$\E[p_i(v_i)] = \E[\phi_i(v_i)x_i(v_i)].$$ If the residual surplus is $$\sum_i \left(v_ix_i - p_i\right) - c({\mathbf x}),$$ where $x_i$ is the probability that agent $i$ will be allocated the item and $c({\mathbf x})$ is the service cost, and so if we want to maximize this, we are maximizing
+
+\begin{align*} \E\left[\sum_i \left(v_ix_i - p_i\right)\right] =  \end{align*}
+
+\section{Exercise 3.4}
+
+\begin{enumerate}[(a)]
+
+\item By the definition of virtual functions, we will maximize revenue when we maximize the virtual surplus (this follows from the definition $\E[p_i(v_i)] = \E[\phi_i(v_i)x_i(v_i)]$. Following Definition 3.5, we can do this via the VSM (VCG) mechanism. 
+
+\end{enumerate}
+
 \end{document}