From 648c77d62ea76b138db8d7a47a889b29e6ee57ab Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 23 Sep 2014 12:57:46 -0400
Subject: more progress on problem 2

---
 ps1/main.tex | 34 +++++++++++++++++++++++++++++++---
 1 file changed, 31 insertions(+), 3 deletions(-)

(limited to 'ps1/main.tex')

diff --git a/ps1/main.tex b/ps1/main.tex
index 60ed947..dfbec82 100644
--- a/ps1/main.tex
+++ b/ps1/main.tex
@@ -162,7 +162,7 @@ non-decreasing, its maximum value is:
     s(1) = \frac{(w_1-w_2)\int_0^1 zf(z)dz}{w_1}
 \end{displaymath}
 Let us show that bids above $s(1)$ are dominated by $s(1)$. The utility of an
-agent with value $v$ when bidding $s(1)$ is $u = w_1(v-s(1))$ since he will be
+agent with value $v$ when bidding $s(1)$ is $u = w_1\big(v-s(1)\big)$ since he will be
 allocated to the first position in this case. Then consider a bid $b>s(1)$; the
 utility in this case will be $u' = w_1(v-b)$ which is less than $u$.
 
@@ -172,9 +172,37 @@ 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
+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:
+\begin{equation}\label{eq:virt}
+    \E[p_i(v_i)] = \E[\phi_i(v_i)x_i(v_i)].
+\end{equation}
+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(v_i) - p_i(v_i)\right)\right] =  
+    \sum_i\E\big[v_ix_i-p_i(v_i)\big]
+\end{align*}
+Using \cref{eq:virt}, this is equal to:
+\begin{displaymath}
+    \sum_i\E\big[x_i(v_i)(v_i-\phi_i(v_i))\big] =
+    \sum_i\E\big[x_i(v_i)h_i(v_i)\big]
+\end{displaymath}
+where we defined $h(v_i)$ to be the inverse of the hazard rate function:
+\begin{displaymath}
+    h_i(v_i) \eqdef \frac{1-F_i(v_i)}{f_i(v_i)}
+\end{displaymath}
+
+Note that by assumption, $h_i$ is non-increasing (since the harzard rate function
+is non-decreasing). Hence, we need to consider $\bar{h_i}$, the function obtained
+by ironing $h_i$. Since $h_i$ is non-increasing, we have to obtain the ironing
+procedure to the whole interval of values, leading to a constant ironed
+function $\bar{h_i} = c_i$. We will call $c_i$ the \emph{ironed constant} of
+agent $i$.
+
+By construction, the mechanism maximizing residual surplus is the VSM mechanism
+where 
+
 
-\begin{align*} \E\left[\sum_i \left(v_ix_i - p_i\right)\right] =  \end{align*}
 
 \section{Exercise 3.4}
 
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