From 17eac00022d428d2898b36c58c5d09d3163bb6be Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Tue, 23 Sep 2014 14:33:05 -0400
Subject: More progress on problem 2

---
 ps1/main.tex | 15 +++++++++------
 1 file changed, 9 insertions(+), 6 deletions(-)

(limited to 'ps1/main.tex')

diff --git a/ps1/main.tex b/ps1/main.tex
index dfbec82..d5290df 100644
--- a/ps1/main.tex
+++ b/ps1/main.tex
@@ -167,8 +167,8 @@ 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 $u$.
 
 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.
+Furthermore we can rule out the possibility of assymetric strategy profiles by
+applying verbatim the proof of Theorem 2.10.
 
 \section{Exercise 3.1}
 
@@ -180,27 +180,30 @@ If the residual surplus is $$\sum_i \left(v_ix_i - p_i\right) - c({\mathbf x}),$
 
 \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]
+    \sum_i\E\big[v_ix_i(v_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:
+where we defined $h_i(v_i)$ 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
+by ironing $h_i$. Since $h_i$ is non-increasing, we have to apply 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 
+where we use the $(c_1,\ldots,c_n)$ as virtual functions. Note that it is easy
+to see in this case that truth telling is a dominant strategy equilibrium: the
+mechanism ignores the bids of the players and only allocates based on the
+ironed constants
 
 
 
-- 
cgit v1.2.3-70-g09d2

