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| -rw-r--r-- | ps1/main.tex | 15 |
1 files changed, 9 insertions, 6 deletions
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 |