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Diffstat (limited to 'myerson.tex')
| -rw-r--r-- | myerson.tex | 18 |
1 files changed, 10 insertions, 8 deletions
diff --git a/myerson.tex b/myerson.tex index bcfedc4..8c80cf6 100644 --- a/myerson.tex +++ b/myerson.tex @@ -10,20 +10,22 @@ other users: We distinguish four cases depending on the value of $s_i(c_i, c_{-i})$ and $s_i'(c_i', c_{-i})$. -Since the mechanism is normalized, if $s_i(c_i, c_{-i})= s_i(c_i', c_{-i})=0$, +\begin{enumerate} +\item $s_i(c_i, c_{-i})= s_i(c_i', c_{-i})=0$. +Since the mechanism is normalized we have $p_i(c_i, c_{-i}) = p_i(c_i', c_{-i})= 0$ and \eqref{eq:local-foobar} is true. - +\item $s_i(c_i', c_{-i}) = s_i(c_i, c_{-i}) = 1$. Note that $i$ is paid her threshold payment when allocated, and since this payment does not depend on $i$'s reported cost, \eqref{eq:local-foobar} is true -(and is in fact an equality) when $s_i(c_i', c_{-i}) = s_i(c_i, c_{-i}) = 1$. - -If $s_i(c_i', c_{-i}) = 0$ and $s_i(c_i, c_{-i}) = 1$, we have $p_i(c_i', +(and is in fact an equality). +\item $s_i(c_i', c_{-i}) = 0$ and $s_i(c_i, c_{-i}) = 1$. + We then have $p_i(c_i', c_{-i}) = 0$ by normalization and \eqref{eq:local-foobar} follows from individual rationality. - -Finally, let us assume that $s_i(c_i', c_{-i}) = 1$ and $s_i(c_i, c_{-i}) = 0$. +\item $s_i(c_i', c_{-i}) = 1$ and $s_i(c_i, c_{-i}) = 0$. By $\delta$-decreasingness of $s_i$, $c_i \geq c_i'+\delta$, and $s_i(c_i, c_{-i}) = 0$ implies that $i$'s threshold payment is less than $c_i$, \emph{i.e.} $p_i(c_i', c_{-i}) \leq c_i$. This last inequality is equivalent to -\eqref{eq:local-foobar} in this final case. +\eqref{eq:local-foobar} in this final case. \qed +\end{enumerate} |