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| -rw-r--r-- | final/main.tex | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/final/main.tex b/final/main.tex index 1a101f5..dae0503 100644 --- a/final/main.tex +++ b/final/main.tex @@ -169,7 +169,7 @@ $\gamma$ is a constant which is at least $\frac{1}{2}$. Given the simplicity of posted-price mechanisms and the fact that they are optimal in the single-agent single-item setting (for a regular distribution $F$), it would be desirable to obtain a mechanism as close as possible to posted-price. -Unfortunately, Hart and Nisan \citep{hart-nisan} showed that even in the +Unfortunately, \citep{hart-nisan} showed that even in the unconstrained, regular case, no posted-price mechanism for the single-agent problem has an approximation ratio better than $\Omega(\log n)$. @@ -223,7 +223,7 @@ is essentially saying that there is a reserve price for each item. The notion of $p$-exclusivity introduced by Yao was crucial in his reduction from the multiple-buyer setting to the single buyer setting. $p$-exclusivity -can easily be enforced in the problem we formulated in Section~\ref{sec:intro}, +can easily be enforced in the optimization we formulated in Section~\ref{sec:intro}, by adding the following non-linear constraints: \begin{displaymath} x_i(t_i - p_i)\geq 0,\quad \forall i\in[m] @@ -234,8 +234,8 @@ Lemma. \begin{lemma} If a mechanism is $p$-exclusive for some vector $p=(p_1,\dots, p_m)$, then - it satisfies the ex-ante allocation constraint defined by $\hat{x} - = \big(F^{-1}(1-p_1), \dots, F^{-1}(1-p_m)\big)$. + it satisfies the ex-ante allocation constraint defined by $$\hat{x} + = \big(F^{-1}(1-p_1), \dots, F^{-1}(1-p_m)\big).$$ \end{lemma} |