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| -rw-r--r-- | ps1/main.tex | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/ps1/main.tex b/ps1/main.tex index 5fd6d6a..0605359 100644 --- a/ps1/main.tex +++ b/ps1/main.tex @@ -72,18 +72,18 @@ For agent 1,$$\Pr[v_1 \geq v_2] = \begin{cases} \int_{\frac{1}{2}}^{v_1} 2\,dx = For agent 1, $P_1^{SP}(v_1)$ equals $\E[v_2|v_1\geq v_2]\cdot \Pr[v_1 \geq v_2]$. By Revenue Equivalence (Corollary 2.5), for $i \in \{1,2\}$, $P_i^{SP}(v_i) = P_i^{FP}(v_i)$. -We directly compute $$\E[v_2|v_1\geq v_2] = \begin{cases} \int_{\frac{1}{2}}^{v_1} 2x\,dx = v_1^2 - \frac{1}{4} &\text{ if } v_1 > \frac{1}{2} \\ 0 &\text{ if } v_1 \leq \frac{1}{2} \end{cases}$$ - -Thus, -$$s_1(v_1) = v_1^2-\frac{1}{4} \text{ if } v_1 > \frac{1}{2}.$$ +We directly compute $$\E[v_2|v_1\geq v_2] = \begin{cases} \frac{1}{2}\big(v_1+\frac{1}{2}\big) &\text{ if } v_1 > \frac{1}{2} \\ 0 &\text{ if } v_1 \leq \frac{1}{2} \end{cases}$$ +Indeed, in expectation a uniform random variable evenly divides the interval it is over, and conditioned on $v_1\geq v_2$, $v_2$ is $U\big[\frac{1}{2},1\big]$. Thus, +$$s_1(v_1) = \begin{cases} \frac{1}{2}\big(v_1+\frac{1}{2}\big) &\text{ if } v_1 > \frac{1}{2} \\ 0 &\text{ if } v_1 \leq \frac{1}{2} \end{cases}.$$ For agent 2, $$\Pr[v_2 \geq v_1] = \int_{0}^{v_2}\,dx = v_2$$ -Computing $$\E[v_1|v_2\geq v_1] = \int_0^{v_2} x\,dx = \frac{v_2^2}{2}$$ +Computing $$\E[v_1|v_2\geq v_1] = \int_0^{v_2} x\,dx = \frac{v_2}{2}$$ +since conditioned on $v_2\geq v_1$, $v_1$ is $U\big[0,v_2\big]$. -So $$s_2(v_2) = \frac{v_2^2}{2}.$$ +So $$s_2(v_2) = \frac{v_2}{2}.$$ -We will show a counterexample to the requirement that the item it always allocated to the agent with the highest value. Specifically, we will exhibit a scenario where $v_1 < v_2$ but $s_1(v_1) > s_2(v_2)$ and so agent 1 will be allocated the item despite having a lower valuation. Let $v_1 = \frac{7}{8}$ and let $v_2 = 1$. Then, $$s_1(v_1) = \frac{49}{64} - \frac{16}{64} = \frac{33}{64}$$ but $$s_2(v_2) = \frac{1}{2.}$$ Therefore, no Bayes-Nash Equilibrium with the desired properties can exist. +We will show a counterexample to the requirement that the item it always allocated to the agent with the highest value. Specifically, we will exhibit a scenario where $v_1 < v_2$ but $s_1(v_1) > s_2(v_2)$ and so agent 1 will be allocated the item despite having a lower valuation. Let $v_1 = \frac{1}{2}$ and let $v_2 = \frac{3}{4}$. Then, $$s_1(v_1) = \frac{1}{2}\left(\frac{1}{2}+\frac{1}{2}\right) = \frac{1}{2}$$ but $$s_2(v_2) = \frac{3}{8}<\frac{1}{2}.$$ Therefore, no Bayes-Nash Equilibrium with the desired properties can exist. \end{enumerate} |