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| -rw-r--r-- | .gitignore | 5 | ||||
| -rw-r--r-- | tex-stuff/Makefile | 7 | ||||
| -rw-r--r-- | tex-stuff/math.tex | 29 |
3 files changed, 41 insertions, 0 deletions
@@ -6,6 +6,11 @@ *.lo *.o +# latex stuff +*.aux +*.log +*.pdf + /.deps/ /.libs/ /Makefile diff --git a/tex-stuff/Makefile b/tex-stuff/Makefile new file mode 100644 index 0000000..c39f286 --- /dev/null +++ b/tex-stuff/Makefile @@ -0,0 +1,7 @@ +TEX_FILES = $(find -iname '*.tex') +IMG_SRCS = $(wildcard img/*) + +all: $(TEX_FILES) $(IMG_SRCS) Makefile + pdflatex math.tex && pdflatex math.tex +clean: + rm -f *.aux *.bbl *.blg *.log *.pdf *.toc *.nav *.out *.snm *.vrb diff --git a/tex-stuff/math.tex b/tex-stuff/math.tex new file mode 100644 index 0000000..4a990cb --- /dev/null +++ b/tex-stuff/math.tex @@ -0,0 +1,29 @@ +\documentclass{article} +\usepackage[a4paper, margin=2cm]{geometry} +\usepackage{amsmath} +\begin{document} +\section{first price auction with tie breaking and private outcome} +\begin{align} + v_{aj} & = \frac{\prod_{i=1}^n \gamma_{aj}^{\times i}}{\prod_{i=1}^n \varphi_{aj}^{\times i}} \\[2.0ex] + & = \frac{\prod_{i=1}^n \gamma_{aj}^{\times i}}{\prod_{i=1}^n \left(\prod_{h=1}^n \delta_{aj}^{\times h}\right)^{x_{+i}}} \\[2.0ex] + & = \frac{\prod_{i=1}^n \left(\left(\prod_{h=1}^n \prod_{d=j+1}^k \alpha_{hd}\right)\cdot\left(\prod_{d=1}^{j-1} \alpha_{ad}\right)\cdot\left(\prod_{h=1}^{a-1} \alpha_{hj}\right)\right)^{m_{aj}^{+i}}}{\prod_{i=1}^n \left(\prod_{h=1}^n \left(\left(\prod_{s=1}^n \prod_{d=j+1}^k \beta_{sd}\right)\cdot\left(\prod_{d=1}^{j-1} \beta_{ad}\right)\cdot\left(\prod_{s=1}^{a-1} \beta_{sj}\right)\right)^{m_{aj}^{+h}}\right)^{x_{+i}}} \\[2.0ex] + & = \frac{\prod_{i=1}^n \left(\left(\prod_{h=1}^n \prod_{d=j+1}^k b_{hd} y^{r_{hd}}\right)\cdot\left(\prod_{d=1}^{j-1} b_{ad} y^{r_{ad}}\right)\cdot\left(\prod_{h=1}^{a-1} b_{hj} y^{r_{hj}}\right)\right)^{m_{aj}^{+i}}}{\prod_{i=1}^n \left(\prod_{h=1}^n \left(\left(\prod_{s=1}^n \prod_{d=j+1}^k g^{r_{sd}}\right)\cdot\left(\prod_{d=1}^{j-1} g^{r_{ad}}\right)\cdot\left(\prod_{s=1}^{a-1} g^{r_{sj}}\right)\right)^{m_{aj}^{+h}}\right)^{x_{+i}}} \\[2.0ex] + & = \frac{\prod_{i=1}^n \left(\left(\prod_{h=1}^n \prod_{d=j+1}^k b_{hd} \left(\prod_{t=1}^n g^{x_{+t}}\right)^{r_{hd}}\right)\cdot\left(\prod_{d=1}^{j-1} b_{ad} \left(\prod_{t=1}^n g^{x_{+t}}\right)^{r_{ad}}\right)\cdot\left(\prod_{h=1}^{a-1} b_{hj} \left(\prod_{t=1}^n g^{x_{+t}}\right)^{r_{hj}}\right)\right)^{m_{aj}^{+i}}}{\prod_{i=1}^n \left(\prod_{h=1}^n \left(\left(\prod_{s=1}^n \prod_{d=j+1}^k g^{r_{sd}}\right)\cdot\left(\prod_{d=1}^{j-1} g^{r_{ad}}\right)\cdot\left(\prod_{s=1}^{a-1} g^{r_{sj}}\right)\right)^{m_{aj}^{+h}}\right)^{x_{+i}}} +\end{align} + +\subsection{outcome function} +\begin{align} + v_a & = \left((2U-I)\sum_{i=1}^n b_i-(2M+1)\mathbf{e}+(2M+2)Lb_a\right)R_a^* \\[2.0ex] + v_{aj} & = \left(\sum_{i=1}^n \left(\sum_{d=j}^k b_{id} + \sum_{d=j+1}^k b_{id}\right)-(2M+1)+(2M+2)\sum_{d=1}^j b_{ad}\right)R_a^* \\[2.0ex] + & \text{switch from additive finite group to multiplicative finite group} \\[2.0ex] + v_{aj} & = \left(\frac{\displaystyle\prod_{i=1}^n \left(\prod_{d=j}^k b_{id} \cdot \prod_{d=j+1}^k b_{id}\right) \cdot \left(\prod_{d=1}^j b_{ad}\right)^{2M+2}}{(2M+1)g}\right)R_a^* \\[2.0ex] +\end{align} + +\subsection{fixes to step 5 in (M+1)st Price auction from the 2003 paper pages 9 an 10} +\begin{align} + \gamma_{ij} = & \frac{\prod_{h=1}^n \prod_{d=j}^k (\alpha_{hd}\alpha_{h,d+1})\left(\prod_{d=1}^j \alpha_{id}\right)^{2M+2}}{(2M+1)Y} \\ + \text{changed to} & \frac{\prod_{h=1}^n \left(\prod_{d=j}^k \alpha_{hd} \cdot \prod_{d=j+1}^k \alpha_{hd}\right)\left(\prod_{d=1}^j \alpha_{id}\right)^{2M+2}}{Y^{2M+1}} \\[2.0ex] + \delta_{ij} = & \prod_{h=1}^n \prod_{d=j}^k (\beta_{hd}\beta_{h,d+1})\left(\prod_{d=1}^j \beta_{id}\right)^{2M+2} \\ + \text{changed to} & \prod_{h=1}^n \left(\prod_{d=j}^k \beta_{hd} \prod_{d=j+1}^k \beta_{hd}\right)\left(\prod_{d=1}^j \beta_{id}\right)^{2M+2} +\end{align} +\end{document} |