project2/main.tex


1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68

\documentclass[10pt]{article}
\usepackage{fullpage}
\usepackage{amsmath,amsfonts,amsthm}
\usepackage[english]{babel}
\usepackage[capitalize, noabbrev]{cleveref}
\usepackage{paralist} 

% these are compressed lists to help fit into a 1 page limit
\newenvironment{enumerate*}%
  {\vspace{-2ex} \begin{enumerate} %
     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
  {\end{enumerate}}
 
\newenvironment{itemize*}%
  {\vspace{-2ex} \begin{itemize} %
     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
  {\end{itemize}}
 
\newenvironment{description*}%
  {\vspace{-2ex} \begin{description} %
     \setlength{\itemsep}{-1ex} \setlength{\parsep}{0pt}}%
  {\end{description}}

\DeclareMathOperator*{\E}{\mathbb{E}}
\let\Pr\relax
\DeclareMathOperator*{\Pr}{\mathbb{P}}

\newcommand{\inprod}[1]{\left\langle #1 \right\rangle}
\newcommand{\R}{\mathbb{R}}
\newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}}
\newcommand{\llbracket}{[\![}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem*{goal}{Goal}

\author{Thibaut Horel \and Paul Tylkin}
\title{Economics 2099 Project}

\begin{document}

\maketitle

\section{Problem}

We are interested in a multi-item auction for agents with submodular
preferences.

If we denote by $m$ the number of items, we look at a specific case where the
type $t$ of an agent is a vector $t=(t_1,\ldots,t_m)$ drawn from a distribution
$F$ over $\mathbb{R}_+^m$ and for an allocation $x$ and price
$p$, the utility of the agent is:
\begin{displaymath}
    u(t, x, p) = \sum_{S\subseteq 2^m} x(S)g\left(1+\sum_{j\in S} t_j\right)-p
\end{displaymath}
where $g$ is a concave function from $\mathbb{R}_+$ to $\mathbb{R}_+$ with
$g(0) = 0$.


\section{Related Work}

\cite{babaioff}
\cite{yao}
\cite{hart}

\bibliographystyle{abbrv}
\bibliography{main}
\end{document}