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

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\maketitle

\section{Motivation and Goals} The motivation for our idea comes from the eBay ascending first-price auction with a \lq\lq Buy It Now\rq\rq option. The auction works as follows:

\begin{itemize}
\item The seller sets a reserve price, a Buy It Now price, and a time duration for the auction.
\item The bidders can submit bids (\$$b_i$) at any time, starting from some minimum bid.
\item At any time during the auction, a buyer can choose to \lq\lq Buy It Now\rq\rq (for $\$B$) and end the auction.
\item At the end of the auction, if none of the bidders have exercised the \lq\lq Buy It Now\rq\rq option, the item is allocated to the bidder with the highest bid, if that bid exceeds the reserve price. If the highest bid does not exceed the reserve price, the item is not allocated.
\item The history of all bids is available to all of the bidders throughout the time of the auction.
\end{itemize}

We believe that this framework, while inspired by eBay auctions, also has applicability to numerous other contexts, such as home buyers bidding for homes (e.g. should they agree to the list price or risk an ascending bid war with other potential buyers), contract negotiations (e.g. should a government contractor agree to a particular value and sacrifice potential additional revenue or engage in negotiations and risk that another contractor will undercut them), etc.

Therefore, our goal is as follows:

\begin{goal} To define a formal model for an eBay-style ascending first-price
auction with a reserve price and a \lq\lq Buy It Now\rq\rq option, relate it to
prior work on dynamic (time-dependent) auctions, analyze its properties and
equilibria, and attempt to find a simpler auction which approximates it.\end{goal}

\section{Desiderata for Model}

In order to formalize the setting we are interested in studying, we need to
make some assumptions on the behaviors of the agents. Finding the right
behavioral axioms is part of the project, but we list a few natural candidates
that we are planning to consider:
\begin{itemize}
    \item each bidder not only has a utility function that expresses his utility at the
        end of the auction, but also maintains a distribution on the
        probability of being allocated.
    \item at each time step, the bidders update their distribution based on
        what has been observed at the previous step. Hence the bidders'
        distribution are a function of the past history up to this point.
    \item at each time step, if a bidder is not currently winning, she is
        willing to sacrifice some expected utility to increase their
        probability of being allocated. That is, the agents' are not fully
        rational with respect to their ``outside-the-auction utility'': while
        the auction is still running, the utility function that they are
        maximizing puts a higher weight on simply winning the auction.
\end{itemize}

Under these behavioral axioms, we want to analyze the eBay-style auctions with
respect to the following properties: truthfulness, envy-freeness, revenue
optimality. One last property that we would like to analyze is \emph{regret}:
we don't know yet how to define it properly, but a good notion of regret should
be a function of the difference in utility induced by the ``non-rational''
behavior of the agents: during the auction they are maximizing a different
utility function (biased by the probability of winning); this might decrease
their true utility.

\section{Relevant Prior Work} We are planning to consider prior work on first-price auctions where the bidders have multiple rounds of bidding over time, and various mechanisms that can capture this idea. One inspiration comes from the \lq\lq bid-your-utility \rq\rq auction described in the following paper: Hoy, D., Jain, K., \& Wilkens, C. A. (2013, June). A dynamic axiomatic approach to first-price auctions. In Proceedings of the fourteenth ACM conference on Electronic commerce (pp. 583-584). ACM.

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