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\begin{document}

% Scalefnt doesn't work on title page elements, so use size commands
\title{{\Huge {The dynamic cost of ex post incentive compatibility}}\\
{\Huge {in repeated games of private information}}}
\author{{\LARGE {David A. Miller}}}
\date{}
\maketitle

\scalefont{1.845} \raggedright

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\pagebreak

\section*{Motivation}

\begin{itemize}
	\item Cooperation with private information requires communication
	\item But folk theorems (FLM, Miller) are not robust to:
	\begin{itemize}
		\item Unstructured communication
		\item No common prior
		\item Spying/Higher order beliefs (Bergemann \& Morris)
	\end{itemize}
	\item Look for equilibria that are robust to these complications
	\begin{itemize}
		\item Each stage must satisfy \emph{ex post incentive compatibility} (EPIC)
		\item Is efficiency attainable? ---No.
		\item Then what is optimal?
	\end{itemize}
\end{itemize}

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\pagebreak

\section*{Contributions}

\begin{itemize}
	\item Formalize a \emph{mechanism design approach} to repeated games
	\begin{itemize}
		\item FLM, ABS
	\end{itemize}
	\item Characterize \emph{ex post perfect public equilibrium} (EPPPE)
	\begin{itemize}
		\item Efficient average utilities not attainable
		\item Computation of an optimum (linear programming)
	\end{itemize}
	\item Characterize optimal EPPPEs in two-player allocation games:
	\begin{itemize}
		\item Optimality $\Longrightarrow $ inefficient allocation
		\item Private valuations: pooling, often stationary
		\item Interdependent valuations $\Longrightarrow $ non-stationary
	\end{itemize}
	\item Provide a new explanation for price wars in collusive equilibrium
\end{itemize}

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\pagebreak

\section{Example: Multicolumns and equations}

\vspace{12pt}

\begin{multicols}{2}
	\noindent Game theory approach:

	\begin{itemize}
		\item Perfect public equilibrium (PPE)
		\item Strategies (best responses)
		\item Arbitrary messages
		\item Actions (observable)
		\item Voluntary transfers
	\end{itemize}

	\noindent Mechanism design approach:

	\begin{itemize}
		\item Recursive mechanism
		\item IC/IR constraints
		\item Direct revelation
		\item Mandatory actions
		\item Mandatory transfers
	\end{itemize}
\end{multicols}

\bigskip

\noindent Out-of-context equation:
\begin{equation*}
	p_{i}^{\ast }(c)=\left\{ 
		\begin{array}{cl}
			1 & \text{if }c_{i}\leq c_{-i} \\ 
			2 & \text{if }c_{i}>c_{-i}%
		\end{array}%
	\right. \text{\quad and\quad }x_{i}(p^{\ast }(c))=\left\{ 
		\begin{array}{cl}
			1 & \text{if }c_{i}<c_{-i} \\ 
			0 & \text{if }c_{i}>c_{-i} \\ 
			\frac{1}{2} & \text{if }c_{1}=c_{2}%
		\end{array}%
	\right.
\end{equation*}

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\pagebreak

\subsection{Example: Theorem}

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	\begin{theorem}["Anti-folk" theorem]
		For $\delta<1$ sufficiently high, the surplus gap of an optimal EPPPE mechanism does not vary with $\delta $.
	\end{theorem}

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\begin{proof}[Proof outline]

Given an outcome function $x$, optimal construction of continuation rewards, 
$w$, is fixed in present value terms; i.e., $\frac{\delta }{1-\delta }w$:
\begin{enumerate}
	\item EPIC constrains $u_{i}(\theta ,\theta ;\left\langle x,t,w\right\rangle
) =\pi_i(\theta,x(\theta))+t_i(\theta)+\frac{\delta}{1-\delta}w_i(\theta)$;
	\item The surplus gap is $\frac{\delta }{1-\delta }\bigl(\max_{\theta }\left[
W(\theta )\right] -\mathbb{E}\left[ W(\theta)\right] \bigr)$;
	\item Individual rationality does not bind for sufficiently high $\delta$. 
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\end{enumerate}

\end{proof}


\end{document}
