Figure~\ref{fig:scalesStimulus} depicts a reference game
scenario. There are three potential referents (A, B, and C), each with
a pre-specified set of properties (wearing glasses, wearing a hat,
having a mustache). The \emph{speaker} is privately assigned one of
the referents. Using a message from a pre-defined vocabulary, the
speaker tries to convey the identity of this referent to the
\emph{listener}. The listener uses the message to choose one of the
referents. Intuitively, the two participants' goals are aligned: they
win just in case the speaker's referent is the hearer's choice.

\begin{figure}[ht]
  \centering
  % \includegraphics[width=\columnwidth]{fig/facesScales.png}
  \includegraphics[width=0.7\columnwidth]{fig/refgame-sample}

  \vspace{-4pt}
  
  A\hspace{55pt}B\hspace{55pt}C
  \caption{Scenario for a simple reference game.}
  \label{fig:scalesStimulus}
\end{figure}

\subsection{Definition}
Formally, a reference game is a tuple $G = (M,T,\sem{\cdot})$, where
$T$ is the set of targets, $M$ is the set of messages, and
$\sem{\cdot} : M \to 2^{T}$ is the semantics of the messages, which
dictates which targets each message is true of.  A speaker $S$ is a
(possibly stochastic) function $G \times T \to M$ from a game and
target to a message. Similarly, a listener $L$ is a function $G \times
M \to T$ from a game and a message to a target.  For a given run of a
game $G$ and target $t$, the speaker produces the message $m =
S(G,t)$, the listener selects a target $t' = L(G,m)$, and they win iff
$t = t'$.

\subsection{Literal Speaker and Listener}
To initialize the learning procedure described in the next section, we
define literal speakers and listeners, who rely entirely on the
semantics of the messages, with no consideration of the other
participant's behavior. The literal speaker $S_{0}$, when given a
target to refer to, picks uniformly at random from messages which are
true of the target:
%
\begin{equation}
S_{0}(G,t)  = \text{Uniform}\left(\{m \vert t \in \sem{m}\}\right)
\end{equation}
%
Similarly, the literal listener $L_0$, when given a message $m$, picks
randomly from targets in the semantics of the message:
%
\begin{equation}
L_{0}(G,m) = \text{Uniform}\left(\{t \vert t \in \sem{m}\}\right)
\end{equation}