Add figure and some text about deal repr

main.tex

@@ -29,6 +29,7 @@
 \titleformat{\subsection}[block]{\large}{\thesubsection.}{1em}{} % Change the look of the section titles
 
 \usepackage{authblk}
+\usepackage{graphicx}
 
 \usepackage{fancyhdr} % Headers and footers
 \pagestyle{fancy} % All pages have headers and footers
@@ -100,24 +101,47 @@ The goal of this whitepaper is to present such an entity --- Agorata.
 \section{Decision-making algorithm}
 In this chapter, the strategy for making decisions will be described.
 
-\textbf{Deal representation}
+\subsection{Deal representation}
 
 A deal proposal is made by the user in the form of a smart contract. A smart contract is considered as an entity with which other entities (users, smart contracts) can interact via messages\footnote{The message concept used here is from The Open Network (TON). A message can include tokens, commands, information, code.}.
 For a state of the contract we can determine the messages that can be sent to the contract and for each of them --- the response messages and the next state. The deal is represented as a tree with states as leafs and actions (messages) as edges. Depending on the actions, the deal has a total outcome --- the number of tokens that were gained/spent by the agent.
 
-\textbf{Agent parameters}
+\begin{figure}[!htb]
+  \centering
+    \includegraphics[width=0.6\linewidth]{graph.png}
+    \caption{Deal representation.}
+    \label{fig:01}
+\end{figure}
+
+Each action (message) has parameters $\theta$, including:
+\begin{enumerate}
+  \item $t$ --- the time at which the message is sent.
+  \item $s$ --- the sender of the message.
+  \item tokens sent with the message.
+  \item any other possible information.
+\end{enumerate}
+
+The parameter space for each action can be constrained --- the smart contract can reject some of the messages. For instance, this can lead to the entire space consisting of a single element (the contract rejects everything except for one specific message, e.g. a specific person sending a specific amount of tokens).
+
+The main parameters of the state are the financial outcomes for the smart contracts.
+
+
+\subsection{Agent parameters}
 \begin{enumerate}
-  \item $T_{0}$ is the maximum deal time --- i.e., the horizon after which the benefits of the deal are not considered
+  \item $T_{0}$ is the maximum deal duration --- i.e., the horizon after which the benefits of the deal are not considered.
-  \item $\tau$ is the charachteristic time of discount --- e.g., $\exp({\frac{1\,year}{\tau}}) - 1$ is the minimal yearly rate for a loan
+  \item $\tau$ is the charachteristic time of discount --- e.g., $\exp({\frac{1\,year}{\tau}}) - 1$ is the minimal yearly rate for a loan.
 \end{enumerate}
 
-Given these parameters, the value function $V((x_{i}), (t_{i}))$ can be determined. It takes the sequence of values $(x_{i}, t_{i})$, where $x_{i}$ are the incoming/outcoming tokens from the agent.
+Given these parameters, the value function $V((x_{i}), (t_{i}))$ (or $V(S)$) can be determined. It takes the sequence of values $(x_{i}, t_{i})$, where $x_{i}$ are the incoming/outcoming tokens from the agent.
 
-\textbf{Deal evaluation}
+\subsection{Deal evaluation}
 
 The agent considers the worst case of the deal from the perspective of game theory: the best (from the perspective of the value function) actions of the agent and the ``worst'' actions of the counteragents.
 This sequence of events is the run through the value function in order to make a decision.
 
+Thus, a deal is considered profitable for the agent ($A$) $\iff$ $p(S_{i}) > 0$ $\iff$
+$$V(S_{i})>0 \vee \\ (\forall a_{ij} (V(S_{ij}) > 0 \vee a.s = A) \wedge a_{ij}: V(S_{ij})>0)$$
+
 
 
 \section{Contract evaluation algorithm}