Improve economic rationality description

README.md

@@ -3,65 +3,62 @@
 Python Generative Autonomous Agents and Multi-Agent Systems aims to evaluate 
 the social behaviors of LLM-based agents.
 
-This prototype allows to analyse the potential of Large Language Models (LLMs) for
-social simulation by assessing their ability to: (a) make decisions aligned
-with explicit preferences; (b) adhere to principles of rationality; and (c)
-refine their beliefs to anticipate the actions of other agents. Through
-game-theoretic experiments, we show that certain models, such as
-\texttt{GPT-4.5} and \texttt{Mistral-Small}, exhibit consistent behaviours in
-simple contexts but struggle with more complex scenarios requiring
-anticipation of other agents' behaviour. Our study outlines research
-directions to overcome the current limitations of LLMs.
-
-## Consistency
-
-To evaluate the decision-making consistency of various LLMs, we introduce an investment 
-game designed to test whether these models follow stable decision-making patterns or 
-react erratically to changes in the game’s parameters.
-
-In the game, an investor allocates a basket \((p_t^A, p_t^B)\) of 100 points between two assets: 
-Asset A and Asset B. The value of these points depends on two random parameters \((a_t, b_t)\), 
-which determine the monetary return per allocated point.
-
-For example, if \(a_t = 0.8\) and \(b_t = 0.5\), each point assigned to Asset A is worth $0.8, 
-while each point allocated to Asset B yields $0.5. The game is played 25 times to assess 
-the consistency of the investor’s decisions.
-
-To evaluate the rationality of the decisions, we use the **Critical Cost Efficiency Index (CCEI)**, 
-a widely used measure in experimental economics and behavioral sciences. The CCEI assesses 
-whether choices adhere to the **Generalized Axiom of Revealed Preference (GARP)**, 
-a fundamental principle of rational decision-making.
-
-If an individual violates rational choice consistency, 
-the CCEI determines the minimal budget adjustment required to make their 
-decisions align with rationality. Mathematically, the budget for each basket is calculated as:
-
-\[
-I_t = p_t^A \times a_t + p_t^B \times b_t
-\]
-
-The CCEI is derived from observed decisions by solving a linear optimization 
-problem that finds the largest \(\lambda\) (where \(0 \leq \lambda \leq 1\)) 
-such that for every observation, the adjusted decisions satisfy the rationality constraint:
-
-\[
-p^_t \cdot x_s \leq \lambda I_t
-\]
-
-This means that if we slightly reduce the budget (multiplying it by \(\lambda\)), 
-the choices will become consistent with rational decision-making. 
-A CCEI close to 1 indicates high rationality and consistency with economic theory.
-A low CCEEI** suggests irrational or inconsistent decision-making.
-
-To ensure response consistency, each model undergoes 30 iterations of the game 
-with a fixed temperature of 0.0.
-
-The results indicate significant differences in decision-making consistency among the evaluated models.
-Mistral-Small demonstrates the highest level of rationality, with CCEI values consistently above 0.75.
-Llama 3 performs moderately well, with CCEI values ranging between 0.2 and 0.74. 
-DeepSeek R1 exhibits inconsistent behavior, with CCEI scores varying widely between 0.15 and 0.83
-
-![CCEI Distribution per model](figures/investment/investment_boxplot.svg)
+This prototype explores the potential of *homo silicus* for social
+simulation. We examine the behaviour exhibited by intelligent
+machines, particularly how generative agents deviate from
+the principles of rationality. To assess their responses to simple human-like
+strategies, we employ a series of tightly controlled and theoretically
+well-understood games. Through behavioral game theory, we evaluate the ability
+of <tt>GPT-4.5</tt>, <tt>Llama3</tt>, <tt>Mistral-Small</tt>}, and
+<tt>DeepSeek-R1</tt> to make coherent one-shot
+decisions, generate algorithmic strategies based on explicit preferences, adhere
+to first- and second-order rationality principles, and refine their beliefs in
+response to other agents’ behaviours.
+
+
+## Economic Rationality
+
+## Evaluating Economic Rationality in LLMs
+
+To evaluate the economic rationality of various LLMs, we introduce an investment game 
+designed to test whether these models follow stable decision-making patterns or react 
+erratically to changes in the game’s parameters.
+
+In this game, an investor allocates a basket $x_t=(x^A_t, x^B_t)$ of $100$ points between 
+two assets: Asset A and Asset B. The value of these points depends on random prices $p_t=(p_{t}^A, p_t^B)$, 
+which determine the monetary return per allocated point. For example, if $p_t^A= 0.8$ and $p_t^B = 0.8$, 
+each point assigned to Asset A is worth $\$0.8$, while each point allocated to Asset B yields $\$0.5$. T
+he game is played $25$ times to assess the consistency of the investor’s decisions.
+
+To evaluate the rationality of the decisions, we use Afriat's
+critical cost efficiency index (CCEI), i.e. a widely used measure in
+experimental economics. The CCEI assesses whether choices adhere to the
+generalized axiom of revealed preference (GARP), a fundamental principle of
+rational decision-making. If an individual violates rational choice consistency,
+the CCEI determines the minimal budget adjustment required to make their
+decisions align with rationality. Mathematically, the budget for each basket is
+calculated as: $ I_t = p_t^A \times x^A_t + p_t^B \times x^B_t$. The CCEI is
+derived from observed decisions by solving a linear optimization problem that
+finds the largest $\lambda$, where $0 \leq \lambda \leq 1$, such that for every
+observation, the adjusted decisions satisfy the rationality constraint: $p_t
+\cdot x_t \leq \lambda I_t$. This means that if we slightly reduce the budget,
+multiplying it by $\lambda$, the choices will become consistent with rational
+decision-making. A CCEI close to 1 indicates high rationality and consistency
+with economic theory. A low CCEEI suggests irrational or inconsistent
+decision-making.
+
+To ensure response consistency, each model undergoes $30$ iterations of the game
+with a fixed temperature of $0.0$. The results shown in
+Figure below highlight significant differences in decision-making
+consistency among the evaluated models. <tt>GPT-4.5</tt>, <tt>LLama3.3:latest</tt> 
+and <tt>DeepSeek-R1:7b</tt> stand out with a
+perfect CCEI score of 1.0, indicating flawless rationality in decision-making.
+<tt>Mistral-Small</tt> and <tt>Mixtral:8x7b</tt> demonstrate the next highest level of rationality. 
+<tt>Llama3</tt> performs moderately well, with CCEI values ranging between 0.2 and 0.74. 
+<tt>DeepSeek-R1</tt> exhibits
+inconsistent behavior, with CCEI scores varying widely between 0.15 and 0.83.
+
+![CCEI Distribution per model](figures/investment/investment_violin.svg)