A theoretical framework to explain non-Nash equilibrium strategic behavior in experimental games

TL;DR

This work tackles the gap between Nash predictions and actual behavior by introducing a per-player temperature that embodies bounded rationality. Grounded in a Boltzmann-weighted decision framework, it predicts full probability distributions over joint strategies and captures non-Nash outcomes by connecting temperature to differences in expected payoffs. The approach yields a practical method to infer players' rationality from observed choices and to forecast behavior across related games, demonstrated via Dictator and Ultimatum experiments and cross-game predictions. By linking temperature to entropy measures, it bridges normative theory and behavioral data, offering a versatile tool for analyzing strategic interactions beyond classical Nash equilibria.

Abstract

Conventional game theory assumes that players are perfectly rational. In a realistic situation, however, players are rarely perfectly rational. This bounded rationality is one of the main reasons why the predictions of Nash equilibrium in normative game theory often diverge from human behavior in real experiments. Motivated by the Boltzmann weight formalism, here we present a theoretical framework to predict the non-Nash equilibrium probabilities of possible outcomes in strategic games by focusing on the differences in expected payoffs of players rather than traditional utility metrics. In this model, bounded rationality is parameterized by assigning a temperature to each player, reflecting their level of rationality by interpolating between two decision-making regimes, i.e., utility maximization and equiprobable choices. Our framework predicts all possible joint strategies and is able to determine the relative probabilities for multiple pure or mixed strategy equilibria. To validate model predictions, by analyzing experimental data we demonstrated that our model can successfully explain non-Nash equilibrium strategic behavior in experimental games. Our approach reinterprets the concept of temperature in game theory, leveraging the development of theoretical frameworks to bridge the gap between the predictions of normative game theory and the results of behavioral experiments.

Figures (10)

• Figure 1: Temperature-based probabilities of possible outcomes of the Prisoner's Dilemma. (A1) Probability distributions that a player chooses the strategy $s_1 = Quiet$ or $s_2 = Betray$ based on her temperature (rationality) assuming that $T_1 = T_2 = T$. (A2) Probability of choosing each strategy when the player is perfectly rational ($T \rightarrow 0$, top) or perfectly irrational ($T \rightarrow \infty$, bottom). (B1) Joint probabilities of possible outcomes of the game. (B2) Joint probability of possible outcomes of the game when the two players are perfectly rational ($T \rightarrow 0$, top) or perfectly irrational ($T \rightarrow \infty$, bottom). In the figure, temperature is dimensionless and $T = 20$ was sufficient to be considered as $T \rightarrow \infty$ where probabilities approach their asymptotic values. Dashed horizontal lines represent asymptotic probabilities at $T \rightarrow \infty$ as a reference.
• Figure 2: Probabilities of possible outcomes of the Prisoner's Dilemma for non-identical temperatures. (A1-A4) Joint probabilities of possible outcomes of the game for non-identical temperatures of the players with two strategies $s_1 = Quiet$ and $s_2 = Betray$. (B1,B2) Examplary joint probabilities of possible outcomes for asymptotic temperatures indicated by a temperature pair ($T_1,T_2$) above each panel.
• Figure 3: Difference between the expected payoffs characterizes Nash equilibrium and non-Nash equilibrium states. (A) Depending on the given joint probability $(p,q)$, the color-coded difference between the expected payoffs ($\Delta \mathcal{E}_1$; for player 1) of the two strategies determines Nash equilibrium (red circle) and non-Nash equilibrium (green circle) states, e.g., according to payoffs given in the Battle of the Sexes in Table \ref{['table1']} (middle). (B) Each arbitrary state in the $p$-$q$ plane representing a $\Delta \mathcal{E}_1$ corresponds to a point in the $T_1$-$T_2$ plane which identifies the temperature pair ($T_1,T_2$) of the players in the game.
• Figure 4: Prediction of the temperature of the players. The joint probability $(p,q)$ of the players is systematically varied and the resultant difference between the expected payoffs is calculated according to Eq. (\ref{['eq:11']}) for player 1 (A) and player 2 (B). The temperature of the players is then calculated via Eq. (\ref{['eq:13']}). The white circles indicate the Nash equilibrium joint probability $(p^*,q^*) = (\frac{2}{3},\frac{1}{3})$.
• Figure 5: Prediction of non-Nash equilibrium probabilities over strategies. Color-coded non-Nash equilibrium probabilities over strategies of player 1 for payoffs given in the Battle of the Sexes in Table \ref{['table1']} (middle) as a function of temperature and expected payoffs. These probabilities correspond to the conditions where the predicted non-Nash equilibrium probability is smaller (A1,B1) or greater (A2,B2) than the Nash equilibrium probability of the player based on Eq. (\ref{['eq:12']}). The two points that are marked represent $\rm a_1$ = (5,0.8) and $\rm a_2$ = (5,0.5) in the $T$-$\Delta \mathcal{E}$ plane. Dashed horizontal lines represent $\Delta \mathcal{E}_1 = 1.5$ which was used to depict Fig. \ref{['fig6']}.