Using SHAP for Feature Importance in Predicting Axelrod Tournament Winners

Introduction

Prisoner’s dilemma is the most known and investigated simple 2x2 game in Game Theory. Rationality demands players to defect, while general common sense implicates that wasting resources is bad for everyone. This gap between an ideal rationality-based theoretical framework and real behavior inspired many ideas on how "to solve" a prisoner’s dilemma. One approach is an iterated setup. Although things do not change for any finite-iterated game, surprisingly, adding uncertainty helps the players cooperate. If the game finishes with probability δ in every round and δ is small enough, cooperation becomes (subgame perfect) Nash equilibrium. In the late 1970s and early 1980s, Robert Axelrod proposed a series of game theory competitions using computers, later called the "Axelrod tournaments." (AT).  Axelrod invited academics to submit computer programs that would compete against each other in a round-robin format, with each pair of strategies playing multiple rounds of the Prisoner’s Dilemma. The first tournament included 14 strategies, and the second - 62 entries. The most interesting outcome was the success of a "Tit-for-Tat" strategy. This strategy begins by cooperating and then copying the opponent’s previous move. The AT fundamentally changed how researchers understood the evolution of cooperation. Still, the exact reasons why Tit-for-Tat was the winner (there are some issues with replicating original tournaments ) and how to predict winners for tournaments with other sample strategies remain unclear. The study of iterated prisoner’s dilemma (IPD) contests is a noteworthy area of research. In real-world cases, interactions are not short-term but ongoing and repetitive, and every choice has a specific outcome for those involved. In this work we consider result of tournament as a black-box model. We perform simulations, gather data and train model to predict the result. Then we are trying to explain the result, using additional context variables.

ML pipeline and SHAP

In this work we explore how machine learning can predict the winning strategy in an Axelrod tournament based on the composition of participating strategies and their pairwise interaction characteristics, and how can we explain the prediction? Our approach can be summarized in the following steps: 1. Data generation. Choose m > 1 random strategies, create a 5000 samples of possible tournaments in form of vectors (n1, n2, ..., nm), run tournaments and gather data about winners. In this work we considered only deterministic strategies, so it is possible for 2 ore more strategies to get exactly the same score. 2. Feature Engineering. Calculate properties for each tournament (level of cooperation for example), and matrix of pairwise matches results for each pair of initially chosen strategies. 3. Train multi-class classification models to predict the winning strategies. 4. Apply SHAP (SHapley Additive exPlanations)  to interpret prediction results We got F1 Score: 0.9283 with simple multilayered model with Adam optimizer. Applying of SHAP allowed us to visualize impact of features Fig. 1) for each particular tournament composition. On this image we see, that winner prediction (Forgiver in this case) based not only on its performance with other strategies but also on performance of some particular strategies against each other. All experiments and result are availible here.

Key Findings of the work 

Tournament outcomes can be predicted with reasonable accuracy when working with deterministic strategies. A higher diversity of participants raises the chances of specific strategies becoming "dominant," which can lead to less balanced outcomes. Despite this additional complexity, both traditional machine learning models and neural networks demonstrate solid performance. By incorporating SHAP values into a classification model we can estimate importance of features. Using matrix of pairwise performance as an additional context features, we can go beyond simply observing which strategy succeeds and start to explain why. Including a disruptor agent helps to distinguish strategies significantly. The tournament dynamics change and show how strong or weak every strategy is by adding a player who acts erratically or attempts to exploit others. In a sense, the disruptor pushes the system to reveal concealed flaws and compel strategies to show how well they can adjust. In settings that are otherwise consistent or too cooperative, this approach helps to reveal the underlying framework of strategic rivalry more clearly.