  Live performances of music are always charming, with the unpredictability of
improvisation due to the dynamic between musicians and interactions with the
audience. Jazz improvisation is a particularly noteworthy example for further
investigation from a theoretical perspective. Here, we introduce a novel
mathematical game theory model for jazz improvisation, providing a framework
for studying music theory and improvisational methodologies. We use
computational modeling, mainly reinforcement learning, to explore diverse
stochastic improvisational strategies and their paired performance on
improvisation. We find that the most effective strategy pair is a strategy that
reacts to the most recent payoff (Stepwise Changes) with a reinforcement
learning strategy limited to notes in the given chord (Chord-Following
Reinforcement Learning). Conversely, a strategy that reacts to the partner's
last note and attempts to harmonize with it (Harmony Prediction) strategy pair
yields the lowest non-control payoff and highest standard deviation, indicating
that picking notes based on immediate reactions to the partner player can yield
inconsistent outcomes. On average, the Chord-Following Reinforcement Learning
strategy demonstrates the highest mean payoff, while Harmony Prediction
exhibits the lowest. Our work lays the foundation for promising applications
beyond jazz: including the use of artificial intelligence (AI) models to
extract data from audio clips to refine musical reward systems, and training
machine learning (ML) models on existing jazz solos to further refine
strategies within the game.
