Optimal strategies in the fighting fantasy gaming system: Influencing stochastic dynamics by gambling with limited resource

In many games and other processes, participants can choose to intervene in some way that does not follow the usual progress of the game (for example, cheating at cards, or spying on rivals) which may provide benefits, but also possibly incur substantial costs. Here, repeated interventions may be more likely to incur negative outcomes – for example, as the chance of getting caught increases. How to optimally employ these risky interventions, trading off potential advantages and disadvantages, can then be challenging to identify. Here, we study such a game, taken from the popular ‘Fighting Fantasy’ gamebook series. This stochastic game involves a series of rounds, each of which may be won or lost. Each round, a unit of limited resource (‘luck’) may be spent on a gamble to amplify benefits from a win or to mitigate deficits from a loss. However, the success of this gamble depends on the number of units of remaining resource, and if the gamble is unsuccessful, benefits are reduced and deficits increased. By choosing to expending resource, a player thus has diminishing probability of positive return, as in the cheating and espionage examples above. We characterise the dynamics of this system using stochastic analysis and dynamic programming, solve the Bellman equation for the complete system with diminishing returns, and identify the optimal strategy for any given state during the game. We use classification tools to distil general principles for this and related problems, informing resource allocation problems with diminishing returns in stochastic decision theory.