Optimal strategies in the Fighting Fantasy gaming system: influencing stochastic dynamics by gambling with limited resource

Fighting Fantasy is a popular recreational fantasy gaming system worldwide. Combat in this system progresses through a stochastic game involving a series of rounds, each of which may be won or lost. Each round, a limited resource (`luck') may be spent on a gamble to amplify the benefit from a win or mitigate the deficit from a loss. However, the success of this gamble depends on the amount of remaining resource, and if the gamble is unsuccessful, benefits are reduced and deficits increased. Players thus dynamically choose to expend resource to attempt to influence the stochastic dynamics of the game, with diminishing probability of positive return. The identification of the optimal strategy for victory is a Markov decision problem that has not yet been solved. Here, we combine stochastic analysis and simulation with dynamic programming to characterise the dynamical behaviour of the system in the absence and presence of gambling policy. We derive a simple expression for the victory probability without luck-based strategy. We use a backward induction approach to solve the Bellman equation for the system and identify the optimal strategy for any given state during the game. The optimal control strategies can dramatically enhance success probabilities, but take detailed forms; we use stochastic simulation to approximate these optimal strategies with simple heuristics that can be practically employed. Our findings provide a roadmap to improving success in the games that millions of people play worldwide, and inform a class of resource allocation problems with diminishing returns in stochastic games.