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Fighting Fantasy is a popular recreational fantasy gaming system worldwide.Combat in this system progresses through a stochastic game involving a seriesof 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 mitigatethe deficit from a loss. However, the success of this gamble depends on theamount of remaining resource, and if the gamble is unsuccessful, benefits arereduced and deficits increased. Players thus dynamically choose to expendresource to attempt to influence the stochastic dynamics of the game, withdiminishing probability of positive return. The identification of the optimalstrategy for victory is a Markov decision problem that has not yet been solved.Here, we combine stochastic analysis and simulation with dynamic programming tocharacterise the dynamical behaviour of the system in the absence and presenceof gambling policy. We derive a simple expression for the victory probabilitywithout luck-based strategy. We use a backward induction approach to solve theBellman equation for the system and identify the optimal strategy for any givenstate during the game. The optimal control strategies can dramatically enhancesuccess probabilities, but take detailed forms; we use stochastic simulation toapproximate these optimal strategies with simple heuristics that can bepractically employed. Our findings provide a roadmap to improving success inthe games that millions of people play worldwide, and inform a class ofresource allocation problems with diminishing returns in stochastic games.

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