Optimal exit strategies of CPT gamblers in unfair gambles

In this paper we study optimal exit strategies of gamblers with cumulative prospect theory (CPT) preferences in games where the expected payoff is strictly negative at each play, and formulate the problem as optimal stopping on asymmetric random walks. Applying a geometric transformation of the underlying cumulative gain/loss process, engaging randomized strategies and changing the decision variable from stopping times to probability distribution of the accumulated gain or loss at exit time, we solve the problem via the Skorokhod embedding. Drastically different from the fair gamble problem studied by He et al. (2019a), we show that the unfair problem in the infinite time horizon has finite values for a wide range of CPT parameter specifications. We then present the analytical solutions in the case of piece-wise power utility and power probability distortion functions. Compared to the strategies used in fair gambling, the CPT gamblers in unfair gambles are less loss-tolerant and choose not to gamble at all when the games are sufficiently unfavorable.