Model
It is interesting to analyze the behavior of different types of gamblers in a casino gambling model. The model was a generalization of the one initially proposed by Barberis (2012). At time $0$, a gambler is offered a fair bet in a casino: win or lose 1 dollar with equal probability. If the gambler takes the bet, the bet is played out and she either gains 1 dollar or loses 1 dollar at time $1$. Then the gambler is offered the same bet and she decides whether to play. The same bet is offered and played out repeatedly until the first time the gambler declines the bet and exits the casino. At that time, the gambler exits the casino with all her prior gains and losses. The gambler needs to determine the best time to stop playing, with the objective to maximize her preference value of the final wealth, while the preferences are represented by the cumulative prospect theory.
When the gambler uses the same CPT preferences to revisit the gambling problem in the future, e.g., at time $t>0$, the optimal gambling strategy solved at time $0$ is typically no longer optimal. This time-inconsistency arises from the probability weighting in CPT. If the gambler is pre-committed, she commits her future selves to the strategy solved at time $0$. If she fails to pre-commit, namely she keeps changing her strategy in the future, then the gambler is said to be naive. The actual strategy implemented by a naive gambler will often be very different from the strategy viewed as optimal at time 0 and which is actually implemented by a pre-committed gambler.
Barberis (2012) formulated and studied a casino gambling model, in which a gambler comes to a casino at time $0$ and is repeatedly offered a bet with an equal chance to win or lose one dollar. The gambler decides when to stop playing. Clearly, the answer depends on the gambler’s preferences. In Barberis (2012), the gambler’s risk preferences are represented by the cumulative prospect theory (CPT). In this theory, individuals’ preferences are determined by an S-shaped utility function and two inverse-S shaped probability weighting functions. The latter effectively overweight the tails of a distribution, so a gambler with CPT preferences overweights very large gains of small probability and thus may decide to play in the casino.
A crucial contribution in Barberis lies in showing that the optimal strategy of a gambler with CPT preferences, a descriptive model for individuals’ preferences, is consistent with several commonly observed gambling behaviors such as the popularity of casino gambling and the implementation of gain-exit and loss-exit strategies. However, the setting of Barberis was restrictive as it assumed that the gambler can only choose among simple (path-independent) strategies.
We call a random time chosen by the gambler to exit the casino a strategy. As in Barberis, we assume CPT preferences for the gambler; so the decision criterion is to maximize the CPT value of his wealth at the time when he leaves the casino. More precisely, the gambler first computes his gain and loss $X$ relative to some reference point and evaluates $X$ by
\begin{equation}
V(X):=\int_0^\infty u(x)d[-w_{+}(1-F_X(x))]+\int_{-\infty}^0 u(x)d[w_{-}(F_X(x))]
\end{equation}
where $F_X$ is the cumulative distribution function (CDF) of $X$. The function $u$, which is
strictly increasing, is called the utility function (or value function) and $w_\pm$, two strictly increasing mappings from $[0,1]$ onto $[0,1]$, are probability weighting (or distortion) functions on gains and losses, respectively. Use the following parametric forms proposed by Tversky and Kahneman (1992) for the utility and probability weighting functions:
\begin{equation}
u(x)=
\begin{cases}
x^{\alpha_+}\qquad x\geq0 \\
-\lambda(-x)^{\alpha_-}\qquad x<0
\end{cases}
\end{equation}
and
\begin{equation}
w_\pm(p)={p^{\delta_\pm}\over (p^{\delta_\pm}+(1-p)^{\delta_\pm})^{1/\delta_\pm}}
\end{equation}
where $\lambda\geq1$, $\alpha_\pm\in(0,1]$ and $\delta_\pm\in[0.28,1]$. Such $u$ is S-shaped and $w_\pm$ are inverse-S-shaped, and thus they are able to describe the fourfold pattern of individuals’ choice under risk that cannot be explained by the classical expected utility theory (EUT).
In the casino gambling problem, as the bet is assumed to be fair, the cumulative gain or loss of the gambler, while he continues to play, is a standard symmetric random walk $S_n$, $n\geq0$, on $\mathbb{Z}$, the set of integers. We further assume that the gambler uses his initial wealth as the reference point, so he perceives $S_n$ as his cumulative gain or loss after $n$ bets. As a result, for any exit time $\tau$, the gambler’s CPT preference value of his gain and loss at the exit time is
\begin{align}
\begin{split}
V(S_\tau)=&\sum_{n=1}^T u(n)(w_+(\mathbb{P}\lbrace S_\tau\geq n\rbrace)-w_+(\mathbb{P}\lbrace S_\tau> n\rbrace)) \\
&+\sum_{n=1}^T u(-n)(w_-(\mathbb{P}\lbrace S_\tau\leq n\rbrace)-w_-(\mathbb{P}\lbrace S_\tau< n\rbrace))
\end{split}
\end{align}
with the convention that $\infty-\infty=-\infty$, so that $V(S_\tau)$ is well-defined. The gambler’s problem at time $0$ is to find an exit time $\tau$ in a set of admissible strategies which maximizes $V(S_\tau)$.
Comparison of Strategies
Barberis consider path independent strategies only, in which for any $t\geq0$, $\lbrace\tau=t\rbrace$ (conditioning on $\lbrace\tau\geq t\rbrace$) is determined by $(t,S_t)$, i.e., by the agent’s cumulative gain and loss at time $t$ only.
