A Counterfactual Analysis of the Dishonest Casino

The dishonest casino is a well-known hidden Markov model (HMM) often used in education to introduce HMMs and graphical models. A sequence of die rolls is observed with the casino switching between a fair and a loaded die. Instead of recovering the latent regime through filtering, smoothing, or the Viterbi algorithm, we ask a counterfactual question: how much of the gambler's winnings are caused by the casino's cheating? We introduce a class of structural causal models (SCMs) consistent with the HMM and define the expected winnings attributable to cheating (EWAC). Because EWAC is only partially identifiable, we bound it via linear programs (LPs). Numerical experiments help to develop intuition using benchmark SCMs based on independence, comonotonic, and countermonotonic copulas. Imposing a time homogeneity condition on the SCM yields tighter bounds, whereas relaxing it produces looser bounds that admit an explicit LP solution. Domain knowledge such as pathwise monotonicity or counterfactual stability can be incorporated through additional linear constraints. Finally, we show the time-averaged EWAC becomes fully identifiable as the number of time periods tends to infinity. Our work is the first to develop LP bounds for counterfactuals in an HMM setting, benefiting educational contexts where counterfactual inference is taught.