A Counterfactual Analysis of the Dishonest Casino

TL;DR

This paper addresses how to attribute winnings in a two-state hidden Markov model (the dishonest casino) to cheating through a counterfactual lens. It embeds the HMM in a structural causal model and derives sharp, linear-programming bounds (EWAC) for the counterfactual winnings attributable to cheating, leveraging a marginals-only formulation over a copula-parameterized joint emission matrix. Time-homogeneity tightens these bounds, while time-inhomogeneity yields separable LPs with analytic solutions; crucially, the time-averaged EWAC becomes fully identifiable as the horizon grows. The work contributes a first-of-its-kind LP-based framework for counterfactual bounds in dynamic latent-state models and offers a tractable educational tool for illustrating partial identification and the role of domain knowledge in causal inference.

Abstract

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.

Figures (8)

• Figure 1: The dishonest casino. The states $H_{1:T}$ represent the fair / biased die and are hidden while the emissions $O_{1:T}$, i.e., die rolls, are observed.
• Figure 2: The SCM underlying the dishonest casino's HMM. The only difference between the SCM here and the HMM of Figure \ref{['fig:SCM-Casino1']} is the addition of (grey) exogenous noise nodes $[\mathbf{U}_t, \mathbf{V}_t]_t$.
• Figure 3: A simple causal graph to illustrate CS.
• Figure 4: EWAC results for Path 1. In Figure \ref{['fig:ewac1a']}, the UB and comonotonic curves coincide (the highest curve in the figure), as do the LB and countermonotonic curves (the lowest curve). In Figure \ref{['fig:ewac1b']}, the UB (CS) and comonotonic curves coincide (the highest curve in the figure). The region shaded gray in Figure \ref{['fig:ewac1a']} corresponds to feasible values of EWAC. The region shaded yellow in Figure \ref{['fig:ewac1b']} corresponds to feasible values of EWAC given that we impose the CS constraints. The independence and countermonotonic SCMs do not satisfy the CS constraints for any value of $\eta$.
• Figure 5: EWAC results for Path 2. The region shaded gray in Figure \ref{['fig:ewac1a']} corresponds to feasible values of EWAC. The region shaded yellow in Figure \ref{['fig:ewac1b']} corresponds to feasible values of EWAC given that we impose the CS constraints. Only the countermonotonic SCM fails to satisfy the CS constraints for all values of $\eta$.

Theorems & Definitions (14)

• Proposition 1: EWAC Characterization
• Proposition 2: EWAC Bounds
• Proposition 3: Time-Inhomogenous Solution
• Proposition 4: EWAC CS Bounds
• Remark 1
• Proposition 5: EWAC Characterization Under Copulas
• Definition 1: Copula
• Theorem 1: Sklar 1959
• Theorem 2: The Fréchet-Hoeffding Bounds
• Definition 2: Comonotonic Copula