First Step Analysis - Gambler's Ruin

Example: First Step Analysis - Gambler’s Ruin

Using the scenario from gambling model, consider a gambler with $N=3$ and win probability $p=0.4$ (lose probability $q=0.6$).

We want to find the probability of reaching the target ($3$) starting from state $1$.

Hitting Probability

Let ${\pi }_i$ be the probability of hitting state $3$ starting from state $i$.

Boundary Conditions:

• ${\pi }_3=1$ (Target reached)
• ${\pi }_0=0$ (Broke)

Equations for Transient States:

• ${\pi }_1=0.4{\pi }_2+0.6{\pi }_0⟹{\pi }_1=0.4{\pi }_2$
• ${\pi }_2=0.4{\pi }_3+0.6{\pi }_1⟹{\pi }_2=0.4(1)+0.6{\pi }_1$

Solution:

${\pi }_1=0.4(0.4+0.6{\pi }_1)=0.16+0.24{\pi }_1$

$0.76{\pi }_1=0.16⟹{\pi }_1\approx 0.2105$

Expected Time to Absorption

Let ${\mu }_i$ be the expected number of steps until reaching state $i$.

Boundary Conditions:

• ${\mu }_0=0$
• ${\mu }_3=0$

Equations:

• ${\mu }_1=1+0.4{\mu }_2+0.6{\mu }_0⟹{\mu }_1=1+0.4{\mu }_2$
• ${\mu }_2=1+0.4{\mu }_3+0.6{\mu }_1⟹{\mu }_2=1+0.6{\mu }_1$

Solution:

${\mu }_1=1+0.4(1+0.6{\mu }_1)=1+0.4+0.24{\mu }_1$

$0.76{\mu }_1=1.4⟹{\mu }_1\approx 1.8421\text{ steps}$

Related

• First Step Analysis
• Gambling Model