Understanding Why the Gambler’s Ruin Equation is a Linear Difference Equation

When studying probability topics such as Gambler’s Ruin, Markov chains, or random walks, we often encounter equations like:

$P_i = pP_{i+1} + qP_{i-1}$

At first glance, this may look confusing, especially for beginners. However, this equation is an example of a linear difference equation. Let us understand why.

What is a Difference Equation?

A difference equation relates values of a sequence at different positions or indices.

In the equation:

$P_i = pP_{i+1} + qP_{i-1}$

the terms involve:

$P_{i+1}$
$P_i$
$P_{i-1}$

These represent values of the same sequence at different steps.

Unlike differential equations, which deal with derivatives and continuous change, difference equations deal with discrete steps such as:

step 1
step 2
step 3

This is why it is called a difference equation.

Rearranging the Equation

To study the equation more systematically, we move all terms to one side:

$pP_{i+1} - P_i + qP_{i-1} = 0$

This is the standard form commonly used while solving recurrence relations and difference equations.

Why is it Linear?

An equation is called linear if:

variables are only raised to power 1
variables are not multiplied together
no nonlinear functions like sine, exponential, or logarithm appear

In the equation:

$pP_{i+1} - P_i + qP_{i-1} = 0$

each term appears only once and to the first power.

There are no terms like:

^2
$P_iP_{i+1}$
$\sin(P_i)$

The coefficients $p$ , $-1$ , and $q$ are constants.

Therefore, the equation is linear.

Examples of Nonlinear Difference Equations

The following would be nonlinear:

$P_{i+1}^2 + P_i = 0$

$P_iP_{i+1} + P_{i-1} = 0$

$\sin(P_i) + P_{i+1} = 0$

These are nonlinear because the variables are squared, multiplied together, or placed inside nonlinear functions.

Connection with Differential Equations

A linear differential equation looks like:

$ay'' + by' + cy = 0$

A linear difference equation looks like:

$aP_{i+1} + bP_i + cP_{i-1} = 0$

Notice the similarity:

derivatives become shifted sequence terms
continuous change becomes discrete change

This is why difference equations are often called the discrete counterpart of differential equations.

Why This Matters in Probability

Linear difference equations appear frequently in:

Gambler’s Ruin
random walks
Markov chains
dynamic programming
reinforcement learning
finance and stock models
queueing systems

The technique used in the lecture screenshot, called First Step Analysis, converts a probability problem into a solvable linear difference equation.

This is one of the foundational ideas behind stochastic processes and modern probabilistic modeling.