Simulating Cumulative Gains and Losses in Repeated Play

From One Play to a Running Total

A game can look favorable at the end and still include a large loss along the way. Or it can cross a loss limit early and later recover. To study these possibilities, record not only each play’s gain or loss but also the cumulative total after every play. Repeating that process for many simulated games lets us estimate probabilities of finishing ahead or reaching a specified loss threshold.

Represent the net gain or loss on play $j$ by $X_j$. A gain is positive, a loss is negative, and a play with no net change has outcome zero. Start the game with total zero. After $k$ plays, the cumulative total is the sum of the first $k$ outcomes. This is a path through time: the sequence of running totals, not just its last value.

Definition: For a game with signed play outcomes $X_1, X_2, \ldots, X_m$, the cumulative total after $k$ plays is $C_k = \sum_{j=1}^{k} X_j$, with $C_0=0$. The final total is $C_m$. A game finishes ahead if $C_m>0$. It reaches or crosses a loss threshold of $L$ dollars if $C_k \leq -L$ for at least one play $k$.

The distinction between the final total and the path matters. The condition $C_m \leq -L$ asks whether the game ends at or below the threshold. The condition $C_k \leq -L$ for at least one $k$ asks whether the threshold was reached at any time, even if later gains lift the final total above it.

State the Play Model Before Simulating

A simulation needs rules for generating each play. For a simple model, suppose each play independently produces a gain of $g$ dollars with probability $p$, and otherwise produces a loss of $\ell$ dollars, where $g>0$ and $\ell>0$. Then an outcome $X_j$ is $g$ with probability $p$ and $-\ell$ with probability $1-p$. Independence means that the outcome of one play does not change the probabilities for another play. The repeated-play model also assumes the same outcome probabilities and amounts apply on every play.

Those assumptions are part of the model, not automatic facts about every game. If rules, stakes, or probabilities change between plays, a model with fixed play probabilities may not represent the situation. For the examples below, the rules are explicitly held fixed, and simulated play outcomes are generated independently according to those rules. Random-number generation and mapping random values to outcomes are standard simulation tools; the emphasis here is on accumulating the resulting outcomes correctly.

To estimate an event probability, simulate many complete games under the stated rules. For each game, keep its cumulative totals and mark whether the event of interest occurred. The estimated probability is the number of simulated games meeting the event condition divided by the total number simulated. This is a Monte Carlo proportion: it describes the simulated runs and estimates the model probability, rather than guaranteeing what will happen in any particular game.

Worked Example: Following One Game’s Cumulative Total

Consider a hypothetical six-play game. Each play, under the game’s rules, independently gives either a gain of $4 or a loss of $3. One simulated game produces the outcomes $+4,-3,-3,+4,-3,+4$, in dollars. Find the cumulative total after each play, the final total, and whether the game reaches a loss threshold of $2.

Start with $C_0=0$, and add each outcome to the previous total:

\begin{aligned} C_1 &= 0+4=4,\\ C_2 &= 4-3=1,\\ C_3 &= 1-3=-2,\\ C_4 &= -2+4=2,\\ C_5 &= 2-3=-1,\\ C_6 &= -1+4=3. \end{aligned}

The final total is $C_6=3$, so this simulated game finishes $3 ahead. However, it reaches the $2 loss threshold at play 3 because $C_3=-2$. Thus “finished ahead” and “never reached a $2 loss” are not equivalent descriptions of this path.

Estimating Probabilities Across Simulated Games

A single simulated game shows one possible path, not the probability of an event. To estimate a probability, repeat the whole game many times, checking the relevant condition separately in each run. The conditions should be written precisely before counting: for example, “final total is positive” or “cumulative total is ever at most $-4$.”

Worked Example: Finishing Ahead or Reaching a Loss Limit

Suppose a hypothetical five-play game pays a $3 gain with probability $0.4$ on each play and otherwise produces a $2 loss. Outcomes are independent and the same rules apply on all five plays. The following table records eight simulated games. The entries in the “Cumulative totals” column are the running totals after plays 1 through 5, all in dollars.

Game	Play outcomes	Cumulative totals	Final total
1	+3, +3, −2, +3, −2	3, 6, 4, 7, 5	5
2	−2, −2, +3, −2, −2	−2, −4, −1, −3, −5	−5
3	+3, −2, +3, −2, +3	3, 1, 4, 2, 5	5
4	−2, +3, −2, −2, +3	−2, 1, −1, −3, 0	0
5	−2, −2, −2, +3, −2	−2, −4, −6, −3, −5	−5
6	+3, +3, +3, −2, +3	3, 6, 9, 7, 10	10
7	−2, +3, +3, −2, −2	−2, 1, 4, 2, 0	0
8	+3, −2, −2, −2, +3	3, 1, −1, −3, 0	0

First estimate the probability of finishing ahead. The qualifying games have final totals greater than zero: games 1, 3, and 6. Game 4 ends at zero, so it does not count as finishing ahead.

\widehat{P}(\text{finish ahead}) = \frac{\text{number finishing ahead}}{\text{number of simulated games}} = \frac{3}{8} = 0.375.

In these eight simulated games, three of eight finished ahead, giving an estimated probability of $0.375$, or 37.5%. Now consider reaching a loss threshold of $4 at any point. Games 2 and 5 have at least one cumulative total at or below $-4$; the other six do not.

\widehat{P}(\text{reach a loss of at least \$4}) = \frac{2}{8} = 0.25.

The estimate for reaching the threshold is therefore 25% from these runs. These estimates summarize only eight simulated games and can vary with a new set of runs. They are not exact probabilities for the model, and they should not be interpreted as guarantees about future play.

Why Counting Paths Gives an Estimate

For a particular event, such as “finish ahead,” each complete simulated game either satisfies the condition or does not. Let $I_i$ equal 1 when game $i$ satisfies the event and 0 otherwise. If $R$ games are simulated, the proportion that satisfy it is the average of these indicators. Each indicator has probability $q$ of being 1, where $q$ is the model probability of the event. Therefore its expected value is $q$, and the expected average of $R$ indicators is also $q$:

\widehat{q} = \frac{I_1+\cdots+I_R}{R}, \qquad E(\widehat{q}) = \frac{E(I_1)+\cdots+E(I_R)}{R} = \frac{q+\cdots+q}{R} = q.

This justifies using the simulated proportion as an estimate of the event probability. The equality describes its expected value over repeated simulation experiments; an actual finite run will generally produce a proportion that differs from $q$. Larger simulation runs usually reduce random variation in the estimate, but they do not repair an inaccurate model or an incorrectly coded event. Earlier material on Monte Carlo error explains how to assess the variability of such estimates.

Worked Example: A Threshold Event Can Occur Before Recovery

Consider a different hypothetical four-play game with a $5 gain on a play with probability $0.3$, and a $2 loss otherwise. Plays are independent and follow the same rules. Six simulated games produce the paths below. Estimate the probability of reaching a cumulative loss of at least $6 at any point.

Game	Play outcomes	Cumulative totals	Reached −$6?
A	−2, −2, −2, −2	−2, −4, −6, −8	Yes
B	+5, −2, −2, −2	5, 3, 1, −1	No
C	−2, +5, −2, −2	−2, 3, 1, −1	No
D	−2, −2, +5, −2	−2, −4, 1, −1	No
E	−2, −2, −2, +5	−2, −4, −6, −1	Yes
F	+5, +5, −2, −2	5, 10, 8, 6	No

The event is “some cumulative total is at most $-6$,” not “the final total is at most $-6$.” Games A and E meet the event: A reaches $-6$ on play 3, while E reaches $-6$ on play 3 and later recovers to $-1$. Thus two of the six simulated games qualify:

\widehat{P}(\text{reach a loss of at least \$6}) = \frac{2}{6} \approx 0.333.

The estimated probability from these six paths is about 0.333, or 33.3%. In particular, game E would be missed if the event were incorrectly checked using final totals alone.

Making the Simulation Useful

Cumulative-play simulations are useful when the order of outcomes matters. A final total alone cannot show whether a player crossed a loss limit, had a temporary large loss, or recovered after a difficult stretch. Tracking $C_1,\ldots,C_m$ makes those questions directly observable in each simulated game.

Before running a simulation, identify the event in a form that can be checked consistently. For example:

Finishing ahead: check whether $C_m>0$.
Ending at least $L behind: check whether $C_m\leq -L$.
Ever reaching a loss of at least $L: check whether any $C_k\leq -L$.

Also state the number of plays, the possible outcomes, and their probabilities. The simulated probability answers a conditional question: if the specified model represents play, how often does the event occur in repeated runs of that model? It does not establish that the model matches real behavior or that one player will experience the estimated frequency over a small number of games.

Common Mistakes

Checking only the final total for a path event. A game can cross a loss threshold and later recover. For an “ever reaches” event, inspect every cumulative total.
Adding outcomes incorrectly. The cumulative total after a play is the previous cumulative total plus that play’s signed outcome. For example, a $2 loss is $-2$, not $+2$.
Counting a zero as finishing ahead. “Ahead” means $C_m>0$; a final total of exactly zero is a tie, not a positive gain.
Treating a small simulated proportion as an exact probability. The proportion is an estimate based on a finite set of random runs. More runs can reduce simulation noise, but only a suitable model can support a meaningful estimate.
Changing assumptions without changing the simulation. If probabilities or payouts depend on previous play, fixed independent play outcomes are not the appropriate model.

Key takeaway: Define each play as a signed outcome, compute the cumulative sum after every play, and specify whether the event concerns the endpoint or any point along the path. Count qualifying simulated games and divide by the number simulated to estimate the event probability under the stated model.

Check Your Understanding

Use cumulative totals to answer each question.

A three-play path is $+5,-2,-4$ dollars. What are the cumulative totals, and does the game finish ahead?
Explain the difference between ending at or below $-\$8$ and reaching or crossing $-\$8$ at any point.
In 200 simulated games, 46 finish with a positive total. What is the estimated probability of finishing ahead?
A game’s cumulative totals are $-3,-7,-2,1$. Did it reach a loss threshold of $6? Did it end at or below $-\$6$?
Name two assumptions that would be needed for a model in which each play independently has the same gain and loss probabilities.