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Introduction

In this article:

• Are $50 better than $1,000?
• How big should a beginner's bankroll be?

Every new member at PokerStrategy.com receives a free $50 starting capital at one of PokerStrategy.com's partner poker rooms. We deliberately choose to refer to the $50 as "starting capital" and not as a "free bankroll" or a "no deposit bonus" as otherwise common.

For many members, this money has formed the foundation of their poker careers. You however have to ask yourself the question: Is the $50 a sufficient amount to start a poker career or would an ambitious player be better off investing a larger sum out of his own pocket?

This article focuses on this question and tries to illustrate the difference between a player who starts out with $50 and one who starts out with $1,000. Does the player with the bigger starting bankroll really have an advantage?

The results of this examination can be applied to every game type, even though the article uses the No Limit Hold'em Big Stack Strategy to represent all existing types.

The Approach

The absolute financial success of a poker player essentially depends on two factors:

His skill: How good a player is he? How big his edge is over his opponents at a given limit? The term "skill" should be taken as rather general, to also include things such as table selection and the psychological aspects of self management.

The limit played: Which limits can he play? The higher the limit, the higher are his absolute winnings based on his win rate - given that his win rate is positive.

In order to find the differences between players with different starting bankrolls, we will create four different player types in the next section of the article. We will make assumptions about the development of their skills, about the limits they play and about their bankroll management.

These player types will then be used in a poker career simulation and we will subsequently examine how they perform when compared to each other.

Initial Assumptions for the Simulation

Let's assume there are four different players: A, B, C and D.

• Player A starts with $50 and is therefore a typical PokerStrategist.
• Players B, C and D start out with $1,000 each.

All four player types have the same prior knowledge of poker strategy and are at the typical beginner level who has studied the basic strategy.

Assumptions on the skill progress curve and limits

In order to perform the simulation, we assign a value to the poker abilities of each player. This value increases with time and it does so at the same rate for all four players. The skill curve is given by a square root function. It therefore climbs very steeply at first and then flattens out progressively.

In addition, we assume that to play break even at each limit requires a certain skill level (break even: +-$0, no profit and no loss). If the skill level of a player is smaller than this value, he will lose money. If the skill level of a player is greater than this value, he will make money. The amount of profit therefore depends on your individual skill, as well as on the limit (a bigger investment also means more leverage). For a specific illustration, we are using limits NL2, NL5, NL10, NL25, NL50 and NL100.

Along with the skill curve, we get the following illustration:

Skill function: Presumed, idealised improvement of skill in relation to time   y-axis: Skill x-axis: Time

The skill values required to move up a given limit are represented as horizontal lines, one above the other.

Assumptions on bankroll management

Player A and player B play with the 20 stack bankroll management. This means they always play the highest limit for which they have a bankroll of 20 stacks available. If they have fewer than 20 stacks available, they will move down a limit.

Player C and player D are more hypothetical players. Player C always plays the highest limit at which his win rate is positive. Player D always plays the limit at which his win rate in $ reaches the maximum.

The following overview is a summary:

	Player A	Player B	Player C	Player D
Starting bankroll	$50	$1,000	$1,000	$1,000
Bankroll management	20 stacks	20 stacks	Always plays at the highest limit that he can play break even.	Always plays at the limit that gives him the highest win rate in $.

No variance

Even though a poker game is always affected by temporary fluctuations, the aspect of variance will not be taken into account to address the questions that this article raises.

How the Simulation Works

This section is aimed at those who are interested in a detailed explanation of how the simulation works. You can carry out the simulation in several different ways. The following approach was chosen here:

For the skill function we assume: skill(t) = t^(0.55)

This function describes the growth of a player's poker skills. We did not use the exact root function so as to provide a more plausible representation of the intersections with the break even values of the various limits.

Alternatively, we could have varied single values. Changing them would only have a small effect on the result.

For the break even skill at the various limits, the following values were used:

NL2	NL5	NL10	NL25	NL50	NL100
1	2	3	4	5	6

If a player has reached the respective skill value at the given limit, he will neither make nor lose money. If his skill value is lower, he will suffer losses, if it is higher, he will make a profit.

The simulation requires a function that returns the played limit for a given bankroll value. This is where players A and B differ from players C and D.

The bankroll will now be calculated in steps, by a process of iteration. With a step width of 1, this is done as follows:

roll(0) = starting bankroll
roll(t+1) = roll(t) + limit( roll(t) ) * ( skill(t) - skill2( limit( roll(t) ) ) )

Depending on the player type, limit( roll(t) ) is given by either answering if the player is a winning player or by applying the 20 stacks rule.

skill2(limit) is the value required for break even play. With every time step, the bankroll grows by the win rate (skill(t)-skill2(limit)) multiplied by the limit played. Poker uses the same principle.

You can basically make this calculation continuous by using a small step width for the time axis and a large number of steps. However, this would have an undesirable effect and wouldn't be very realistic.

Let's assume a player's combination of skill level and bankroll puts him between two limits. If this player reaches the bankroll value for the higher limit, he immediately moves down again because he loses a small amount. He then immediately moves up again because he wins a small amount. After many calculation steps, we obtain a line that precisely overlaps the line representing skill level required for break even play at the higher limit, until the player beats the higher limit. Curves of this kind don't convey a very realistic image, which is why we used a rather wide step width. This allows small steps to the next higher or lower limit, until the higher limit is beaten.

All important functions have now been illustrated. The method of iteration allows you to construct a tuple of numbers and represent it graphically for different starting bankrolls, as well as for varying step widths and a varying number of steps.

Absolute Results

We let the four proposed player types play poker for a while, expand their poker skills based on their skill function and move up the limits depending on their bankroll management. What does their bankroll look like after a certain amount of time?

Let's take a look at the growth of the bankroll of player A, who started with $50.

Bankroll player A   y-axis: Bankroll in $ x-axis: Time

We would expect the graph to look like this, or at least similar, given we don't take variance into account. The profit per time unit, i.e. the slope of the curve, rises due to the growing skill and due to the higher limit played. Therefore the slope keeps increasing with time. In comparison, see the results for player B.

Bankroll player B   y-axis: Bankroll in $ x-axis: Time

As the skill curve indicates, the player will lose some of this money at first because he starts out at a limit that is too high for his current poker skills.

While player A only spends a short time as a losing player and turns into a consistent winning player, from the moment he beats NL2, player B moves down from NL50 to NL25 until his skill level increases to the extent that he can beat the limit.

This is when player B also starts making money. After this point, his curve shows similar characteristics to that of player A. This is logical: both players have the same skills as well as the same bankroll management.

We then have player type C. He always plays at the highest limit he can beat (otherwise he plays at the smallest available limit). His graph looks as follows:

Bankroll player C   y-axis: Bankroll in $ x-axis: Time

In contrast to player B, he doesn't lose any large amounts of money, but he is not able to reach a much better final result either. You can also recognise an obvious mistake that this player makes: His win rate tends towards 0 per time unit because he moves up as soon as he plays break even. He still needs time to gain an edge and make a profit at the higher limits.

Finally we will take a look at the ideal player D. He plays with the highest win rate there is at all times and will perform better than all of his opponents. The question is: How well does he perform? Here's the answer:

Bankroll player D   y-axis: Bankroll in $ x-axis: Time

The curve shows a much lower number of fluctuations . This is because his win rate never dramatically changes when he moves up a limit, as he always moves up to the ideal spot. His result is clearly higher than that of the other players.

The comparison between player types B, C and D with player A is particularly interesting, as they have a large starting capital whereas player type A has a $50 starting capital.

Comparison of the Results

As the sizes of the starting capitals differ for the various players, an examination of the differences between the bankroll sizes should shed some light on the financial success of the players, within the simulated period of time. How much more money does a player have at a given point in his poker career when compared to another player?

Player A vs. Player B

Let's first compare player A and player B. Player A started with $50 while player B started with $1,000. Both of them used the same bankroll management of 20 stacks.

In the illustration, player A's bankroll is subtracted from player B's bankroll at every stage. The graph therefore starts at a value of $950 ($1,000 - $50).

Difference between the bankrolls of player A and player B   y-axis: Difference between the bankrolls of player A and player B in $ x-axis: Time

As player B initially loses a lot of money at the high starting limit, both values quickly approximate each other until the difference is down to around $300. Only when player B increases his skill enough to make a profit does the difference begin to grow again, due to the bigger financial leverage that he has by playing at a higher limit.

The difference between both bankrolls eventually level off at a steady value of $700. In this rather simple model, this value will persist because both players play the highest available limit with the same win rate per time unit.

Let's recap: Player B, who had a financial advantage of $950 over player A at the beginning of his career, has not been able to use this advantage. The opposite is true; if he had only played with $50 instead of $1,000, he would now have around $150 more than at the beginning. This is due to the fact that he played at limits that were too high for him and which cost him a lot of money when starting out.

Player A vs. Player C

What follows now is a comparison between player C and player A. Player type C is quite theoretical in nature, although were he real, at least he would avoid losing major sums of money.

We also see considerably fewer fluctuations than in the first comparison. After just over the first half of time has passed, player C has an edge because player A would already beat NL25, but can't play it yet because his bankroll is too small.

When player A moves up, the edge becomes relative again. Towards the end, the difference is down to around $900. Player A performed better again even though he only started out with $50: $950 less than player C.

Difference between the bankrolls of player A and player C   y-axis: Difference between the bankrolls of player A and player C in $ x-axis: Time

Player A vs. Player D

The final comparison is between player A and the ideal player D, who started out with $1,000 and always plays the limit where he can make the best profit. The difference looks as follows:

Difference between the bankrolls of player A and player D   y-axis: Difference between the bankrolls of player A and player D in $ x-axis: Time

Over time, player D manages to establish an advantage of around $1,500. Compared to the starting situation, this is a profit of $550. Player D uses the advantage that, due to his bigger starting bankroll, he can move up to a higher limit as soon as he has reached the necessary skill level. Player A, on the other hand, has to earn the bankroll first, but on the smaller limits.

Does this mean that the ideal start into your poker career includes paying $1,000 out of your own pocket?

Discussing the Results

There are some other factors we have to consider here. On the one hand, players B and C each act "stupidly" in their own way. Player B uses bad bankroll management and seems to be willing to lose big parts of his money instead of moving down a limit.

Every time player C moves up to a limit at which he is a break even player, he surrenders part of his win rate in $. This is clear to see from the fact that the curve that represents his bankroll shows many spots with a slope of 0 (parallels to the time axis). If he had continued playing at the lower limit, he would have won more.

Both players, the possibly realistic player B as well as theoretical player C, perform worse than the $50 beginner, despite their higher starting capital. Only the ideal player D is able to use his bigger initial bankroll to his advantage over time. Although this advantage does exist, at the end of the simulation it still only amounts to around 20% of player A's bankroll.

From this discussion you can conclude that from the assumptions we have made, a beginner does not benefit from starting out with a high sum. Only if he was able to make perfect decisions regarding his win rate in $ at different limits, would he establish a slight advantage of 20%. Although this would still be with a starting capital that is 2000% bigger.

This is simply because a player with a low starting capital will rarely have a higher win rate at a limit that his bankroll is not big enough for than at a limit where he can play with his bankroll.

The system usually regulates itself quite well. A bigger bankroll would have one advantage that this discussion hasn't taken into account: it could counterbalance swings. It would be best for a beginner with a high starting capital to begin at the smallest limits and move up after certain periods of time, as soon as he considers himself ready.

This is how the influence of variance on the bankroll of the $50 beginner could be counterbalanced. This player could definitely use his higher starting capital to his advantage, if he is able to responsibly judge his own skill level.

However, this advantage is not at all necessary to play successful poker. It is quickly relativised by the fact that you receive the $50 starting capital for free while you would have to pay the $1000 out of your own pocket. This amount as an investment, in order to absorb swings on NL10, should be considered very carefully.

Conclusion

Starting out with $50 has turned out to be quite ideal. Under certain circumstances, using a considerably higher starting capital might lead to a slightly better result. How often this ideal case is reproduced in reality however, remains questionable.

With an initial bankroll that is bigger by 2000%, a final bankroll advantage of 20% is rather small. Obviously the relative advantage you would get with a starting bankroll that's smaller than $1,000 but bigger than $50 would be lower than 20%. At best, you can attain a very slight advantage by investing a lot more capital.

The only advantage, which can't be quantified in this article, is the avoidance of variance and of down swings. However, if you consider that a much larger initial investment is required and that there is a greater number of risks, such as the incorrect assessment of your own skills or playing limits that are too high (see player type B), we come to the conclusion that a starting bankroll of $50 (or $100 to reduce variance), is ideal for the beginner.

To finalise, let's take another look at the winnings of all four players, summed up by this graph:

• Yellow: Player A - $50 starting bankroll, 20 stacks bankroll management
• Green: Player B - $1,000 starting bankroll, 20 stacks bankroll management
• Red: Player C - $1,000 starting bankroll, plays at the limit to at least break even
• Blue: Player D - $1,000 starting bankroll, plays the limit at which he makes maximum profit

Development of the winnings of all four players   y-axis: Winnings in $ x-axis: Time

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