This article discusses and analyzes the ROI maximization of buying in late from my post Rebuy Turbo Tournament Strategy – Maximizing ROI – Single buy in late and add on. It is very poker math heavy as a fair warning.

We are comparing the two strategies:

- Buy in early: Buy into a turbo rebuy from start
- Buy in late: Buy into a turbo rebuy with a single buy in straight to add on

We want to find a proportion of the two ROI’s to decide what strategy suits us better. We introduce several variables to succeed in our endeavor:

Assume your ROI if you play from start is: ROI_start

And ROI to play from end is: ROI_end

### Buy in early

Also assume you if you play from start you use 4 buy ins total plus the add on. (this is a good average) => total number buy ins 5

All turbo rebuys I saw in TCOOP gave 10x as many chips for add on.

Lets just say for simplicity your buy in is 1k and the add on 10k.

So for 14k chips you pay 5 buy ins. Chips_start = 14k

### Buy in late

Now let us turn to the buy in late strategy:

With same numbers, you spend 2 buy ins for 11k chips. Chips_end = 11k

So you get almost 70% of chips as if you played from start but only pay 40% of the price. This alone should be enough, but I will continue and get to ration of the ROI’s.

### Poker Math

We are ready to tackle the math part:

Assume there are N chips in play. We will model our mtt equity as

MTTEquity = Chips/N

This is a good enough approximation very far from the money. The closer we came to payouts we would have to switch to ICM.

With the Equity we can calculate the average return as:

AvgReturn = MTTEquity * PrizePool

substitute the MTTEquity:

AvgReturn = Chips/N * PrizePool

Now lets also look at the average return from a ROI perspective:

AvgReturn = ROI * BuyIn

What we end up is **two ways** to calculate the average return. Moreover they all have to be equal.

AvgReturn = Chips/N * PrizePool = ROI * BuyIn

Lets calculate the average return for _start and _end

AvgReturn_start = Chips_start/N * PrizePool = ROI_start * BuyIn_start

AvgReturn_end = Chips_end/N * PrizePool = ROI_end * BuyIn_end

Lets take out the two last equations and solve for ROI_start and ROI_end respectively:

ROI_start = Chips_start/N * PrizePool * 1/BuyIn_start

ROI_end = Chips_end/N * PrizePool * 1/BuyIn_end

There are many variables we don’t know, but we are not interested in the absolute value of those, we only want to know a proportion, so lets calculate ROI_start/ROI_end:

ROI_start/ROI_end = (Chips_start/n * PrizePool * 1/BuyIn_start)/(Chips_end/n * PrizePool * 1/BuyIn_end)

= Chips_start * BuyIn_end /( Chips_end * BuyIn_start)

= 14k * 2 /(11k * 5) = 0.51

Solve above for ROI_end

<=> ROI_end = 1.96 * ROI_start

Voila, we can see that ROI_end is almost twice as big as ROI_start.

### Discussion of results

1) We probably have more than 14k chips after 1 hour of play on average. Assume for example you had 16k on average, than that factor would go down to 1.72 which is still very high.

2) Even though the ROI of the buy in late strategy is almost twice as big, we will make more money with the buy in early strategy! But we have to invest a whole lot more. This means our variance is higher.

You can see this by looking at the average return and knowing that 2.5 * BuyIn_end = BuyIn_start.

Start with

AvgReturn_start = ROI_start * BuyIn_start

substiturte ROI_start and BuyIn_start with equations from above

ROI_start = 1/1.96 * ROI_end

BuyIn_start = 2.5 * BuyIn_end

AvgReturn_start = ROI_start * BuyIn_start = .51 * ROI_end * 2.5 * BuyIn_end = 1.27 * ROI_end * BuyIn_end

Compare this to

AvgReturn_end = ROI_end * BuyIn_end

and you see that:

AvgReturn_start = ROI_start * BuyIn_start = .51 * ROI_end * 2.5 * BuyIn_end = 1.27 * ROI_end * BuyIn_end > ROI_end * BuyIn_end = AvgReturn_end

<=>

AvgReturn_start > AvgReturn_end

### Summary

- If you are trying to make most money, buy in early!
- If you are trying to minimize variance and maximize your ROI, buy in late.

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