My ROI simulator gave me a very counter-intuitive result, please help.

I was playing around with a ROI simulator today, and I ran a simulation of 100000 sets of 5 games. It gave the chance for a +10% ROI player making a loss over these 5 games as 52%.

Whaaaaaat?

Am I reading the results correctly? What is going on here?

[elitegimp]

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I was playing around with a ROI simulator today, and I ran a simulation of 100000 sets of 5 games. It gave the chance for a +10% ROI player making a loss over these 5 games as 52%.

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What does it mean to be a +10% ROI player? My take on it (correct me if I'm way off) is that there is a probability p1 that the player makes X1 above his buy-in, p2 that the player makes X2, ... pN probabliity that the player makes XN with the numbers chosen so that p1*X1+...+pN*XN = 1.1*buy-in. So there is a 1-p1-p2-...-pN chance that the player loses his buy-in. So here's a simple case with a 10% ROI:

* This is an example, not an offer!!!*

You give me a dollar. I pick a [real] number between 0 and 10. If the number is between 0 and 5.5, I'll give you $2, otherwise I'll keep your dollar. In the long run, you make $0.10 for every dollar you play with, for an ROI of 10%. However, if you only play 5 times, you need to win 3, 4, or 5 times to profit. (If you don't win at all, you lose $5. If you win once, you lose $3. If you win twice, you lose $1.)

So what is the probability that you lose money? Well, you have a 55% chance of winning each particular round. Using the Binomial Distribution, we see that the probability of you not winning at all is 0.45^5. The probability of only winning once is 5*0.55*(0.45^4). The probability of only winning twice is 10*(0.55^2)*(0.45^3). Summing these up, you will lose money 40.6% of the time!

If you change the game so that 27.5% of the time you win $3 (i.e. I give you $4) and 72.5% of the time you lose $1, then your ROI is still 10%. In this case, you come out ahead provided you win twice. The probability of winning only once (or not at all) is 58%!!

As you can see, if you make it more difficult to win (but with much higher payouts) then the odds of not being profitable after just 5 attempts increases. This is variance at work. (The other way to look at it is that the times you _do_ come out ahead, you'll likely come out WAY ahead)

[BruceZ]

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I was playing around with a ROI simulator today, and I ran a simulation of 100000 sets of 5 games. It gave the chance for a +10% ROI player making a loss over these 5 games as 52%.

Whaaaaaat?

Am I reading the results correctly? What is going on here?

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That's certainly possible. You haven't given enough information. You have to give the probability distribution of the payoff of the game, not just the ROI. For example, suppose that 52% of the time you lose 8.5% of your initial investment, and 48% of the time you win +30% of your initial investment. That would be a +10% ROI since 52%*(-8.5%)+ 48%*30% =~ +10%. You could have a +10% ROI and lose almost 100% of the time, as long as you win big enough when you win, and lose small enough when you lose, so that your rare but big wins offset your frequent but relatively smaller losses.

[pzhon]

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I was playing around with a ROI simulator today, and I ran a simulation of 100000 sets of 5 games. It gave the chance for a +10% ROI player making a loss over these 5 games as 52%.

What is going on here?

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I presume you are simulating SNGs. The distribution is not symmetric about the average. There is a large chance of a small loss, and a small chance of a large win. In that type of distribution, the median result is lower than the expected result. This tendency persists when you combine the results of a few tournaments.

Suppose you are first, second, or third with probability 12.1% each. That gives you a 10% ROI if the rake is 10%. The probability of being ahead after 5 tournaments is 0.482752, but when you are ahead, you average a gain of 3.51947 buy-ins. You are behind with probability 0.517248, but when you are behind, you average a loss of only 2.3181 buy-ins. As a check, .517*(-2.32)+.483*(3.52)~0.5 buy-in, the expected result from playing 5 tournaments with a 10% ROI.

I did the exact calculations by expanding ((1 - 3p)x^-1 + p x^(9/11) + p x^(19/11) + p x^(39/11))^5 with p = .121. This is a generating function for the results after 5 tournaments. The coefficient of x^n is the probability of finishing ahead by n buy-ins.

this gets confusing in my mind.

but you are talking about 5 games... your odds of being up or down after 5 games is pretty close to 50/50.

i think you are confusing 1,000,0000 simulations of 5 games with testing 5 million games, where your odds of being behind at 1 in a billion or more. not the same thing at all.