Game Entry Statistics

[bpbenda]

I am trying to figure out the method of calculating the number of times that a certain game must be entered in order to advance to a specific level based on average results. Here is what I am trying to calculate:

Game = Multiple levels with nine (9) players per level.

// LEVEL 1
1st & 2nd Place = Go to level 2.
3rd Place = Replay level 1.
4th - 9th Place = Out.

// LEVEL 2
1st & 2nd Place = Go to level 3.
3rd Place = Replay level 2.
4th Place = Go back to level 1.
5th - 9th Place = Out.

// LEVEL 3
1st - 2nd Place = Go to level 4.
3rd - 5th Place = Replay level 3.
6th - 9th Place = Out.

... Okay, there are more levels past this, but this should be enough info to get me started in the right direction. Say, for instance, that I want to know how many times I have to enter the game on average in order to get to level 3, how do I calculate this?

On average, I know that I will get to Level 2 once in every 4.5 attempts. I know this because 1st and 2nd are two (2) out of the nine (9) places, which is 2:9 or 1:4.5. However, once I get to Level 2 the calculation gets much more complicated because a certain percentage of the time at Level 2 I will get 3rd place and be able to replay the level or get 4th and be able to replay the game back at Level 1 without having to re-enter.

Given the aforementioned, what is the process or equation for figuring out how many times I would need to enter this game on average in order to get to Level 3? I am more interested in the process and equation than the number of times. I am more interested in this because I am trying to represent these statistics in an Excel spreadsheet.

Thanks in advance for the help! Regards, Brian.

[Dason]

I didn't really read through this completely but I know for tough problems where it's hard to analytically get a result it's sometimes really easy to just use simulation. So what you could do is program into R or some other package how to play the game (or how to play the game to a certain level) and return how long/how many turns it took. Then just have it play the game a couple thousand/million times and take the mean. Walla! You've got a good estimate of what you want.

[terzi]

Hi bpbenda,

Although I'm not an expert in the subject, I think that this problem can be analyzed using Markov Chains. Passing from one level to another would be the transition. With these techniques you can get means, probabilities and a lot of useful information, you can even perform some cool simulations:)

[bpbenda]

Thanks, terzi! I have started looking into Markov Chains and believe this might very well enable me to model what I am trying to do here. Apparently, there is a famous Markov Chain called the "drunkard's walk". This sounds very similar in theory to the multi-tiered game (previously described) for which I am trying to understand mathematically. Thanks for the direction, and I hope that I am able to wrap my head around this enough to figure it out.