HCDefence13
05 Jun 2005, 06:59 PM
This approach is different than was presented in another (similar) thread, so I thought I'd post the results. As opposed to simulating matches based on a rankings system, the idea of this method is to actually generate every possible outcome (win, loss, or tie) in every single game of each round. You then inspect all the permutations of possible tables and see what proportion have team 'X' in a certain place (1st, 2nd, etc.). In this way there is no bias based on subjective rankings...it's just all about counting the possibilities. Unfortunately, some drawbacks exist, but I'll first give the results.
RESULTS:
Assuming the U.S. defeats Panama in round 5, here are probabilities of where we would stand in the final table (after round 10)**
1st - 46.72%
Tied for 1st - 9.58%
2nd - 27.26%
Tied for 2nd - 5.42%
3rd - 6.81%
lower than 3rd - 4.21%
CONCLUSION:
Although the U.S. cannot mathematically clinch a spot with a win in the next round (i.e., it is possible to end up tied for 3rd place or lower), the chances of not advancing to WC '06 are miniscule. I haven't thoroughly inspected the actual tables where the U.S. doesn't advance with a win in round 5, but my guess is that the vast majority (maybe even all???) would require losses in each of the last 5 rounds of qualifying - and on top of that, probably even a specific series of outcomes occuring in matches we don't play in. Obviously, the likelihood of that happening is remote.
METHODOLOGY (for those interested):
The major drawback at this point in time is that the number of possible tables that still exist is ridiculously large. Consider that there are 3 possible outcomes for each game played: Home team wins, away team loses; teams tie; home team loses, away team wins. Therefore, in each round there are 3*3*3=27 possible tables that can occur. In turn, this means that the number of outcomes (tables) possible grows exponentially with each round you look to generate. To generate 2 rounds worth, there are 27^2 possible tables; 3 rounds have 27^3, etc. Assuming the U.S. wins in round 5 (leaving 9 possible tables for round 5, based on the results of the other two games), I would have to generate 9*(27^5)=129,140,163 tables!
(**)Unfortunately, my computer/software/program won't handle that much information, so the results above are not the EXACT probabilities. As we progress into later rounds, the number of tables will decrease and I can update my program and hopefully look at the exact probabilities. But for now, I am forced to sample from the possible tables and look at the outcomes. For those of you that know what I'm talking about, this would imply that there is an associated variability for the percentages given in the results sections. I have not gotten far enough to calculate those yet, but based on a number of simulations I've run, all look to have that last category being at less than 5%. The other 5 categories vary a bit more, but since it doesn't really matter if we end up 1st, 2nd, or 3rd that is of less importance to me.
One other thing to note - The "tied for X place" categories above might include situations where >2 teams are even on points. For instance, in the "Tied for 2nd" category, a table(s) might exist where the U.S. is even in the standings with Costa Rica AND Guatemala. Obviously, the tiebreakers would be applied at this point to determine who gets automatic bids and who is sent to the playoff. Other such similar situations are mathematically possible, too. So, although the "tied for X place" categories will include only 2 teams tied on points the majority of the time, you have to look at those categories with a bit of caution.
WRAP-UP:
I plan to keep running simulations to verify the results I posted above. I also update my program based on the actual outcomes after each qualifying round and re-run it. If there is any interest here, I can give updated percentages after each round is played. As I said above, at some point in the (near) future, I should be able to generate the exact percentages.
RESULTS:
Assuming the U.S. defeats Panama in round 5, here are probabilities of where we would stand in the final table (after round 10)**
1st - 46.72%
Tied for 1st - 9.58%
2nd - 27.26%
Tied for 2nd - 5.42%
3rd - 6.81%
lower than 3rd - 4.21%
CONCLUSION:
Although the U.S. cannot mathematically clinch a spot with a win in the next round (i.e., it is possible to end up tied for 3rd place or lower), the chances of not advancing to WC '06 are miniscule. I haven't thoroughly inspected the actual tables where the U.S. doesn't advance with a win in round 5, but my guess is that the vast majority (maybe even all???) would require losses in each of the last 5 rounds of qualifying - and on top of that, probably even a specific series of outcomes occuring in matches we don't play in. Obviously, the likelihood of that happening is remote.
METHODOLOGY (for those interested):
The major drawback at this point in time is that the number of possible tables that still exist is ridiculously large. Consider that there are 3 possible outcomes for each game played: Home team wins, away team loses; teams tie; home team loses, away team wins. Therefore, in each round there are 3*3*3=27 possible tables that can occur. In turn, this means that the number of outcomes (tables) possible grows exponentially with each round you look to generate. To generate 2 rounds worth, there are 27^2 possible tables; 3 rounds have 27^3, etc. Assuming the U.S. wins in round 5 (leaving 9 possible tables for round 5, based on the results of the other two games), I would have to generate 9*(27^5)=129,140,163 tables!
(**)Unfortunately, my computer/software/program won't handle that much information, so the results above are not the EXACT probabilities. As we progress into later rounds, the number of tables will decrease and I can update my program and hopefully look at the exact probabilities. But for now, I am forced to sample from the possible tables and look at the outcomes. For those of you that know what I'm talking about, this would imply that there is an associated variability for the percentages given in the results sections. I have not gotten far enough to calculate those yet, but based on a number of simulations I've run, all look to have that last category being at less than 5%. The other 5 categories vary a bit more, but since it doesn't really matter if we end up 1st, 2nd, or 3rd that is of less importance to me.
One other thing to note - The "tied for X place" categories above might include situations where >2 teams are even on points. For instance, in the "Tied for 2nd" category, a table(s) might exist where the U.S. is even in the standings with Costa Rica AND Guatemala. Obviously, the tiebreakers would be applied at this point to determine who gets automatic bids and who is sent to the playoff. Other such similar situations are mathematically possible, too. So, although the "tied for X place" categories will include only 2 teams tied on points the majority of the time, you have to look at those categories with a bit of caution.
WRAP-UP:
I plan to keep running simulations to verify the results I posted above. I also update my program based on the actual outcomes after each qualifying round and re-run it. If there is any interest here, I can give updated percentages after each round is played. As I said above, at some point in the (near) future, I should be able to generate the exact percentages.