Game Theory

I'm not sure if this qualifies as Game Theory as studied by John Nash, but it seems like it to me..

I've been playing online 8-Ball billiards at Miniclip.com. What makes it so cool, is the fact that you can login with Facebook and it will keep track of your stats.
You start out as a Rookie, and the more games you win, the more you move up, to Amateur, Professional, Expert, Master, and finally Grand Master. As you play, you can earn different rewards, like Denial (Running the table on your opponent) Or Perfect Win (your opponent didn't pot any balls) along with countless others.

You can also, use points earned to buy various different cues, different table styles and different cloths. These points can also be used to purchase "skill levels" (increased spin, power or aim--extended "aim lines" showing where the object ball will go)

Now to earn points, you could simply buy them. You can also earn them by entering into tournaments. There are three levels of tournaments, 8, 16 and 32 players. Winning an 8 player tourney gets you 25 points, 50 points for 16 and 100 points for 32. I was trying to figure out, ignoring the time it takes, which level is best. At first glance it seems obvious, 32 player only requires 5 wins to achieve 100 points. A 16 requires 8 wins to achieve 100, and 8 requires 12 wins to achieve 100. However, that really depends on how good you are. So I came up with a formula.

X = Win percentage Y = Number of wins to win tourney Z = Number of points earned for winning tournament

X^Y * Z The higher the value of that expression, the better the "value" of entering that tournament.

My win percentage is 55%, and the highest value is the 32 player tourney. My actual win percentage in a tournament is higher however, since I could be playing someone of ANY skill level.

So, using 55% win percentage, the value of each tournament is:

8 -- 4.159
16 -- 4.575
32 -- 5.033

Am I right in my thinking?

Also, I think if your win percentage is below 50% then 8 is the way to go, if it's above 50% 32 is the way to go, and right at 50%, it doesn't matter, the value is the same for all of them...

I agree with your mathematical reasoning were this some sort of bet, but I think that you are thinking too much like a gambler.

You will theoretically encounter stiffer competition the further you advance in each tournament, (especially if the better players are also playing the Round of 32) and I don't think your Math takes into consideration the potentiality of that nor of losing in the Round of 32 after you have won three or four in a row. I would think the value of your time should play a factor in your decision, and once you have won three-in-a-row in the Round of 8 or four-in-a-row in the Round of Sixteen, those are points that you cannot lose. The potentiality of winning two then losing exists in all three and is, therefore, irrelevant.

What do you get if you redo your Math based on one, two and three wins, given that you must win the first two regardless? Do you have any idea what the average winning percentage is of each tournament participant to factor that in?

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Quote: Mission146

I agree with your mathematical reasoning were this some sort of bet, but I think that you are thinking too much like a gambler.

You will theoretically encounter stiffer competition the further you advance in each tournament, (especially if the better players are also playing the Round of 32) and I don't think your Math takes into consideration the potentiality of that nor of losing in the Round of 32 after you have won three or four in a row. I would think the value of your time should play a factor in your decision, and once you have won three-in-a-row in the Round of 8 or four-in-a-row in the Round of Sixteen, those are points that you cannot lose. The potentiality of winning two then losing exists in all three and is, therefore, irrelevant.

What do you get if you redo your Math based on one, two and three wins, given that you must win the first two regardless? Do you have any idea what the average winning percentage is of each tournament participant to factor that in?

You make a couple of great points. I agree there's some factors that aren't considered. I deliberately didn't account for time. It is absolutely true, that time plays a big role, waiting time (for your next opponent to finish their game) tends to be longer in 32 player tourneys. Plus playing five takes longer than three, obviously. So if it's about making more points per hour so to speak, 8 player tourneys are probably the way to go.

Now, as you advance, you face stiffer competition, which would lower your chance of winning, but you also sometimes have a better chance of advancing, by playing much lower competition, since everybody is pooled together in these tourneys. It's also possible I play someone that's not very good, but still the best of several lower caliber players.

I'm not sure if you could figure out the average win percentage of other tournament players. Some have played just a few games, some of played hundreds.

Quote: ewjones080

You make a couple of great points. I agree there's some factors that aren't considered. I deliberately didn't account for time. It is absolutely true, that time plays a big role, waiting time (for your next opponent to finish their game) tends to be longer in 32 player tourneys. Plus playing five takes longer than three, obviously. So if it's about making more points per hour so to speak, 8 player tourneys are probably the way to go.

Thank you for the compliment. I would think that you would want to find some way of maximizing your points per hour. You will have a limited sample size, but perhaps if you were to win five of each kind of tournament, and time them, you would get an idea of where you need to be to maximize the value of your time. Win/Loss ratio of Tournaments (based on size) will also be a factor, except I think the sample size would be way too low to come up with anything relevant, here.

Quote:

Now, as you advance, you face stiffer competition, which would lower your chance of winning, but you also sometimes have a better chance of advancing, by playing much lower competition, since everybody is pooled together in these tourneys. It's also possible I play someone that's not very good, but still the best of several lower caliber players.

That's a good point, your first match in the Round of 32 is probably often garbage, but if it is randomized, could you not also be pitted against the best player in the first round?

Quote:

I'm not sure if you could figure out the average win percentage of other tournament players. Some have played just a few games, some of played hundreds.

I would just do some kind of weighted system to determine the average. For example, you could just go for the mean winning percentage, but use a basis of 50 games and a player with more than 100 games would count as two players, 150 would count as three and so on. If a player has less than fifity games, then he counts as one player. For example:

Player A: 124 Games .589

Player B: 41 Games .675

Player C: 217 Games .523

Player D: 17 Games .481

Player A counts as two players, Player C counts as four players and the others count as one.

(.589 + .589 + .675 + .523 + .523 + .523 + .523 + .481)/8 = .55325 or 55.325% winning percentage.

You could even weight it more specifically by giving a weight of + .02 for each game over 50 Games and accepting anything under 50 (but more than 20, I would say) as one player. Player A, for example, would then be 2.48 * .589 in the calculation. You would also add 2.48 to the total to be divided from to get the mean, of course.