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Message: [QUOTE="maltz, post: 637758, member: 26654"] [b]more models coming[/b] [COLOR=black]After working out the Pro's weak coin, I was wondering about one thing obvious - how can we all beat the pro by that much? There must be [B]something to be improved[/B] in the model.[/COLOR] [COLOR=black]When we use our own examples in the coin model, we [B]start from our BB/100 hand[/B]. However, when we apply the coin model to the pro, we [B]start from his standard deviation[/B]. Now, if our coin model [B]underestimates[/B] the variance, we are going to give the pro a poorer coin than he deserves.[/COLOR] [COLOR=black]Actually, it is very likely that the coin model is underestimating the variance of NL hold'em. This is because:[/COLOR] [COLOR=black]- People [B]don't play every hand[/B]. [B]By playing fewer hands there are more variance[/B]. Our coin model assumes that you do play every hand.[/COLOR] [COLOR=black]- The frequency of [B]all-ins[/B] (either doubling up or lose everything) is not negligible, and when that happens the pots are usually HUGE.[/COLOR] [COLOR=black]The next closest thing that comes up in my mind is [B]DICE[/B]. A standard dice has 6 faces, numbering from 1 to 6. When you roll the dice you can expect 1-6 to show up with equal chances.[/COLOR] [COLOR=black]A standard dice would have a higher variance than the coin model. By rolling the dice 100 times, you are expecting an average of (1+6)/2 = 3.5 (or, 0 gain). Your standard deviation would be [B]1.71[/B]. That's 70% higher than flipping a coin.[/COLOR] [COLOR=black]Yet the good thing about dice is that we can change its numbers on the faces. Let's now [B]exaggerate the dice[/B] a bit to reflect:[/COLOR] [COLOR=black](1) [B]Most of the times you don't play a hand, hence the gain/loss is minimal [/B][/COLOR] [COLOR=black](2) [B]Sometimes when you do get heavily involved, your either win a lot or lose a lot[/B][/COLOR] [COLOR=black]Let's paint the dice with:[/COLOR] [COLOR=black]5[/COLOR] [COLOR=black]1[/COLOR] [COLOR=black]0[/COLOR] [COLOR=black]0[/COLOR] [COLOR=black]-1[/COLOR] [COLOR=black]-5[/COLOR] [COLOR=black]This is the amount you are going to win/lose.[/COLOR] [COLOR=black]The next interesting concept is the average pot size.[/COLOR] [COLOR=black]For one approach we can [B]average the dice numbers[/B]. (5+1+1+5)/6=2[/COLOR] [COLOR=black]For another approach, in reality both you and your opponents (as one entity) [B]still each contribute $1[/B] to roll, so the pot is still $2.[/COLOR] [COLOR=black]Actually, I purposely designed the dice to be this way, so the two approaches are equivalent. Now we don't have to worry about which approach is correct. (In fact I have no idea which one is correct. :D)[/COLOR] [COLOR=black]Now let's roll this 5,1,0,0,-1,-5 dice for 100 times. For an average player:[/COLOR] [COLOR=black][B]Expected win: $0[/B][/COLOR] [B][COLOR=black]Standard deviation: $2.94[/COLOR][/B] [COLOR=black]Now we have successfully elevated the variance of our model, from [B]$1 (coin) [/B]to [B]$1.71 (standard dice)[/B] to [B]$2.94 (our customized poker dice)[/B].[/COLOR] [COLOR=black]That's for an average player. How about a good player? The dice is now [B]biased towards the positive side[/B]. We can imagine the positive side of the dice (0, 1, 5) as [B]one side of the coin[/B], and the negative side (0, -1, -5) as the other side of the coin. Therefore, a 60-40 dice has [B]20%[/B] (60/3) chance to hit 0, 1 or 5, and [B]13.33%[/B] (40/3) chance to hit 0, -1, -5.[/COLOR] [COLOR=black]Now let's revisit our examples:[/COLOR] [COLOR=black][B][Case 1][/B] Professional with his hourly rate:standard deviation = 1:6[/COLOR] [COLOR=black]Let's still assume he plays 100 hands per hour.[/COLOR] [COLOR=black]It turns out that the Pro is using a poker dice of [B]50.122 vs. 49.878[/B].[/COLOR] [COLOR=black](Compare to our previous coin model of [B]50.083 vs. 49.917[/B].)[/COLOR] [COLOR=black]You might think there isn't a lot of difference at all - [B]indeed there isn't[/B]! Our pro, using our new poker dice model, is just having a coin (dice) that is [B]50% better[/B].[/COLOR] [COLOR=black][B][Case 2][/B] You earn 10BB / 100 hands.[/COLOR] [COLOR=black]Everything remains the same. Your poker dice is [B]50.417 vs 49.583[/B].[/COLOR] [COLOR=black]Your poker dice is still much better than the Pro! But if you pay a closer attention, the difference between the Pro and you have reduced a bit, due to the increased variance of our new model.[/COLOR] [COLOR=black]***[/COLOR] [COLOR=black]It seems to me the poker dice is doing ok but not that great. I further improved the model with [B]an imaginery dice of 20 faces[/B].[/COLOR] [COLOR=black]Face 1 -- 15[/COLOR] [COLOR=black]Face 2 -- 4[/COLOR] [COLOR=black]Face 3 -- 1[/COLOR] [COLOR=black]Face 4-17 -- 0[/COLOR] [COLOR=black]Face 18 -- -1[/COLOR] [COLOR=black]Face 19 -- -4[/COLOR] [COLOR=black]Face 20 -- -15[/COLOR] [COLOR=black]Now that sounds like poker even more! When I run the same calculation, the [B]standard deviation becomes 4.92[/B]. This is the turbo version of our poker dice (stdev = 2.94).[/COLOR] [COLOR=black]So what kind of Turbo dice is our pro using? I will save you the trouble of reading and just tell you the result: [B]50.205 vs. 49.795[/B]. Our pro's true edge is getting more obvious now. [/COLOR] [COLOR=black]Yes, this is still about twice as weak as your dice (10BB/100 hands). We are doing better probably because our opponents are much weaker at low/micro limits.[/COLOR] [COLOR=black]***[/COLOR] [COLOR=black]To show that you didn't waste your past 5 minutes, here comes the summary:[/COLOR] [COLOR=black](1) You can design multiple [B]models[/B] to describe Poker (or anything). Some models are closer to reality.[/COLOR] [COLOR=black](2) Even though we approach the ideal model of poker, [B]your edge is still tiny[/B] (way less than 51:49 per hand) no matter how good you are.[/COLOR] [COLOR=black](3) [B]Multi-tabling micro-stake tables[/B] may be [B]more profitable[/B] than simply [B]moving up the limit[/B] and play single table. Your edge is greater down there, and your variance is reduced by playing more hands.[/COLOR] [COLOR=black]p.s. My BB here means Big Blind. I just read that people use BB (big bet) as 2x big blind. So in the 10xBB example, you are actually earning 20 big blinds. Your dice would be twice as good![/COLOR] [/QUOTE]

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