Dooglus "AMA"

[dooglus]

What does generally a pretty bad idea mean ? As far as I can understand , the house edge should be equal at all levels and that would just be a lot less fun (as you have no chance of winning something substantial) , with the same amount of risk.

Is my understanding wrong ?

If you want to win 1 BTC, you could:

a) risk 0.1 BTC at 9% for an 11x payout, or you could
b) risk 10 BTC at 90% for a 1.1x payout.

In each case you expect to lose 1% of the amount you risk, so b) costs about 100 times more than a) in terms of the amount you expect to lose, while only having a 10 times better chance of success.

That makes 90% seem like a bad deal compared to 9%, even though they both have a 1% edge.

Does that make sense?

[jjc326]

I do understand OP and it's worth a shot.  You're thinking that even though people might be "winning" with an edge of like 1% or something, they'll keep playing until they lose.  It is risly but if you could make it owrk that'd be great.  You would have to stop bots because they wouldn't have "emotion" so that was a great point too.

[dooglus]

Someone have a simulator to help me fully realize I'm wrong? I dunno how to create the formula for doing this quickly. 1,000 users have a $1,000 bankroll and must make 1,000 $25 bets with a 51% chance of 2x return, 49% chance of 0x return. The casino has an unlimited bankroll. The casino only loses if someone cashes out their winnings, which in this simulator, requires someone to make 1,000 $25 bets without their bankroll ever hitting $0. They get to keep whatever's left at the end of that.

I'll run that simulation and let you know how it goes.

[dooglus]

Someone have a simulator to help me fully realize I'm wrong? I dunno how to create the formula for doing this quickly. 1,000 users have a $1,000 bankroll and must make 1,000 $25 bets with a 51% chance of 2x return, 49% chance of 0x return. The casino has an unlimited bankroll. The casino only loses if someone cashes out their winnings, which in this simulator, requires someone to make 1,000 $25 bets without their bankroll ever hitting $0. They get to keep whatever's left at the end of that.

I'll run that simulation and let you know how it goes.

OK, maybe I misunderstood your specification, but here are the results:

I coded a simulation, ran it 6 times.  The first time I ran it the first player busted and the 2nd player made 1000 bets without busting.  The other 5 times the first player made 1000 bets without busting:

player    1  won bet    1.  balance = 1025
player    1  won bet    2.  balance = 1050
player    1 lost bet    3.  balance = 1025
player    1 lost bet    4.  balance = 1000
player    1  won bet    5.  balance = 1025
player    1 lost bet    6.  balance = 1000
player    1 lost bet    7.  balance =  975
player    1  won bet    8.  balance = 1000
player    1  won bet    9.  balance = 1025
player    1 lost bet   10.  balance = 1000
[...]
player    1  won bet  991.  balance = 2475
player    1 lost bet  992.  balance = 2450
player    1 lost bet  993.  balance = 2425
player    1 lost bet  994.  balance = 2400
player    1 lost bet  995.  balance = 2375
player    1  won bet  996.  balance = 2400
player    1 lost bet  997.  balance = 2375
player    1  won bet  998.  balance = 2400
player    1  won bet  999.  balance = 2425
player    1  won bet 1000.  balance = 2450
player 1 made 1000 bets without busting

Here's the simulation code:

#!/usr/bin/env python

import random, sys

players = 1000
bets_per_player = 1000
starting_balance = 1000
stake = 25

def roll():
    return random.random() < 0.51

player = 1
while player <= players:
    balance = starting_balance
    bets = 0
    while True:
        bets += 1
        if roll():
            balance += stake
            print "player %4d  won bet %4d.  balance = %4d" % (player, bets, balance)
        else:
            balance -= stake
            print "player %4d lost bet %4d.  balance = %4d" % (player, bets, balance)
            if balance < stake:
                print "player %d bust after %d bets" % (player, bets)
                break
        if bets == bets_per_player:
            print "player %d made %d bets without busting" % (player, bets)
            sys.exit(0)

    player += 1

print "all the players busted"

[a1choi]

What does generally a pretty bad idea mean ? As far as I can understand , the house edge should be equal at all levels and that would just be a lot less fun (as you have no chance of winning something substantial) , with the same amount of risk.

Is my understanding wrong ?

If you want to win 1 BTC, you could:

a) risk 0.1 BTC at 9% for an 11x payout, or you could
b) risk 10 BTC at 90% for a 1.1x payout.

In each case you expect to lose 1% of the amount you risk, so b) costs about 100 times more than a) in terms of the amount you expect to lose, while only having a 10 times better chance of success.

That makes 90% seem like a bad deal compared to 9%, even though they both have a 1% edge.

Does that make sense?

i guess my risk preferences are more risk averse than yours.  i'd gladly risk the 100x for the 10x more chance of winning.  90% just seems a much better shot than 9%.

but personally i'd be limited to only acting on the 0.1BTC bet because i don't have the bankroll for 10BTC bets.

[dooglus]

I coded a simulation, ran it 6 times.  The first time I ran it the first player busted and the 2nd player made 1000 bets without busting.  The other 5 times the first player made 1000 bets without busting:

OK, so let's give the casino a better chance.  Each player starts with just $25, and has to flat-bet $25 over and over until they either bust or make 1000 bets.

Now the casino is bound to win, right?

Wrong:

player    1 lost bet    1.  balance =    0
player 1 bust after 1 bets
player    2 lost bet    1.  balance =    0
player 2 bust after 1 bets
player    3 lost bet    1.  balance =    0
player 3 bust after 1 bets
player    4 lost bet    1.  balance =    0
player 4 bust after 1 bets
player    5 lost bet    1.  balance =    0
player 5 bust after 1 bets
player    6  won bet    1.  balance =   50
player    6 lost bet    2.  balance =   25
player    6 lost bet    3.  balance =    0
player 6 bust after 3 bets
player    7  won bet    1.  balance =   50
player    7 lost bet    2.  balance =   25
player    7  won bet    3.  balance =   50
player    7 lost bet    4.  balance =   25
player    7  won bet    5.  balance =   50
player    7  won bet    6.  balance =   75
player    7 lost bet    7.  balance =   50
player    7 lost bet    8.  balance =   25
player    7 lost bet    9.  balance =    0
player 7 bust after 9 bets
player    8 lost bet    1.  balance =    0
player 8 bust after 1 bets
player    9  won bet    1.  balance =   50
player    9  won bet    2.  balance =   75
player    9  won bet    3.  balance =  100
player    9  won bet    4.  balance =  125
player    9 lost bet    5.  balance =  100
player    9  won bet    6.  balance =  125
player    9 lost bet    7.  balance =  100
player    9  won bet    8.  balance =  125
player    9  won bet    9.  balance =  150
player    9  won bet   10.  balance =  175
player    9 lost bet   11.  balance =  150
player    9 lost bet   12.  balance =  125
player    9 lost bet   13.  balance =  100
player    9 lost bet   14.  balance =   75
player    9 lost bet   15.  balance =   50
player    9  won bet   16.  balance =   75
player    9 lost bet   17.  balance =   50
player    9 lost bet   18.  balance =   25
player    9  won bet   19.  balance =   50
player    9 lost bet   20.  balance =   25
player    9 lost bet   21.  balance =    0
player 9 bust after 21 bets
player   10 lost bet    1.  balance =    0
player 10 bust after 1 bets
player   11  won bet    1.  balance =   50
player   11 lost bet    2.  balance =   25
player   11 lost bet    3.  balance =    0
player 11 bust after 3 bets
player   12  won bet    1.  balance =   50
player   12  won bet    2.  balance =   75
player   12 lost bet    3.  balance =   50
player   12 lost bet    4.  balance =   25
player   12  won bet    5.  balance =   50
player   12 lost bet    6.  balance =   25
player   12 lost bet    7.  balance =    0
player 12 bust after 7 bets
player   13 lost bet    1.  balance =    0
player 13 bust after 1 bets
player   14  won bet    1.  balance =   50
player   14 lost bet    2.  balance =   25
player   14  won bet    3.  balance =   50
player   14  won bet    4.  balance =   75
player   14  won bet    5.  balance =  100
player   14  won bet    6.  balance =  125
player   14  won bet    7.  balance =  150
player   14  won bet    8.  balance =  175
player   14 lost bet    9.  balance =  150
player   14  won bet   10.  balance =  175
[...]
player   14  won bet  224.  balance =  125
player   14 lost bet  225.  balance =  100
player   14 lost bet  226.  balance =   75
player   14  won bet  227.  balance =  100
player   14  won bet  228.  balance =  125
player   14 lost bet  229.  balance =  100
player   14 lost bet  230.  balance =   75
player   14 lost bet  231.  balance =   50
player   14 lost bet  232.  balance =   25
player   14 lost bet  233.  balance =    0
player 14 bust after 233 bets
player   15  won bet    1.  balance =   50
player   15  won bet    2.  balance =   75
player   15  won bet    3.  balance =  100
player   15 lost bet    4.  balance =   75
player   15  won bet    5.  balance =  100
player   15  won bet    6.  balance =  125
player   15 lost bet    7.  balance =  100
player   15  won bet    8.  balance =  125
player   15  won bet    9.  balance =  150
player   15 lost bet   10.  balance =  125
[...]
player   15 lost bet  991.  balance =  850
player   15 lost bet  992.  balance =  825
player   15 lost bet  993.  balance =  800
player   15  won bet  994.  balance =  825
player   15  won bet  995.  balance =  850
player   15  won bet  996.  balance =  875
player   15  won bet  997.  balance =  900
player   15 lost bet  998.  balance =  875
player   15 lost bet  999.  balance =  850
player   15 lost bet 1000.  balance =  825
player 15 made 1000 bets without busting

[dooglus]

OK, so let's give the casino a better chance.  Each player starts with just $25, and has to flat-bet $25 over and over until they either bust or make 1000 bets.

OK, still too easy for the player.  What if we don't let them cash out until they've made a million bets, and still only let them start with enough to make a single bet?

Still no good.  It takes a few more players before one goes to the moon, but it still happens, and when it does he breaks the bank:

player    1 lost bet    1.  balance =    0
player 1 bust after 1 bets
player    2  won bet    1.  balance =   50
player    2  won bet    2.  balance =   75
player    2  won bet    3.  balance =  100
[...]
player    2  won bet   18.  balance =  225
player    2  won bet   19.  balance =  250
player    2  won bet   20.  balance =  275
player    2 lost bet   21.  balance =  250
player    2 lost bet   22.  balance =  225
[...]
player    2 lost bet   51.  balance =   50
player    2 lost bet   52.  balance =   25
player    2 lost bet   53.  balance =    0
player 2 bust after 53 bets
player    3 lost bet    1.  balance =    0
player 3 bust after 1 bets
player    4  won bet    1.  balance =   50
player    4 lost bet    2.  balance =   25
player    4 lost bet    3.  balance =    0
player 4 bust after 3 bets
player    5  won bet    1.  balance =   50
player    5 lost bet    2.  balance =   25
player    5 lost bet    3.  balance =    0
player 5 bust after 3 bets
player    6 lost bet    1.  balance =    0
player 6 bust after 1 bets
player    7 lost bet    1.  balance =    0
player 7 bust after 1 bets
player    8  won bet    1.  balance =   50
player    8 lost bet    2.  balance =   25
player    8  won bet    3.  balance =   50
player    8 lost bet    4.  balance =   25
player    8  won bet    5.  balance =   50
player    8 lost bet    6.  balance =   25
player    8  won bet    7.  balance =   50
player    8 lost bet    8.  balance =   25
player    8 lost bet    9.  balance =    0
player 8 bust after 9 bets
player    9  won bet    1.  balance =   50
player    9 lost bet    2.  balance =   25
player    9  won bet    3.  balance =   50
player    9 lost bet    4.  balance =   25
player    9 lost bet    5.  balance =    0
player 9 bust after 5 bets
player   10 lost bet    1.  balance =    0
player 10 bust after 1 bets
player   11 lost bet    1.  balance =    0
player 11 bust after 1 bets
player   12  won bet    1.  balance =   50
player   12  won bet    2.  balance =   75
player   12  won bet    3.  balance =  100
player   12  won bet    4.  balance =  125
player   12 lost bet    5.  balance =  100
player   12  won bet    6.  balance =  125
player   12 lost bet    7.  balance =  100
player   12  won bet    8.  balance =  125
player   12  won bet    9.  balance =  150
player   12  won bet   10.  balance =  175
player   12  won bet   11.  balance =  200
player   12  won bet   12.  balance =  225
player   12  won bet   13.  balance =  250
player   12 lost bet   14.  balance =  225
player   12 lost bet   15.  balance =  200
player   12 lost bet   16.  balance =  175
player   12 lost bet   17.  balance =  150
player   12 lost bet   18.  balance =  125
player   12  won bet   19.  balance =  150
player   12  won bet   20.  balance =  175
player   12 lost bet   21.  balance =  150
player   12 lost bet   22.  balance =  125
player   12 lost bet   23.  balance =  100
player   12  won bet   24.  balance =  125
player   12  won bet   25.  balance =  150
player   12 lost bet   26.  balance =  125
player   12 lost bet   27.  balance =  100
player   12 lost bet   28.  balance =   75
player   12 lost bet   29.  balance =   50
player   12 lost bet   30.  balance =   25
player   12  won bet   31.  balance =   50
player   12 lost bet   32.  balance =   25
player   12  won bet   33.  balance =   50
player   12 lost bet   34.  balance =   25
player   12 lost bet   35.  balance =    0
player 12 bust after 35 bets
player   13 lost bet    1.  balance =    0
player 13 bust after 1 bets
player   14 lost bet    1.  balance =    0
player 14 bust after 1 bets
player   15 lost bet    1.  balance =    0
player 15 bust after 1 bets
player   16  won bet    1.  balance =   50
player   16  won bet    2.  balance =   75
player   16 lost bet    3.  balance =   50
player   16 lost bet    4.  balance =   25
player   16 lost bet    5.  balance =    0
player 16 bust after 5 bets
player   17 lost bet    1.  balance =    0
player 17 bust after 1 bets
player   18 lost bet    1.  balance =    0
player 18 bust after 1 bets
player   19 lost bet    1.  balance =    0
player 19 bust after 1 bets
player   20  won bet    1.  balance =   50
player   20 lost bet    2.  balance =   25
player   20  won bet    3.  balance =   50
player   20 lost bet    4.  balance =   25
player   20 lost bet    5.  balance =    0
player 20 bust after 5 bets
player   21 lost bet    1.  balance =    0
player 21 bust after 1 bets
player   22  won bet    1.  balance =   50
player   22  won bet    2.  balance =   75
player   22 lost bet    3.  balance =   50
player   22  won bet    4.  balance =   75
player   22 lost bet    5.  balance =   50
player   22 lost bet    6.  balance =   25
player   22  won bet    7.  balance =   50
player   22  won bet    8.  balance =   75
player   22 lost bet    9.  balance =   50
player   22 lost bet   10.  balance =   25
player   22  won bet   11.  balance =   50
player   22 lost bet   12.  balance =   25
player   22  won bet   13.  balance =   50
player   22 lost bet   14.  balance =   25
player   22 lost bet   15.  balance =    0
player 22 bust after 15 bets
player   23 lost bet    1.  balance =    0
player 23 bust after 1 bets
player   24  won bet    1.  balance =   50
player   24 lost bet    2.  balance =   25
player   24  won bet    3.  balance =   50
player   24  won bet    4.  balance =   75
player   24  won bet    5.  balance =  100
player   24  won bet    6.  balance =  125
player   24  won bet    7.  balance =  150
player   24 lost bet    8.  balance =  125
player   24 lost bet    9.  balance =  100
player   24 lost bet   10.  balance =   75
player   24 lost bet   11.  balance =   50
player   24 lost bet   12.  balance =   25
player   24 lost bet   13.  balance =    0
player 24 bust after 13 bets
player   25  won bet    1.  balance =   50
player   25 lost bet    2.  balance =   25
player   25 lost bet    3.  balance =    0
player 25 bust after 3 bets
player   26 lost bet    1.  balance =    0
player 26 bust after 1 bets
player   27 lost bet    1.  balance =    0
player 27 bust after 1 bets
player   28 lost bet    1.  balance =    0
player 28 bust after 1 bets
player   29 lost bet    1.  balance =    0
player 29 bust after 1 bets
player   30 lost bet    1.  balance =    0
player 30 bust after 1 bets
player   31  won bet    1.  balance =   50
player   31  won bet    2.  balance =   75
player   31  won bet    3.  balance =  100
player   31  won bet    4.  balance =  125
player   31 lost bet    5.  balance =  100
player   31  won bet    6.  balance =  125
player   31 lost bet    7.  balance =  100
player   31 lost bet    8.  balance =   75
player   31 lost bet    9.  balance =   50
player   31 lost bet   10.  balance =   25
[...]
player   31 lost bet 999991.  balance = 502750
player   31 lost bet 999992.  balance = 502725
player   31 lost bet 999993.  balance = 502700
player   31 lost bet 999994.  balance = 502675
player   31 lost bet 999995.  balance = 502650
player   31  won bet 999996.  balance = 502675
player   31  won bet 999997.  balance = 502700
player   31 lost bet 999998.  balance = 502675
player   31  won bet 999999.  balance = 502700
player   31 lost bet 1000000.  balance = 502675
player 31 made 1000000 bets without busting

[Kluge]

Yay! Thank you, dooglus. I'll run it 50 times, average what the house earns, then do that set of fifty 9 more times. Tonight's my last night of free time for a while. Time well spent, IMO. ETA: Actually, I'll do EV neutral, since there's still a fair chance I'll get some doubt benefit if variance is with me.

[Kluge]

Game rules:
1000 players bet 1000 times in a game where they have a 50% chance of winning each bet (actually I think they have a 49.999999999% chance or whatever - Idunno exactly how random works, tbh). They each have a starting balance of 1 unit and must bet 1 unit. (changed from original so there's less 0s to write)

Set rules:
One game must be played 50 times.

Results of 10 sets: https://docs.google.com/spreadsheets/d/1KR_Wd9Z7K1YfWCsOgiq0KyRF7lD4dYaBnbW99BkSbSY/pubhtml

Conclusion:
House ended up very slightly (each game and set, the house seemed likely to lose, but rarely won huge) - nothing abnormal. Theory's definitely bunk, and I've now paid my debt to rationality by manually typing all those numbers in.

Script run (only "real" change is to print "casino profit" after each game so I don't have to think):

Code:

#!/usr/bin/env python

import random, sys

players = 1000
bets_per_player = 1000
starting_balance = 1
stake = 1

def roll():
    return random.random() < 0.50

player = 1
while player <= players:
    balance = starting_balance
    bets = 0
    while True:
        bets += 1
        if roll():
            balance += stake
            print "player %4d  won bet %4d.  balance = %4d" % (player, bets, balance)
        else:
            balance -= stake
            print "player %4d lost bet %4d.  balance = %4d" % (player, bets, balance)
            if balance < stake:
                print "player %d bust after %d bets" % (player, bets)
                break
        if bets == bets_per_player:
            print "player %d made %d bets without busting" % (player, bets)
            print "casino profit is %d" % (player-balance)
            sys.exit(0)

    player += 1

print "all the players busted"

[jambola2]

What does generally a pretty bad idea mean ? As far as I can understand , the house edge should be equal at all levels and that would just be a lot less fun (as you have no chance of winning something substantial) , with the same amount of risk.

Is my understanding wrong ?

If you want to win 1 BTC, you could:

a) risk 0.1 BTC at 9% for an 11x payout, or you could
b) risk 10 BTC at 90% for a 1.1x payout.

In each case you expect to lose 1% of the amount you risk, so b) costs about 100 times more than a) in terms of the amount you expect to lose, while only having a 10 times better chance of success.

That makes 90% seem like a bad deal compared to 9%, even though they both have a 1% edge.

Does that make sense?

Well , I first expected it to still work , because you could just bet at 90% with 0.1 BTC over and over.

0.1 * (1.1)^25 = 1.083 BTC

So , taking the same percentage of 90% chance of winning,

(90/100)^ 25 = 7.17% of winning

What you said makes sense now that I have calculated it , but why does this happen ?

You have a lesser chance of winning and also a lesser amount of winnings , does this mean a bet with lesser chances of winning is always better ?

[Light]

Well , I first expected it to still work , because you could just bet at 90% with 0.1 BTC over and over.

0.1 * (1.1)^25 = 1.083 BTC

So , taking the same percentage of 90% chance of winning,

(90/100)^ 25 = 7.17% of winning

What you said makes sense now that I have calculated it , but why does this happen ?

You have a lesser chance of winning and also a lesser amount of winnings , does this mean a bet with lesser chances of winning is always better ?

It's because when you accept those odds and gamble you effectively reduce your bet by 1% (this is true over an n case scenario where n approaches infinity). Hence, mathematically it is better to bet in smaller amounts as you 'lose' less from the edge. You don't notice this as in comparison to the number of bets you make the losses you get from the 1% are tiny. Maybe after something like 50 million bets that 1% would be clearer but over the short term variance wins out.

[jambola2]

Well , I first expected it to still work , because you could just bet at 90% with 0.1 BTC over and over.

0.1 * (1.1)^25 = 1.083 BTC

So , taking the same percentage of 90% chance of winning,

(90/100)^ 25 = 7.17% of winning

What you said makes sense now that I have calculated it , but why does this happen ?

You have a lesser chance of winning and also a lesser amount of winnings , does this mean a bet with lesser chances of winning is always better ?

It's because when you accept those odds and gamble you effectively reduce your bet by 1% (this is true over an n case scenario where n approaches infinity). Hence, mathematically it is better to bet in smaller amounts as you 'lose' less from the edge. You don't notice this as in comparison to the number of bets you make the losses you get from the 1% are tiny. Maybe after something like 50 million bets that 1% would be clearer but over the short term variance wins out.

So , if you are betting in an attempt to win something , it is best to bet all you can afford to bet at once ?

Does this imply that the martingale strategy is worse than a normal strategy ?

[Salmon1989]

Well , I first expected it to still work , because you could just bet at 90% with 0.1 BTC over and over.

0.1 * (1.1)^25 = 1.083 BTC

So , taking the same percentage of 90% chance of winning,

(90/100)^ 25 = 7.17% of winning

What you said makes sense now that I have calculated it , but why does this happen ?

You have a lesser chance of winning and also a lesser amount of winnings , does this mean a bet with lesser chances of winning is always better ?

It's because when you accept those odds and gamble you effectively reduce your bet by 1% (this is true over an n case scenario where n approaches infinity). Hence, mathematically it is better to bet in smaller amounts as you 'lose' less from the edge. You don't notice this as in comparison to the number of bets you make the losses you get from the 1% are tiny. Maybe after something like 50 million bets that 1% would be clearer but over the short term variance wins out.

So , if you are betting in an attempt to win something , it is best to bet all you can afford to bet at once ?

Does this imply that the martingale strategy is worse than a normal strategy ?

Yes and yes.
With "1% house edge", statistically speaking, you should expect to lose 1% of your total wagered amount.

[dooglus]

So , if you are betting in an attempt to win something , it is best to bet all you can afford to bet at once ?

Does this imply that the martingale strategy is worse than a normal strategy ?

Yes and yes.

I hesitate to mention this, but if you carefully pick your martingale strategy it is more effective than a single large bet at achieving your goal (whatever that goal is).

[Salmon1989]

So , if you are betting in an attempt to win something , it is best to bet all you can afford to bet at once ?

Does this imply that the martingale strategy is worse than a normal strategy ?

Yes and yes.

I hesitate to mention this, but if you carefully pick your martingale strategy it is more effective than a single large bet at achieving your goal (whatever that goal is).

Say, my target is to double my initial deposit.
Is it really possible to use martingale (or whatever strategy) to achieve the goal with higher than 49.5% success rate (one single bet)?

[dooglus]

Say, my target is to double my initial deposit.
Is it really possible to use martingale (or whatever strategy) to achieve the goal with higher than 49.5% success rate (one single bet)?

Yes.

And you're the first person who responded in such an open manner.

Everyone else just tells me I'm wrong.  

The trick is to split up your bet (the amount you were going to risk in a single bet) into a series of amounts which sum to a the same, and which form a sequence such that you can bet the smallest amount, and if it wins, you make the same as if you bet the whole amount at 49.5% (so you'll be betting with a smaller chance, and higher payout multiplier).  And if it loses, you want betting the 2nd amount to cover the first loss and make the same net profit.  Etc.

If you can find such a sequence (and you always can, though it can involve some hairy math depending on the length of the sequence you're looking for) then the amount you expect to risk is less than your whole amount (since there's a non-zero chance that you will win before the last bet, and stop at that point), and so the amount you expect to lose, being 1% of the amount you risk, is less than when you make the single bet.

Here's a very simple example:

you have 1 BTC and want to double it.

* you could bet it all at 49.5%, and succeed in doubling up with probability 0.495

* or you could bet 0.41421356 BTC at 28.99642866% with payout multiplier 3.41421356x, and if you lose, bet the rest at the same chance.  If you win either bet, you double up, else you lose.  Your chance of doubling up is 0.4958492857 - a little higher than the 0.495 you have with the single bet.

Cool, huh?

That's breaking the single bet up into a sequence of length 2.

If you break it up into more, smaller bets, then the probability of success increases further.

The more steps, the closer to 0.5 your probability of success gets.

You'll be limited by real-life barriers, like the invisibility indivisibility of the satoshi, and the limit of 4 decimal places on the chance at JD.  But in theory you can get arbitrarily close to 0.5.  I think.  

[Rigon]

Only EV plus games are PVP games where players can win and not the house. 

[a1choi]

You'll be limited by real-life barriers, like the invisibility of the satoshi, and the limit of 4 decimal places on the chance at JD.  But in theory you can get arbitrarily close to 0.5.  I think. 

cool! i didn't know there was a 4 dec limit on chance.  good to know.

Also, can you explain more about the invisibility of the satoshi?  never heard that term until now.

[dooglus]

Only EV plus games are PVP games where players can win and not the house. 

Except https://bitcointalk.org/index.php?topic=639787.msg7141055#msg7141055

[dooglus]

You'll be limited by real-life barriers, like the invisibility of the satoshi, and the limit of 4 decimal places on the chance at JD.  But in theory you can get arbitrarily close to 0.5.  I think.  

Also, can you explain more about the invisibility of the satoshi?  never heard that term until now.

Oh, I meant indivisibility.  As is there's nothing between 0 BTC and 1 satoshi.  Sorry - it's been a long day...

I expect at some point you're going to be wanting bet 0.01 satoshi at 0.00001% if you keep breaking down the martingale into ever more steps.  Neither of which (tiny stake nor tiny chance) is possible.