What is your strategy?

[nonlinear5]

But what about this crazy synergy? Even adding a negative expectation bet can increase performance! :eek:

Well, we are measuring the risk-adjusted performance, as signified by the rate of growth (rather than by the absolute profit). By including the bets with negative expectation (such as Green-0), the expected profit is reduced, but the risk is reduced by a larger degree. The ratio of the two (profit to risk) then goes up, until we hit the point where any further reduction of expected profit is no longer justified by the reduced risk. Kinda makes sense to me.

 

[Visaria]

Doesn't make any sense to me whatsoever. Adding a neg expectation bet increases the overall ev, wtf?

 

[Visaria]

Well, we are measuring the risk-adjusted performance, as signified by the rate of growth (rather than by the absolute profit). By including the bets with negative expectation (such as Green-0), the expected profit is reduced, but the risk is reduced by a larger degree. The ratio of the two (profit to risk) then goes up, until we hit the point where any further reduction of expected profit is no longer justified by the reduced risk. Kinda makes sense to me.

ok, fair explanation, but will need to do my own investigation of this.

 

[kut2k2]

Doesn't make any sense to me whatsoever. Adding a neg expectation bet increases the overall ev, wtf?

Like the old saying goes, the math is the math. Sometimes the math forces you to accept counter-intuitive synergy when adding a negative-expectation bet.

 

[nonlinear5]

Just deucedly hard to calculate k for three or more bets is all.

In one of your other threads, SplawnDarts gave a recipe for calculating Kelly from any series of trade returns:

Any reasonable (read: subject to integration, does not result in a negative ending bankroll in any case) distribution of trade returns allows the computation of a Kelly fraction. You solve:

MAX(across f, INTEGRAL(across PMF range, (PMF*ln(EBR)))

Where EBR is your ending bankroll: starting bankroll - (f*trade result)
with trade results reported in units of bankrolls.

Let's verify this recipe. Suppose that the rules of our roulette game are simplified. The only allowed bet is that on R16. How much do we bet? The precise solution is given by discrete Kelly:

DK = 4/37 - (1 - 4/37) / 35 = 0.0826 = 8.26%

Now let's think of this roulette game as if it is a trading system, and the outcomes are the trade returns. The distribution of returns (given the uniform distribution of results over 37 trades) would look like this:

R = {+3500%*s, +3500%*s, +3500%*s, +3500%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s)

where s is the fraction of he bankroll. Note that the order of gains and losses does not matter, because the product of returns is the same.

Based on this distribution of returns, I started with $1000 and calculated the ending bankroll (EBR) in Excel.

Finally, I solved for s which maximizes the following:
F(s) = log(EBR) - log(1000)

Guess what s turned out to be? Exactly 0.0826!

We've just calculated continuous Kelly, and it's the same as the discrete Kelly. The same methodology can be used with any arbitrary series of returns, and the calculation of Kelly is simply figuring out (numerically) where the maximum of the following occurs:

F(s) = log(ending bankroll) - log(starting bankroll)

 

[kut2k2]

In one of your other threads, SplawnDarts gave a recipe for calculating Kelly from any series of trade returns:

Let's verify this recipe. Suppose that the rules of our roulette game are simplified. The only allowed bet is that on R16. How much do we bet? The precise solution is given by discrete Kelly:

DK = 4/37 - (1 - 4/37) / 35 = 0.0826 = 8.26%

Now let's think of this roulette game as if it is a trading system, and the outcomes are the trade returns. The distribution of returns (given the uniform distribution of results over 37 trades) would look like this:

R = {+3500%*s, +3500%*s, +3500%*s, +3500%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s, -100%*s)

where s is the fraction of he bankroll. Note that the order of gains and losses does not matter, because the product of returns is the same.

Based on this distribution of returns, I started with $1000 and calculated the ending bankroll (EBR) in Excel.

Finally, I solved for s which maximizes the following:
F(s) = log(EBR) - log(1000)

Guess what s turned out to be? Exactly 0.0826!

We've just calculated continuous Kelly, and it's the same as the discrete Kelly. The same methodology can be used with any arbitrary series of returns, and the calculation of Kelly is simply figuring out (numerically) where the maximum of the following occurs:

F(s) = log(ending bankroll) - log(starting bankroll)

Yeah now try doing that for two or more simultaneous bets.

I'll stick with discrete Kelly because it's always exact. CK is only an approximation in most cases. Thorp admits that in the paper you linked to.

It took me a while to figure out how to solve discrete Kelly for multiple simultaneous bets but now I know what to do. It's just very laborious for 3 or more bets.

 

[nonlinear5]

CK is only an approximation in most cases. Thorp admits that in the paper you linked to.

It's an approximation only when it takes the analytic form of (r / s^2), because it makes an assumption that the returns are normally distributed. If you solve it numerically as I outlined above, no such assumptions are made, so the CK is exact, in the sense that it maximizes the compounded rate of growth.

I can calculate the CK for any distribution. Pick any sequence of returns, and we'll compare the results.

 

[SplawnDarts]

V suspicious that.

If correct, why not stick something on all the other (black) numbers as well?

Assuming it's allowed you would do exactly that, since it creates a pure arbitrage (which as you approach it Kelly would tell you to be arbitrarily big).

That of course gets to the point that Kelly tells you how much to risk, but says nothing about actually having the cash in hand to place the bets. As something gets closer to an arbitrage, those two become increasingly decoupled. So you would then have to add an extra constraint bet1+bet2.... <= 1000. The result is a non-linear program to solve (maximization, but now with constraints). Excel's solver looks (at least in this case) up to the task as well if someone wants to set it up.

 

[nonlinear5]

Assuming it's allowed you would do exactly that, since it creates a pure arbitrage (which as you approach it Kelly would tell you to be arbitrarily big).

So, bet a small percentage on all black numbers, with the bulk bet on R16, R14, and Red?

 

[SplawnDarts]

So, bet a small percentage on all black numbers, with the bulk bet on R16, R14, and Red?

If you were allowed to borrow money to make the bets (ie. sum of the bets > 1000 was OK as long as max loss was < 1000) then I think there's no unique solution. For example, a layout of 1 unit on all black and green numbers (14 spaces/units total) and 15 on red would be profitable for all outcomes - a perfect arbitrage. That would also be true if you put 20 units on red. Any of those solutions you'd leverage infinitely, giving you the same infinite amount of money after 1 bet.

So the more interesting case is where sum of bets < 1000.

This basically re-thinks Kelly, which is looking for a maxima of the N-surface representing bankroll growth. In traditional Kelly that maxima is always in the middle, away from constraint edges. But here there is no maxima because of the perfect arb - just an ever rising "ridge" along the ratio of bets that makes the arb safe. As a result what was previously a non-linear programing problem with a local maxima now becomes akin to a linear programming problem in that the solution must lie on the constraint edge.