This isn't true. But there might be a new account restriction.

Yes.

However, betting what you guys call Kelly is the asymptote (as number of trials approaches infinity).

Assume a case with 2:1 payoff and p=.5. The expected growth optimal fraction to bet if you are only going to play once, is 100%....

If you are going to quite after 2 plays, growth is maximized by betting 50%...

If you are going to quite after 3 plays, about 37%....

It keeps getting smaller before stabilizing at an amount the would be equal to Kelly in the special case. You can see this graphically:



(The horizontal axis is the percent you risk, the vertical axis, the amount you make on your stake as a multiple.)

Kelly is the same as the asymptotic expected growth-optimal fraction(which ultimately is .25 or 25% of your stake to risk in this example) that maximizes the growth function, the value that is approached as the number of plays approach infinity.

This is VERY important to gamblers. You can sit and wait until the shoe gives you a certain edge, and that edge will last for likely a rather small horizon -- you are not maximizing what you can make by betting Kelly. Ever.

Note in the graphic the curious point where they all intersect, i.e. make the same amount of money even though each is a different number of plays that you quit at. For example, betting about 52% of your stake, it would appear, in this 2:1 payoff with p=.5 makes the same regardless of how many plays you decide to quit at!

Since I have a program to handle simple cases like this, here are the Expected Values and Median Values under the assumptions of Mr. Vince's example.

Suppose you are a gambler with a starting bankroll of 1. Suppose you are faced with a series of 10 independent binary bets each of which pays 2:1 and each of which wins 50% (and loses 50%). Suppose further that you must wager a fixed fraction of your then-current bankroll on each of the ten bets. Call this fixed fraction F.

As shown above, the Kelly fraction given by the formula (bp-q)/b becomes (2*.5-.5)/2 = .25 = 25%.

Here is the table of how the Expected Value and Median Value vary depending upon the Fixed Fraction chosen.

Fixed Fraction	Expected Value	Median Value
0%	1.00	1.00
10%	1.63	1.47
20%	2.59	1.76
25%	3.25	1.80
30%	4.05	1.76
40%	6.19	1.47
50%	9.31	1.00
60%	13.79	0.53
70%	20.11	0.19
80%	28.93	0.04
90%	41.08	0.01
100%	57.67	0.00

As in the previous simple numerical example posted above, EV is maximized at F=100% while Median is maximized at F=Kelly=25%.

No, it;s NOT 25%, that's my point. That;s the asymptotic value. If you adjust your program to rn things in finer increments, you will see this, it is ALWAYS > 25%

My program's level of precision is unable to prove/disprove your statement.

However, the following paper seems to state that Kelly indeed maximizes the median value in this case (where np is an integer in the binary case).

http://hariseshadri.com/docs/kelly-b...lly-median.pdf

I am not anywhere near expert enough to offer my own opinion on the issue.

Are you saying, in a normative sense, that a random should bet 100% if presented with the opportunity to make that wager once? If not, what is your number supposed to be used for?

What you are using, the Kelly fraction given by the formula (bp-q)/b gives you the asymptotic growth-optimal fraction, and as such, it is never the growth optimal fraction(!) always slightly less, the degree of which gets ever smaller as the number of trials increases.

As an interesting aside, these so-called Kelly formulas actually do equal the asymptotic expected growth-optimal fraction, and therefore don;t necessarily satisfy the Kelly Criterion (as specified in his 1956 paper) (in cases precisely as mentioned earlier up this thread where the solution that satisfies he Kelly Criterion is a value of f > 1.0).

If you want to solve or the real expected growth-optimal fraction, for a specified horizon, for one or many simultneously-played games, where there is a time-carry involved which can be +, - or zero, I have it up at

https://papers.ssrn.com/sol3/papers....act_id=2577782

It gives the exact fraction, not the asymptotic approximation others use. Not that you cannot use the asymptoic approximation -- it is very close with even a low number of trials. the point is, to understand this stuff, it;s critical to understand what happens to the actual expected growth optimal fraction:
-For a (classical) positive expectation game, f is 1.0 for quitting at 1 play ( regardless of how big or small the edge is)
-For an infinite number of plays, it converges to the expected growth-optimal fraction, and this convergence happens quite quickly.

Yes, if you have a classical positive expectation,then you maximize expected growth by wagering 100% (because there is no reinvestment at one play, just as with EV an any number of plays).

And this is regardless of the size of the edge -- which is disconcerting. In recent years, I have opted for other important points on the growth curve other than just the peak, realizing that the peak doesn't always match up with my criteria. (Recall the phi point, alluded to earlier in this thread.) That point, along with the peak, or the Kelly point as you call it, are important points on the curve. There are other points that maximize things for other criteria, but these too, are a function of the number of trials. I was involved ina pretty deep paper on this topic with two others:

https://papers.ssrn.com/sol3/papers....act_id=2364092

Which is still under peer review awaiting publication. A similar paper on using these ideas in blackjack is a paper that was published as "Optimal Betting Sizes for the Game of Blackjack." Journal of Investment Strategies 4(4), 53-75. I don have a copy of it anymore but it regards wagering at the other critical points that maximize gain with respect to risk on the curve (whereas the peak, or Kelly, only seeks to maximize gain and all else be damned) For one who seeks to maximize risk, and all else be damned, you can be confronted with the unpalatable proposition of an edge of, say, .00001 and be thus required, if your horizon is 1 play, to put up the entire stake.

This has turned into a great thread. Great threads cause me to do extra reading I wasn't planning to do.

That's basically implying that people have a linear utility function, which they don't.

Here's more detail in case anyone is interested in my attempt at trying to numerically zero-in on where the median actually occurs (see post 104). My program written in R using double-precision calculations reports the following is the last level of precision where it recognizes a difference in the median value.

.......F........... MEDIAN

.2499999 1.8020324707030
.2500000 1.8020324707031
.2500001 1.8020324707030

So my program, as far as I can tell, thinks that the median value of the probability distribution of the end bankroll is maximized at or very near .2500000.

True, and Im not so certain that reinvestment is even avoidable. First, if you start wit ha stake of $100, and it grows to $1,000,000 are you going to wager it in the same size ou did at first?

Second, there IS a time-value to money, and

Third, since the base currency is a currency relative to something else, (gold or euros or cocnuts or ping pong balls or poontang) the value of those dollars we are wagering is changing through time, and I believe this introduces and implicit reinvement.

I think you are using hte Kelly formula though, whcih will always give you the asymptici value, regardless of number of trials.

I am merely (via brute force computer program) traversing the entire 1024-branch outcome tree. Nothing to do with "Kelly".

For any inputted value for the Fixed Fraction (F), the program calculates the exact probability distribution of ending bankroll values (1024 probability-$ couplets).

The program then calculates the Expected Value and Median Value of that exact probability distribution of ending bankroll values.

Sorry if I was not clear in describing this previously.

Hmmm, without seeing the code....I suspect you are not traversing through each of those 1024 branches though, from play 1 through play 10. Your values are too close to asymptotic.

so, for example, if a the first two nodes of the branch are +2,-1,...and the f value being attempted is .256, the values would be
1.0 (the zero-eth or starting node)
*1.0 * (1+2*.256)
*1.0 * (1-1*.256)
etc.

It's not even a repeated gambling question. Take a 75-year-old retiree whose assets far exceed his ability to earn more money from work. The reasonable one-time bet is nowhere near 100% of assets. Interest, inflation, etc are nothing but noise- no reasonable values are going to suggest any bet anywhere near all-in. The utility of money isn't (just, or primarily) the utility of being able to maximize some Scoreboard! function of it after a series of wagers. It also includes being able to consume what the money can buy, the utility of which is quite nonlinear, and I don't see where your approach really acknowledges that.

Tom,
It does acknowledge it though. For starters, we're talking about a criterion of maximizing expected-growth (all else be damned) strategies. These strategies are clearly not for everyone -- only for those who know what risks go with these strategies and truly have as their criterion that of maximizing expected growth.

It may be that our gambler cannot afford to lose more than 10% on any single play, and that he is wagering his social security check. We can see then where he is on the chart of no reinvestment, the chart of the red, straight line going up and to the right on the chart of eight lines I posted earlier.

He may have a different criterion, that of maximizing what his expected gain is to his expected risk, If we look at risk as being the percent of his stake to bet, then the following chart (forty plays, 2:-1 with p=.5):



We can see the slope of the green line, which is the relationship of gain to risk at the peak, at Kelly, is less than the slope of the blue line -- which represents a point where the gain to the risk is maximized at the tangent to the curve. So for some, this point, this tangent point on the curve is a more important point to them than the peak (Kelly point).

Here's another important point. Notice how, after enough trials have occurred, the curve begins to develop a bell-shape (this bell shape is necessary in order to have a tangent point, as well as this next point).



If the horizontal axis is risk, and the vertical is gain, and we traverse this line from the far left, it is concave up -- it is picking up gain for every marginal increase in risk faster with each marginal increase in risk. At some f value before the peak, it goes from concave up to concave down. Thus, the point where it flips, this inflection point, represents that point where marginal increase in gain is accruing faster with each marginal increase in risk, up to the inflection point. Thus, in terms of risk-adjusted return, perhaps the inflection point satisfies one's criterion.

Both the inflection point and the tangent point require the bell-shape to begin developing, and both are a function of horizon. In the graphically shown cases, the horizon is 40 trials. As the horizon gets ever-larger, both of these points migrate towards the peak, and all three are the same at an infinite number of trials.

Do you see why the notion of "Half-Kelly" is simple and arbitrary? The points I am speaking about on the other hand, are significant geometric points along the curve, not arbitrary points.

Shown next, the phi point we discussed earlier, on the same chart.



And next, a chart of the same game showing, graphically, why you don't want to wager beyond the peak, as we discussed earlier in this thread:



At f=.4 you make the same as at f=.1 but your risk is four times as high!

So I dont want to give the impression you or anyone should wager at the peak, or at the expected growth optimal spot (100% for a single play) unless your criterion is to maximize expected growth, all else be damned.

Rather, I want to provide illumination, and get you to see that this framework isnt; merely the growth optimal spot (as misunderstood as it is, ie. what Kelly really is, the asymptotic nature of all of this, etc).

Instead, what we have here is a framework for examining all money management strategies, in gambling, in capital markets, and in all other disciplines where we may be seeking the diminishment of growth.

Look at the following graph, which represent 2 of these 2:-1 coin tosses after 20 simultaneous plays.



Everyone who plays these two coin tosses, simultaneously, is, de facto, upon this surface somewhere (even your old guy you were writing about!) they just don't now it. They are at some location on a given (multiple simultaneous, in this case) bet. Often, they are travelling on little trails upon this surface from one bet to the next, again, unwittlingly. They are moving about this surface oblivious to the points of geometric consequence (peak, tangent, inflection, even phi) and my point is that, sine this is something we actually have control over (how much we wager) as opposed to the outcomes themselves, we should use this framework to satisfy our own criteria in wagering - including your old guy!

Everyone, for years, has been hung up on "Kelly, Kelly Kelly," without even truly understanding it, but look at the surface, look at paths along it. For example, 1.1. and as you lose, your path takes you towards the peak, as you win, your path takes you back towards 1.1. This is a means, a basic shape of movement about this surface of what I call "The Leverage Space Manifold" that would keep you at and closer to equity highs. There are innumerable criteria you can make from this, and all money management strategies fit i this manifold. So we should use it for our benefit, being that we have absolute control over it.

Finally, then I am going to evaporate for a while, look at all that grey area on he chrt. The blue area is profitable, the grey area, those loci where the growth function becomes <1. Look how little blue there is in this wildly profitable pair of games! If you are off, on only one axis, to where you are in the grey, you're in trouble. This is why, if you have a 100 stock portfolio, and 100 axes, if you are off on only one axis, over- allocated on just one stock, you are in trouble. Previous portfolio models never illuminated this, but here we can see it, graphically and clearly.

So don;t think in terms of a single "point" or percent to risk. Define what your criteria is, then, using the framework, craft your money management to satisfy your criteria.

Sorry if I am a bit long-winded.

Which, for all intents and purposes, is nobody, almost literally nobody capable of wagering anywhere near their entire worth (somebody with little money and great employment income expectation is going to be more aggressive than their bankroll-on-hand would call for, but that's basically semantics- if they COULD wager against their future employment income stream, they wouldn't go all-in). I can't believe you're seriously arguing that anybody, outside of a handful of contrived or pathological situations, should act this way, and if you're not, you can't really be defending any model that says you should.

You literally just told me I should.
http://forumserver.twoplustwo.com/sh...&postcount=109


You're just reinventing the purpose of utility functions. Kelly maximizes U=E[ln(roll)]. Other strategies do other things. I don't see why you would try to derive behavior by arbitrary cutoffs of risk/gain or maximizing arbitrary numbers instead of determining a utility function and going from there.


It's obvious that you can be horribly misallocated and still profitable/growing. Cash in an inflationary environment is a straight loser, but you can still be plenty profitable/growing with some percentage of your money stuck in a mattress. You can have 10% of your money in a straight turd -EV stock and still be profitable/growing. You can literally light money on fire every day and still be growing, even though the proper allocations to all of those things is zero (so you're overallocated to all of them). Sure, if you've overconcentrated (or overleveraged), you can run your growth negative, but there's a huge range where you're just hurting your growth before you move on to running yourself into the ground. It's so hard it basically takes effort to do with 100 major stocks not on margin.

I cannot believe I am posting this, but I can promise that my program correctly traverses all 1024 branches (that is quite child's play for any computer). The Expected Value figures posted above could indicate that the complete outcome tree is being visited.

I know guys who have traded to be expected growth-optimal, but for finite time periods -- see Larry William Robbins trading championship 1987. But I agree, expected growth optimal, all else be damned, is not a criteria for anyone ecxept the occasional freelance madman.

And Kelly seeks to be expected growth-optimal, but, as I said, Kelly is asymptotic, as the number of trials approach infinity. At one trial, if you are going to play once and quit,the fraction to risk (if one seeks expected growth maximization) is 100%.

Your push back is not with the answer - but with the criteria here, and I agree with you (which is why I introduced the other two important geometric points that maximize gain with respect to risk, i.e. the tangent point and the inflection point as far more germane to satisfying real-world criteria).

Only for the criteria of expected growth-maximization. I don;t know what hte criteria you want solved here is.Fr some criteria, of course, the amount to risk might be 0.

No. You can still use your utlity functions if you like. Regardless of what you use however, you are somewhere in the leverage space manifold (which could just be a 2D manifold as pictured in the charts for the single-case game, or a 3D manifold for the example given of two multiple simultaneous coin tosses, or a 501D manifold if you are trading a portfolio of 500 stocks). But regardless of what you use to determine your risked quantities, you are somewhere on the surface in that manifold, and those important geometric points (i have enumerated three others in addition to the peak, what you call Kelly) and the topology of the curve is at work upon you whether you acknowledge it or not.

Not true at all Tom, that is the misconception I am speaking of. Remember, we're talking reinvestment, so the surface curves on the second simultaneous multiple play (or period of reinvestment) and approaches an asymptotic shape as the number of trials or periods increases.

Now, lets go back to the single 2:-1 coin toss case where f=.25 asyptotically. Now let;s do tow of these simultaneously (which was pictured as the 3D chart with the blue space the profitable space) where each f = .23, such that the sum of the f values is .46.

If we lost on both games simultaneously, we would be down 46%.

In multiple simultaneous situations, I call the sum of the f values of all the components the "Sigma f."

But remember, this .46 is asymptotic. Initially, the surface would have two planes making a V shape where, if I bet all on one coin, it wold be a straight line going up to 1.0 on each axis, and in between each axis, a straight line to .5 (so a to risk, Sigma f, 100% by wagering 50% on each coin and quitting after the single, multiple-simultaneous play)

As the number of plays continues, as there is reinvestment, that shape morphs into the shape shown for 20 plays in my previous post with the blue section as the profitable section.

As the number of multiple simultaneous propositions increases, Sigma f approaches 1.. Thus, if we were to play 100 simultaneous 2:-1 coin tosses, the f for each game would be a little less than .01.

For a portfolio of 500 stocks, the allocation to each individual stock, though not necessarily uniform, is very small. Thus, the ph point is very small as well. Remember, we are talking about reinvestment here, not single-play cases.

So yes, if you are trading 500 stocks, a small amount, in an entirely unlevered position, could, conceivably, make the growth function beyond one reinvestment period be negative in the aggregate.

Kelly doesn't seek to be anything any more than 2+2 seeks to be 4. It maximizes U=E[ln(roll)].

These points are essentially chosen at random.

The problem is that your answer to "how do you value various different amounts of money" is "depends on how many bets it takes me to get there", which most people would consider a non sequitur. If your prescriptions are mapped to utility functions, then your strategies range from linear utility to ~log utility SIMULTANEOUSLY OVER THE SAME RANGE OF NUMBERS. It's not a reasonable framework for behavior or anything remotely resembling one, and since all the strategies are Kelly or more aggressive, it's never even randomly closer to correct, and not to be rude, but not anything anybody should take seriously as a behavioral recommendation.

You've solved a niche part of a problem, but it's just not something that's generally of any use (you have shown that repeatedly overbetting is bad, but that's not remotely new)

I have a change bucket. The combination of my portfolio and the change bucket is positive growth relative to inflation. I'd collapse all my floors long before I could allocate enough coins to change buckets to flip to negative. This is obvious.

Trust me, I've solved plenty of way more complicated versions of this.

If your portfolio is barely expected to grow, ****ing it up a little could flip you. For anything remotely resembling major US stock indices, it can fade a lot before going negative. The amount you can fade is similar whether you have 100, 500, 1000, or 5000 stocks- it's mostly a function of the performance of the rest of your portfolio, and slightly one of your average position size, once you've included a large number already (adding stocks to decrease portfolio performance variance has diminishing returns).

I hope that everyone realizes that the expected value numbers need no computer to calculate. You simply take the number that the first bet turns your one dollar bankroll into on average and raise it to the tenth power.

What does "citation" mean. In any case could you perhaps use the same technique to get other VIPs to these forums?

Dave,
I am not a VIP. (I prove that to myself in the markets daily, usually at least once before noon). I am always scouring forums and lists and blogs for ideas along these lines which is how I got here.