Another Kelly conundrum

[puff]

Hi All. Been reading a number of threads on this site, with great interest.

I'm a big fan of betting theory, and I'm very much sold on the Kelly staking plan (half-Kelly actually). I have question though that I simply cannot get my head around. I consider myself a pretty good mathematician, but this one's got me stumped. Very much in awe of Ganchrow's work, I think he may be the man for this one! Apologies if it may have already been covered somewhere along the way.

My question is this...

Let's say you've established that a betting proposition has a 50% chance of ocurrence. And you're therefore happy to bet +110. Given a bank of $10,000 and an average bet size of $1000, I think Kelly suggests a $455 stake on this occasion. However, the odds on that selection for some reason get bigger rather than shorter, such that you can now get +120. For the sake of argument, let's say this market move doesn't change my assessment of the betting proposition (at this point, I should say that, as an economist, I'm also a great believer in market forces and therefore generally have due respect for the efficiency of the market). Now I've always thought that the bigger odds requires a supplementary bet. All things being equal, at +120, Kelly suggests a $833 stake, but since there's already an established position on the proposition (+500, -455), I cannot find an additional stake which is justified by Kelly ($378 @ +120 is simply too much, given the bank constraints, even $340 given the reduced bank size from the previous bet, but there also seems no incremental amount that fits the criteria). Am I therefore to believe that Kelly suggests no further bet at the greater value odds?? Doesn't sound right to me, but I don't seem to be in a position to prove it!

[Data]

Welcome to the forum!

http://forum.sbrforum.com/handicapper-think-tank/26617-overbets-line-changes-kelly-spreadsheet.html

[RickySteve]

I'm guessing that spreadsheet does the trick, but you need to risk just a little less than $400 at +120, given the established position. It is a bit counterintuitive that your total risk is larger than it would be just betting +120.

[Justin7]

On a very serious note... If the market moved against you, you should look very closely at you assumption that "this is a 50% play". You'll find that 90% of the time, the line will move towards Pk instead of +120. When it moves "the wrong way", someone is betting big and knows something you don't.

[puff]

Many thanks, guys.

Just trying to play around with the spreadsheet, putting in all the relevant parameters of my particular conundrum, but unless I'm missing something (and it's very possible, in fact probable, that I am!), it looks like the relevant read-only cell ("Stake", for "Position after Line Move") has been overwritten with an arbitrary number. Or perhaps it's just that I don't have Solver, and cannot Calculate Stakes. Either way, I still can't seem to make sense of it!

I thought the answer was just less than $400 too, RickySteve (833-455=$378). But I'm not sure that's right, since that gives you a new position of $833 @ average odds of ~ 2.14, and at those average odds, that would seem to be overstaking, according to Kelly.

Rest assured, Justin, I have every respect for market forces, just wanted to get a handle on the theory.

[Ganchrow]

Many thanks, guys.

Just trying to play around with the spreadsheet, putting in all the relevant parameters of my particular conundrum, but unless I'm missing something (and it's very possible, in fact probable, that I am!), it looks like the relevant read-only cell ("Stake", for "Position after Line Move") has been overwritten with an arbitrary number. Or perhaps it's just that I don't have Solver, and cannot Calculate Stakes. Either way, I still can't seem to make sense of it!

If you don't have Solver installed it will spit out an error and not do anything else.

Otherwise, the number overwriting the cell isn't arbitrary ... it's your answer.

I thought the answer was just less than $400 too, RickySteve (833-455=$378). But I'm not sure that's right, since that gives you a new position of $833 @ average odds of ~ 2.14, and at those average odds, that would seem to be overstaking, according to Kelly.

I just edited the spreadsheet to include your desired data by default. Try clearing your cache and downloading again (I also fixed a bug such that the Expected Growth figure display was actually the Expected Return).

The answer is 3.9773% (or $398.73 assuming a starting bankroll of $10,000). If you were actually betting at odds of 2.14 then the 8.5227% total stake would indeed be an over stake. But that's not what you're doing. You're betting at odds of 2.10 given a boundary condition (the initial position).

[Ganchrow]

There's a different economic interpretation of a payout odds (aka "price") change that I often find intuitively useful. It involves marking all positions to the current market price. I find this interpretation particularly illustrative when market making.

The idea behind it is simple. When a market price change occurs on a bet in which a player already has a position, then if the player marks that position to market there will be two economic effects (in addition to the stated price change) on his ledger:

1. The position will have accrued unrealized trading profit or loss; and
2. His exposure to the event in question will change

Specifically, if a price moves in a player's favor (meaning that the available market payout odds have decreased, e.g. a move from -160 to -170, or from +120 to +110), then the player will realize trading profit and will have his effective exposure to the position at the new price increased.

Conversely, if a price moves against a player (meaning that the available market payout odds have decreased, e.g. a move from -170 to -160, or from +110 to +120), then the player will realize trading loss and will have his effective exposure to the position at the new price decreased.

To wit:

Let o = decimal payout odds prior to the price change
Let n = decimal payout odds after the price change
Let x = bet stake prior to the price change
Let Y = bet stake after the price change
Let Π = unrealized profit after price change

So given o, n, and x, we're looking to solve for Y and Π.

Now obviously, regardless of the price change that occurs, the economic result will be the same given a particular event resolution.

So in the case of a win we'd have:

(o-1)*x = (n-1)*Y + Π

And in the case of a loss:

-x = -Y + Π

Solving these two simultaneous equations for Y and Π gives us:

Π = x/n * (o-n)
Y = x/n * o

So let's plug in the numbers from the OP's initial question:

o = 2.1
n = 2.2
x = 4.545%

whcih then gives us:

Π = x/n * (o-n) = -0.207%
Y = x/n * o = 4.339%

This means that after the price change the player's new economic situation will be a position of 4.339% of initial bankroll on the bet +120 and a new bankroll equal to 1-0.207% = 99.793% of initial.

So now let's consider this in terms of Kelly.

At +120, the player's full Kelly stake would be 8.333% of current bankroll, which is 99.793% of initial. Hence, his desired position as a percentage of initial bankroll should be 99.793%*8.333% = 8.316% at +120.

However, the player already has an exposure of 4.339%, so to get back in line with Kelly he'd only need to purchase additional exposure equal to 8.316%-4.339% = 3.977% of initial bankroll, which is of course the same answer as that obtained from performing a full optimization.

[puff]

Great stuff, Ganchrow, I thought you might be the man for the job! Very relieved to hear that there is an answer after all, I was at a loss there for a while. In the absence of Solver, are you able to tell me what the relevant calculations are??

[puff]

Oops, you seem to have snuck in another post while I was typing, let me read that first...

[puff]

There's a different economic interpretation of a payout odds (aka "price") change that I often find intuitively useful. It involves marking all positions to the current market price. I find this interpretation particularly illustrative when market making.

-snip-

OK, got it, Ganchrow. Thanks! Is that your recommended method for solving this problem??

[Ganchrow]

OK, got it, Ganchrow. Thanks! Is that your recommended method for solving this problem??

Either way works.

[Data]

This explains how to make Kelly stakes in perfectly efficient zero-vig markets. Make sure you make adjustments according to the market reality. For instance, due to market inefficiency, one can find a better Y value while Π=0. This will result in a smaller additional stake.

[puff]

This explains how to make Kelly stakes in perfectly efficient zero-vig markets. Make sure you make adjustments according to the market reality. For instance, due to market inefficiency, one can find a better Y value while Π=0. This will result in a smaller additional stake.

So am I to understand that in the case of a completely inefficient market (far-from-true in reality, of course), Π=0, and the answer is simply 8.333 % (Kelly stakes at the new odds) less 4.545% (Original Kelly stake at the previous odds)??

[Data]

I think that we need to average the decimal odds after making the additional bet. The simple calculations show that the odds increase from 2.1 to 2.2 justifies for an additional bet of 0-size.

[Ganchrow]

This explains how to make Kelly stakes in perfectly efficient zero-vig markets. Make sure you make adjustments according to the market reality. For instance, due to market inefficiency, one can find a better Y value while Π=0. This will result in a smaller additional stake.

Realize that "profit" as described above only really corresponds to "accounting profit". It provides for no consiertations of expectations or present value.

Rather than simply marking-to-market we could also mark-to-open, mark-to-fair, mark-to-liquidation-price, or mark-to-3-year-old-nephew's-best-guess. The neat thing about this, however, is that regardless of the price to which we mark, no matter how arbitrarily defined it is, then given a new buy/sell price and exogenously determined win probability. the implied change to position size will be fully invariant.

So as such, as long as we express all prices in the same terms (be they mark-to-market, or mark-to-whatever), then conditioned on our own expectations and on an available transaction price, neither vig nor market inefficiency are of any consequence to our results whatsoever.

Remember that results using this methodology are economically indistinguishable from results obtained using a full optimization for every possible outcome.

So am I to understand that in the case of a completely inefficient market (far-from-true in reality, of course), Π=0, and the answer is simply 8.333 % (Kelly stakes at the new odds) less 4.545% (Original Kelly stake at the previous odds)??

I think that we need to average the decimal odds after making the additional bet. The simple calculations show that the odds increase from 2.1 to 2.2 justifies for an additional bet of 0-size.

Neither of these interpretations are really correct. No matter how efficient the market (even, I'd assume, were it "completely inefficient", whatever that means ), one's optimal Kelly stake would be a function only of price, win probability (for a binary-outcome bet), current bankroll, and preexisting positions (if any).

So if one's view were a win expectation of 50%, the best available market price were +120, and the bettor's only other position were 4.545% of bankroll at +110, then for full-Kelly optimality the additional stake would 4.339% of bankroll irrespective of either vig or market inefficiency. (Note that this does not take into account "trading decisions" where a bettor believed he might later be able to buy or sell at lower or higher prices, respectively.)

If I have the time and the desire is there, then at some later point I might encapsulate this all into a spreadsheet (or into my JavaScript Kelly calculator). This will allow for a fully computationally-precise Kelly solution (i.e., no optimization required) given initial positions at prices different from prevailing (although without employing a branch-and-bound we'd need to assume zero-vig in most cases of multiple assets so as to handle the Boolean bet/no-bet boundary conditions for parlays). In other words, imagine the JavaScript Kelly calculator with an additional form in which to enter preexisting positions. It seems to me that this enhancement would have negligible impact on the speed of the algorithm. Actually this is starting to sound pretty cool ...

[Data]

Realize that "profit" as described above only really corresponds to "accounting profit". It provides for no consiertations of expectations or present value.

Rather than simply marking-to-market we could also mark-to-open, mark-to-fair, mark-to-liquidation-price, or mark-to-3-year-old-nephew's-best-guess. The neat thing about this, however, is that regardless of the price to which we mark, no matter how arbitrarily defined it is, then given a new buy/sell price and exogenously determined win probability. the implied change to position size will be fully invariant.

I am having hard time realizing this. I am really trying to find a flaw in my reasoning but you were not very convincing above. I think you need to define "profit" better because I do not immediately believe in profits gained due to "arbitrary defined" prices. On a side note, I was talking about buy prices, not buy/sell prices.

one's optimal Kelly stake would be a function only of price, win probability (for a binary-outcome bet), current bankroll, and preexisting positions (if any)

That is precisely what I was using. I calculate how much I need to add to my position based on the newly discovered availability of a better price and assuming no change in bankroll. This all is "neat" as well and in accordance with Kelly. However, this is also in accordance with the common sense that tells me that if I made a bet at +110 and the line was -130/+110 and then I found -140/+120, say, in a book for "recreational players", then that does not change the size of my bankroll. Here is a follow up question, does my bankroll change if -140/+120 line exists but I never found it?

[Ganchrow]

I am having hard time realizing this. I am really trying to find a flaw in my reasoning but you were not very convincing above. I think you need to define "profit" better because I do not immediately believe in profits gained due to "arbitrary defined" prices. On a side note, I was talking about buy prices, not buy/sell prices.

I was actually hoping that you'd be able to convince yourself by examining the above equations.

"Profit", you see, just corresponds to accounting profit which is merely the negative of the difference between the buy price and the "current" price, however you choose to define it. As (I suspect you're aware, this is how P&L reporting exists within the financial trading industry.) If the mark-to price also happened to be the "fair" price then the accounting profit will also equal the "expected" profit. If the mark-to price also happened to be the market transaction price (as in the example I gave above) then the benefit that the Kelly result can be obtained using only the standard single-bet Kelly formula without any further calculus. From an accounting perspective neither one of these options are necessary.

I think you're also getting hung up on the concept of a changing bankroll. This again is simply an accounting construct used to satisfy the Law of Conservation of Unrealized Bet Capital (a phrase coined by yours truly not 15 seconds ago). To wit:

Y = x + Π

or in plain English:

bet stake after price change =
bet stake prior to price change +
unrealized profit after price change

Anyway, let's go back to the original example. The player acquires 4.545% of bankroll worth of a binary bet at +110 that is expected to win with probability 50%. The market price then moves to +120.

You choose, however, to mark-to-the-first-number-uttered by cousin-Steve. This number is -1000.

I'm now going to demonstrate that using the mark-to-the-first-number-uttered by cousin-Steve method, yields the same additional bet (3.977% of original bankroll) as both mark-to-market and a full optimization using the line change spreadsheet.

So here's what we have so far:

o = 2.1
n =
x = 4.545%

which then gives us:

Π = x/n * (o-n) = 4.132%
Y = x/n * o = 8.678%

So in other words, if we mark to a price of -1000 we have unrealized accounting profit of 4.132% of initial bankroll and a current position at -1000 of 8.678% initial bankroll.

We're presented with the opportunity to increase our bet size at +120. However, since, we're marking to -1000 each unit at +120 yields:

o₂ = 2.2
n =
Π₂ = x*(2.2-1.1)/1.1 = 100%*x profit
Y₂ = 2.2/1.1 = 200%*x exposure to the bet at -1000

So in other words, after marking to -1000, each dollar wagered at +120 yields an immediate $1 of accounting profit and an immediate $2 of exposure to the bet at -1000.

So if δ represents our additional stake at +120 then our full-Kelly utility function looks like this:

U(δ) = 50%*log(1+(4.132%+δ)+(1.1-1)*(8.678%+2*δ))
U(δ) + 50%*log(1+(4.132%+δ) - (8.678%+2*δ))

which simplifies to:

U(δ) = 0.5*log(0.95454 - δ) + 0.5*log(1.05 + 1.2*δ)

Differentiating wrt δ and setting to zero then yields:

0.5/(0.95454 - δ) = 0.6/(1.05 + 1.2*δ)

Solving for δ we find, lo and behold ...

δ = 3.977%

Imagine that!

[3put]

I think the problem we try to solve in this thread can, and should be generalized and indeed simplified.

We want to adjust an existing position using a Kelly wager.

How this existing position was established is irrelevant. We used a line, a bankrol, a Kelly-fraction when the position was taken. Maybe we even have adjusted the original position before.
All that matters is the position, bankroll, line, Kelly-fraction and probabilities of winning, pushing and losing NOW.

So we need a simple spreadsheet that calculate the Kelly stake as a percent of the actual bankroll under the condition of the actual position.

[Ganchrow]

I think the problem we try to solve in this thread can, and should be generalized and indeed simplified.

We want to adjust an existing position using a Kelly wager.

How this existing position was established is irrelevant. We used a line, a bankrol, a Kelly-fraction when the position was taken. Maybe we even have adjusted the original position before.
All that matters is the position, bankroll, line, Kelly-fraction and probabilities of winning, pushing and losing NOW.

So we need a simple spreadsheet that calculate the Kelly stake as a percent of the actual bankroll under the condition of the actual position.

See http://forum.sbrforum.com/handicappe...readsheet.html

[3put]

See http://forum.sbrforum.com/handicappe...readsheet.html

This is the spreadsheet I am talking about.

To use it you have to input many details about the first wager.
My point is that ALL that matters NOW is the actual position, not have you got it.

[Ganchrow]

This is the spreadsheet I am talking about.

To use it you have to input many details about the first wager.
My point is that ALL that matters NOW is the actual position, not have you got it.

Which piece of information are you suggesting is unnecessary?

Specifying the payout odds of the initial bet is important because the payout odds impact bankroll in the event of a win. Knowing the Kelly multiplier is important because it impacts the final form of the utility function.

For full generality the spreadsheet allows for line changes as well as odds changes. In this manner the initial bet can have a different win probability than the follow up bet. Hence, it's also necessary to specify a win/push probability for the initial separately.

For example, if you got in to your quarter-Kelly NFL position at +2½ -120, and the current line is now ±3½ -120/+110, what's the optimal staking at the new price? After assigning probabilities to the spreadsheet may then be used to solve this general problem.

In short, for full generality in optimizing bet size after mis-bets and changes to line/payout odds, all the requested information is necessary.

[3put]

I just want to clarify that by 'position' I mean Risk/ToWin, eg
$100 risked to win $60.

If my initial wager was a future bet 6 months ago and I have adjusted it 10 times since at different lines, I still think that the only thing that is important is my position now.

But I know I am wrong!

[Data]

accounting profit which is merely the negative of the difference between the buy price and the "current" price

First, accounting profit is NOT that. Second, a difference in buy prices does not necessitate a profit at all.

I do not think you hear what I am saying because what you said does not address my questions and arguments. I guess your just get used to (nearly)efficient financial markets that do not want to see how the existance of off-market prices in inefficient markets calls for slightly different approaches. Again, I might be wrong, but thus far common sense does not create any contradictions and seems true.

Think of the Matchbook interface. We place $454.55 at 2.1. Then, we place $1 at 2.2. Now, we have $455.55 at 2.100219. Kelly stake at these odds is $455.45 which is LESS than our stake of $455.55. Thus, by adding to our original position at these odds we deviated from Kelly stake. Therefore, adding at +120 is not justified, we need to get the odds above +121 to start adding here.

If we get +122 we add $37.20 at these odds. Our total stake will be $491.75. Our average odds are 2.109078. The full Kelly stake at these odds is $491.75. And the best part? There is no bankroll decrease!

[tomcowley]

You're missing a key point.

I am with you that the kelly stake at 2.1 is 454.54(repeating).

I am with you that the kelly stake at 2.100219 is 455.55.

Each marginal dollar that you wager at +120 not only increases your stake, but ALSO increases your net odds. Even though you cannot bet the full-kelly stake (for the net odds) by adding on over $1 at +120, that does NOT mean that your expected growth is decreased by doing so- you would never add on beyond $1 if your odds stayed the same (2.1000219(, but they don't stay the same.

Using Excel, when you have 454.55 wagered at 2.1 (to win 954.555, screw rounding), your expected growth is .11357%

When you add on $1 at 2.2, your growth goes up to .11405%.

Adding on optimally at +120 means adding on $397.72 for a total expected growth of .20833%. In this case, your stake is 852.27 to win 1829.539, at net odds of 2.14666- which is still above the kelly stake for those odds (639.53).

In the absence of information that changes your opinion of the true line, it is always +EG to add on at a better price.

[Ganchrow]

First, accounting profit is NOT that. Second, a difference in buy prices does not necessitate a profit at all.

I'm not saying that this corresponds to accounting profits according to GAAP, but rather this is how I'm defining it for the purposes of this framework. If as an accountant you object to the terminology that's fine ... you may call it as you wish.

Let's not lose focus. My entire purpose with post #7 was to define a framework through which Kelly optimal positions may be determined following a change in payout odds given some preexisting set of positions at differing payout odds. This is exactly what I believe I've done.

Nevertheless, as always, I may well be wrong. If you can find a single situation where the solution obtained using the above methodology following a change in payout odds yields different results than those obtained using the spreadsheet (or indeed if you believe that the results obtained are themselves faulty) I'd certainly need to drastically rethink what I've written.

But remember ... the point here is not to get into a debate over semantics, but rather to solve the OP's initial question without appealing to any calculus (or optimization engine) other than that implicit in the standard single-bet full-Kelly solution. Again, if you believe I've not done this please provide a counterexample yielding greater expected utility than that implied by the solution using the above framework.

I do not think you hear what I am saying because what you said does not address my questions and arguments. I guess your just get used to (nearly)efficient financial markets that do not want to see how the existance of off-market prices in inefficient markets calls for slightly different approaches. Again, I might be wrong, but thus far common sense does not create any contradictions and seems true.

If I haven't addressed your questions and arguments I do apologize. Nevertheless the framework I've espoused has nothing whatsoever to do with market efficiency. Indeed, by assuming that odds of +120 are offered on a bet winning with 50% probability, we're clearly acknowledging market inefficiency.

Think of the Matchbook interface. We place $454.55 at 2.1. Then, we place $1 at 2.2. Now, we have $455.55 at 2.100219. Kelly stake at these odds is $455.45 which is LESS than our stake of $455.55. Thus, by adding to our original position at these odds we deviated from Kelly stake. Therefore, adding at +120 is not justified, we need to get the odds above +121 to start adding here.

This is incorrect.

Without adding to the current position (which we're taking to be 1/22 = 4.545% of bankroll) we have expected utility of:

E(U) = 50%*ln(1+4.545%*1.1) + 50%*ln(1-4.545%) ≈ 0.1135%

Adding 0.01% of bankroll to the current position yields expected utility of:

E(U) = 50%*ln(1+4.545%*1.1 + 0.01%*1.2) + 50%*ln(1-4.545%-0.01%) ≈ 0.1140%

While adding 3.9773% of bankroll to the current position yields expected utility of:

E(U) = 50%*ln(1+4.545%*1.1 + 3.9773%*1.2) + 50%*ln(1-4.545%-3.9773%) ≈ 0.2081%

which, because this represents the Kelly optimal solution, is the highest expected utility attainable following the line change given the initial position.

Note, however, that had we acquired our entire full-Kelly position at +120 (representing 8.3333% of bankroll) our expected utility would be 0.4149%.

So yes, if we could go back in time we could indeed do better than our 0.2081% utility, but we can't so we're forced to transact at the new price with the added baggage of the old position. Each dollar we add on that new price also raises our payout odds, so we're not simply adding on to our position at some static average, but rather we're also increasing our possible as we expand our position.

If we get +122 we add $37.20 at these odds. Our total stake will be $491.75. Our average odds are 2.109078. The full Kelly stake at these odds is $491.75. And the best part? There is no bankroll decrease!

Were the line to move to +122 rather than +120 the additional Kelly optimal stake assuming an initial bankroll of $10,000 would be about $469.45, yielding expected full-Kelly utility 0.2475%.

[Ganchrow]

kelly stake at 2.1 is 454.54(repeating).

That's why I had added in the "overline" [OL][/OL] vBulletin tag.

So 4.5[OL]45[/OL]% when parsed yields: 4.545%.

[Data]

Thank you, gentlemen. I can see your point about maximizing EG. That point comes perfectly within Ganchrow's framework on this topic. Forgive me, but I still think there is something fishy here, I just cannot put my finger on it. Perhaps that is just me, still, help me resolve the following difficulty.

Say, same as in the example above, we took an initial bet at 2.1. Then, the price went up to 3 and we placed an additional bet. The problem is, that was not the end of it. The price then went up to 5.5, 19, 50, 233, 839, and we took additional bets at every step. What stake do you end up with following this scenario?

[Ganchrow]

Say, same as in the example above, we took an initial bet at 2.1. Then, the price went up to 3 and we placed an additional bet. The problem is, that was not the end of it. The price then went up to 5.5, 19, 50, 233, 839, and we took additional bets at every step. What stake do you end up with following this scenario?

You'd just need to repeatedly apply the above methodology.

Price	Bet Added
2.1	4.545%
2.2	3.977%
3	18.295%
5.5	20.328%
19	19.820%
50	10.449%
233	8.907%
839	4.945%

This yields a total risk amount of 91.268% in order to win 7217.2% for an average price of roughly .

Expected utility would be 92.730%.

[Ganchrow]

I just noticed you didn't explicitly include odds of 2.2 in your question, but rather went from 2.1 directly to 3. Given that price progression we'd have:

Price	Bet Added
2.1	4.545%
3	21.477%
5.5	20.549%
19	20.036%
50	10.563%
233	9.004%
839	4.999%

This yields a total risk amount of 91.173% in order to win 7296.7% for an average price of .

Expected utility would be 93.811%.

[Data]

This yields a total risk amount of 91.173%

I got the same numbers. Don't you think that this is just wrong to put almost entire bankroll on 50/50 proposition? I do. The full Kelly bet maxes at 50%. This is the reason I feel uneasy about those stakes.

[Ganchrow]

I got the same numbers. Don't you think that this is just wrong to put almost entire bankroll on 50/50 proposition? I do.

If by "wrong" you mean the bet poses unacceptable risk given your own personal preferences then that's a perfectly legitimate complaint regarding full Kelly.

The full Kelly bet maxes at 50%. This is the reason I feel uneasy about those stakes.

I hear you. Nevertheless, if the witnessed price volatility were unforeseeable then the player's actions were fully in keeping with the maximization of expected bankroll growth.

[Data]

If by "wrong" you mean the bet poses unacceptable risk given your own personal preferences then that's a perfectly legitimate complaint regarding full Kelly.

I am OK with full Kelly. The risk here is much larger than full Kelly. It is so large it makes me think that there is something wrong with the framework that calls for it.

[Ganchrow]

I am OK with full Kelly. The risk here is much larger than full Kelly. It is so large it makes me think that there is something wrong with the framework that calls for it.

Well like it or not this indeed represents full-Kelly staking.

You're certainly more than welcome to try to find a problem with the framework and prove otherwise. Within this context, however, I feel exceedingly confident in predicting that you won't be able to find any such problem insofar as none exist.


(But hey ... you never know. You have a dinner waiting for you at the steakhouse of your choice in NYC if you somehow manage to prove me wrong.)

[puff]

Say, same as in the example above, we took an initial bet at 2.1. Then, the price went up to 3 and we placed an additional bet. The problem is, that was not the end of it. The price then went up to 5.5, 19, 50, 233, 839, and we took additional bets at every step. What stake do you end up with following this scenario?

I had no idea what I meant by "completely inefficient market" either, but this looks something like what I had in mind!

I think we're all agreed that risking more than 90% of your bank on a 50/50 propositon is an awful lot! This is obviously an extreme set of circumstances, however (and of course, for the sake of this argument, we're putting aside market efficiencies).

Every odds increase (and its additional edge) should necessitate an incremental bet. Any previous position on the proposition cannot be ignored, but also shouldn't inhibit us from making the correct, objective trading decision (which is essentially a mathematics problem). I believe even if I'd stopped betting at the odds of 233, in the event of a win I'd be annoyed with myself for not taking the monstrous 839 (in the event of a loss, I think could accept the additional 5% damage to my bank).

I have every confidence in my odds-making and I understand the efficiencies of the market, and given a sound mathematical basis, I tend to trade quite aggressively, despite an intrinsic risk-aversity! This one had me stumped though, so I'm enjoying the discussion!