[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!

Overbets and Line Changes -- A Kelly Spreadsheet

[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]

Originally Posted by 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!

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.

Originally Posted by puff

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]

Originally Posted by 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.

-snip-

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

[Ganchrow]

Originally Posted by puff

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]

Originally Posted by 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.

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]

Originally Posted by 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.

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.

Originally Posted by puff

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)??

Originally Posted by 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.

Neither of these interpretations are really correct. No matter how efficient the market (even, I'd assume, were it "completely