The speed with which one's wealth grows, and its absolute level, are tied to one's life style (one's consumption of one's income). A useful simplification is to assume one's income derives largely from one's wealth. Economically, this is almost completely unreasonable, since it applies only to a vanishingly small fraction of humanity. One then needs to spend to live from this wealth. There are however minimal quality of life spends which may imply several modalities in the relation between the wealth growth process and the spend process. I assume for simplicity that wealth is sufficiently large that the income spent can be made in a way which still leaves wealth growing. Put another way, there is an assumption that the wealth process grows faster than both inflation and the daily consumption of your lifestyle. A second critical threshold is for now also ignored - as with the case where the lifestyle spend significantly impacts the wealth process, transaction costs also can incur a third hurdle to overcome. These assumptions clear away much of the thrust of the Darst book on asset allocation.
Next, an implicit starting assumption is that wealth at time $t$ may be considered as residing in one or more currency (short term fixed income) buckets. One then imagines that the mean value theorem can be applied to the act of taking financial risk above this risk free (globalist) position. That is to say, in equilibrium, the entirety of the job of strategy allocation and capital deployment can be waved away as solved for now, and modelled as a single 'bet' over an appropriate time frame, whose outcome can be a win or a loss. One then determines the ideal bet size, per unit of time, based on the mathematics of Gambler's Ruin. That is to say, that the average bet size can be no bigger than some fraction $\delta$ of wealth at point $t$ if volatility (and long term, ruin) is to be avoided.
Of course, in reality, the complete opposite applies with titubit. Bet sizing is often ignored and instead one's lifestyle generates the major driving constraint to investment returns variance tolerance. In short, our lack of funds makes us bet too big - this together with transaction costs, destroys our wealth.
The whole space of strategy allocation is shaped by two massively important risks - inflation risk and gambler's ruin. The first bites your wealth from below, when your allocation strategy overall is too focused on principal protection, where the return can be below the inflation rate, and the second bites your wealth from above, when your allocation strategy overall is focused on principal growth and your 'bet on green' at the roulette table of life stops you out and you go home early. Each fate is ugly, you either dying in dog food penury or dying young in a bloody accident. It ought to be the goal of an ideal strategy allocation approach to avoid both outcomes and instead enjoy healthy lunches - free and paid for - over as long a stretch of your life as you can.
This blog post is in general a post for everyone, but of course poor people first need to arrive somehow at a pool of investable capital (separate from their day to day living costs and the capital they have for investing in their business). I think it is a fair statement, at this early stage, to suggest that younger folks with capital can afford to be closer to gambler's ruin than older folks, since they can trade their labour, brain, body, time for paying their today costs, whereas post-retirement oldies have less flexibility and hence have a big income draw-down demand.
How close a young investor gets, of course, to gambler's ruin, is a cultural question as well as an economic one. Their appetite for risk ought to be higher, insofar as making the tilt for wealth growth over wealth protection is paid for by their greater expected lifespan. I've heard it said that private equity / startup investors like to hear from a founder that they failed once or twice in the past. Secondly, bankruptcy law is all about buying back in gamblers who have reached ruin with their firm. Our culture of long term GDP growth has some of this risk taking burnt in. There's a sense that the fable of Icarus is seen not only as a warning but also as admirable somehow. Back through human history, we have moved forwards in time by combining our prudence with our spirit of adventure.
And vigour, life, vitality, novelty, creativity, growth, these are all inter-connected concepts culturally. As opposed to self-sustainability, entropy, predictability, familiarity, maintenance. But the ideal strategy allocation algorithm must partake in all of those concepts. Unfortunately, all too often, both of these existentially definitive risks are under-emphasised on behalf of investors. But wealth generation is in the limit a lifetime activity (longer, for companies, or for aristocrats or for anyone who plans to leave an inheritance for their loved ones). It is a common fate for us collectively to understand the importance of long term planning only as we get to be old.
In so far as odds are products of a book maker, they reflect not true chances but bookie-hedged or risk-neutral odds. So right at the birth of probability theory you had a move from risk-neutral odds to risk neutral slices, in the sense of dividing up a pie. The odds, remember, reflect the betting action, not directly the likelihood of respective outcomes. If there's heavy betting in one direction, then the odds (and the corresponding probability distribution) will reflect it, regardless of any participant's own opinion on the real probabilities. Those subjective assessments of the real likelihood start, at their most general, as a set of prior subjective probability models in each interested party's head. Ongoing revelation of information may adjust that probability distribution. If the event being betted on is purely random (that is, with no strategic element, a distinction Cardano made), then one or more participants might correctly model the situation in a way which is as good as they'll want, that is immune to new information. For example, the rolling of two dice and the relative occurrence of pips summing to 10 versus the relative occurrence of pips summing to 9 is the basis of a game where an interested party may well hit upon the theoretical outcomes implied by Cardano and others, and would stick with that model.
Another way of putting this is to say that probability theory only co-incidentally cares about correspondence to reality. This extra property of a probability distribution over a sample space is not in any way essential. In other words, the fair value of these games, or the various actual likelihoods are just one probability distribution of infinitely many for the game.
Yet another way of putting this is to say that the core of the theory of probability didn't need to require the analysis of the fair odds of a game. The discoverers ought to have been familiar with bookies odds and how they may differ from likely outcome odds. Their move was in switching from hedge odds of "a to b" to hedge probabilities of $\frac{b}{a+b}$. That it did bind this up with a search for fair odds is no doubt partly due to the history of the idea of a fair price, dating back in the Christian tradition as far back as Saint Thomas Aquinas.
Imagine two players, Pascal and Fermat, playing a coin tossing game. They both arrive with equal bags of coins which represent their two wagers. They hand these wagers to the organisers, who take care of the pair of wagers. Imagine they each come with 6,000,000 USD. The organisers hand out six tokens each , made of plastic and otherwise identical looking. Then the coin is brought out. Everyone knows that the coin will be very slightly biassed, but only the organisers know precisely to what degree, or whether towards heads or tails. The game is simple. Player 1 is the heads player, player 2 tails. Player 1 starts. He tosses a coin. If it is heads, he takes one of his opponent's plastic coins and puts it in his pile. If that happened, he'd have 7 to his opponent's 6. If he's wrong, then he surrenders one of his tokens to his opponent. Then the opponent takes his turn collecting on tails and paying out on heads. The game ends when the winner gets to have all 12 tokens and the loser has 0 tokens. The winner keeps the 12,000,000 USD, a tidy 100% profit for an afternoon's work. The loser just lost 6,000,000 USD. Each player can quit the game at any point.
Meanwhile this game is televised and on the internet. There are 15 major independent betting cartels around the world taking bets on the game. In each of these geographic regions, the betting is radically different, leading to 15 sets of odds on a Pascal or a Fermat victory.
Totally independent to those 15 cartels of betting, there are a further 15 betting cartels which have an inside bet on, which pays out if you guessed who would see 6 victories first, not necessarily in a row.
Now this second have is inside the first, since you can't finish the first game unless you collected 6 points too. Pascal and Fermat don't know or care about the inner game. They're battling it out for total ownership of the tokens, at which point their game ends. The second betting cartel are guaranteed to finish in at most 11 tosses every time, and possibly as few as 6 tosses.
Just by coincidence, Fermat, player 1, gets 4 heads in a row, to bring him to 10 points of total ownership of all the tokens. He only needs 2 more heads to win. At this point Pascal decides to quit the game. To betters in cartel 1 it looks like Pascal and Fermat are playing gambler's ruin, to cartel 1 it looks like they're playing 'first to get six wins', which is the game the real Pascal and Fermat analyse in their famous letters.
Soon after, Pascal's religious conversion wipes out his gambling dalliance, and Fermat, only partly engaged with this problem, withdraws his interest. Both men metaphorically enacting gambler's ruin and the problem of points.
There's a famous, famously lousy betting strategy which is referred to as the Martingale. The simplest characterisation is as follows. You're faced with a betting game based on tossing a coin. You place a bet of any size you care on the outcome of the coin toss. If you're right, you get your original stake back, doubled. If you lose, you lose your stake. You'll see it referred to as 'doubling down' also in the context of trading - this is a looser variant where you're raising your bet size as the market goes continually against you.
With the Martingale algorithm assisting you with placing your best size, then if you are infinitely wealthy and are prepared to toss the coin infinitely often, you can construct a winning strategy. Interestingly, the strategy is nothing whatsoever to do with actually predicting the outcome of the coin toss. It is all about how big a bet you place on any of the sequence of coin tossing games you participate in. You bet an initial stake on the first toss. if you win, you're richer by the initial stake. if you lose, you play the game again, doubling the best size. If you win, you get back your second stake, plus the same again. That extra second iteration stake fully makes up for the loss of the initial stake, and you're left with an initial stake's worth of profit. Repeat and become rich beyond your wildest dreams.
In practice, the gambling institution (or your own finite wealth) will impose bet size limits, which dramatically increases the chances of gambler's ruin in a short losing streak.
Lying under it is the gambler's fallacy, the belief that in an evens game, wins and losses even out. That is, people dramatically underestimate the likelihood of a long string of losses. As you burn exponentially though your cash pile, each independent evens bet doesn't care where it is in the history of your losses or wins. Each future toss is either just a win or a loss.
In the context of trading, the poorest reasonable assumption to be made about you is you are no better than a random process when making a call on a binary market outcome. However this is not always going to be true, since markets do exhibit runs and reversals.
The doubling down strategy might have its origin in that part of Kahneman's prospect theory which states that when you've made a big loss, you're more likely to roll the dice to break even, rather than the rational path, which is to reduce your risk sizing and wait for the market to pick up again. Your bets get exponentially bigger in order to 'pay' for the thrill of reversing luck in the very next bet. But in practice with real trading, you might be happy with recovery of your losses over a number of bets.
Surely there are circumstances when a trading strategy somewhat like the Martingale one makes sense? First of all, assume you're putting on a trade with no better than evens odds, as per the original Martingale theory. Now, suppose you had 1,000,000 currency units to allocate to this bet. Well, you could just put the lot on this bet. But you don't know the future, so you could foresee a couple of trading periods where the investment moves against you. Why not put on 100 units initially. If the market moves against you, continue to buy into the position in 100 unit chunks - I assume that nothing changes in your trading thesis. As long as your trading thesis looks OK, you're getting to buy in at a price even better than at the starting period, and you were happy to buy in then. You wouldn't need to increase the bet exponentially since you don't need to pay for the thrill of a dramatic single jump to profit in one trading period. You'd be able to last for a much longer bad run, each time buying in at a lower average price. Again, assuming nothing changed in your trading thesis, you could at worst imagine 10,000 trading periods all going against you, before you exhausted your overall trade limit of 1,000,000 units.
The above scenario concentrated on a use case which exhibited extreme downside behaviour against you and had a distinct Martingale-like feel to it, but certainly doesn't strike me as a prospect theory like bias.
So if an alien came down and examined the set of trades on a market and saw Martingale-like trading patterns, they couldn't really say this was because of a prospect theory bias reason or because of a more healthy conservative opening strategy.
If this is the case, then any time you come across a trading book with dismisses Martingale-like trade sizing algorithms out of hand as wrong, or as evidence of a prospect theory kind of bias, then you know they're not telling you the full picture. On the flip side anyone suggesting a Martingale like sizing strategy is probably giving you bad advice.
Two more points to make on this. First, the theoretical Martingale is marked theoretical due to the house limits or the wealth of the player. Another angle on this is to say that the real unspoken criterion here is the minimal bet size (and the frequency of bets). If there was a market where the minimal bet size was a tiny fraction of a cent, and you could trade it hundreds of thousands of times a second, then you're moving a lot closer to getting it to work as a winning trade sizing strategy.
Second, this dovetails with a piece of mathematics called the gambler's ruin, which is often seen in probability textbooks showing how long you have got before any given fixed outcome gamble exhausts your initial wealth.
A Martingale is also the name given to the strap which attaches around a horse's neck and to its body, keeping its head in a narrow, froward looking position. The analogy was in the classical bet sizing strategy which calculated the expected profit or loss at time $t$ to be the current size of the winnings pot at time $t, W_t$. Apparently there was a French village Martique which had famously miserly inhabitants.
Paul Levy in the 1930s took the word and applied it to one of the two basic properties of randomly generated numbers which would result in them being normally distributed. The other was finite variance.
