Imagine two racetrack bettors. The first is a cautious accountant, terrified of losses and willing to risk only a small sum. The second is a daring entrepreneur, used to half of his ventures failing while the other half makes him a fortune. Both are looking at the same board with the same horses and the same odds. The question is: will they bet on the same horses?
Intuition suggests: of course not. The cautious accountant will pick the favorite with minimal risk. The entrepreneur will bet on a long shot with tempting odds. Different personalities, different bets.
But the math says otherwise. And this is one of the most elegant discoveries in decision theory, which I want to explain to you today.
Why Think of Betting as Economics at All?
Before we dive into the details, allow me to take a step back. Betting isn't just about racetracks and casinos. It's a model for any decision made under uncertainty, where we have a limited resource (money, time, attention), several competing options, and a certain probability that each will succeed.
An investor allocating capital among stocks is a bettor. A farmer deciding how much land to devote to wheat versus corn is one, too. Even a manager distributing a budget among projects is solving the exact same mathematical problem.
And at the heart of this problem is always one key question: what exactly to spend the resource on? Not how much – that's the second question. First, «what».
In mathematics, this set of “whats” is called the portfolio's support. The support is simply the list of options to which you allocate a non-zero portion of your budget. You ignore everything else.
Where Does 'Taste for Risk' Come From?
In economic theory, a person's attitude toward risk is described by what's called a utility function. It sounds dry, but the idea is simple and even poetic.
Imagine you're offered a choice: receive €100 for sure, or flip a coin where heads gives you €200 and tails gives you zero. Mathematically, both options are equivalent: the expected payoff is the same. But most people choose the guaranteed €100. Why? Because the pain of losing €100 feels more intense than the joy of winning that same €100. This is risk aversion.
Economists describe this with a curve: for a cautious person, the utility curve is concave – each additional euro brings slightly less joy than the one before it. For someone who loves a gamble, the curve has a different shape. Each person has their own curve, their own “utility function.”
It would seem that since the curves are different, the optimal decisions should be different too. Those who are risk-averse bet on what's reliable. Those who are risk-takers bet on what's profitable. Makes sense, right?
Logical. But, as it turns out, only partially.
The Main Surprise: 'What' Doesn't Depend on 'Who'
Here is what the research I want to tell you about proves. It's a mathematical result obtained within the theory of optimal allocation of bets on simultaneous, independent events.
“Independent events” are the key words here. Imagine you're simultaneously placing bets on several different races in different cities, which are in no way connected. The result of the first race has no effect on the second.
Well, here it is: the set of outcomes that are rational to bet on is the same for everyone – regardless of how cautious or daring they are.
This sounds almost provocative. How? Do the timid accountant and the bold entrepreneur really have to bet on the same horses? Yes. Exactly. On the same ones. The size of their bets will differ – but the choice of “who to include on the list” will be identical.
The only thing that determines whether an outcome makes it onto this list is one simple ratio.
The Magic Ratio: Probability to Price
This ratio looks like this: the real probability of an event divided by its market price (or, in bookmaker's terms, the inverse of the odds).
Let's call this the “edge” – the degree to which reality diverges from what the market says. If the market underestimates a horse (the odds on it are high, but its real chances of winning are also high), then the probability-to-price ratio will be large. If the market overestimates a favorite (everyone is betting on it, the odds are low, but it has no real advantages), the ratio is small.
Mathematically, this is written as pℓi/πℓi, where pℓi is the objective probability of outcome i in event ℓ, and πℓi is its market price.
The algorithm turns out to be surprisingly simple:
- For each event, sort all possible outcomes in descending order of their “probability-to-price” ratio.
- Include in your list (the support) the outcomes from the top of the list – just enough to achieve a mathematical balance.
- Ignore everything else.
And this list will be the same – for the accountant, for the entrepreneur, for the retiree, and for the student. The shape of the utility curve only affects how much to bet on each chosen outcome, not what to choose.
Why Does This Work? The Intuition Behind the Formulas
Let me try to explain this without math – through an analogy.
Imagine you're going to a market with a limited budget. In front of you are various goods with different prices and different values to you. Your task is to buy the most useful stuff you can.
How will you choose? Obviously, you first take what offers the greatest value per unit of price. Then the next most advantageous item. And so on, until your budget runs out or the next item is no longer worth its money.
Now for the important part: this “value-for-price” principle doesn't depend on your personality. Whether you love luxury or prefer modesty, the ranking principle is the same. The only thing that changes is the cutoff point – which item you reach before you stop.
The exact same thing happens with betting. The pℓi/πℓi ratio is the value of a bet per unit of its “market cost.” A rational player always starts with the most advantageous options. The cutoff point differs from person to person – but the sorting order is the same.
Independent Events and the Separation Effect
Now let's add another beautiful element. What happens when you bet on several independent events at once?
It would seem the task becomes incredibly complex. All the events get entangled in your budget: money spent on the first race is unavailable for the second. A huge system of interdependencies arises. Mathematicians call such problems the “curse of dimensionality” – with each new event, the complexity grows exponentially.
But the research shows that this “curse” doesn't exist when the problem is framed correctly. Because what works is something we can call the principle of sequential separation.
It goes like this: the optimal list of bets for the entire portfolio is simply the union of the optimal lists for each event individually.
That is: first, you solve the problem for the first race – find its “good horses.” Then, separately, for the second. Then for the third. And the final list is simply the sum of these individual lists. No complex cross-calculations needed.
It's like how a good chef designs a menu: first, they choose the best appetizers, then the best main courses, then the best desserts. The final menu is simply the sum of these choices. They don't try to optimize the entire meal as a single mathematical object.
A Bit of History: Where It All Comes From
This result doesn't come out of nowhere. Researchers Smoczynski and Miles once described a similar pattern for so-called risk-free parimutuel markets – a specific form of betting, popular at racetracks, where the odds are shaped by the betting market itself, not by a bookmaker.
In that setup, they had already shown that the choice of “what to bet on” does not depend on the utility function. But that was a special case – a single event, a specific market structure.
The new result takes a step forward: the same principle applies to any number of simultaneous, independent events under much more general conditions. The key to extending the theorem was the concept of a “continuation factor” – a special value that describes how uncertainty about inactive outcomes affects the overall calculation. Once this factor is correctly identified, the entire structure of the proof can be reproduced in the more general case.
It's a good example of how mathematical discoveries work: not through revolutions, but through precise expansions. Like a cartographer who refines a map, adding detail to previously schematic areas.
What Does This Mean in Practice?
Let me translate all of this into practical terms – because abstractions are fine, but life is concrete.
For portfolio managers. If you're allocating investments among several independent assets or projects, the choice of “which ones to participate in” is determined by a single principle: the ratio of expected return to the market price of risk. This is independent of your risk appetite. Your risk profile only affects the size of the positions.
For managers and decision-makers. When you decide which of several independent projects to include in a plan, you are facing a task of ranking by the “expected return to cost” ratio. Your personal risk preferences only change the aggressiveness of the investments, not the list of candidates.
For anyone just thinking about money. The next time you're choosing between several investment options – be it a deposit, a stock, or a small business – ask yourself one question: what is the real probability of success compared to what the market is offering? That ratio is more important than your feelings of anxiety or excitement.
The Space Between the Formula and the Person
I want to end not with math, but with an observation.
We tend to think our decisions are deeply individual – that our character, our fear of loss, and our appetite for risk make us unique market participants. And in a way, that's true. Our emotions, our biases, our history – all of this affects how we act.
But mathematics sometimes uncovers surprisingly universal structures beneath all this diversity. It turns out that the rational choice of “what to pay attention to” follows a single principle – regardless of who is making the choice.
This doesn't make us all the same. It makes us human – people who, for all our different temperaments and histories, ultimately respond to the same signals from reality. The probability-to-price ratio isn't just a formula. It's the language that rationality uses to speak with uncertainty.
And this language, as it turns out, is understood by everyone – no matter how cautious or bold you consider yourself to be.
