Funny and crazy mathematical beings

This post is just a small collection of mathematical concepts that I thought were crazy when I first heard about them. These are some level of abstraction that are sometimes purely aesthetic per se. Enjoy! I might basically update regularly this post along the way I discover some other crazy mathematical beings.

Metrics spaces

Imagine you’re on a trip. You’ve got a map, you’re trying to figure out how far you are from your destination, or perhaps you’re looking for the shortest path to it. Well, in a way, that’s exactly what a metric space does - it generalizes the concept of ‘distance’ or ‘closeness’ between points in a set. It’s not just for our usual three-dimensional world, but for more abstract spaces as well.

Thing is, we’ve always been used to taking as a reference that a distance between two points is basically a segment between those two points. It’s just super intuitive, you take Pythagore theorem and done. But in fact we could very well imagine another definition of distance. You could imagine, for example, that the distance could be a curve instead of a straight line. I mean why not? Well, that’s where the usefulness of defining what we mean by distance comes in, i.e. establishing a metric. A metric space, fundamentally, is a set together with this ‘distance function’ (also called a metric then) that, much like our trusty road map, gives us the ‘distance’ between any two points in the set. But just realise that this is something that is quite funny in itself because a set is an ultra-vast mathematical concept and can contain almost anything and everything in terms of properties or mathematical beings contained within that set. So like defining the metric is basically the key starting point, otherwise you can’t do that much… The notion of distance is like the very basic and probably most important thing in maths, I mean, look at the definition of continuity. You can’t handle that much mathematical concepts if you don’y understand properly why distance is such a usefull thing to define I guess. Without distance, the beautifully intricate world of calculus would simply collapse.

Functional operators

Simply put, it’s a kind of operator that takes one or more functions as input and produces a new function as output. It’s like a function for functions! It’s a step up in the level of abstraction from our usual functions that take numbers or other values and return numbers or other values. You might thus recognize easily that derivatives and integralss are functional operators. Ones of the most important results here are several fixed point theorems. The best-known fixed point theorem is probably the Brouwer fixed point theorem, which applies to functions from a compact convex set to itself. Without fixed point theorem, game theory will basically collapse since almost every theorems Nash showed involves a fixed point theorem at some point (lol).

Fuzzy topological space

To understand Fuzzy Topological Spaces, we need to start with traditional topology. In classic topology, we look at mathematical ‘spaces’ and their properties - think of how things connect, cluster, and interact. We’re interested in concepts like continuity, compactness, and convergence, among others. In these spaces, something either definitely belongs to a set or definitely does not - it’s binary, a world of crisp, clear distinctions. Thing is, we often deal with ambiguity, vagueness, and uncertainty in life right? Fuzzy logic allows us to mathematically handle ‘degrees of truth’, that is, we can say that something belongs to a set to some degree, between 0 and 1. A fuzzy topological space is just like a regular topological space, but instead of classic sets, we deal with fuzzy sets, where membership is a matter of degree. This new kind of space is defined by a fuzzy topology, a collection of fuzzy sets satisfying properties similar to those in classic topology, but nuanced by the fuzziness.

Fourier Series

Imagine a vast, tranquil lake. Now toss a pebble into it. What happens? Ripples, right? Waves start spreading out from the point where the pebble hit the water. These ripples are basically a beautiful metaphor for understanding the magic of Fourier Series.

Fourier discovered that you can break down any periodic function (which repeats itself over time) into a sum of simple sine and cosine waves, each ‘playing’ at a different frequency and amplitude. It’s as if your complex function is a grand musical symphony, and the Fourier Series reveals each individual instrument playing its part!

Formally, for a function $f$ with period $2\pi$, $f(x) = a_0 + \sum_{n=1}^{\infty}(a_n\cos(nx) + b_n\sin(nx))$

$a_0$ is the constant term in the Fourier series. It is equal to the average value of the function over one period. $a_n$ is the nth coefficient of the cosine terms in the series. It measures how much of the cosine wave with frequency $n$ is present in the function $f(x)$. If $a_n$ is large, then the cosine wave with frequency $n$ is a significant part of $f(x)$. Likewise for $b_n$ and sine.

Ok let’s better undertsand why I think Fourier series is an incredibly powerful concept. As you can see above, the function $f$ has been decomposed in an infinite sum of cosin and sin. You can literally write any periodic function into an infinite sum of cosine and sine. It’s like having a mathematical X-ray machine that lets us see inside a function and understand its intricate structure in terms of simpler, harmonic components.

Decision trees with continuous action spaces in extensive form game

In a traditional extensive form game, we generally deal with discrete strategy spaces (and usually, to make things even esaier, only 2 strategies are considered like, idk, defect or cooperate). At every decision node, a player has a finite set of actions they can choose from. Thing is, we usually face a lot more strategies so that the set of strategy can eventually be considered as infinite. How do take thta into account? Well, in an extensive form game, a continuous action space (contrary to the discrte case we mentionned above) is typically drawn using an arc connecting two branches representing the upper and lower bounds of the action space.

The use of an arc to represent a continuous strategy space in an extensive form game is a powerful way to capture a whole spectrum of strategic possibilities that a player can choose from. It’s a marvelous way to represent and analyze complex strategic interactions where the players have an infinite number of choices.

This difference is incredibly powerful as it allows us to model and understand a broader range of real-world strategic interactions. Whether it’s negotiations, pricing strategies, or any scenario where decisions aren’t merely discrete choices, the use of continuous strategies in extensive form games truly broadens the horizons of strategic decision making in GT.

Stochastic differential equations

One day I was wandering around my university library, glancing casually at some of the book titles on the shelves. Suddenly, one title struck me. ‘Stochastic differential equation’. I stopped immediately, shocked by the title, which seemed to me to be an oxymoron. I had always learned that differential equations were, by definition, something purely deterministic: given the initial state of the system (the initial conditions), the differential equation provides a precise and purely deterministic formula for how the system evolves over time, i.e the future state of a system is entirely determined by its current state and the laws of motion.

Thus, how on earth do you have the audacity to add the qualifier ‘stochastic’ to a differential equation?? Well, thing is, uncertainty and variability and like almost everywhere in a lot of world models. An SDE is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process, capturing the eventual uncertainty or variability. This means the future state of the process isn’t determined solely by the current state, but also involves a random element. SDEs are instrumental in many fields, especially in physics, economics, and engineering, where systems are subjected to random influences. In finance, for instance, they’re used to model assets or instrument prices because they capture both the general trend (like interest rates) and the random market fluctuations. One canonical example of SDEs is the Brownian motion.

example :

This graph shows five sample paths $X_t\left(\omega_1\right), X_t\left(\omega_2\right), X_t\left(\omega_3\right)$ and $X_t\left(\omega_4\right)$ of a geometric Brownian motion $X_t(\omega)$, i.e of the solution of a (1-dimensional) stochastic differential equation of the form:

$\frac{d X_t}{d t}=\left(r+\alpha \cdot W_t\right) X_t \quad t \geq 0 ; X_0=x$

where $x, r$ and $\alpha$ are constants and $W_t=W_t(\omega)$ is white noise. This process is often used to model ‘exponential growth under uncertainty’ (example coming from the book ‘Stochastic Differential Equations: An Introduction with Applications’, Bernt Øksendal, 2003)

Multi-objective Optimization

OK, so we’re used to solving optimisation problems that involve maximising or minimising a single function. Like simply finding the max or the min. Whether it’s a multivariate, static, dynamic, discrete or continuous optimisation problem. But in my bachelors and even first year of master in economics, we’ve hardly talk about, well, how do we do if we basically want to optimize several functions? Like finding the max or the min of several functions almost simultaneously. Each of these objective functions to optimize typically represents a different goal that needs to be satisfied.

As you may probably intuitively understand, in such scenarios, it’s usually very hard to find a solution that optimally satisfies all objectives simultaneously. Hence, rather than finding a single “optimal” solution, multi-objective optimization aims to find a set of “Pareto optimal” solutions. As you may intuitively understand, a solution is Pareto optimal if there’s no other solution that improves one objective without worsening at least one other objective.

General form: Let’s assume that we have $n$ decision variables, $k$ objective functions, $j$ inequality constraints, and $m$ equality constraints.

Minimize/Maximize: $f_i(x)$, for all $i$ in ${1, 2, …, K}$

Subject to:

$g_j(x) <= 0$, for all $j$ in ${1, 2, …, J}$

$h_m(x) = 0$, for all $m$ in ${1, 2, …, M}$

With bounds:

$L_i <= x_i <= U_i$, for all $i$ in ${1, 2, …, n}$

Just as said before, the solution to this optimization problem may not be a single point, but a set of Pareto optimal points.

Multi-objective decision making is a facinating field arising from multi-objective opti problems. For multi-objective games, each player doesn’t have just one single objective (or utility function); instead, they have multiple objectives (multiple utility functions) they want to optimize. As such, the payoff for each player for each strategy is not a single number, but a vector, where each element represents the payoff for one objective! Here’s an example with a specific 2x2 game with vector payoffs:

As you might guess, finding the Nash equilibrium in this kind of game is more complicated. How do you know if (3,4) is better than (1,6)? Determining these kinds of equilibria typically requires more advanced techniques. But for the record, typically, we might say that a payoff vector is better if it’s better in at least one objective and no worse in any other objective.

Randomized Social Choice

Randomized Social Choice comes into play when deterministic social choice functions just can’t cut the mustard, or when there’s simply no one-size-fits-all solution. Rather than forcing a single choice, we allow for a distribution of choices – like tossing a weighted dice that can land us in different outcomes, each with a probability reflecting the players’ preferences. Instead of each person voting for one alternative, they vote for a probability distribution over the alternatives.

Vector fields

A vector field in its simplest form is an assignment of a vector to every point in a space. Super usefull in physics as you may understand (e.g through visualizing flows and forces). Formaly, a vector field on a subset $U$ of $\mathbb{R}^n$ is a function that assigns to each point in $U$ a vector in $\mathbb{R}^n$. In mathematical notation, this can be expressed as:

$F: U \rightarrow \mathbb{R}^n$

Vector fields can have interesting topological features, such as zeros (where the vector field is zero), sources and sinks (where vectors are diverging or converging, respectively), and vortexes and saddles (where the behavior of the field is more complex). Vector operators, also known as differential operators, are tools used to analyze vector fields and scalar fields. Two of the most commonly used vector operators are the divergence (a scalar field that provides a measure of the vector field’s source or sink at a given point) and the curl (a vector field that describes the infinitesimal rotation of the field in three dimensions. In other words, it measures how much and in what direction the field ‘rotates’).

And here’s the truly mind-bending part: you can have vector fields not just in two or three dimensions, but in any number of dimensions. For instance, in the field of differential geometry, vector fields on manifolds (which can be thought of as multi-dimensional surfaces) are a central object of study.

I thank chatGPT for helping me to write (hopefully) this blog post more clearly.