The Paradox of Simulations - Sports

This is a newer type of paradox based on an old idea. One part is about whether we exist in a computer simulated environment. You may recognize this concept from the film Matrix. However, previously there were science fiction novels along these lines. This type of paradox can easily be searched online, and we will not take it up at present. Every resolution establishes the simulated universe concept is erroneous or even ridiculous. In the appendix, we include some of the simulated universe paradoxes.

Our intent is to discuss an alternative paradox of simulations for our own simulations of realistic situations. As discussed earlier, computer simulations, such as for the Monte Hall Paradox (#16) can be used to resolve it. They are also used to study other contingency possibilities. Even war games can be computer simulated. Currently, computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics (computational physics), astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering.

We take up another human activity, sports, where we give detailed results giving almost no precise details (perhaps paradox in itself). In particular, we consider soccer league standings for the English Premier League (EPL) [1]. This simulation involved assigning skill levels of each team, from least skilled to most skilled. Then, using these skills levels, the relative probability of one particular team winning against another was computed. This was done for each game of a typical 38 game season for every team, the results totalled to give the league standings at the end of the season. This was the consequence of one run. Think of this as a weighted coin flipping experiment for many unfair coins and multiple flips. In coin flipping, anything can happen, so multiple runs are needed to get grand averages. For example, you must roll a pair of dice many times to see what the numbers come out to be through averages. 

The next step was to simulate many seasons and average the results. That is, we averaged the number of wins for the best team (by assigned skill), the second-best team, and so on for all the teams. This gives our simulated but average league standing for, say, a million seasons. This would be out specimen for simulated league standings, and our task was to compare the simulated results with actual results. Little do most sports observers don’t notice is that league standings (not teams) are numerically almost the same year after year. So, we compared our simulated standings with actual standings with the statistical chi-squared test.  The results were that the comparisons exceeded the highest levels of statistical tests. This means an “outstandingly good agreement.”

At first, I thought I had discovered something fundamental about league teams, namely that skill levels were essentially equally spaced in a typical league. But then, there occurred an alternative viewpoint – the paradox. So far, the paradox has been unmentioned. What is it?  It concerns the skill levels assigned to the teams. For the simulations, we assigned a linear order with equal skill divisions for each team. This is virtually untrue. The better teams should be and probably are bunched toward the top of the scale, while the poorer teams are bunched at the bottom of the scale. Yet, with skill levels equally spaced from 0.45 to 0.55, all very close together, we derived league standings with outstanding accuracy. This is the paradox of simulations. Using rather inaccurate, even reasonably guessed input, it was possible to predict extremely accurate final league standings. The story is not quite over, in that, we did a similar simulation for American baseball, with very similar agreements to actual final league standings.

So, why should this be a paradox? Why can this not be evidence of causality, that team skills are truly equally spaced, or nearly so? It is because equal spacing of skill levels between teams is so utterly unreasonable as to be discounted from the onset of the experiment[2]. Our conclusion is to ask the question as to how many simulations are made with entirely wrong input can give virtually correct or certainly reasonable results? Even further, making predictions using incorrect input can lead to thinking casually where there is none – nada. Bottom line? Simulations can come out as expected or hope even with incorrect data input.

For completeness, we note flaws in the coin-flipping nature of our model. In sports, circumstances change weekly, as does the health of key players and even the team morale. In season, the acquisition of new players is a significant factor. In reality, we should have used a variation of Prospect Theory[3] to build the model. However, prospect models do have various constants that need defining, and there is not much known about how to do this.

Applications. This may be an issue with subjects like war games such as geopolitical conflicts, with the desired outcome a consequence of the assumed input, especially when wars of the past with known outcomes and assumed input are available. But is the input correct and complete? The same can be said about Climate. If input used is from a century ago and it accurately predicts climate today, does this imply causality sufficiently to be applicable into the future? Similarly, the prediction problems for new pandemics are very fragile, with few reliable results available. 

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Appendix. For completeness, we include a few of the paradoxes about a simulated universe. This paradox challenges our fundamental assumptions about reality, existence, and consciousness, leading to deep philosophical and scientific questions about the nature of the universe, which suggests that our perceived universe may be an artificial construct rather than a fundamental reality. Several paradoxical elements emerge from this idea:

1. The Reality-Illusion Paradox. If we are in a simulation, then everything we perceive as real is merely an illusion, yet we have no way to determine the difference between the simulated and the real. Example: If all sensory experiences are simulated, then distinguishing between a "true" reality and the simulation is impossible because our perception is entirely defined by the simulation itself.

2. The Observer Paradox. If we discover we are in a simulation, does that change the nature of the simulation itself? Example: If simulated beings realize their existence is artificial, would the simulation’s creators alter or reset the simulation, or would the awareness itself be part of the designed experiment?

3. The Infinite Regression Paradox. If our universe is a simulation, then the entities that created it may also exist within a simulated reality, leading to an infinite chain of simulated worlds. Example: If an advanced civilization simulates our reality, who simulated them? And who simulated the ones before them? This creates an infinite nesting of simulated realities.

4. The Self-Destruction Paradox. If civilizations eventually develop the ability to simulate conscious beings, then there should be countless simulations, making it overwhelmingly likely that we are in one. But if civilizations also tend to destroy themselves before reaching such capability, then the probability of us being in a simulation drastically decreases. This tension between technological progress and self-destruction creates an unresolved paradox about whether advanced civilizations ever reach a stage where large-scale simulations are common.

5. The Free Will Paradox. If we exist in a simulation, do we actually have free will, or are our decisions predetermined by the simulation’s rules and algorithms? Example: If our thoughts and choices are influenced by programmed parameters, then what we perceive as "free will" may just be an illusion created within the simulation.

6. The Escape Paradox. If we could prove we are in a simulation, what then? Could we escape it, and if so, what would that mean? Example: If reality is simulated and we "exit" the simulation, do we enter a higher level of reality, or do we cease to exist entirely?

7. The Magnitude Paradox. If we are in a simulated universe, the hardware required to simulate it must exceed the total mass and energy of the universe. This is reasonable to see because in a simulated universe every single particle we can perceive must be simulated. This required code for each particle and this code will require more mass than the particle, not to mention the simulated motion, charge, spin, gravity, and whatever else a particle needs. As well, the programmer would be the equivalent of a god.

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[1] Allen, G. Donald, Simulations for the EPL Using Competitive Balance Models, Journal of Sports and Physical Education, e-ISSN: 2347-6737, p-ISSN: 2347-6745, Volume 4, Issue 2, (Mar. – Apr. 2017), PP 33-43. http://www.iosrjournals.org/iosr-jspe/papers/Vol-4Issue2/G04023343.pdf

[2] More generally, this could also be called the Paradox of False Causality.

[3] Kahneman, Daniel; Tversky, Amos (1979). "Prospect Theory: An Analysis of Decision under Risk" (PDF). Econometrica. 47 (2): 263–291. See also Wikipedia at https://en.wikipedia.org/wiki/Prospect_theory

©2025 G Donald Allen