Research Note: PoP

Introduction

This post is my attempt to formalise and prove an argument relating the the so-called ‘Paradox of Predictability (PoP hereafter). An overview of the PoP can be found in (Rummens and Cuypers 2009).

A specific version of the paradox can be seen as there are events that are unpredictable. Suppose that you are playing a game against an opponent. You must make a binary decision at time \(t_{1}\) and your opponent makes a binary decision at \(t_{2} > t_{1}\). You goal is to make your binary decision the same as your opponents. The Paradox of Predictability (though I’m not sure it is a paradox in the traditional sense) holds that there is a strategy your opponent can play that makes the game unwinnable for you. The strategy procedes as follows.

1). Observe the binary decision, \(x\), made by the player who decides at time \(t_1\). 2). Set your decision \(x_{2}\) at time \(t_{2}\) to be \(\neg x\).

Given that the first player’s aim is to make \(x_1\) equal \(x_2\) and \(x_2 = \neg x_1\), it is impossible for the first player to ever win as this would require \(x_1 = \neq x_1\).

The only assumptions here that I think could be quarrelled with is that there is enough time between \(t_{1}\) and \(t_{2}\) to observe \(x_1\), negate its value, and set it to be the value of your predictor, and that events can happen instantaneously. I don’t think these assumptions are limiting but here is where picking a coherent mathemtical structure for events and time is important.

In my view, what the PoP really states is that there cannot exist a Universal Embedded Predictor. Unpacking this statement, a predictor is a function that, given a time and an event, will make a guess as to whether the event occurs at that time. Universality of the predictor is with respect to some determined timeline of events. The predictor is universal if its predictions are consistent with the timeline i.e. whatever it says happens at a given time does indeed happen at that time. The predictor is embedded with respect to a timeline if the predictions that the predictor makes are events within the timeline.

My interest in this problem stems from the structure of this argument and the game itself. This result has the natural duality that many (seemingly self-referential) paradoxical results have in the foundations of computer science and logic i.e. the liars paradox, halting problem, godel’s incompleteness theorem etc. This is due the fact that we can derive some limitation on the ability of the system i.e. the lack of a Universal Embedded Predictor but we can also find some pathological object that leads to this i.e. the game we describe above. This reminds me of how in (Yanofsky 2003) many of the aformentioned paradoxes are subsumed into Lawvere’s Fixed Point theorem from category theory. The goal of this note from hereon will be to formalise clumsily what needs to be formalised and investigate whether there are mathematical objects that enable me to view this paradox through the lens of Lawvere’s FPT.

Clumsy Mess

We begin with lazy definitions of a Universal Embedded Predictor.

I work with discrete time but am confident that an appropriate order structure could be found to fix the problems

We begin with some abstract collection of Events \(E\) supra to all timelines. I have no confidence I will be able to provide a coherent notion of this so I will leave it abstract. A timeline could be a total function \(f: \mathbb{N} \rightarrow \mathcal{P}\) which simply enumerates what events occur at each time point (here is where event cracks will start to appear. What does it mean for no events to occur? Do we need some kind of $σ$-algebra like structure for the notion that the opposite of events can happen?).

For a given timeline we can ask the inverse question of what times an event happens at \(g: E \rightarrow \mathcal{P}(\mathbb{N})\) which returns all the times at which an event occurs in the timeline (and returns the empty list if it does not occur). From \(f\) we can define a function \(\textrm{OccursAt} : E \times \mathbb{N} \rightarrow \mathbf{2}\) defined as

\begin{align*} \textrm{OccursAt}(w, i) = 1, & \text{if } w \in f(i) \\\
0, & \text{otherwise} \begin{cases} \end{cases} \end{align*}

The type of \(\textrm{OccursAt}\) appears to make a statement about what events occur at different times. This is precisely what we wanted predictors to do and therefore all functions of that type will be called predictors. A universal predictor was a predictor that was “correct” about the timeline. Given that \(\textrm{OccursAt}\) is generated from the timeline we presume that this will itself be a universal predictor. Giving a precise definition of this, a predictor \(p\) for a timeline \(f\) is universal if for all \(e \in Events\) and for all \(i \in \mathbb{N}\) \(p(e, i) = 1 \Leftrightarrow e \in f(i)\). From this definition it is now clear that \(\textrm{OccursAt}\) is a universal predictor for any timeline.

Now we come to the sticky part, embeddedness. My intuition for embeddedness procedes as follows: A predictor is embedded within a timeline if an evaluation of the predictor for a given set of arguments is observable in the timeline along with its result. It is at this point that it becomes clear that we need a notion of events. What is needed now is a criterion for embedding predictions into an event.

The statement that there is no universal embedded predictor becomes more interesting perhaps. It is something more like, for every predictor there exists a timeline in which it is embedded such that it is not universal.

Stupid speculation

This work could then move into establishing what the necessary conditions are for a timeline function to be able to beat an assocated predictor i.e. you must be able to observe the outcomes of the prediction and it must be physically possible to invalidate the prediction i.e. the invalidation procedure must be a realisable operation. I see that it is impossible for this to be something we can prove about our own timeline. This perhaps creates a hole in the notion of timeline. Perhaps it would be better to instead postulate an update function and derive a timeline function from it which may generalise better to arbitrary algebraic structures. There are also some interesting things about intentionality and extensionality here. We often have this notion that an outside observer will “know” what will happen if they have access to the global state but clearly timeline functions can internally reference itself. This means that it would require the timeline function to be fully evaluate for anything to actually be able to state all the facts about the world. Your actions are determined in-so-far as they could be defined by a timeline function but only something capable of knowing all outputs of a function that calls other parts of the function in some prespecified order without evaluating it in some order will be able to tell you all facts about the world without being in some way constrained by causality.

Bibliography

Rummens, Stefan, and Stefaan E. Cuypers. 2009. “Determinism and the Paradox of Predictability.” Erkenntnis 72 (2). Springer Science and Business Media LLC:233–49. https://doi.org/10.1007/s10670-009-9199-1.

Yanofsky, Noson S. 2003. “A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points.” Bulletin of Symbolic Logic 9 (3). Cambridge University Press:362–86.