## Time Paradox

Another good idea with a weak exposition. But these are just notes, not a real essay, so I'm okay. I think Feynman wrote about this.February 6, 2002

There used to be a show on TV that was based on an interesting premise. The main character for some reason would receive the next day’s edition of the *New York Times* at his door every morning. He would scan it for the news of any disasters which he then would avert.

I only saw one episode, so I don’t remember what the name of the show was. I do remember that this guy had a friend who was salivating over the chance to get his hands on one of the papers. The friend’s goal was rather more mundane than saving the world: it was making millions in the stock market.

Thinking about this show got me thinking about time travel in general and time paradoxes in particular. One paradox, of course, is the fact that the paper can report disasters that the main character is supposed to have prevented by the time it goes to print (we assume, by the way, that the *New York Times* always gets the facts straight and won’t report something that didn’t happen, or fail to report something that did). I do have a theory about how this paradox works out but I’ll leave it for another time. Right now, I want to focus on the Friend (let’s capitalize him for ease of reference) and his financial aspirations.

If the Friend manages to get the paper and use it, then his actions will generate a process consisting of a series of events that revolve in a loop. The steps in the series are roughly:

- the
*Times*gets printed and sent to previous day; - Friend reads it;
- Friend speculates on the stock market and possibly changes prices from what they would have been otherwise;
- the resulting set of prices is given to the
*Times*; - return to step (1), repeat.

Every time the loop runs through steps (1) - (6), our process completes one iteration.

An iterative process can develop in many ways. The simplest case is where the Friend’s machinations are so small-scale that they don’t affect stock prices. If the prices stay the same, the *Times* will report exactly the same thing from one iteration to the next and thus all iterations will contain identical events. History will remain static, with only one possible future. Nice and easy.

Now assume that the Friend’s actions *do* change prices from one iteration to the next. The result is that the iterations, at least in the beginning of the process, will not contain identical events. That is, since prices change, the Friend’s speculative strategies also change, which in turn changes the prices and again changes the strategies, *ad infinitum*.

Let’s say that the set of stock prices that results from step (4) is the *outcome* of an iteration. Over the span of multiple iterations, outcomes can either follow some pattern or be completely random. I don’t even want to think about all the possible patterns—with one exception! It is possible that outcomes will converge on what a scientist might call a steady state. What’s a steady state exactly? Well, in our example it’s a state of affairs where existing stock prices, when plugged into the loop, return themselves as the outcome. Those versed in game theory should immediately recognize the similarity of this steady state to a Nash equilibrium. The similarity is not coincidental: the steady state *is* a Nash equilibrium in the game where each iteration is a round, the rules are given by the laws of history (or perhaps nature), and the Friend is the sole player. Achieving the steady state brings history back into balance, where it develops into just once possible future rather than sprout offshoots of alternative futures all over the place. (To those familiar with statistics: there will be some probability distribution of historic outcomes and their resulting futures; in case of convergence, whatever outcome yields the steady state will lie at the distribution’s median).

If convergence doesn’t happen, the process will continue returning different outcomes forever. Each of these outcomes will spawn a multitude of additional outcomes. This is where most people get confused, because it seems that various versions of history take place simultaneously, and worse still, the number of versions is constantly multiplying. Actually, the notion that “the number of versions is constantly multiplying” is itself confusing because “constantly multiplying” means increasing steadily from the past into the future whereas in our process the future merges right into the past and keeps going in circles.

I won’t tackle the explanation of what’s going on except suggest that in addition to the fourth dimension (time), one needs to pay attention to a fifth dimension here—the dimension along which individual iterations will be ordered. If you studied multivariable calculus, you should know exactly what I’m talking about—not just history, but hyperhistory, with multiple versions (and versions of versions, etc.), measured along axes additional to time.

Have I lost you yet? If you made it this far, I guess heavy artillery is in order to throw you off. Here are two things off the top of my head. First, up until now, I was discussing the implications of stock market manipulation by the Friend alone. Yet where one man hath trod, a crowd shall follow: imagine several Friends rolling up their sleeves and kneading the future like Play-Doh? Even the simple static-history scenario above will then become convoluted, although probably not impossible. Without the benefit of rigorous examination, I shall speculate that of the multiple-player cases the easiest ones will lie at the extremes: the first involving only one player (that is, the case I've been discussing all this time) and the second one involving everybody. Now try to generalize to cover the in-between range!

Second, if stock price behavior is chaotic (a “random walk”) then even if the Friend is alone and uses exactly the same strategy in iteration *n* and in iteration *n + 1*, the outcomes of these iterations won’t be identical. Then (a) permanent convergence will be impossible, and (b) the number of alternative courses of history will multiply way beyond what we had previously imagined. Tracing the dynamics of chaotic time loops is an exercise for another day. Or eternity.

Enjoy the headache!

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