Tuesday, April 25, 2023

The light and clap game

Suppose a light turns on at a uniformly chosen random time between 10 and 11 am, not including 10 am, and Alice wins a prize if she claps her hands exactly once after 10 am but before the light is on. Alice is capable of clapping or not clapping her hands instantaneously at any time, and at every time she knows whether the light is already on.

It seems that no matter when the light turns on, Alice could have clapped her hands before that, and hence if she does not clap, she can be rationally faulted.

But is there a strategy by which Alice is sure to win? Here is a reason to doubt it. Suppose there is such a strategy, and let C be the time at which Alice claps according to the strategy. Let L be the time at which the light turns on. Then we must have P(10<C<L) = 1: the strategy is sure to work. But let’s think about how C depends on L. If L ≤ 10 + x, for some specific x > 0, then it’s guaranteed that C < 10 + x. But because the only information available for deciding at a time t is whether the light is on or off, the probability that we have C < 10 + x cannot depend on what exact value L has as long as that value is at least 10 + x. You can’t retroactively affect the probability of C being before 10 + x once 10 + x comes around. Thus, P(C<10+x|L∈[t,t+δ]) will be the same for any t ≥ 10 + x and any δ > 0. But P(C<10+x|L∈[10+x,11]) = 1. So, P(C<10+x|L∈[t,t+δ]) = 1 whenever t ≥ 10 + x and x > 0. By countable additivity, it follows that P(C≤10|L∈[t,t+δ]) = 1, which is impossible since C > 10. Contradiction!

So there is no measurable random variable C that yields the time at which Alice wins and that depends only on the information available to Alice at the relevant time. So there is no winning strategy. Yet there is always a way to win!

I don’t know how paradoxical this is. But if you think it’s paradoxical, then I guess it’s another argument for causal finitism.

1 comment:

Walter Van den Acker said...

This seems to depend on whether time is discrete or not. If time is discrete, then the first moment at which Alice can clap her hands is at 10+1, that is the first possible moment of time after 10 am. But that is also the fist time the light can turn on.
In that case, it is not true that Alice could have clapped her hands before the light is turned on.
The problem I see here is not with causal finitism or infinitism, the problem is that if Alice is capable of clapping here hands intstantaneously, she can at the same time both clap her hands and not clap her hands and the light can, at the same time be on and out.
Now, if time is continuous instead of discrete, then you might say that before any moment the light is possibly not on, there is also a moment when it is possibly on, at infinitum, so the only strategy for Alice would be to clap her hands at the very first moment the light is not on, but if causal infinitism is true, there is no absolute first time unless she decides to clap at exactly 10 am.
But if causal infinitism is true, there is no such thing as exactly 10 am, so I don't believe that, if causal infinitism is true, it is even possible to instanteaously do anything at alL;