Squid Game asymptotic probability

Difficulty:   ★★★☆☆   undergraduate

We continue with the bridge-crossing scenario from Squid Game called “Glass Stepping Stones”. Here I analyse the probabilities for late contestants on a very long bridge, and the expectation value for a player’s progress. Last time I found an exact expression for the probability P(i,n) that player number i is still alive on the nth step. Now seems a good place to mention there are equivalent expressions, such as:

    \[P(i,n) = 1 - \binom{n}{i}\cdot{_2F_1}(i,n+1,i+1;-1),\]

where {_2F_1} is called the hypergeometric function, and the other term is the binomial coefficient which is read as “n choose i”. The factor 2^{-n} seen previously has been absorbed. We listed several special cases of the probabilities last time. Another is:

    \[P(i,2i-1) = \frac{1}{2}.\]

So remarkably, the chance player i will be alive on step 2i – 1 is precisely 50%! For fixed large i, if we plot the probability distribution as a function of n it looks smooth, remaining near 1 for early steps before rolling down to near 0. Qualitatively this looks like a \tan^{-1}, tanh, or erf (“error function”). We reflect these curves, centre them on the value 1/2 at n = 2i – 1, and scale them linearly: so they have the appropriate bounds and match the slope at the centre point. See the Figure below.

probability for player 400
Figure: The probability player number i = 400 is still alive at step n. The scaled tanh function is close to the exact curve, while the scaled erf function nearly overlays it.

In fact the slope used in the Figure is only an approximation as described next, but this is a deliberate choice to show it still gives a good fit. The exact slope \partial P/\partial n evaluated at n = 2i – 1 seems a little too complicated to be useful. It contains a derivative of the hypergeometric function, which appears to approach -1/2 in the limit of large i, hence the slope at the centre point is asymptotic to -1/\sqrt{4\pi i}. Another approach is to consider the subsequent bridge step, for which:

    \[P(i,2i) = \frac{1}{2} - \frac{\Gamma(i+1/2)}{2\sqrt\pi\,\Gamma(i+1)},\]

which uses the Gamma function. The difference P(i,2i) – P(i,2i-1) approximates the slope, and is also asymptotic to -1/\sqrt{4\pi i} as i \rightarrow \infty. Hence our approximation for late players is:

    \[P(i,n) \approx \frac{1}{2}\Big( 1-\operatorname{erf}\frac{n+1-2i}{2\sqrt i} \Big).\]

Now the error function by definition is the integral of a gaussian curve. The derivative with respect to n of our approximation is precisely the righthand side below, which itself approximates the chance a late player dies on that step:

    \[P(i\textrm{ dies on }n) \approx -\frac{1}{\sqrt{4\pi i}}e^{-(n-2i+1)^2/4i}.\]

For fixed i this is a gaussian distribution with centre n = 2i – 1 and standard deviation \sqrt{2i}. It is normalised in the sense its integral over n \in \mathbb R is exactly 1, but physically we want the discrete sum over n \in \mathbb N^+. For the 10th player this is approx. 0.9991, which is already close. The exact chance for dying on a given step was determined in the previous article to be b_{i,n} = \binom{n-1}{i-1} \cdot 2^{-n}. The Figure below shows some early values. As before, we can extend the function beyond integer parameters.

probability of death
Figure: The probability player i will die on step n, given no foreknowledge (that is, before the game begins). The blue dots correspond to integer values of the parameters, which are physical. Contestants face a near-gaussian “hill of death” so to speak, which peaks at n = 2i – 2 and 2i – 1. I have included the “spires” at the back for sake of interest, as a peek into the rich structure for negative n, though this is unphysical.

The ratio b_{i,n}/b_{i,n+1} = 2(n-i+1)/n precisely. Hence for a given player, the adjacent steps n = 2i – 2 and 2i – 1 are equally likely locations their game will be “discontinued”. This is surely the maximum assuming integer parameters, apart from the first player for whom step 0 is safe but step 1 is their most likely “resting place”. Hence the reader might prefer to translate our gaussian approximation by half a step or so; apparently there are various approximations to a binomial coefficient. The subsequent step n = 2i is a more likely endpoint than the earlier step 2i – 3.

The expectation value for a given player’s death is:

    \[\sum_{n=1}^\infty n \cdot b_{i,n} = \binom{0}{i-1} \cdot {_2F_1}(1,-i,2-i;-1).\]

This function is very close to 2i, apart from a small oscillatory wiggle. At integer i it is singular, but from inspection of its plot it may be extended to a continuous function with value precisely 2i on physical parameter values (that is, integer i). Finally, for a given step n, the probability that some player breaks a tile is:

    \[\sum_{i=1}^n b_{i,n} = \frac{1}{2}\]

precisely, which is unsurprising. (Better terminology would be the nth “column”, as Henle+  use.) This assumes that n or more players have finished their run, otherwise the step is less likely to be broken.

Update, May 19: The death chance b_{i,n} is ½ times a binomial distribution in n. We previously found a gaussian curve for a given player i. Now. for a fixed step n, the de Moivre-Laplace approximation is a gaussian over the player number i:

    \[\frac{1}{\sqrt{2\pi(n-1)}}e^{-\frac{(i - n/2 - 1/2)^2}{(n - 1)/2}.\]

🡐 general solution | Squid Game bridge | conditional probabilities 🠦

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