Show HN: Betting game puzzle (Hamming neighbor sum in linear time)

It is definitely the right place!

Following your suggestion, I’ve implemented the PSA algorithm (the optimized one) in C++. It might be tricky to understand without an explanation, especially since (1) it’s generalized for any dimensionality, and (2) I manually implemented the basic transformation ("oscillating shift") instead of using NumPy’s reshape+roll+flatten, which I believe makes it easier to grasp. But you can verify that it returns the same results as the naive algorithm. For the quiniela case, it shouldn’t take more than 2-3 seconds to compute the sum of neighbors for all points. https://github.com/petopello/PSA/blob/main/PSA.cpp

The key point is that your code takes as input ‘probas’, a 3x14 table with probabilities of independent events. From this, you assume the structure of the space by multiplying result of 14 events. Here, you can apply the trick of grouping negative probabilities. In fact, some of these calculations can be done only once, as many points share the same multiplication path. This is what I implemented in my SHN_BOXES function.

The issue is that our problem isn’t exactly like that. I haven’t gone into too much detail, but you should be able to create a function that takes a pre-built space as input and sums over it, even if you don’t know how the space was built or whether it was constructed with random values.

For details on why this is the case:

Step 1.1: First, calculate, based on a table 3x14 like ‘probas’, which represents how many people have bet on event j for match i, the number of winners in each category if a certain prediction occurs (I assume homogeneity here). This is also a Hamming neighborhood sum, but you can use the boxes algorithm (~1 sec).

Step 1.2: The prizes depend (inversely) on the number of winners. So, apply a formula like this to the previous space: bet_price * coefficients / (winners + bet_price / revenue)

Where 'winners' are the values calculated in 1.1, and the rest are inputs: REVENUE = 1000000.0 PRICE = 0.75 COEFFICIENTS = [0.16, 0.075, 0.075, 0.075, 0.09] //percentaje of revenue correspondy to each category follow game rules

Step 1.3: Now we have the prizes for each prediction. To calculate the value of betting on a specific prediction, we need to do a sum product of the prizes corresponding to that bet (distance <= 4), each multiplied by its probability. So, we multiply the results from the previous formula by the probability of occurrence (using another table similar to ‘probas’). So it only least make the summation.

Step 2: Now that the space is constructed, we need to sum Hamming neighbors, and this is where there’s no shortcut that I know of. You have to assume the space contains randomly generated values. This is the computational bottleneck, and this is where the algorithm I mentioned applies. In fact, as you can see, it doesn’t only go to one space but five, one for each category. So, the sum shouldn’t go to r <= x but should sum the neighbors exactly at x in the corresponding space.

My algorithm reduces the number of summations by computing all expected values at once, so I can't compute just a single value individually. For a single value, I think the naive approach is the only option. In my first naive implementation, computing 5,000–10,000 values took around 5 minutes, and I didn’t improve it much beyond that.

Once all values are computed, one approach is to search for the highest expected value, but that’s not the only criterion to manage: (1) there's a huge variance issue (you’ll win a lot, but with very low probability), (2) if you're placing multiple bets overlap reduces their combined value.

Great work on your approach! I’ll try to understand the code you linked, but I suspect it’s not doing exactly the same thing (or only for a very specific case). In my code, there's a function called SHN_boxes (which takes ~1s in Colab for this problem), and it’s a shortcut applicable only in some cases (not in this one). Did you use a similar approach?