Wednesday, October 23, 2013

First Simulation Results


What do these numbers mean?  100,000,000 random games of Football Squares were simulated with squares filled out with players A-Z as shown above. The astute reader will recognize that Players A-P correspond to the 16 possible ways one can uniquely choose four squares (see the Post on "Player Selections"), Q-V are the 6 ways one can choose 3 squares, W-Y, 2 squares, and Player Z has only one square.

The first column with numbers around 16,000,000 are the total number of squares that a player wins in all the simulations. The second column with numbers between 1.9 and 2.3 million represent the number of simulations where the player wins two squares in a single simulation. The third column (varies between 272 and 500 thousand) has the number for each player winning three squares, and the fourth column (varies between 14 and 66 thousand) represents the player winning all four squares in the simulation.

So with a single square (Player Z), you have a probability of 2432/100,000,000  or 0.0024% that you pick the right square and the game is such that you win all four squares. Based on this number and the fact that there were 4953 simulated games, I can predict that 12 out of those 4953 games are such that the same ending digit appears in the scores after all Q1-Q3 quarters and the final score. Indeed, I verified that there were 12 such games where in Q2-Q4, both teams scored either 0, 10, or 20 points such that the ending digit never changed.

Key Finding 1: There is no advantage to your overall expectation value (how many squares one wins divided by how many simulations) no matter what you do. Indeed, for a single square, Player Z wins about 4 million times, all the two squares players win 8 million times, 12 million for the 3 square players, and 16 million times for picking four squares. The small differences are consistent within statistical uncertainties (sometimes when you flip a coin, you'll get four heads in a row). Chance of winning is 4% per square as there are four winners on a 100 square grid.

So, it does not matter whether you pick squares all in a row or all in a unique row and column in terms of your overall long term probability of winning squares. There is no strategy that can make you more money over the long term.

Here's a simple example that convinced me that this is true. Imagine playing a smaller squares game with a 2x2 grid and imagine a game played with a first half and a second half where you knew 100% that the visiting team would have the same digit and the home team would have 2 unique digits (the two winning squares would be one on top of the other (denoted by "X"). Let the player pick 2 squares in the three ways that they can uniquely do this. This is taking the non-random factors present in correlated scores between quarters to the extreme. All the possibilities are shown below:

Notice that if the Player picks two side-by-side squares or picks squares on a distinct row and column, they will always win one square no matter what. If the Player picks two squares on top of each other, the player will win two squares 50% of the time and win no squares 50% of the time. In either case, the Player is on average, over a long time, expected to win one square per game.

Key Finding 2: In the simple example above, one can see that the strategy of picking squares will result in a large variation of what's possible and likely to happen in any given game (either you always win one, or you can have a strategy that gives you some chance of winning both squares - though you now have a chance of  not winning any). In the first simulation results, you can observe significant variations in the 2nd, 3rd, and 4th column of numbers representing the number of times of winning two, three, or all four squares. It's this Key Finding that the following posts will study in great detail to develop strategies that can be employed for Football Squares.

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