NBC’s game show The Wall debuted at the New Year and is effectively a mix of Who Wants to Be a Millionaire and Plinko rolled into one. With its high suspense and random action, it’s been a relative hit for NBA, garnering a 1.5/5 to 1.6/6 rating/share in the Nielsen polls.
Note: For those who are unfamiliar with Nielsen ratings, a rating/share is a sampling methodology to estimate the number of viewers tuning in to a particular show. For the 2017 season, Nielsen estimated approximately 1.156 million television equipped households in the U.S. Therefore a rating of 1.6 would suggest that 1.6% of all TV equipped households are tuning in for The Wall; while 6% of all TV equipped households that are actively viewing TV at that time slot are watching The Wall. Compare the 1.6 to Seinfeld’s run of 20.6 to 22.0 in its final four years and you can see The Wall is a relatively low-key event in comparison.
Now, the game is relatively simple. A series of balls are colored red or green, dependent on some rule set and dropped from a selected spot at the top of the board. Each ball heads through a series of twelve transitions of offset guides, causing the balls to bounce left and right as they fall. At the bottom, there are fifteen slots for the balls to fall in. Each slot is valued at varying values. Green balls that land in the slot gain those values; while red balls subtract from those values. There are only seven initial starting spots at the top of the wall.
The rule set for determining a ball color is based on the ball position, and whether a player gets a series of questions correct. This process is broken into three varying steps.
It should also be noted that the contestants can never fall below zero dollars.
Step 1: Free – Fall
The first part of the game is free-fall, where contestants are given two choice questions and have to answer as a trio of balls fall from spots 1, 4, and 7. If the contestants are right, all balls are colored green. If the contestants are wrong, the balls turn red.
The minimum number on the board is 1 dollar; the maximum is 25,000. Therefore on each question, the minimum possible value is 3 dollars while the maximum possible value is 75,000 dollars. Therefore, a contestant can walk away with between 0 and 375,000 dollars at this round.
if the contestants finish above zero, they are able to go on to Steps 2 and 3.
Step 2: Little Money
The second round of the game, the contestants are split up. One goes into a silent room and answers questions when prompted, while the other chooses initial starting points for the balls, as well as the number of balls used. To start the round, two green balls are dropped from the locations of interest to the outside contestant.
Afterwards, there are a series of three questions. The outside contestant sees the three choices of answers and then picks an initial starting point for a yet-to-be-colored ball. The inside person is then asked the question. If the inside contestant is correct, the ball turns green and is dropped. Otherwise the ball is colored red and dropped. For questions 2 and 3, the outside contestant is given the option to “double-up” and “triple-up” respectively. Those balls are then multiplied and placed into the same initial starting point.
Finally, two red balls are dropped from the same initial points as the two green balls that started the round and subtracted from the total.
The fifteen slots are updated to range from 1 dollar up to 250,000 dollars. This means that the contestants can walk away with a minimum of 0 dollars; or a maximum of 1,999,998 dollars for the round. To determine this maximum, there are 2 green balls, a correct question, a double-up (if selected) correct question, a triple-up (if selected) correct question, and two red balls; with all green balls falling into the 250,000 slot and the red balls falling into the 1 dollar slots. This would give 8*250,000 – 2*1 = 1,999,998.
Step 3: Big Money
The third round plays out with four green balls, three questions in the same format as above, and four red balls. This time, instead of three choices per question, the contestants are given four choices per question. Similarly, the slots are now changed from having a minimum of 1 dollar to a maximum of 1,000,000 dollars. From this stage, the contestants can walk away with between 0 and 9,999,996 dollars.
This means that a pair of contestants can walk away with at most 12,374,994 dollars.
The Game Board
The game board itself starts with a pre-determined seven slot initial state. Each possible movement through the guides afterwards can be viewed as a Markov Process as the next step can only be preceded by the previous step (thanks to gravity)!
The second state has eight entry points. The third state has 11 entry points. The fourth state has 12 entry points. The fifth state has 13 entry points. Afterwards, the states rotate between 14 and 15 entry points until the final 15 point state.
It is easy to interpret that balls starting in initial state 7 will tend to complete their traversal on the right half of the board while balls starting in initial state 1 will tend to complete their traversal on the left half of the board. The question then is… how likely?
To estimate the probabilities of movement based on initial positions, we can do one of two things. The first is to treat the Markov process as a black-box and just look at the pairs: (initial state, final state) and just do a regression. This isn’t exciting.
Instead, we form the transitions and then build a traversal function that approximates the transition probabilities from one state to the next. For instance, a ball in initial state 7 has say a 50/50 chance of going left or right; but not two or more slots left or right.
In the image, we see that the transition movement must be a probability or moving from the current state to the next state. For this 50/50 (and nowhere else) movement, believe it or not, a ball placed in the number 7 slot leads to a highest probability of landing in the 11th spot on the board. This is a 10 dollar value in the Big Money round!! The million dollar slot has a probability of 5.6641% of being landed in!
What if we dropped from initial spot 6? Well, we move the highest probability over to the 10th spot, which is the 400,000 dollar slot. The million dollar slot falls to a 1.6357% of being landed in.
The expected value for a ball dropped from the 6 spot is 206,526 dollars. How about from the 7 spot? 247,377 dollars. However, the variance with the 6 spot is 32,359 dollars versus 37,726 dollars for the seven spot. That’s roughly a 200 dollar difference on the standard deviation. This only suggests that we expect more money from the 7 spot, but have a higher variation (only by a marginal amount).
Movement is not 50/50…
Instead of imposing some fixed model, suppose we instead use results of ball drops to estimate the transition matrix. Over a series ten episodes, we recorded a sample of transitions across all games. We say a sample as we are limited as viewers of the show. We do not see all transitions. However, we were able to obtain a large number of samples and can express a model for transitions.
First, we consider the current state of the ball. Then the ball can traverse into the next state up to the distance away from its current state. If it falls within the two slots directly below its current state, then the ball travels 0.5 slots in distance. Remember the slots are offset.
At most a ball can travel 14.5 slots. At least a ball can travel 0.5 slots. Note that the top levels are not 15 slots wide, so we have to handle this with care. Fortunately, of the thousand or so transitions from he show, no balls traveled farther than 5 slots; which is covered at every transition state.
Hence we model this movement with an exponential decay at each transition. We observe the linear movement and then find a coefficient that smooths the observations to match the rate of decay of the probabilities. First, our observations:
The exponential smoother looks at the distance traveled and multiplies by a value, beta, which weights the distance traveled. This value is then the argument of the inverse-exponential function; a common function that describes rate of decay. Based on the sampled data, beta is 1.1545851.
Hence, we can no start simulating the probabilities of hitting each slot in the final version!
Probabilities of Each Slot Given an Initial Point
OK, now that we have our Markov model and smoothing function between states, we are able to write out the probabilities!
Example: Village Idiot Plays
OK, to see how this works, let’s consider the village idiots team up to play this game. Their rules are simple. They guess at everything and play big every chance they get. And then they refuse the contract. We are interested in two probabilities: Probability that they win money. Probability that they win a million or more dollars.
The idiot portion of the village idiot is that the players do not known any answers and blindly guess at options. On top of that, they will double up and triple up at every opportunity and place the balls are far right as possible. This will set them up for potentially big gains and big losses.
With this hostile methodology of attacking the game, we run one million simulations of the Village Idiot game. We find that of the one million trials, the contestants manage to escape free-fall at 75.2851 percent!
Similarly, the contestants have a 25.0691% chance of taking away money in the game. This a relatively low percentage. Finally, the contestants have an 8.8937% chance of taking away a million or more dollars! Let that sink in. Gambling every time and guessing every question, we expect that roughly one out of every 11-12 pairs of contestants should win a million or more dollars.
Remember, however, these contestants do run roughly a seventy-five percent chance of walking away with zero dollars. So the strategy is very high risk.
Now you see how the game works and the probabilities of obtaining each slot. How do you think you’d fair in the game? If you interested in the code (.m files) feel free to drop a comment or send an e-mail to firstname.lastname@example.org!