Episode 11 – Beautiful Math

Today at Save Vs. Rant, we’re going to be discussing math and logic, the building blocks of games! Remember: if you think you hate math but you like games, what you really don’t like is the presentation of the math, because all games are, at their core, math.

Monty Hall, the famous host of Let’s Make a Deal, passed away this week. His memory will live on, however, in several ways relevant to our podcast. The first, and most obvious, is the infamous “Monty Haul Campaign,” wherein a DM gives away an inordinately large amount of treasure or inappropriately powerful magic item. This is named for the fabulous prizes to be won on Let’s Make a Deal. If you listened to our DMing 101 episode, you know that one of our warnings is not to be afraid of the player characters becoming too powerful, but there are limits to these things. Every game has some guidelines for what sort of treasure or magic items should be available to the player characters at any given level of play (whether the system uses levels, as such, or some other indication to measure power), and as with any rule, it’s important that before you consider breaking the rules and giving them more, you understand the purpose and parameters of the rule as written.

Monty Hall is also known for the infamous Monty Hall Problem in mathematics, an example of a counter-intuitive mathematical outcome. On the show Let’s Make a Deal, there was a classic scenario wherein the player had to choose between 3 doors (numbered, conveniently enough, Door #1, Door #2 and Door #3). Behind two of the doors was a dummy prize – a goat, for example – and behind the third door was the real prize – a brand new car, in our example. These were arranged randomly.

Upon choosing a door, instead of revealing what was behind the door you chose, Monty would open one of the other doors, revealing one of the goats. He would then offer to let the player switch doors* to the other still-closed door. The question, therefore is, should you switch doors? Does it even matter? And the answer to both of these questions, surprisingly, is “absolutely yes.” Switching doors increases your odds of getting the car.

Now, most of you are scratching your heads trying to make sense of what difference it makes. The difference is all in the information you are given. You now know where one of the goats is and, confident in this knowledge, you change your answer to capitalize on this newfound knowledge.

“Aha!” some of you might proclaim! “Since you’ve had one door opened for you, your chances have raised to 50%! I get it now!” Well… not exactly. Switching doors actually increases your odds of having the right door to 2/3. “Two thirds!?” you exclaim in disbelief, “How could this be?” The answer is, switching doors will always give you the right door, provided you chose wrong in the first place. Since your chances of choosing wrong are 2/3, your odds of getting the right door upon switching are, consequently, 2/3.

Consider the scenario that the car is behind Door #1. There are three possible outcomes of your first choice:

  • If you chose Door #1, Monty will open either Door #2 or Door #3 – it’s irrelevant, but let’s say just for simplicity’s sake that he opens Door #2. If you switch, you get Door #3 and lose. That’s a shame.
  • If you chose Door #2, Monty must open Door #3. If you switch, you will, therefore, get Door #1. You win!
  • If you chose Door #3, Monty  must open Door #2. If you switch, you will, therefore, get Door #1. You win!

So, out of 3 scenarios, 2 of them end with you winning if you switch doors. And this holds true, of course, regardless of which door the car is behind. There. Simple illustration. And now you understand!

… or not. The problem has confused actual mathematicians, primarily because the pattern is difficult for the human mind to follow as presented. Let’s flip the way we think of this. Instead of saying that Monty is going to show us where one of the goats is, we can say that Monty is going to show us where all the goats except one are, with the stipulation that if we picked a goat, that is the goat he will not reveal. This rewording doesn’t change anything to our perception with the 3 door example (although it is, to be clear, a more accurate description of the rule by which Monty is playing), but the problem becomes obvious if you increase the number of doors and continue to have only one prize.

Let’s say, for example, there are 10 doors, 9 goats, and 1 car. When you choose a door, Monty will show you where all the unpicked goats except one are, with the stipulation that if you picked a goat, that is the goat he will not reveal. So if the car was in Door #1,  and you picked Door #1, Monty will open all but one of the other doors – doesn’t matter which one – and switching will make you lose. If, however, you chose Door #2, Monty opens Doors #3 – #10, leaving only Door #1 and Door #2 (which you picked) unopened. If you switch, you win.  The same holds true for every door with a goat – Doors #2 – #10 will yield a winning result if your switched! If you picked a goat originally – and odds are good you did – switching will always make you win.

This is a great example of how math and logic can be counter-intuitive, but still demonstrable in practical terms. A simple thing like giving you one more piece of information might do a lot more than eliminate one wrong answer, as anyone who is really good at social deduction games will tell you – when playing werewolf, the discovery that one player is a werewolf will almost always give the opportunity to see who they were collaborating with, and a big-picture approach might make give the whole picture from a single piece of information!

Mathematical discoveries like this can be one of the joys of gaming. Discovering the connections between seemingly unconnected aspects of a game, or noting the hidden consequences of a piece of information can give you a feeling of satisfaction like no other. Try to find other examples of hidden information that, if discovered, has greater consequences than are immediately apparent. Chances are, this sort of scenario exists in games you already play!

* I have read that this never actually happened on the show Let’s Make a Deal, and is just a thought experiment concocted by a creative mathematician, but let’s suspend our disbelief for a moment here.

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