Recently I’ve been pondering the Monty Hall problem and have conceived of what I believe to be an intuitive explanation of its highly counterintuitive solution.
The Monty Hall problem takes its name from the host of the game show Let’s Make a Deal. The game goes as follows:
The contestant has the opportunity to win a new car by guessing which of three doors it’s behind (the other doors have goats behind them). The contestant selects a door. Then Monty opens a door among the other two that has the goat behind it. The contestant then has the option of switching his selection to the remaining door or sticking with his initial selection. Monty then opens that door and the contestant goes home with whatever’s behind it.
The purpose of this post isn’t to weigh whether it’s better to switch or stay, because simple math has already proven the superiority of switching. If the contestant switches he has a 2/3 chance of winning the car. If he stays the chance is 1/3. What is far more fascinating than the math itself is how extremely counterintuitive the answer it produces is. Despite the simplicity of the math behind it, a reader of Marilyn Vos Savant’s “Ask Marilyn” column felt it worthwhile to reassure himself with the opinion of the world’s highest tested IQ about the conundrum. To the credit of smartest persons on Earth everywhere Marilyn supplied the correct answer: switch. What was even more remarkable than the fact of the conundrum being posed to her, however, was that she received in response to her published opinion on it a flurry of contravening commentary from numerous “experts” on economics and game theory, some of whom were downright indignant, outraged even, that Marilyn had allegedly fallen for some sort of con. Their commentary was to the effect of “You, Miss Savant, are far too intelligent to make such a childishly simple error in judgment” or “You may think you’re clever Marilyn, but I’m an expert in post-Nashian meta game theory and let me tell you…”. Marilyn’s response to the reader had been in lay terms, but even after the superiority of switching was demonstrated in formal mathematical proof by Marilyn and experts on the correct side, some of Marilyn’s critics refused to believe it.
The Monty Hall problem is one of the most striking instances of human ineptitude with probability. Our brains have deeply ingrained heuristics that were evolved under conditions requiring objectives quite deviant from that of maximizing the expected value of a raw economic quantity. Marilyn’s outlier cognitive ability notwithstanding, the tenor of response to her shows there is little correlation between intelligence and intuition for probability. Explaining the dynamics of probability in the Monty Hall problem can be done in a number of ways, but I think in terms of explaining the problem in a manner attuned to primitivist, and also right-wing reactionary, sensibilities, mine, which takes the form of an allegory, is hard to beat:
A professional sports gambler of the moniker Big Game Al has three sons, Stephen, Mark, and Robert. By the end of this story his sons will have acquired the monikers Secluded Stephen, Monty’s Mark, and Robert the Robber. Here’s how they acquire them: Al has amassed an impressive fortune in winnings, especially from having won a large bet on longshot tennis prospect Emma Raducanu to win the 2021 US Open. Nearing the end of his life, he writes a will for his sons in which, to circumvent death tax, he officially leaves them only $1.00 in taxable assets. This asset takes the form of lottery tickets to win his antique Red Barchetta sports car. Egalitarian cuck that he is, Al bequeaths his to each of sons a ticket with a value of $1.00/3.
When Al dies the first son to hear of his passing is Stephen. At this, Stephen immediately preps for the possibility that the Feds will selectively target him for tax evasion, especially in light of him amassing a large fortune of his own through speculation on blockchain assets of ambiguous tax status. He builds a compound in northern Idaho called Ruby Ridge Revenge, complete with drones, pitbulls, fully automatic, laser-automated machine gun turrets and a private army called the Blockchain Gang comprising nothing but ex-Navy SEALs of the copy pasta persuasion. Libertarian ubermensch Stephen then retreats with his lottery ticket to the solitude of his fortress, posting behind him “trespassers will be shot” sign, which for him is not an empty threat. Stephen, erring on the side of paranoia, is willing go so far as shooting his brothers on sight assuming them to be Federal imposters. As far as Stephen is concerned the outside world is dead to him with the sole exception of possibly winning the Barchetta.
The next son to hear of his father’s passing is Mark. Mark is a naive Trustafarian communist and is indifferent to the grubby materialistic concern of inheritance. He’d like to check his white privilege in the form of giving his ticket away to an aspiring dubstep rapper but he’s too stoned and lazy to do so. Being this way, he makes no effort to safeguard his father’s legacy either.
The last son to get the ticket is Robert. Robert is a national socialist and hates Mark, whom he regards as a race traitor. Upon hearing that his father put them on equal footing in his will, Robert flies into a rage and hires a hitman named Monty Hollowpoint to take Mark out. Monty, in the easiest job he’s ever had, carries it out and gives Mark’s ticket to Robert. Robert is a nice-guy national socialist and considers Stephen based despite his differing views on economics. He considers sharing the ticket with Stephen in some way but realizes this is now impossible. While watching Dexter on Showtime for ideas on disposing of Mark’s body he muses to himself “Stephen at times invokes the notion of ‘inalienable rights’. In secluding himself in the way that he has he’s created what is now an inalienable ticket”.
In this story each brother represents a different door on the show and his ticket the initial probability of winning the car. Stephen the Secluded is the contestant’s initial selection. By rule, Monty Hall cannot do anything with that door. Its contents and chance of winning of 1/3 might as well exist in a different dimension. The fact that Monty does not open the initially selected door is determined entirely by rule and has absolutely nothing to do with whether initial selection is correct or not, as Monty always has a non-car door to open no matter which door the contestant initially selects.
Monty’s Mark is the door Monty opens. Mark’s assassination in the story and the taking of his ticket is analogous to Monty revealing one of the goats. The winning brother/door is now down to two, but the counterintuitive core of the conundrum is that whether the contestant switches or not, his odds of winning are at that point not 50/50!
Robert the Robber is the remaining door. In killing Mark and taking his ticket, Robert has doubled his own chances of winning the Barchetta and reduced Mark’s by 100% – from 1/3 to 2/3 and from 1/3 to 0/3 respectively. Winnings and losses from games of chance (and most games in general) are from the player’s point of view zero sum. The corresponding odds must sum to one (hence Al setting the official value of the Barchetta to $1). A relatable real world instance is racial quotas. Taking the size of a desired incoming class as fixed, a college admission department’s putting a floor on the number of dark-skinned students is necessarily putting a ceiling on the number of light-skinned ones.
The error that contestants (and some “experts”) make is partial but no less fatal for that. Most people will intuit that Monty’s opening one of the goat doors increases the chance that the remaining door is a winner. Where they err, however, is in believing that Monty’s revelation also increases the chance of their door being the winner! For the reasons associated with Stephen the Secluded described above, this is literally 100% wrong. The Monty Hall problem elicits a potent manifestation of the possession heuristic, where people who form a connection to an entity, no matter how superficial, tenuous, transitory or even perverse (e.g. Stockholm Syndrome) tend to value that entity more. The possession heuristic is actually a distillation of human spirituality itself – the desire to connect with something external to ourselves. So compelling is this instinct that such sentimental preference can extend even to a fake door on a game show whose only connection to us is a random guess. The notion of “their” door being on a desert island in a vast sea of the statistical sample space leaves neurotypicals feeling cold.
My readers who are not nice-guy national socialists (or just not very bright) might read into my allegory the moral that national socialism is a superior political economy to libertarianism. The true social lesson (this essay is not on morality; I hate moralism actually) here is in fact, albeit a recondite one, the opposite: the importance of reconciling one’s economic model to certain facts of reality, no matter how spiritually unsatisfying (or simply irrelevant in my view) they may be. Just as those who want to maximize their chances of winning the zero sum game of Let’s Make a Deal have to set aside their possessive inclination and accept the statistically atomized nature of their position in the context of their final decision, those who want to maximize their chances of winning the zero sum stakes in the fight against globalism need to reconcile themselves to the fact of our neurally atomized nature in the context of economic decision making. The only sound model of such behavior can be one that recognizes that we are in this sense no less atomized from our brother nationals than the 1/3 chance of contestant’s initially selected door being the winner from the chances of his brother doors at the moment of Monty’s reveal. This model is called “methodological individualism”. The individuation of your value set in its totality, which informs your equally individualized decision making throughout your life, is absolute and belongs only to you. Put most simply, you cannot live anyone’s life but yours. The spiritual connective instinct is most potent in the human need to belong to and further a family, which in my story manifested in Robert’s desire to share his ticket with his brother. For this reason many nationalists allow this instinct toward their extended family in particular to compromise their thinking on economic issues, even though to do so is tantamount to rejecting math itself. In formulating a model to underpin a political economy nationalist leaders need in their thinking to stand cold, hard and alone, and the correct conclusion regarding the market drawn from methodological individualism is to treat the market as Stephen wished to be:
Left the hell alone.
