From my jacket pocket I produce three playing cards: an Ace of Spades, a Two of Hearts and a Two of Diamonds. Three props for demonstrating a probability puzzle, the solution to which provides a window through which we may catch a glimpse–however fleeting–of the American soul. This of course begs the question: Does that soul has an afterlife? If so, where is it likely headed–heaven or hell? By reflecting on three separate but related cultural phenomena–what I call The Three Montys–we can perhaps hazard an answer these questions.
The first of my Three Montys is the classic short-con game, Three Card Monte. Played throughout the world on street corners and public busses, in alleyways and even dorm rooms, players come alive with the drama of a ‘game of chance’ or the prospect of a good hustle. The game goes by many names: ‘Find the Lady’ (when you are seeking to spot a Queen rather than an Ace as in my example), ‘Bola-bola’, ‘Thimble-rig’ or ‘shell game’–the latter substituting cups and a ball instead of cards. Regardless of what it’s called or how it’s played it always operates according to the same unwritten, but widely accepted, rules. We’ll get to those.
The second Monty is The Monty Hall Problem which derives it’s name from Monty Hall, the onetime host of the American television game show Let’s Make A Deal. Strictly speaking The Monty Hall Problem is not really a problem at all and shouldn’t be named after Monty Hall–but we will get to that, too.
The last Monty, The Full Monty, is my touchstone reference that contains within it the solution to the contradictions involved with the other two Montys.
So, in order, we have: Three Card Monte, The Monty Hall Problem and The Full Monty.
First, Three Card Monte.
You may have an image of a young man throwing dice against a wall, or leaning over a small fold-up table. You’re in the right territory. Three Card Monte involves three people, although the game relies on the illusion that only two are playing at any given time.
First there is the Con, or host of the game, who shuffles three playing cards and will lay them face down. Two of these cards are alike and undesirable. In my example above the Two of Diamonds and Two of Hearts, both red cards of low value, are to be avoided while the Ace of Spades, the only black card among the three and of the highest value, is the card a player wants to pick. When the shuffling is finished, the Con invites players to guess where the Ace of Spades lies among the three downturned cards, and wager a bet that they know where it is hidden.
The second person in this game is the Shill, a seemingly disinterested player, who often appears hapless unable to guess where the one card of value keeps getting on to.
The last person is the Mark, who doesn’t understand that the Shill is not an independent player, but rather an assistant to the Con. The Shill’s staged play is designed to lure a player with money–the Mark–into the game and secure large bets. If done correctly, the Mark will become convinced that they can better ascertain where the Con is hiding the desired card than the Shill and thereby gain an advantage. The Mark will believe they are competing against the Shill, all the while there is no real chance of winning at all. Another variation has the Shill convince the Mark that they can team up against the Con when in fact the opposite will occur. The Shill should lose as many rounds as it takes to get the Mark to place a bet–preferably a large one–that can be won by the Con, who will judge as to how long the Mark can be kept in the game and how much money can be obtained. In fact, although money will pass between the Shill and the Con, no money is actually exchanged. There is only the appearance that it has done so, all for the benefit of the Mark.
The Con and Shill use misdirection (“look at the birdy”), charm (“you have beautiful hair”), distraction (the Shill bumps the Mark at an appropriate time) and outright deception (palming the correct card instead of actually placing it among the three so that there is no way the Mark can correctly guess where the desired card is placed, as it is not there) to alternately allow or disallow the Mark to win a round of betting. As in all ‘con games’ the Con and Shill gain the confidence of the Mark, (hence a ‘con game’) encouraging higher bets by intentionally losing, thereby building up the Mark’s confidence and wagers, until the Con lowers the boom and takes a large bet, or all the Mark’s money and the Con and Shill suddenly have to leave (bus to catch!). In fact the name of the game–Three Card Monte–references not only the fact that there are three cards but, ironically, that there are three players, not two, as an uninitiated player would think.
In this formulation the game is fixed through cheating in order to secure an outcome favorable to the Con and Shill. It’s a game that only works if the Mark doesn’t understand the difference between how the game appears to function and how it actually functions. It is no surprise that in most locales it is illegal.
Perhaps it is also no surprise that some casino poker rooms employ Shills, although if asked, they generally must identify themselves as such. The continuity in terminology is suggestive here, but in a casino Shills are not ‘cheating’, per se, but engaged in a type of team play that has a hidden benefactor–usually the House (casino). A casino Shill works for the House as a ‘street’ Shill works for the Con, but with an important difference. Casino Shills are hired to stimulate more frequent play and larger bets because the House is making a small percentage–what’s called ‘the rake’–off each round of betting. The more rounds and higher the bets, the more the House makes. The fact that Shills are allowed to play in casinos, but must identify themselves as such if asked, signals that the House is in a position of power; poker is the one game in a casino that is not, strictly speaking, a con game. The House can afford to identify Shills because Casino games rely on something other than cheating to secure a favorable outcome. And it’s that ‘something other’ that concerns us here.
The Monty Hall Problem
Monty Hall was the avuncular host of Let’s Make A Deal, a television game show that aired in its original form from 1963 into the mid-1970s. It has since inspired various sequels and imitations, such as The New Let’s Make A Deal which currently airs on CBS. The 1963 pilot featured well-behaved, buttoned-down studio audience members chosen by Hall to become contestants who would compete, most often against each other, for the opportunity to “buy, sell or trade” their way up to a “Big Deal” held at the end of each episode. Sometimes contestants seemed to compete against themselves–wringing their hands and furrowing their brows as they wrestled with their better natures trying to decide whether to keep whatever prize they had won from Hall or risk it all for something of greater value.
By 1970, the dress and mores of so-called ‘middle America’ had begun to morph under the sustained influences of the 60’s counterculture and Monty Hall’s “Marketplace of America” began to traffic in new cultural references–in one episode Monty suggests divorce to a female contestant, something inconceivable in1963. By the early 1970s contestants had started to dress up in costumes–Raggedy Ann, a sailor, a scarecrow, a police officer, etc.–to get Hall’s attention and make the hallowed transition from passive (if boisterous) audience member to contestant and therein (hopefully) to big winner.
The props included the perfect embodiment of commodity fetishism–the iconic “Instant Cash Machine”–that prefigured our now ubiquitous ATMs (can you even remember what that acronym stands for?) together with a constant flow of booby prizes that amused and entertained: behind curtain #2 is…a camel! Or lion cub or bargain item of little value. Contestants were from all-American locales such as Simi Valley, California or Elgin, Illinois and competed against one another to guess the “West Coast Suggested Manufacturer’s Retail Price” for a can of Hilton’s Oyster Stew or a pair of Mother Goose women’s shoes. A contestant who wins some money at the cash machine may be called back as a “Trader” and given the opportunity to risk what they had won up to that point for something of greater value; but they would always compete against another trader.
Let’s Make A Deal pioneered now-standard game show set-pieces such as estimating a “manufacturer’s retail price” and progressive risk-taking through ‘trading up’. Such elements would later define such game shows as The Price Is Right. Let’s Make A Deal was also an early and successful example of advertising through product placement, as brand name items–often furniture, household appliances, and leisure culture staples such as sailboats and cruises–were featured prominently, over and over again, on the program’s stage, in the audience, even emerging from Monty Hall’s pockets. The pace of Let’s Make A Deal, rapid fire and intense with it’s carnival atmosphere helped suppress critical thinking in exchange for immediate gratification; the relaxation or abandonment of skepticism in order to ‘scratch that itch’. All of this reflected the post World War II American explosion in domestic consumption and change in America’s self-identity from a nation of people(s) into a nation of consumers. With this development also coincided the notion that rights such as ‘liberty’ and ‘freedom’ were inherently bound up with one’s ability to exercise ‘choice’ in the marketplace. Freedom, even democracy, became inseparable from the marketplace whereas prior to this historical juncture capital was more often considered in competition with democracy and freedom. The show is also an excellent example of commodity fetishism: how the magic of the marketplace (where products are conjured up from seemingly nowhere, or everywhere) obscures the true character of human relations that make possible the production, exchange, and consumption of those products.
In some respects the original Let’s Make A Deal resembled a televised casino game; the program took off in popularity about the same time the city of Las Vegas was being established as the world’s epicenter for so-called ‘games of chance’. In its own way Let’s Make A Deal reflected a marriage between two innovations that were coming into their own in the post WWII era: mass marketing and the cult of the consumer, on the one hand, and novel ways of using probabilistic sophistry to part people from their money, on the other.
All casinos have ‘bells and whistles’ that come with their games. The most impressive ones can count on a battery of field-tested social-psychological accoutrements designed not so much to obscure or deceive or misdirect the Mark as to lard their experience with critical thinking suppression devices–alcohol and other drugs, sensual pleasures, entertainment and toys. In other words, what happens around casino games is of at least as much importance as what goes on inside the games. Besides, casino games share much in common with Three Card Monte, but much more with Let’s Make A Deal.
A Letter to The American Statistician
As a game show Let’s Make A Deal was innovative, but in at least one respect it was ground breaking. It wasn’t until 1975, 12 years after the show’s debut–that someone noticed what that something was. Let’s Make A Deal did not become ‘The Monty Hall Problem’ until 1975 when Steve Selvin, a professor at Berkeley published two letters in The American Statistician. (Vol. 29, No.s 1 and 3). In Silven’s first letter he cleverly reworks (without attribution) the Three Prisoners Dilemma, a well-known probability paradox that itself is indebted to Bertand’s Box, which dates back to 1889. Selvin changes the format of Let’s Make A Deal to illustrate how veridical probability paradoxes work–they produce results that appear absurd but can be mathematically demonstrated to be true. Through his letter Selvin was trying to show his readers that Let’s Make A Deal helps us understand how probability paradoxes work.
I’m trying to show that Let’s Make A Deal and it’s association with probability paradoxes can help us understand how our economy and politics work.
Instead of using The Three Prisoner Dilemma format whereby one prisoner is to be pardoned and two others executed, Selvin substitutes a hypothetical contestant on Let’s Make A Deal who faces a similar choice, though without the added stress of potential execution: three boxes, one containing the keys to a 1975 Lincoln Continental, the other two, nothing.
Silven’s hypothetical contestant chooses a box, let’s say ‘Box B’. The host (Monty Hall) offers the contestant $100 for the box (this is a nod to the format of the show, but superfluous to his reconstruction of the paradox). The contestant says, “no.” Hall then offers $200 in exchange for the box–telling the contestant to remember that the probability of ‘his’ box being the correct 1/3 while the probability of it being empty still 2/3. “Still want to keep it?” Asks Hall. How about $500?” The contestant still replies no!
Hall then says he’ll do the contestant a favor and open Box A, which is shown to be empty. He then tells the contestant his probability of picking the correct box is now 1 in 2, but offers him $1000 for his box. But the contestant says, “no”.
Silven then writes “WAIT!!! Is Monte right?”
Silven asks whether Monte has done the contestant a favor by opening one of the remaining boxes and whether his probability improved from 1/3 to 1/2. Again it appears as though the contestant’s probability of guessing the right box has improved from 1/3 to 1/2, but Silven will show it hasn’t. But his example doesn’t end there–there is a kicker. Silven offers up a role reversal wherein the contestant offers Monte a deal, saying he’ll “trade you my box B for the box C…”
Silven has Hall respond: “That’s weird!?”
But the contestant has done the math: Silven proceeds to consider the possible outcomes, after which two things are established: After Monty opens one of the empty boxes the contestant faces no improved probability of success, as it remains 1/3, even though it appears to have improved to 1/2. Furthermore, on the program Let’s Make A Deal there are always two contestants, and no box switching is allowed. Just because one of the boxes has been eliminated doesn’t change the fact that the hypothetical contestant chose his box when there where 3 boxes up for grabs. This the host tries to muddy by offering money, deliberately trying mislead the contestant that their odds had changed (an homage to the original Three Card Monte). What changes is if the contestant has the ability to stay or switch after the host has opened an empty box. While it doesn’t appear to improve the chances of success for the contestant, the option to switch improves their probability from 1/3 to 2/3. This is the counterintuitive, hidden play within The Monty Hall Problem that is so interesting.
That’s the original statement of the problem. It illustrates both the counterintuitive answer of either variation of the dilemma and as such is a better statement than practically any follow-up since then (and there have been hundreds, most of which do not grasp the nuances between the variations, including numerous iterations by The New York Times). Silven followed up with his second letter “The Monty Hall Problem” presumably because there was a great deal of criticism leveled at his conclusion that the contestant was correct in trying to switch. Silven restated his proposition, provided a mathematical proof to support it, then reprinted a section of a telling May 12, 1975 letter to him from Monty Hall:
“Although I am not a student of…[statistics], I do know that these figures can always be used to one’s advantage…The big hole in your argument…is that once the first box is seen to be empty, the contestant cannot exchange his box…after one is seen to be empty, his chances are no longer 50/50 but remain what they were in the first place, one out of three. It just seems to the contestant that one box having been eliminated, he stands a better chance. Not so. It was always two to one against him [1/3] And if you ever get on my show, the rules hold fast for you–no trading boxes after the selection.
Through his letter Monty Hall acknowledges both that the probability a contestant will choose the correct box does not change after the host (surprisingly) opens an empty box and that if he were to allow a contestant to switch choices after the fact (which he does not ever do) that would change their chances from 1/3 to 2/3.
Selvin’s second 1975 letter “On The Monty Hall Problem” is the first use of that term in print. It is amusing that Selvin’s first letter uses the ‘Monte’ spelling so closely associated with the low-brow con game, perhaps an unconscious nod to the unfairness at the heart of both Montys. In his second letter, Selvin uses the spelling “Monty”, perhaps because he had by that time received the letter from the game show host himself.
A second iteration of the paradox comes in 1990 with the publication of “Game Show Problem” by the celebrity intellectual and columnist Marilyn vos Savant in, of all places, Parade Magazine. Savant correctly interpreted the Monty Hall Problem as a probability puzzle with one solution: Switch. Savant published a letter by a reader from Maryland who is the first to structure the paradox using the now familiar two goats and one car formulation. Here the paradox is boiled down to “switch or stay”, and Savant correctly answers “Yes; you should switch.” But it doesn’t matter, as she reportedly received a mountain of angry letters denouncing her stupidity, many from PhDs.
Savant gladly reprinted some, including these gems:
“You’re in error, but Albert Einstein earned a dearer place in the hearts of people after he admitted his errors.–Frank Rose, Ph.D., University of Michigan.”
“Maybe women look at math problems differently than men.–Don Edwards, Sunriver Oregon”
“You are the goat!–Glenn Calkins Western State College”
And my favorite:
“You made a mistake, but look at the positive side. If all those Ph.D’s were wrong, the country would be in some very serious trouble.–Everett Harman, Ph.D., U.S. Army Research Institute”
Savant was correct and generally too polite in the face of unremitting, vicious criticism. Of course what is rarely discussed, and what Monty Hall referenced in his letter to Selvin, is that Let’s Make A Deal never allowed for a construction of the problem in the way it is now set up. In other words, the now familiar two-goats and a car construction of the so-called Monty Hall Problem, which the New York Times uses, is apocryphal. It does not come from Let’s Make A Deal, but directly from Selvin’s letter of 1975, which itself is a reworking of the Three Prisoner’s Dilemma (1959) which goes back to Betrand’s Box (1889).
It was only after Savant published her column and her letter-writer articulated the two goats and a car format together with the option to switch for the contestant that the Monty Hall Problem would achieve the status of a celebrity paradox, although in actuality since 1963–a span of over 51 years–the Monty Hall Problem has only ever been a reworking of a probability paradox that itself owed its origins to another dating back more than a century. It is somewhat ironic that in the almost 40 years since Selvin published his letter the very discussion of the meaning of The Monty Hall Problem, with it’s origin’s in Parade Magazine and later with the New York Times has also illustrated how marketing and probabilistic sophistry go so well together without explaining why the so-called ‘problem’, or the show itself, or any of it should matter at all.
Games of Chance
The term ‘game of chance’, by the way, seems calculated to obscure what these games really are: specialized con games that have supplanted their crude predecessors, such as Three Card Monte with its primitive accumulation and reliance on brute deception, with ever more sophisticated forms of probability schemes–innovative long cons. What do structured asset-backed securities such as collateralized debt obligations (CDO), predatory lending and other exotic financial products owe to these probability puzzles? Quite a bit, I suspect. What to someone else is a ‘game of chance’ seems more properly rendered ‘probabilistic sophistry’: a long con based on stable structures of aggregate accumulation. While that may be a mouthful, each word is important for understanding what is actually happening when you put your money into that slot machine, or are forced to purchase that insurance policy or ‘variable-rate mortgage.’ Stable, as in probability doesn’t change, whether the machine “paid” a jackpot yesterday or last year. Structured, as in an intelligently designed, closed system that God doesn’t care about and that misleads through appearance. Aggregate, as in large amounts of money collected over time, not one-time plays; and lastly, Accumulation rather than “pay out” as in the machine gathers your money. So another (more realistic) way to think about a slot machine that has a “97% pay out rate” is that it accumulates at a 3% rate while whoever plays it will play at a 3% rate of loss. And that’s if you never put back in your ‘winnings’, which you will. Over time, of course, which is all that matters to the casino, all players lose. Whether it’s poker, roulette or slot machines, all casino games work on this same principle. Unfortunately, more and more of our modern economy and political system appear to work like this as well.
Because The Monty Hall Problem is a pure example of this principle, understanding the mechanism at work within it helps illustrate what is actually happening, rather than the magical thinking involved with marketing that has rechristened all this business as ‘gaming’. Additionally, a re-examination of the game show and the probability puzzle it inspired helps us understand the profit motive at the center of the American Dream–or Nightmare, whichever you prefer.
But to really get a sense of why all of these variations on a theme are so difficult to understand–so counterintuitive and hard to grasp; why an explication of its play format can still elicit such passionate opposition, and how it suggests American political economy you need to see it, step by step, from the perspective of the uninitiated. Then, after your head cannot seem to wrap itself around this paradox, you can accompany me into its inner sanctum and see it from the only perspective from which it makes any sense: that of the House.
Let’s re-work Selvin’s Monty Hall scenario to better suit our purposes: As a thought experiment, let’s pretend we are a contestant, one of the uninitiated, and see how play proceeds.
Let’s Make A Deal — Two Goats and a Car
First, in this scenario we are the only contestant playing against the House–remember that in Monty’s actual version there are always two contestants, or ‘traders’, playing against each other who are never given the option to switch boxes. That was the point of Monty’s letter to Steve Selvin. But in our pure scenario it is just us against the House. For a high production value experience of the game go to the New York Times webpage at: http://www.nytimes.com/2008/04/08/science/08monty.html?_r=0 . Don’t press ‘How It Works’ yet.
We get to choose one of three doors behind which there are two goats and one car. In choosing a door there is a devious exploitation of bias set to work. It’s our door. We chose it. Once we’ve chosen a door, we are loathe to second guess ourselves. So we pick a door, which we really want to be the correct door, and wait expectantly for what comes next.
So we have our door, let’s say #1. Then the host ‘helps’ us by revealing one of the ‘goat’ doors, let’s say #2. Cool! That’s a prize I don’t want, and the host just eliminated it! It appears as though our odds were 1/3, but now they are 1/2. Then the host does us another favor! He says we can stay with our original choice or switch to the other door. Confident that nothing has occurred that would make us switch, as the odds, as it were, remain 50-50 you roll it over in your minds eye once, twice, –staying and switching are the same, because it’s still two doors, but somehow…
It doesn’t seem to make any difference, so we will stay. It’s the polite thing to do. And besides, the humiliation of switching away from the winning door and thereby losing when we were right (“you switched? Ouch!”) comes into play here. More reason to stay. It would be counterintuitive to think otherwise, and there doesn’t appear to be any compelling reason to switch. So we stay. When I play the game with the uninitiated I always frame the choice as 1) Stay or switch? 2) Does it matter? and, 3)Why or why not? Everyone says it doesn’t matter, so they stay as their probability of guessing the correct door appears to have improved from 1/3 to 1/2, and staying or switching doors doesn’t appear to impact that. Everyone I’ve ever had play the game who was uniformed about it stays. No one switches.
What we now know (all those Ph.Ds notwithstanding) is that in this format you should always switch. What appears to be an act of revealing a door in your favor also reinforces your desire to stay (because it appears as though the probability of choosing the correct door has improved) and to do otherwise would be, well, rude. I mean the guy did show you one of the doors! Switching strikes one as somehow a breach of trust with the avuncular host. The act of switching would somehow be going against both yourself and the host, who has done you a favor. Somewhere in your noggin is a well-intentioned but ignorant little fairy that does not help us here; it hurts us. All of this is compounded by that wholly irrational but nagging feeling that we are somehow better, more worthy people if we are right and somehow assholes if we are wrong.
The experience of choosing our door and staying with our initial choice involves the appearance of balance and fairness. There is so much that mitigates against the correct choice here that it begs the question as to whether there is something hard-wired into humans or perhaps culturally shaped (or both) that makes most people uniquely unsuited to ferret out this probability paradox. It is very difficult to grasp, much less explain, even after having it laid bare dozens of times. The game conceals a clear advantage for one choice of action (switching) and seems to promote the other (staying). Mathematically staying involves a 33% chance of guessing the car while switching will increase one’s probability to 66%. There is mathematical certainty in the advantage to switching, which the game, through probabilistic sophistry, suppresses.
Let’s Make A Deal–House Perspective
Now go back to the The New York Times web page The Monty Hall Problem http://www.nytimes.com/2008/04/08/science/08monty.html?_r=0 and play the game, but click on ‘How It Works’. Notice that it only makes sense when viewed from the perspective of the House, or dealer. It appears one way, but actually functions another.
With it’s roots in the Three Card Monte tradition of con games and it’s clever exploitation of probabilistic sophistry, did Let’s Make A Deal foreshadow the rise of casino culture economics and ever more sophisticated computer-based efforts to swindle people–and make them happy about being swindled?
But all of this is ‘free will’, is it not? It is not. And this is where I part ways with most social psychologists and economists. Free will and free choice presuppose a community that has relatively equal access to and acquisition of information and education; our society seems exquisitely constructed to prevent just that. Furthermore, our financial institutions increasingly utilize scams such as these to prey upon those of us with reduced access to information, education and just plain old credit.
But perhaps there is something of even greater importance that Let’s Make A Deal illuminates. Did the marriage between the suppression of critical thinking at the heart of modern marketing combined with new techniques of probability sophistry that Let’s Make A Deal pioneered foreshadow the development of our current casino economy? That would make the ‘Monty Hall Problem’ and Let’s Make A Deal both ingenious and insidious.
The Full Monty
There is a third Monty, of course, The Full Monty by which I mean ‘showing it all’. That is my Monty of choice. I suspect that this term has its roots in the Three Card Monte con game, although I have yet to see that documented. It is the perfect solution to both Three Card Monte and the so-called ‘Monty Hall Problem’. Both rely on concealment, as one doesn’t ‘see’ everything in either con game. But when you do The Full Monty, as with the film of the same name, you are stripping the problem down to its essence, forcing the House to show all. And that’s what both of these con games need–exposure, sunlight, understanding.
After all, sometimes our intuition is correct as when we get that feeling in our gut that ‘the game is fixed’, our horizons limited, our choices winnowed down, our effective parameters of action squeezed shut. Perhaps because there is no other game around, we play. Our political system, with its complete dominance by money, resembles a probability paradox. Our economy is now entirely dependent upon such games, a situation as unstable as it is unsustainable.
Am I suggesting a curtailment of free will? Yes, if you equate the rule of free markets with free will, or money with free speech. In my mind ‘free will’ or ‘free speech’ become moot points in a world where you have no real options other than the cheating version or the sophisticated bullshit version–both empty your pockets. My problem with these ‘problems’ is that within a world of increasing social inequality the last thing we need are more sophisticated scams that suck our wealth and income upwards; that said, I’d welcome one that did just the opposite! And if “money is the root of all evil” we have a provisional answer to the question I asked at the beginning of this essay.
August 1, 2014