These Games Are Rigged—Like Our Economy
November 16, 2020 — One evening last year, in the days when people could gather indoors to socialize, I played a game of Yahtzee with my wife and another couple. Monopoly has a bit of strategy to it, whereas Yahtzee is mostly luck, but both of these classic American games are idealized representations of capitalism—idealized in that each player starts with the same likelihood of success. Yet in both of them, once a player achieves a significant lead, the rewards multiply so fast that it becomes impossible for the rest to catch up. How long are you gonna last against the guy who owns all the railroads and has hotels on Broadway and Park Place?
Yahtzee, for those who haven’t played in a while, involves rolling five dice. You hang onto the ones you want and keep rolling the rest, up to three times, as you try to hit certain combinations. The most valuable roll is five of a kind—a “Yahtzee”—worth 50 points. Assuming a player only tries for a Yahtzee when they get three-of-a-kind on their first roll, the odds of that player rolling one or more Yahtzees in a game (13 turns) are just over 1 in 4.
But once you roll a Yahtzee, every subsequent Yahtzee is worth double: 100 points. The chance of a player rolling two or more Yahtzees in a game is roughly 1 in 23. The chance of rolling three or more is 1 in 143. On this particular night, I rolled four!
It was like winning the lottery—well, except that the odds of rolling four or more Yahtzees in a game are 0.4 percent, or 1 in 250, which is just slightly better than the Powerball grand prize odds of 1 in 292 million. Still, by my third Yahtzee, the other players were basically peasants, whereas I was the Dice King, reaping the rewards of my earlier dumb luck. In Paul Piff’s Monopoly experiment, whose details I describe in my upcoming book, Jackpot, he rigged a Monopoly game and watched the “rich players” become increasingly arrogant as they amassed wealth. With this in mind, I made a concerted effort to remain humble. But the peasants, my own wife included, turned on me. They threw shade in my direction, and when I tried to respond, they rolled their eyes: “Yeah, whatever, dude.” The pitchforks were out and sharpened. This is one of the paradoxes of wealth in America. We all want it, yet we feel resentment toward those who have it—even in a silly game.
From a competitive perspective, Yahtzee’s scoring make no sense. If you roll one Yahtzee, you’re already way ahead of the game. If you roll two or three, game over. Why create rules that give a huge advantage to the player who’s already ahead? (One might ask Congress that very question.)
I read up on the game’s origins. Yahtzee, originally called “The Yacht Game,” was invented in 1954 by a wealthy Canadian couple to pass the time aboard—yes—their yacht! The game became so popular within their elite circles that they approached Edwin Lowe, a man who made his fortune in the 1920s by transforming Beano, an old carnival game, into the best-selling board game we know as Bingo. The Canadian yachters would have been well aware of the advantages society bestows upon those who have already rolled a few Yahtzees in life.
Monopoly, however, has anticapitalist origins. It’s predecessor, The Landlord’s Game, was patented in 1904, circa the Gilded Age, by one Elizabeth Magie. A stenographer and typist by trade, Magie was smitten with the ideas of author Henry George, whose 1879 best-seller, Progress and Poverty, sold millions of copies. Though not a politician, George was the Bernie Sanders or Elizabeth Warren of his day. Among other things, he supported a “land value tax”—a wealth tax meant to curb the power of greedy landlords. Magie hoped her game would educate the public about George’s beliefs.
In her version of Monopoly, the square we know as “GO” contained a world map and the phrase, “Labor Upon Mother Earth Produces Wages.” Her “go to jail” square was a critique of class disparities, reading, “Owner, Lord Blueblood, London England, No Trespassing, Go to Jail.” Who’d have guessed that two games that underscore the unfairness of our society would become some of our most enduring family pastimes, selling hundreds of millions of sets—and still selling strong today? Americans, apparently, are gluttons for punishment.
See you in the parlor.
I relied on a pro for this: friend and neighbor Matthias Beck, a math professor at the University of California San Francisco. These are his calculations:
Let’s assume a player has a very simple strategy, namely they will “go for” a Yahtzee precisely when they have three or more equal dice in their first roll… Then the odds of rolling a Yahtzee in a given turn is 6*(5 choose 3)*(1/6)^3*(5/6)^2*((5/6)^2*(1/6)^2 + 2*(1/6)*(5/6)*(1/6) + (1/6)^2) + 6*5*(1/6)^4*(5/6)*(1/6 + (5/6)*(1/6)) + 6*(1/6)^5 = 2.47%.
Then the chance of no Yahtzee in 13 turns is (.975)^13 = 72%. => the chance to get at least one Yahtzee is 28%.
The chance of exactly one Yahtzee is 13*(.0247)*(.975)^12 = 23.7%. => the chance to get at least two Yahtzees is 4.3%.
The chance of exactly two Yahtzees is (13 choose 2)*(.0247)^2*(.975)^11 = 3.6%. => the chance to get at least three Yahtzees is 0.7%.
The chance of exactly three Yahtzees is (13 choose 3)*(.0247)^3*(.975)^10 = 0.33%. => the chance to get at least four Yahtzees is 0.4%.