Superrationality: Understanding Decision-Making in Dilemmas
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Chapter 1: The Gamble of Deal or No Deal
This chapter serves as a continuation from the previous one.
In the popular television game show Deal or No Deal, the sole contestant encounters 26 boxes containing different amounts of money. While the contestant is aware of the amounts available, the specific allocation of each amount to its respective box remains unknown.
Throughout the game, the contestant eliminates boxes, ultimately leading to a final decision on the last remaining box. Alternatively, at various intervals, the contestant can choose to accept a cash offer, effectively ending the game.
One particularly noteworthy episode featured a contestant who managed to narrow the options down to two boxes: one containing $1 and the other $1,000,000. At this stage, he was presented with a cash offer of $416,000. Opting for "No deal," he decided to gamble, taking a 50/50 chance between winning either $1 or $1,000,000 instead of securing the guaranteed $416,000. Unfortunately, he opened the box with $1.
Was this a wise decision?
Did the contestant act rationally? Some argue that he did because he chose the option with the highest expected monetary value. The calculation shows that with a 50% chance of winning $1,000,000 and the same chance of winning $1, the expected value calculates to 0.50 * $1,000,000 + 0.50 * $1 = $500,000 + $0.50 = $500,000.50, which exceeds the cash offer of $416,000.
However, while this math checks out, the argument is flawed. It overlooks the fact that the worth of money varies for the player based on how much they already possess. One must consider: does winning $1,000,000 truly feel more than twice as rewarding as winning $416,000?
Personally, I would argue that it does not. Winning $416,000 would be an incredible achievement, and while I would cherish winning $1,000,000 even more, the leap from winning nothing to securing $416,000 is significantly more impactful than the difference between $416,000 and $1,000,000.
This subjective assessment of value is crucial; for instance, a person in financial hardship would likely value an additional $100 far more than a billionaire would.
To understand this better, we can use the concept of utility, which quantifies the perceived value of an outcome. Similar to measuring temperature, utility can be quantified in "utils."
Consider utils as a measure of satisfaction: receiving a Christmas card might be worth 5 utils, enjoying a beer with friends could be valued at 50 utils, while winning a car might hold a value of 1000 utils for you.
Returning to our unfortunate contestant: whether his choice of "No deal" was rational depends significantly on his individual circumstances and his perception of money's value.
If his viewpoint resembles mine, he might evaluate the amounts as follows: $1 is of minimal significance, worth merely 2 utils. Winning $416,000 would be life-changing, valued at 10,000 utils. The $1,000,000 prize, while even more significant, may not be valued as highly compared to the immediate life improvements offered by $416,000—perhaps worth only 14,000 utils.
In this case, the deal would equate to 10,000 utils, while choosing "No deal" translates to 0.50 * 14,000 utils + 0.50 * 2 utils = 7,001 utils. Thus, based on this analysis, accepting the deal may actually be the superior choice.
Of course, we cannot accurately gauge the contestant's personal valuations. If he is wealthy, he may have already fulfilled his desires, making the prospect of winning $1,000,000 feel much more significant than the $416,000.
In the subsequent chapters of this Superrationality series, we will delve deeper into dilemmas framed around the concept of utility. Since monetary values are more relatable than utils, we will often translate utility into dollar amounts. After this chapter, you will gain a clearer understanding of how utility functions!
Join us in Chapter III, where we will explore the misconceptions surrounding Eve’s choices. Thank you for reading!
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