文章探讨了两个公平硬币翻转的打赌情境及相关的心理学现象。多数本科生会拒绝第一个打赌，若拒绝第一个却接受第二个打赌，会出现悖论。文章还分析了其中的效用函数、折扣率等问题，并提到了损失厌恶现象及相关讨论。

🎯在第一个打赌情境中，若硬币翻面为 tails，需给对方$100，若为 heads，对方给自己$110，约90%的本科生会拒绝此打赌。这反映出人们在面对风险时的一种决策倾向，可能存在损失厌恶心理。

💥在第二个打赌情境中，若硬币翻面为 tails，需给对方$1000，若为 heads，对方给自己$1,000,000,000。若拒绝第一个打赌却接受第二个，会出现Rabin悖论，涉及效用函数和极端折扣率等问题。

🔍文章通过数学计算和分析，探讨了在不同打赌情境下的效用、折扣率等概念。例如，计算得出若持续拒绝第一个打赌，在一定范围内，每增加$110/$ - $100的区间，正美元的价值平均低于负美元的10/11，这体现了指数折扣现象。

🧐文章还提到了一些注意事项和反注意事项，如在讨论悖论时应考虑在潜在广泛的财富范围内持续拒绝第一个打赌的情况，以及进行实验心理学实验来测试百万富翁在小赌注中的损失厌恶等。

Published on August 14, 2024 5:40 AM GMT

Quick psychology experiment

Right now, if I offered you a bet that was a fair coin flip, on tails you give me $100, heads I give you $110, would you take it?

Got an answer? Good.

Hover over the spoiler to see what other people think:

Second part now, if I offered you a bet that was a fair coin flip, on tails you give me $1000, on heads I give you $1,000,000,000, would you take it?

Got an answer?

Hover over the spoiler to reveal Rabin's paradox^[2]:

What? How?

The general sketch is to suppose there was some utility function that you could have (with the requisite nice properties), and show that if you reject the first bet (and would keep rejecting it within a couple-thousand-dollar domain), you must have an extreme discount rate when the betting amounts are extrapolated out.

If you reject the first bet, then the average utility (hypothesizing some utility function U) of winning / losing the bet is less than the status quo: U, 0.5$\cdot$U(W+$110) + 0.5$\cdot$U(W-$100) < U(W). In other words, the positive dollars are worth, on average, less than 10/11 as much as the negative dollars.

But if you keep rejecting this bet over a broad range of possible starting wealth W, then over every +$110/-$100 interval in that range the positive dollars are worth less than 10/11 the negative dollars. If every time you move up an interval you lose a constant fraction of value, that's exponential discounting.

How to turn this into a numerical answer? Well, just do calculus on the guess that each marginal dollar is worth exponentially less than the last.

$f\equiv e^{-kw}$

$\frac{1}{200}{\int }_{-100}^{0}f+\frac{1}{220}{\int }_0^{110}f=f\left(0\right)$

$\frac{1}{200k}(e^{100k}-1)+\frac{1}{220k}(1-e^{-110k})=1$

The step of the calculation where you plug it into wolframalpha

$k\approx 0.0013$

Some numbers, given this model:

The benefit from gaining the first $1 is about 1 utilon.

The maximum benefit from gaining $1,000,000,000 is a mere seven-hundred and sixty nine utilons. This is also the modeled benefit from gaining infinity dollars, because exponential discounting.

The minimum detriment from losing $1000 is over two thousand utilons.

So a discount of 100/110 over a span of $210 seems to imply that there is no amount of positive money you should accept against a loss of $1000.

Caveat and counter-caveat

When this paradox gets talked about, people rarely bring up the caveat that to make the math nice you're supposed to keep rejecting this first bet over a potentially broad range of wealth. What if you change your ways and start accepting the bet after you make your first $10,000,000? Then the utility you assign to infinite money could be modeled as unbounded.

This suggests an important experimental psychology experiment: test multi-millionaires for loss aversion in small bets.

But counter-caveat: you don't actually need a range of $1,000,000,000. Betting $1000 against $5000, or $1000 against $10,000, still sounds appealing, but the benefit of the winnings is squished against the ceiling of seven hundred and sixty nine utilons all the same. The logic doesn't require that the trend continues forever.

The fact of the matter is that not accepting the bet of $100 against $110 is the sort of thing homo economicus would do only if they were nigh-starving and losing another $769 or so would completely ruin them. When real non-starving undergrads refuse the bet, they're exhibiting loss aversion and it shouldn't be too surprising that you can find a contrasting bet that will show that they're not following a utility function.

Is loss aversion bad?

One can make a defense of loss aversion as a sort of "population ethics of your future selves." Just as you're allowed to want a future for humanity that doesn't strictly maximize the sum of each human's revealed preferences (you might value justice, or diversity, or beauty to an external observer), you're also "allowed" to want a future for your probailistically-distributed self that doesn't strictly maximize expected value.

But that said... c'mon. Most loss aversion is not worth twisting yourself up in knots to protect. It's intuitive to refuse to risk $100 on a slightly-positive bet. But we're allowed to have intuitions that are wrong.

1. ^{^}

   Bleichrodt et al. (2017)

2. ^{^}

   Which, as is par for the course with names, was probably first mentioned by Arrow, as Rabin notes.

3. ^{^}

   plus rejecting the first bet even if your total wealth was somewhat different

4. ^{^}

    (concave, continuous, state-based)

5. ^{^}

   Rabin (2000)

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