# Rabin's Paradox

Quick psychology experiment
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Right now, if I offered you a bet[^n7f4fxrkisq] 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:

About 90% of undergrads will reject this bet[^jxj38sr3lz].

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[^fib2q97a2xw]:

If you rejected the first bet and accepted the second bet, just that[^uebxuam0bc] is enough to rule you out from having any[^k4cwkyrptp] utility function consistent with your decisions.[^3ofk44j1alt]

What? How?
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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 engaging in the bet is less than the status quo: 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 on average.

> $f\equiv e^{-kw}$
> 
> $\frac{1}{200}\int_{-100}^0 f + \frac{1}{220}\int_{0}^{110} f = f(0)$
> 
> $\frac{1}{200k}(e^{100k}-1) + \frac{1}{220k}(1-e^{-110k}) = 1$
> 
> The step of the calculation where you plug it into wolframalpha
> 
> $k\approx0.0013$

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 $100,000? 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 to refuse the bet up to $1,000,000,000. The mathematical argument above says that on the domain you reject the bet, your hypothetical marginal of utility of money would be decreasing at least as fast on average as $e^{-0.0013w}$. If you start accepting the bet once you have $100,000, then what we infer is that this hypothetical maximum average marginal utility decreases exponentially up to $100,000.

We even know a little bit more: above $100,000 the marginal utility of money is still bounded to be less than what it was *at* $100,000 (assuming there's no special amount of money where money suddenly becomes more valuable to you). If your starting wealth was, say, $50,000, then the exponential decay up to $100,000 has already shrunk the marginal utility of money past that by $5\cdot 10^{-29}$!$$

So the paradox still works perfectly well if you'll only reject the first bet until you have $5000 or $10,000 more. 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?
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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 probalistically-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.

[^n7f4fxrkisq]: If you're curious about the experimental details, suppose that you've signed up for a psychology experiment, and at the start of it I've given you $100, had you play a game to distract you so you feel like the money is yours, and then asked you if you want to bet that money. 

[^jxj38sr3lz]: Bleichrodt et al. (2017) (Albeit for smaller stakes and with a few extra wrinkles to experimental design) 

[^fib2q97a2xw]: Which, as is par for the course with names, was probably first mentioned by Arrow, as Rabin notes. 

[^uebxuam0bc]: plus rejecting the first bet even if your total wealth was somewhat different 

[^k4cwkyrptp]:  (concave, continuous, state-based) 

[^3ofk44j1alt]: Rabin (2000)