# money ≠ value

The relationship between money and value is not linear or consistent. This means betting odds don’t necessarily correspond to the probability of an event.

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Converting money to value is done by exchanging it for things you want. The value of increasing your wealth by dollars is bounded by the most valuable thing you might use more dollars for. The most valuable thing you might do with more dollars is affected by what you want and what you know how to get.

People’s efficiency at converting money to value changes on short-term and long-term scales, and depends on how much money they have in total.

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We can graph the value to a person of increasing their wealth by dollars from a certain point. This graph is different for different people because different people want different things and can get different things.

In a simplified example, where a person starts with 0 dollars and will never make more money in the future, their graph might look like this:

The slope of the graph increases in the range where the person can barely afford housing, because increasing how much money they are given within this range makes a bigger difference than increasing how much money they are given within other ranges.

In real life people don’t start with 0 dollars and only receive money once. Each increase in money shifts by a little the probability they will buy something they are not sure they can afford. It does not determine with certainty what they can afford.

In real life the function will also be less simple; there will be many places where the rate of value increase per money increase changes, rather than just one that occurs when the person can afford housing.

The function will still be bounded by the most cost-efficient thing they might buy or save to buy, which changes depending on who they are and how much money they are gaining.

People’s intuition about how much they value an increase in money is different from how much they actually value that increase in money. However, this intuition is also not linear*.*

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The value of money becomes linear when the thing you prefer to buy gives you the same incremental value every time no matter how many you bought already.

For example, when saving up to buy something, each increase in money affects the ability to buy it the same amount. This makes the graph of money value linear during the range where extra money will be put towards saving for that thing.

For another example, certain charities take the same action every time they receive new money and that action has the same impact each time it is done. A person who gives to that charity has a graph of money value which is linear during whatever range of wealth that charity is the best use of their resources.

Circumstances like this, which make the graph of the value of money linear within a certain range, do not cause the whole graph to be linear. They also do not cause people’s intuition about the value of money to be linear.

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The idea that betting odds correspond directly to the probability of an event assumes that the graph of money vs value looks like this:

This is not the case.

If a person would be willing to gain $10 if an event occurs and lose $1 if it does not occur, this does not necessarily mean they think the event is 10% likely to occur. They might value a $10 increase 100 times as much as a $1 increase, in which case they must think the event is 1% likely to occur.

The same principle applies for any other two amounts of money someone could bet. Betting odds don’t necessarily correspond to the probability of an event.

In order to determine whether betting odds correspond to the probability of an event, determine whether, given the better’s priorities and circumstances, the ratio between the amount gained/lost in the bet is the same as the ratio between the value gained/lost. This is the same as determining if the graph of the value of wealth increases is linear inside the range where the bet falls.

Most bets of very small amounts of money will not correspond to event probability because people prefer to use their intuition about money value, which is rarely linear. Additionally, for small amounts of money, other factors may dominate the valuation of the money increase, such as looking impressive to an audience.

If someone will bet $1 against your $5 that something will occur, it is often not the case that they think the event is 20% likely to occur. They likely don’t value $5 exactly 5 times as much as $1, because $5 doesn’t give them exactly 5 times as many bragging rights.

Most bets of very large amounts will not correspond to event probability because giving people large sums of money drastically changes the efficiency of trades they can make with their money.

If someone will bet at 1,000,000:1 odds, it is not likely that their probability for the event is 0.000001%. Receiving a million dollars causes them to be able to purchase a different set of things than they could save for previously.

Other bets will not correspond to event probability if they might alter the efficiency with which the better can trade money for value, for example by making a purchase possible sooner in time.

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But wait! If a person valued an increase of $ differently from half as much as $, their preference is inconsistent! They value an increase of $ differently from two increases of $ in a row, and you can exploit this to make money off of them!

No, they don’t. It is possible for them to value the increase of $ differently from half as much as the increase of $ because they don’t expect to receive another $ immediately. They are valuing their ability to use $ by itself, without saving it to purchase something else. If they did expect to use the $ to contribute to a $ purchase, then each $ would be worth exactly half as much to them as the $.

The actual bets I have seen discussed on LW have all been a small proportion of the players’ financial worth. Money behaves linearly on that scale, because everything is linear to first order.

I agree that most of the bets here are accurate indications of probability; most are in the range of 20-1000 dollars, and also there is a culture of honesty that seems like it would prevent offering a bet that didn’t reflect someone’s probability of an event while representing that it did.

The most common case I see where betting odds and probability don’t match is with really small values.

It seems to be encouraged here to spontaneously do friendly betting in person. From my experience, this usually involves pretty small values (0-10 dollars) paid in cash. If I won a small bet like that I might buy a bag of chips, or something, but if I won twice as much I wouldn’t buy two bags of chips, and it probably wouldn’t be worth the effort to save it, so I would mostly just forget..

Another time this comes up is when people bet with fairly high odds ratios, for example $100 to $1. This ends up with pretty low winnings on the $1 side because the alternative is the $100 being way higher, less of a casual amount to bet. A lot of times this might as well be $100 to $0. the median hourly wage is like $17 (in the US) so it’s not even worth 4 minutes of time to get the $1 into your account. what you mostly win is pride.

I also have seen some cases on here of people discussing really big bets, with amounts in the ten thousands/higher, and assuming that the betting odds will still correspond directly to probability.

Mostly I’m trying to say that it seems pretty automatic here to equate between betting odds and probability, and really there are some very common circumstances where this is not the case. (in the spirit of https://www.lesswrong.com/posts/zhRgcBopkR5maBcau/always-know-where-your-abstractions-break)

Similar gripe I have: utility≠money in any fashion. You can have non-differentiable utility out of money. You can value only prime-numbered amounts of money. Your utility function can be negative in money.

Under vNM, utility need only be linear in probability, nothing else.