1. Money wants to be linear, but wants even more to be logarithmic.

• People sometimes talk as if risk-aversion (or risk-loving) is irrational in itself. It is true that VNM-rationality implies you just take expected values, and hence, don't penalize variance or any such thing. However, you are allowed to have a concave utility function, such as utility which is logarithmic in money. This creates risk-averse behavior. (You could also have a convex utility function, creating risk-seeking behavior.)
  • Counterpoint: if you have risk-averse behavior, other agents can exploit you by selling you insurance. Hence, money flows from risk-averse agents to less risk-averse agents. Similarly, risk-seeking agents can be exploited by charging them for participating in gambles. From this, one might think a market will evolve away from risk aversion(/seeking), as risk-neutral agents accumulate money.
• People clearly act more like money has diminishing utility, rather than linear utility. So revealed preferences would appear to favor risk-aversion. Furthermore, it's clear that the amount of pleasure one person can get per dollar diminishes as we give that person more and more money.
  • On the other hand, that being the case, we can easily purchase a lot of pleasure by giving money to others with less. So from a more altruistic perspective, utility does not diminish nearly so rapidly.
• Rationality arguments of the Dutch-book and money-pump variety require an assumption that "money" exists. This "money" acts very much like utility, suggesting that utility is supposed to be linear in money. Dutch-book arguments assume from the start that agents are willing to make bets if the expected value of those bets is nonnegative. Money-pump arguments, on the other hand, can establish this from other assumptions.
  • Stuart Armstrong summarizes the money-pump arguments in favor of applying the VNM axioms directly to real money. This would imply risk-neutrality and utility linear in money.
• On the other hand, the Kelly criterion implies betting as if utility were logarithmic in money.
  • The Kelly criterion is not derived via Bayesian rationality, but rather, an asymptotic argument about average-case performance (which is kinda frequentist). So initially it seems this is no contradiction.
  • However, it is a theorem that a diverse market would come to be dominated by Kelly bettors, as Kelly betting maximizes long-term growth rate. This means the previous counterpoint was wrong: expected-money bettors profit in expectation from selling insurance to Kelly bettors, but the Kelly bettors eventually dominate the market.
  • Expected-money bettors continue to have the most money in expectation, but this high expectation comes from increasingly improbable strings of wins. So you might see an expected-money bettor initially get a lot of money from a string of luck, but eventually burn out.
    • (For example, suppose an investment opportunity triples money 50% of the time, and loses it all the other 50% of the time. An expected money bettor will go all-in, while a Kelly bettor will invest some money but hold some aside. The expected-money betting strategy has the highest expected value, but will almost surely be out in a few rounds.)
• The kelly criterion still implies near-linearity for small quantities of money.
  • Moreover, the more money you have, the closer to linearity -- so the larger the quantity of money you'll treat as an expected-money-maximizer would.
  • This vindicates, to a limited extent, the idea that a market will approach linearity -- Kelly bettors will act more and more like expected-money maximizers as they accumulate money.
  • As argued before, we get agents with a large bankroll (and so, with behavior closer to linear) selling insurance to Kelly agents with smaller bankroll (and hence more risk-averse), and profiting from doing so.
  • But everyone is still Kelly in this picture, making logarithmic utility the correct view.
• So the money-pump arguments seem to almost pin us down to maximum-expectation reasoning about money, but actually leave enough wiggle room for logarithmic value.
• If money-pump arguments for expectation-maximization doesn't apply in practice to money, why should we expect it to apply elsewhere?
  • Kelly betting is fully compatible with expected utility maximization, since we can maximize the expectation of the logarithm of money. But if the money-pump arguments are our reason for buying into the expectation-maximization picture in the first place, then their failure to apply to money should make us ask: why would they apply to utility any better?
  • Candidate answer: utility is defined as the quantity those arguments work for. Kelly-betting preferences on money don't actually violate any of the VNM axioms. Because the VNM axioms hold, we can re-scale money to get utility. That's what the VNM axioms give us.
    • The VNM axioms only rule out extreme risk-aversion or risk-seeking where a gamble between A and B is outside of the range of values from A to B. Risk aversion is just fine if we can understand it as a re-scaling.
    • So any kind of re-scaled expectation maximization, such as maximization of the log, should be seen as a success of VNM-like reasoning, not a failure.
    • Furthermore, thanks to continuity, any such re-scaling will closely resemble linear expectation maximization when small quantities are involved. Any convex (risk-averse) re-scaling will resemble linear expectation more as the background numbers (to which we compare gains and losses) become larger.
    • It still seems important to note again, however, that the usual justification for Kelly betting is "not very Bayesian" (very different from subjective preference theories such as VNM, and heavily reliant on long-run frequency arguments).

2. Money wants to go negative, but can't.

• Money can't go negative. Well, it can, just a little: we do have a concept of debt. But if the economy were a computer program, debt would seem like a big hack. There's no absolute guarantee that debt can be collected. There are a lot of incentives in place to help ensure debt can be collected, but ultimately, bankruptcy or death or disappearance can make a debt uncollectible. This means money is in this weird place where we sort of act like it can go negative for a lot of purposes, but it also sort of can't.
  • This is especially weird if we think of money as debt, as is the case for gold-standard currencies and similar: money is an IOU issued by the government, which can be repaid upon request.
  • Any kind of money is ultimately based on some kind of trust. This can include trust in financial institutions, trust that gold will still be desirable later, trust in cryptographic algorithms, and so on. But thinking about debt emphasizes that a lot of this trust is trust in people.
• Money can have a scarcity problem.
  • This is one of the weirdest things about money. You might expect that if there were "too little money" the value of money would simply re-adjust, so long as you can subdivide further and the vast majority of people have a nonzero amount. But this is not the case. We can be in a situation where "no one has enough money" -- the great depression was a time when there were too few jobs and too much work left undone. Not enough money to buy the essentials. Too many essentials left unsold. No savings to turn into loans. No loans to create new businesses. And all this, not because of any change in the underlying physical resources. Seemingly, economics itself broke down: the supply was there, the demand was there, but the supply and demand curves could not meet.
    • (I am not really trained in economics, nor a historian, so my summary of the great depression could be mistaken or misleading.)
  • My loose understanding of monetary policy suggests that scarcity is a concern even in normal times.
• The scarcity problem would not exist if money could be reliably manufactured through debt.
  • I'm not really sure of this statement.
  • When I visualize a scarcity of money, it's like there's both work needing done and people needing work, but there's not enough money to pay them. Easy manufacturing of money through debt should allow people to pay other people to do work.
  • OTOH, if it's too easy to go negative, then the concept of money doesn't make sense any more: spending money doesn't decrease your buying power any more if you can just keep going into debt. So everyone should just spend like crazy.
    • Note that this isn't a problem in theoretical settings where money is equated with utility (IE, when we assume utility is linear in money), because money is being inherently valued in those settings, rather than valued instrumentally for what it can get. This assumption is a convenient approximation, but we can see here that it radically falls apart for questions of negative bankroll -- it seems easy to handle (infinitely) negative money if we act like it has intrinsic (terminal) value, but it all falls apart if we see its value as extrinsic (instrumental). 
  • So it seems like we want to facilitate negative bank accounts "as much as possible, but not too much"?
• Note that Dutch-book and money-pump arguments tend to implicitly assume an infinite bankroll, ie, money which can go negative as much as it wants. Otherwise you don't know whether the agent has enough to participate in the proposed transaction.
  • Kelly betting, on the other hand, assumes a finite bankroll -- and indeed, might have to be abandoned or adjusted to handle negative money.
  • I believe many mechanism-design ideas also rely on an infinite bankroll.