I only read up to the end of the "Bayesian Updating" part because the next section introduced a few terms I didn't know and seemed to be expected to be familiar with.
So on that section that I read, I can say that I didn't manage to follow it with all the numbers. This might just be because I'm a mathematician and I hate numbers. But formulas would have greatly helped.
This post is going to mostly be propaganda for Kelly betting. However, the reasons presented in this post differ greatly from the reasons people normally use to argue for Kelly betting.
The curse of uncertainty is that I must make decisions that simultaneously affect many different versions of myself. When I close my eyes and then flip a coin, there are two potential versions of me: one sitting in front of a coin showing heads, the other sitting in front of a coin showing tails. Both of these potential versions of me are stakeholders in my current decisions. How can I make decisions on behalf of these multiple stakeholders?
If it is a fair coin, then we can think of these two potential selves as equal stakeholders in my decisions. However, I know that it is not a fair coin. It has a 60 percent chance of coming up heads. Thus, heads-me is a 60 percent stakeholder in my current decisions, and tails-me is a 40 percent stakeholder. They amount of each one's stake is naturally in proportion to the probability that they actually exist.
You, however, do not know if it is a fair coin, and are offering me a fair bet. I only have 100 dollars to my name, and I am can bet as much as I want (up to 100 dollars) in either direction at even odds.
If I bet 100 dollars on heads, heads-me gets 200 dollars, and tails-me gets nothing. If I bet 100 dollars on tails, tails-me gets 200 dollars, and heads-me gets nothing. If I bet nothing, both versions of me get 100 dollars.
However, every dollar in the hands of heads-me is worth 1.5 times as much as a dollar in the hands of tails-me, since heads-me exists 1.5 times as much. (I am ignoring here any diminishing returns in my value of money.)
Thus, to maximize value I should bet 100 dollars on heads. However, maybe it is better to think of tails-me as the rightful owner of 40 percent of my resources. When I bet 100 dollars on heads, I am seizing money from tails-me for the greater good, since heads-me has the (proportionally greater) existence necessary to better take advantage of it.
Alternatively, I could say that since 60 percent of me is heads-me, heads-me should only control 60 dollars, which can be bet on heads. Tails-me should control 40 dollars, which can be bet on tails. These two bets partially cancel each other out, and the net result is that I bet 20 dollars on heads.
If you are especially fast at maximizing expected logarithms, you might see where this is going.
Now, I am ready to introduce my friend, Kelly. Kelly also has her eyes closed, also has 100 dollars, and is sitting in front of the same coin. However, Kelly has different beliefs. Kelly believes that the coin has a 90 percent chance of coming up tails, and Kelly also has 100 dollars.
I bet 20 dollars on heads, for the reasons described above. Kelly bets 80 dollars on tails for similar reasons (90 dollars on tails, partially nullified by 10 dollars on heads).
I have another friend, Marge. Marge is sitting on the other side of the table with her eyes closed. Marge has 200 dollars. Marge doesn't know much about coins, but knows my and Kelly's beliefs, and thinks Kelly and I are equally likely to be correct. Thus, Marge assigns a 65 percent chance that the coin comes up tails. Marge thus bets 60 dollars on tails (130 dollars on tails, partially nullified by 70 dollars on heads).
Note that the 60 dollars bet by Marge is the same as the net 60 dollar bet you get if you draw a box around me and Kelly. This is representing the compositionality of this betting policy. When you draw a box around me and Kelly, you can think of us as one agent whose wealth is the sum of our wealths, and whose beliefs are the weighted (by wealth) average of our beliefs.
If Kelly, Marge and I all implemented the other strategy, of putting all our money on the outcome we thought was most likely, this would not have happened. Marge would have put 200 dollars on tails, while Kelly and I would have, on net, bet nothing.
This should not be surprising. When you implement a majoritarian policy, it matters where you draw the boundaries. When you instead implement a proportional representation policy, It does not matter where you draw the boundaries. When you have an internal voting block, you have to be careful who you let into your voting block, since it might swing the whole block in the other direction. I think many phenomena that get labeled as politics are actually about fighting over where to draw the boundaries. Wouldn't it be nice if we didn't have to worry about where we draw the boundaries?
We all open our eyes, and see that the coin came up heads. I am given 20 dollars, and now have 120 dollars. Kelly loses her 80 dollars, and is left with 20 dollars. Marge loses her 60 dollars, and is left with 140 dollars. Yay! Sorry, Kelly and Marge.
We all close our eyes, and the coin is flipped again, and we are offered the same bets.
I only had one hypothesis, that the coin was a biased coin with a 60 percent chance of coming up heads, so I do not update at all, and will bet similarly again. I have two potential selves, sitting in front of different coins: heads-me has 72 dollars, which are bet entirely on heads, while tails-me has 48 dollars, which are bet entirely on tails. These bets partially cancel out, and on net I bet 24 dollars on heads (20 percent more money than last time, since I have 20 percent more money). Note that these different versions of me are not the same as the ones from last round. There is a new coin flip, so there's a new branch in my future than before. Similarly, Kelly has 20 dollars, and so bets 16 dollars on tails (18 dollars on tails, partially nullified by 2 dollars on heads).
Marge's situation is more complicated. Marge had two different hypotheses about the coin: one in which I am right, and one in which Kelly is right. Marge has observed some Bayesian evidence that I am right, with an odds ratio of 6 to 1. This evidence that I am right translates into evidence that the coin will come up heads. Marge thus updates her 65 percent probability the coin will come up tails to an approximately 53 percent probability the coin will come up heads. (a 37/70 chance of coming up heads, to be exact). Marge then bets exactly 8 of her 140 dollars on heads (74 dollars on heads, partially nullified by 66 dollars on tails).
Again, Marge's bet is exactly the same as the net bet of me and Kelly.
Indeed, whenever Kelly and I bet, you can break this up into a net bet with the house, together with an internal bet that determines how much control we will each have over whatever money our collective ends up with. The internal bet will always exactly implement Bayesian updating on how much the collective trusts each of us.
As a Bayesian agent, you can think of yourself as a collection of bettors that implement this proportional representation betting strategy and bet with each other. Instead of betting with money, they are betting with a currency that represents your posterior beliefs. When used internally, it recreates Bayesian updating. Maybe as a society, we could get some pretty cool results if we also followed this strategy collectively.
The above analysis was a weird case because you were offering both sides of the bet at a fair price. In practice, this is unrealistic. Let's instead look at what happens if I only win 95 cents for each dollar I stake. Heads-me wants to put his 60 dollars on heads, and will win 57 dollars, so I end up with 117 dollars if the coin landed heads. Tails-me put his 40 dollars on tails, so I end up with 78 dollars if the coin landed tails. The fact that I am betting on both sides is wasteful. There is a Pareto improvement where I bet less on both sides, and end up only betting on heads. And, I want to pick up this Pareto improvement.
There was no such Pareto improvement before, because my two selves were essentially in a zero sum game. Every dollar one of them got corresponded to a dollar the other one didn't get. Now, they are in a positive sum game and need to split the gains they get not betting on both sides. (However, if I am Bayesian updating, they might internally bet with each other without paying the 5 percent fee.)
How should I split the gains from trade between my two potential selves? Hmm, if only I had some strategy for fairly distributing utility in a Pareto optimal way when I have uncertainty about who I am.
I will have my different selves Nash bargain! The 0 utility point will be no money. The utility functions will be linear in money, and the distribution on my potential selves will come from the uncertainty I already have.
When I Nash bargain, I end up maximizing the expected logarithm of expected utility. In this case, the outer expectation is over who I am, which I am thinking of as including the state of the coin. Since we moved our uncertainty about the world into our uncertainty about identity, the only thing left in the inner expectation is randomness coming from our action. However, since we can bet continuous amounts of money, and we are treating utility of my various selves as linear in money, we don't have to ever actually randomize, we can just mix between strategies by mixing between our betting amounts. Thus, I end up maximizing the expected logarithm of wealth.
This betting strategy, where you maximize the expected logarithm of your wealth, is known as Kelly betting. In the simplifying example where you can bet on anything, and fairly take either side of any bet (which should be approximately true given sufficiently large markets), it is equivalent to treating your various hypotheses as owning proportional portions of your wealth, which they bet entirely on the world that they are in.
I will leave it as an exercise to try to get an intuitive understanding for why maximizing the expected logarithm might be deeply entangled with proportional representation. *coughlogscoringrulecough* *coughminimizingcrossentropycough*
Again, this is not the standard argument for Kelly betting. The standard argument is very good, and is basically that (roughly) if you don't Kelly bet, then after enough time, you will with probability approaching 1 have less money than if you did Kelly bet.
There is a nice parallel between what happens when you don't Kelly bet and when you don't Nash bargain. When you maximize expected wealth, you end up with more money in expectation, unfortunately all that money ends up in the same world, which over time has smaller and smaller probability. In all other worlds, you are left with nothing. This would be fine if you had some channel to transfer the wealth from the one tiny world to all the other worlds, but you don't, so you just end up broke with probability 1.
Similarly, when you maximize total utility, rather than Nash bargaining, you end up with more total utility, but you might end up devoting all of your resources to one utility monster. This would be fine if you could transfer that utility to everyone else, but you can't, so almost everyone might end up with nothing.
Many claim that even Kelly betting is not risk averse enough. One major alternative considered is fractional Kelly betting. For example in half Kelly betting, you bet half as much as you would if you were Kelly betting. This may seem like a hack, but I think it kind of makes sense.
Let's say that I maintain two different probability distributions. Society has their market probability distribution P, which is updated using who-knows-what. I have my inside view probabilities Q, which I try to update Bayesianly, but am obviously not perfect. However, I also have my outside view probabilities Q′. My inside view might be right, or the market might be right, so let's average between them. Q′=12Q+12P. I want to keep my inside view and my outside view separate. I use my inside view to think, and I use my outside view to bet.
What happens when I Kelly bet according to my outside view? If you think of Kelly betting as maximizing an expected logarithm, you might start doing some crazy computations, but if you have been following this post thus far, you can just say:
I am the sum of two agents each with half my wealth. The first Kelly bets according to my inside view, and the second doesn't bet at all. This I bet half as much as I would if I were Kelly betting according to my inside view. Isn't compositionality nice?
So, why isn't half Kelly betting just thought of as Kelly betting with different beliefs? The difference is in the updating. I do not update my outside view in a Bayesian way. I update my inside view in a Bayesian way, but I maintain the fact that I think there is a 50 percent chance the market is right instead of me. This is in spite of the observation that I seem to be making money. If on round one I make money, and on round two, I still only make half a Kelly bet, I am choosing to defer to the market more than a Bayesian update on my outside view would suggest.
I am subsidizing my deference to the market by doing a wealth transfer, from my inside view to my deference to the market. Given that I do not fully trust my reasoning and my ability to update my inside view correctly, this seems not entirely crazy to me, and I think it makes more sense when thought of as market deference than when thought of as just cutting my bet in half to be conservative.
People sometimes get confused looking at the standard argument for Kelly betting, and say "My utility is already logarithmic in dollars, Shouldn't I bet so as to maximize my expected log log wealth?" Firstly, your utility is not logarithmic in dollars. Utilities are bounded. But secondly, according to the standard argument, the answer is no. If you make enough bets, and continue disagreeing in the market, in the long run, you will, with probability approaching 1, wish you maximized expected log wealth.
However, the arguments in this post are not about repeated bets. They are about respecting your epistemic subagents, and apply even if you only make one bet. If you have utility proportional to the logarithm of 1 dollar plus your wealth, and you Nash bargain across all your possible selves, you end up approximately maximizing the expected logarithm of the logarithm of 1 dollar plus your wealth. (I had to add in the dollar to avoid negative infinity madness.)
I would be careful here, though. I am not sure I endorse going this far. You are sacrificing Bayesian compositionality niceness, and I am not sure exactly what kind of introspecting I would have to do to verify that I actually have preferences logarithmic in wealth, and do not just think that I do because I have Kelly betting intuitions hard coded into me. Anyway, be careful, but again, not entirely crazy to me.
Just wanted to say that I thought this post was very interesting and I was grateful to read it.