The log-returns are not linear in the bits. (They aren’t even constant for a given level of bits.)
For example, say the market is 1:1, and you have 1 bit of information: you think the odds are 1:2, then Kelly betting you will bet 1⁄3 of your bankroll and expect to make a ~20% log-return.
Say the market was 1:2, and you had 1 bit of information: you think the odds are 1:4, then Kelly betting, you will bet 1⁄5 of your bankroll and expect to make a ~27% log-return.
We’ve already determined that quite different returns can be obtained for the same amount of information.
To get some idea of what’s going on, we can plot various scenarios, how much log-returns does the additional bit (or many bits) generate for a fixed market probability. (I’ve plotted these as two separate charts—one where the market already thinks an event is evens or better,
For example, we can spot where the as the market thinks an event is more likely (first chart, 0 bits in the market, 1 bit in the market, etc) and looking at the point where we have 1 bit, we can where we expect to make 20%, then 27% [the examples I worked through first], and more-and-more as the market becomes more confident.
The fact that 1 bit has relatively little value when the market has low probability is also fairly obvious. When there are fairly long odds, 2^(n+1):1 and you think it should be 2^n:1, you will bet approximately(!) 1/2^n, to approximately(!) either double your bankroll or leave it approximately(!) unchanged. So the log-return of these outcomes is capped, but the probability of success keeps declining
There’s also a meta-question here which is—why would you expect a strict relationship between bits and return. Ignoring the Kelly framework, we could just look at the expected value of 1 bit of information. The expected value of a market currently priced at q, which should be priced at p is p-q.
When the market has no information (it’s 1:1) a single bit is valuable, but when a market has lots of information the marginal bit doesn’t move the probabilities as much (and therefore the expected value is smaller)
This is similar to the answer I got from o1-preview in ChatGPT when I originally asked with OP’s post as the text, so that’s pleasant to see. (I didn’t post anything here because I was unsure and wasn’t checking it in enough detail to repost, and so didn’t believe in publishing it without being able to improve it.)
I thought there might be some relationship at first with an appropriate transformation, but when I recalled how Kelly requires both edge and net worth, and the problem of frequency of payoffs, I lost my confidence that there would be any simple elegant relationship beyond a simple ‘more information = more returns’. Why indeed would you expect 1 bit of information to be equally valuable for maximizing expected log growth in eg. both a 50:50 shot and a 1,000,000,000:1 shot? Or for a billionaire vs a bankrupt? (Another way to think of it: suppose you have 1 bit of information on both over the market and you earn the same amount. How many trades would it take before your more informed trade ever made a difference? In the first case, you quickly start earning a return and can compound that immediately; in the second case, you might live a hundred lives without ever once seeing a payoff.)
The log-returns are not linear in the bits. (They aren’t even constant for a given level of bits.)
For example, say the market is 1:1, and you have 1 bit of information: you think the odds are 1:2, then Kelly betting you will bet 1⁄3 of your bankroll and expect to make a ~20% log-return.
Say the market was 1:2, and you had 1 bit of information: you think the odds are 1:4, then Kelly betting, you will bet 1⁄5 of your bankroll and expect to make a ~27% log-return.
We’ve already determined that quite different returns can be obtained for the same amount of information.
To get some idea of what’s going on, we can plot various scenarios, how much log-returns does the additional bit (or many bits) generate for a fixed market probability. (I’ve plotted these as two separate charts—one where the market already thinks an event is evens or better,
For example, we can spot where the as the market thinks an event is more likely (first chart, 0 bits in the market, 1 bit in the market, etc) and looking at the point where we have 1 bit, we can where we expect to make 20%, then 27% [the examples I worked through first], and more-and-more as the market becomes more confident.
The fact that 1 bit has relatively little value when the market has low probability is also fairly obvious. When there are fairly long odds, 2^(n+1):1 and you think it should be 2^n:1, you will bet approximately(!) 1/2^n, to approximately(!) either double your bankroll or leave it approximately(!) unchanged. So the log-return of these outcomes is capped, but the probability of success keeps declining
There’s also a meta-question here which is—why would you expect a strict relationship between bits and return. Ignoring the Kelly framework, we could just look at the expected value of 1 bit of information. The expected value of a market currently priced at q, which should be priced at p is p-q.
When the market has no information (it’s 1:1) a single bit is valuable, but when a market has lots of information the marginal bit doesn’t move the probabilities as much (and therefore the expected value is smaller)
This is similar to the answer I got from
o1-previewin ChatGPT when I originally asked with OP’s post as the text, so that’s pleasant to see. (I didn’t post anything here because I was unsure and wasn’t checking it in enough detail to repost, and so didn’t believe in publishing it without being able to improve it.)I thought there might be some relationship at first with an appropriate transformation, but when I recalled how Kelly requires both edge and net worth, and the problem of frequency of payoffs, I lost my confidence that there would be any simple elegant relationship beyond a simple ‘more information = more returns’. Why indeed would you expect 1 bit of information to be equally valuable for maximizing expected log growth in eg. both a 50:50 shot and a 1,000,000,000:1 shot? Or for a billionaire vs a bankrupt? (Another way to think of it: suppose you have 1 bit of information on both over the market and you earn the same amount. How many trades would it take before your more informed trade ever made a difference? In the first case, you quickly start earning a return and can compound that immediately; in the second case, you might live a hundred lives without ever once seeing a payoff.)