Your definition of the Heaviside step function has H(0) = 1.
Your definition of L has L(0) = 1⁄2, so you’re not really taking the derivative of the same function.
I don’t really believe nonstandard analysis helps us differentiate the Heaviside step function. You have found a function that is quite a lot like the step function and shown that it has a derivative (maybe), but I would need to be convinced that all functions have the same derivative to be convinced that something meaningful is going on. (And since all your derivatives have different values, this seems like a not useful definition of a derivative)
SimonM
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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 decliningThere’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)
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.
This is exactly the first thing I bring up when people talk about this.
But counter-caveat: you don’t actually need a range of $1,000,000,000. 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.
I don’t think so? The 769 limit is coming from never accepting the 100⁄110 bet at ANY wealth, which is a silly assumption
Thus, an attacker, knowing this, could only reasonably expect to demand half the amount to get paid.
Who bears the cost of a tax depends on the elasticities of supply and demand. In the case of a ransomware attack, I would expect the vast majority of the burden to fall on the victim.
I wrote about exactly this recently- https://www.lesswrong.com/posts/zLnHk9udC28D34GBB/prediction-markets-aren-t-magic
I don’t give much weight to his diagnosis of problematic group decision mechanisms
I have quite a lot of time for it personally.
The world is dominated by a lot of large organizations that have a lot of dysfunction. Anybody over the age of 40 will just agree with me on this. I think it’s pretty hard to find anybody who would disagree about that who’s been around the world. Our world is full of big organizations that just make a lot of bad decisions because they find it hard to aggregate information from all the different people.
This is roughly Hanson’s reasoning, and you can spell out the details a bit more. (Poor communication between high level decision makers and shop-floor workers, incentives at all levels dissuading truth telling etc). Fundamentally though I find it hard to make a case this isn’t true in /any/ large organization. Maybe the big tech companies can make a case for this, but I doubt it. Office politics and self-interest are powerful forces.
For employment decisions, it’s not clear that there is usable (legally and socially tolerated) information which a market can provide
I roughly agree—this is the point I was trying to make. All the information is already there in interview evaluations. I don’t think Robin is expecting new information though—he’s expecting to combine the information more effectively. I just don’t expect that to make much difference in this case.
Hiring decisions are not suitable for prediction markets
So the first question is: “how much should we expect the sample mean to move?”.
If the current state is , and we see a sample of (where is going to be 0 or 1 based on whether or not we have heads or tails), then the expected change is:
In these steps we are using the facts that ( is independent of the previous samples, and the distribution of is Bernoulli with . (So and ).
To do the proper version of this, we would be interested in how our prior changes, and our distribution for wouldn’t purely be a function of . This will reduce the difference, so I have glossed over this detail.
The next question is: “given we shift the market parameter by , how much money (pnl) should we expect to be able to extract from the market in expectation?”
For this, I am assuming that our market is equivalent to a proper scoring rule. This duality is laid out nicely here. Expending the proper scoring rule out locally, it must be of the form , since we have to be at a local minima. To use some classic examples, in a log scoring rule:
in a brier scoring rule:
Whoops. Good catch. Fixing
x is the result of the (n+1)th draw sigma is the standard deviation after the first n draws pnl is the profit and loss the bettor can expect to earn
Prediction markets generate information. Information is valuable as a public good. Failure of public good provision is not a failure of prediction markets.
I think you’ve slightly missed my point. My claim is narrower than this. I’m saying that prediction markets have a concrete issue which means you should expect them to be less efficient at gathering data than alternatives. Even if information is a public good, it might not be worth as much as prediction markets would charge to find that information. Imagine if the cost of information via a prediction market was exponential in the cost of information gathering, that wouldn’t mean the right answer is to subsidise prediction markets more.
If you have another suggestion for a title, I’d be happy to use it
Even if there is no acceptable way to share the data semi-anonymously outside of match group, the arguments for prediction markets still apply within match group. A well designed prediction market would still be a better way to distribute internal resources and rewards amongst competing data science teams within match group.
I used to think things like this, but now I disagree, and actually think it’s fairly unlikely this is the case.
Internal prediction markets have tried (and failed) at multiple large organisations who made serious efforts to create them
As I’ve explained in this post, prediction markets are very inefficient at sharing rewards. Internal to a company you are unlikely to have the right incentives in place as much as just subsidising a single team who can share models etc. The added frictions of a market are substantial.
The big selling points of prediction markets (imo) come from:
Being able to share results without sharing information (ie I can do some research, keep the information secret, but have people benefit from the conclusions)
Incentivising a wider range of people. At an orgasation, you’d hire the most appropriate people into your data science team and let them run. There’s no need to wonder if someone from marketing is going to outperform their algorithm.
People who actually match and meetup with another user will probably have important inside view information inaccessible to the algorithms of match group.
I strongly agree. I think people often confuse “market” and “prediction market”. There is another (arguably better) model of dating apps which is that the market participants are the users, and the site is actually acting as a matching engine. Since I (generally) think markets are great, this also seems pretty great to me.
Sure—but that answer doesn’t explain their relative lack of success in other countries (eg the UK)
Additionally, where prediction markets work well (eg sports betting, political betting) there is a thriving offshore market catering to US customers.
Prediction Markets aren’t Magic
This post triggered me a bit, so I ended up writing one of my own.
I agree the entire thing is about how to subsidise the markets, but I think you’re overestimating how good markets are as a mechanism for subsidising forecasting (in general). Specifically for your examples:
Direct subsidies are expensive relative to the alternatives (the point of my post)
Hedging doesn’t apply in lots of markets, and in the ones where it does make sense those markets already exist. (Eg insurance)
New traders is a terrible idea as you say. It will work in some niches (eg where there’s lots of organic interest, but it wont work at scale for important things)
I’m excited about the potential of conditional prediction markets to improve on them and solve two-sided adverse selection.
This applies to roughly the entire post, but I see an awful lot of magical thinking in this space. What is the actual mechanism by which you think prediction markets will solve these problems?
In order to get a good prediction from a market you need traders to put prices in the right places. This means you need to subsidise the markets. Whether or not a subsidised prediction market is going to be cheaper for the equivalent level of forecast than paying another 3rd party (as is currently the case in most of your examples) is very unclear to me
A thing Larry Summers once said that seems relevant, from Elizabeth Warren:
He said something very similar to Yanis Varoufakis (https://www.globaljustice.org.uk/blog/2017/06/when-yanis-met-prince-darkness-extract-adults-room/) and now I like to assume he goes around saying this to everyone
No, it’s fairly straightforward to see this won’t work
Let N be the random variable denoting the number of rounds. Let x = p*w+(1-p)*l where p is probability of winning and w=1-f+o*f, l=1-f the amounts we win or lose betting a fraction f of our wealth.
Then the value we care about is E[x^N], which is the moment generating function of X evaluated at log(x). Since our mgf is increasing as a function of x, we want to maximise x. ie our linear utility doesn’t change
The original post also addresses this suggestion