# $500 Bounty/Contest: Explain Infra-Bayes In The Language Of Game Theory

Here’s my current best guess at how Infra-Bayes works:

We want to get worst-case guarantees for an agent using a Bayesian-like framework.

So, let our agent be a Bayesian which models the environment as containing an adversary which chooses worst-case values for any of the things over which we want worst-case guarantees.

That’s just a standard two-player zero-sum game between the agent and the adversary, so we can import all the nice intuitive stuff from game theory.

… but instead of that, we’re going to express everything in the unnecessarily-abstract language of measure theory and convex sets, and rederive a bunch of game theory without mentioning that that’s what we’re doing.

This bounty is for someone to write an intuitively-accessible infrabayes explainer in game theoretic language, and explain how the game-theoretic concepts relate to the concepts in existing presentations of infra-bayes. In short: provide a translation.

Here’s a sample of the sort of thing I have in mind:

Conceptually, an infrabayesian agent is just an ordinary Bayesian game-theoretic agent, which models itself/its environment as a standard two-player zero-sum game.

In the existing presentations of infra-bayes, the two-player game is only given implicitly. The agent’s strategy solves the problem:

In game-theoretic terms, the “max” represents the agent’s decision, while the “min” represents the adversary’s.

Much of the mathematical tractability stems from the fact that is a convex set of environments (i.e. functions from policy to probability distributions). In game-theoretic terms, the adversary’s choice of strategy determines which “environment” the agent faces, and the adversary can choose from any option in . Convexity of follows from the adversary’s ability to use mixed strategies: because the adversary can take a randomized mix of any two strategies available to it, the adversary can make the agent face any convex combination of (policy → distribution) functions in . Thus, is closed under convex combinations; it’s a convex set.

I’d like a writeup along roughly these conceptual lines which covers as much as possible of the major high-level definitions and results in infra-bayes to date. On the other hand, I give approximately-zero shits about all the measure theory; just state the relevant high-level results in game-theoretic language, say what they mean intuitively, maybe mention whether there’s some pre-existing standard game-theory theorem which can do the job or whether the infra-bayes version of the theorem is in fact the first proof of the game-theoretic equivalent, and move on.

Alternatively, insofar as core parts of infrabayes *differ* from a two-player zero-sum game, or the general path I’m pointing to doesn’t work, an explanation of how they differ and what the consequences are could also qualify for prize money.

## Bounty/Contest Operationalization

Most of the headache in administering this sort of bounty is the risk that some well-intended person will write something which is not at all what I want, expecting to get paid, and then I will either have to explain how/why it’s not what I want (which takes a lot of work), or I have to just accept it. To mitigate that failure mode, I’ll run this as a contest: to submit, write up your explanation as a lesswrong post, then send me a message on lesswrong to make sure I’m aware of it. **Deadline is end of April**. I will distribute money among submissions based on my own highly-subjective judgement. If people write stuff up early, I might leave feedback on their posts, but no promises.

I will count the “sample” above as a submission in its own right—i.e. I will imagine that three-paragraph blurb were instead a three-paragraph post in its own right, and someone submitted it. That will provide a baseline for prizes to be paid out at all: if no submission adds value not already included in the three-paragraph blurb, then the three-paragraph blurb gets the prize money, i.e. I don’t pay anyone.

Note that the $500 prize is probably not enough to fully pay for the amount of effort which I expect will be involved in doing this well. Others are welcome to add to the prize pool; please leave a comment if you’d like to do so.

- Result Of The Bounty/Contest To Explain Infra-Bayes In The Language Of Game Theory by 9 May 2023 16:35 UTC; 79 points) (
- A very non-technical explanation of the basics of infra-Bayesianism by 26 Apr 2023 22:57 UTC; 62 points) (
- Untangling Infrabayesianism: A redistillation [PDF link; ~12k words + lots of math] by 1 Aug 2023 12:42 UTC; 29 points) (

Adding $200 to the pool. Also, I endorse the existence of more bounties/contests like this.

I’ll add $500 to the pot.

adding $200 to the pool.

I wonder what other math distillations are worth funding.

Hmm, update: I’ve heard from someone who looked closely at infrabayes that it might not be that insightful anyway? I do not retract this bounty, but I am now expecting to be disappointed and find that infra-bayes basically simply boils down to game theory in general, rather than identifying new general shapes in the implications.

Adding $80 to the pool.

Edited from $50 to $80 after realizing market price I’m usually willing to pay for excellent math distillations.

Aside: Vanessa mentioned in person at one point that the game-theoretic perspective on infra-bayes indeed basically works, and she has a result somewhere about the equivalence. So that might prove useful, if you’re looking to claim this prize.

Are there any specific results of infra-Bayesiansm that you would like to see derived using the worst case game-theoretical approach?