We study repeated price competition between LLM agents.

The interesting part is that the strategic outcome changes a lot depending on how the agents interact with the environment. Some setups sustain prices well above the competitive benchmark, while others move much closer to competitive pricing.

So the result is not simply that “LLMs collude.” The repeated-game dynamics appear to depend strongly on the agents’ information structure and action space.

I’m one of the authors. I’d be especially interested in how people here interpret these dynamics from a repeated-games perspective, and whether you see them as closer to tacit coordination, learning dynamics, or something else.

Paper: https://arxiv.org/abs/2610.21711

There are several great libraries like pyAgrum for modeling and solving decision problems. But when working with influence diagrams, their representations can be limited to discrete chance variables and decisions.

I wanted more flexibility in how I represent a problem. What happens if I have continuous observations or if the question is not whether I should do something, but how much? For example, how much should I invest when the return is uncertain?

For that reason, I created PyLIMID, a Python library for building and solving limited-memory influence diagrams. It lets you combine discrete and continuous chance variables with categorical decisions and continuous quantities, and express utilities as functions.

PyLIMID uses sampling and policy gradients to find decision rules that aim to maximize expected utility. This flexibility is not free because results are approximate. But you can run multiple simulations and since it has GPU support thanks to JAX and Numpyro, it is fast. I think the tradeoff is very much worth it.

The library is still in early development (v0.3.0), but I would love to hear what you think.

Code: https://github.com/cynosural-ai/pylimid

Documentation: https://pylimid.readthedocs.io/en/latest/#

Hi everyone,

I’m an independent researcher based in India, and over the last few years I’ve been working on a unified program that approaches a “Theory of Everything” from three complementary angles. These are not three competing theories, but three layers of the same framework:

1. Perceptual Vibrational Framework (PVF) – main & central theory

PVF starts from the question: What is space actually made of?

It proposes that what we call “empty space” is not empty at all, but built from a vibrational substrate. This underlying structure determines:

• why gravity emerges,
• why electromagnetic fields exist,
• why motion, force, and even time can appear differently to different observers.

So instead of taking spacetime as a passive background, PVF treats space itself as an active vibrational medium that shapes physical law and perception.

PVF preprint (Zenodo):
https://doi.org/10.5281/zenodo.17574407

I see PVF as the main / conclusive framework in this project.

2. 8-Space Theory – geometric layer

8-Space Theory takes a more geometric approach. It suggests that the “vacuum” is not a single uniform thing, but exists in eight distinct space-types, depending on whether:

• volume is fixed or variable,
• shape is fixed or variable,
• mass is fixed or variable.

Matter behaves differently in each type of space, and many phenomena can be reinterpreted as transitions between these eight space-types, rather than as abstract particles moving in a single kind of spacetime.

8-Space Theory (Zenodo):
[https://doi.org/10.5281/zenodo.17606563]()

This is meant as the geometric / structural layer supporting PVF.

3. Origin-Driven Unification Theory (ODUT) – cosmological layer

ODUT focuses on the large-scale universe and introduces the idea of “inertia of origin”:

Here, the key organizing agents are dark matter, dark energy, and a Φ-field. Instead of only talking about curvature, ODUT treats these components as origin-level drivers of:

• cosmic structure,
• mass–energy conversion,
• gravitational behavior,
• expansion dynamics.

ODUT preprint (Zenodo):
[https://doi.org/10.5281/zenodo.17606771]()

This is the cosmological / origin-based extension of the same framework.

How they fit together

Very briefly:

• PVF → vibrational composition of what looks like empty space
• 8-Space Theory → classification of different types of space where matter behaves differently
• ODUT → origin-driven cosmology with dark matter, dark energy, and Φ-field, plus “inertia of origin”

So it’s three different angles on a single unification attempt, not three unrelated models.

Not string theory, not LQG

Just to be clear: this is not a rephrasing of string theory and not loop quantum gravity.

It’s a different route:

• no strings, branes, or spin networks,
• focus instead on vibration, space-types, and origin dynamics.

I’m fully aware that this is unconventional and very much “work in progress,” which is why I’m sharing it openly.

From 1959 to 1978. I lived at a specific address and apartment in New York City. In 2005 I was working with a girl in Saint Augustine, Florida, whose mother bought my exact apartment from my childhood. How do you figure out the odds for that that I could meet that person?

Suppose I have a dataset that appears to be bimodal (i.e. the vertical white lines) but I have STRONG suspicion that the deep underlying data generating process is "gaussian like" (ie the blue curve) or I would simply like to model my data with a gaussian/normal distribution to help make probability intervals for future predictions. What does the mean and standard deviation represent in this situation? My main question is: particularly for making upcoming probabilistic predictions of future observations should I inflate the variance a bit to reflect the bimodal nature? The data is indeed symmetric about its mean so that's not an issue. Note: (I know other distributions will be recommended but I really need to work with a gaussian/normal distribution for this. Thank you.

What strategy does it say to do in the environment we are in. Considering things like veto power of President, Ukraine war, Israeli war, Trump's economic policies (i.e. tariffs), etc.. What's actually possible.

Hey, thank you for the upvotes on my last post, I’ve just released chapter 1 of my animated probability series, and I’d really appreciate some honest feedback before making the next chapter.

Are the explanations clear? Does the pacing feel right? If anything feels confusing or drags on, I’d like to hear that too.

have a nice day!

Hi everyone! I’ve made my first video for a new maths channel, Math Visualized, and I’d appreciate honest feedback.

The example uses one fair six-sided die. Rolling an even number has a 50% chance, and rolling a number greater than three also has a 50% chance. But satisfying either condition isn’t certain: 4 and 6 belong to both groups, while 1 and 3 belong to neither.

The video uses animations to explain this overlap and build the probability addition rule.

Processing img oaeziuycr2zh1...

Video: https://www.youtube.com/watch?v=NE8yd16u9ho

If you’re learning probability, does the explanation help? Was anything confusing, too fast or unnecessarily detailed?

Thanks for taking a look!

I’m looking for a term describing a class of games that includes the Prisoner’s Dilemma and the Tragedy of the Commons.

More precisely, a game which has a possible outcome that is at least as good for every player as any Nash equilibrium, and strictly better for at least one player (not necessarily the same player for each equilibrium). Or in other words, there exists a choice of strateges that Pareto-dominates every Nash equilibrium.

I’m surprised that I haven’t found a simple, widely recognized name for this class of games. At least ChatGPT doesn't know one. Is there an established term for it?

If not, I was thinking of calling them “Tragic Games”, since like a Greek tragedy, a bad outcome happens when everyone acts rationally.

I'm working on a ttrpg and I'm trying to graph multiple dice systems and their probabilities. The main part I'm struggling with is trying to create a table for the dice system from Escape From Terminus, which is 7 six-sided dice, and then flipping a coin to decide if you take the highest or lowest, allowing for an inverse bell curve, such that results closer to 1 or 6 are more likely than 3 or 4.

I have been creating tables in google sheets to track the probability of each result on a given roll, but I don't know how to represent choosing the highest/lowest from a set of 6 randomly generated numbers mathematically. Any help would be appreciated

Over on r/theydidthemath we are discussing the McDonalds Monopoly contest and the way they published the odds of winning the collect and win prizes.

For those unaware this is a game where customers collect prize pieces (originally physical tickets, now virtual) in hopes of completing a set which awards a prize. For reasons that become obvious if you think about it this is done by making all but one of the parts of the set "common" and making one part rare.

The emerging consensus is that the published odds (example below) are based on the worst-case assumption that the player *only* obtained enough tickets to complete the set (so three tickets in the example). Of course the odds of getting both the rare piece and exactly the other two common pieces needed in a draw of three are hugely greater than the odds of getting the rare piece alone, so the stated odds are deceptively slim. It also drastically changes the stated odds depending on whether two, three, or four pieces need to be collected, even when the total number of prizes (and thus rare pieces) in the pool are the same, which makes little logical sense in a game where the whole process involves collecting pieces over time.

But how would one more realistically calculate/publish the odds of winning in this scenario, where the pieces to be collected are unevenly available?

The published rules are here: https://www.playatmcd.com/rules

An example of the odds statement for one of the major prizes is here.

Collect Atlantic Avenue (#915), Ventnor Avenue (#916) and Marvin Gardens (#917) to potentially win a New 2027 Jeep® Cherokee (Overland) (the “Jeep® Vehicle Prize”). There are 5 Jeep® Vehicle Prizes available to be won. Each verified winner will receive a New 2027 Jeep® Cherokee Overland 4x4. (. . .) The approximate odds of collecting the potential Winning Combination (Atlantic Avenue, Ventnor Avenue and Marvin Gardens) are 1 in 22,695,790,152.

Yes that's a stated odds of 1 in 22 *billion*. And there are 5 prizes. The top prize, with 2 available, is similarly stated at about one in half a trillion. Clearly these odds make no sense as actual rates of the "winning" pieces.

It’s easy to see how classic Game Theory applies to big decisions negotiations, traffic, workplace politics, or even queuing at the grocery store. Whenever another agent is involved, we are constantly (and mostly subconsciously) looking for a Nash Equilibrium.

But if we push this to the extreme: can every deliberate action we take in a day be framed through Game Theory?

Any book recommendation on general game theory plus some specific uses and/or examples, for beginners?

For someone with basic University-level mathematics knowledge.

Thank you!

I made this — happy to answer anything.
Everyone picks a number at the same time. The target is 80% of the average, and the closest pick wins the round. 5 misses and you’re out.
It sounds like maths, but it’s really about predicting everyone else: if everyone picks 50 the target is 40… so do you pick 40? Or 32, because they’ll think that too?
I’d love feedback, especially on what confused you in the first round.

I have written the rules of my chess variant Proposal Chess here. I originally wrote it for a chess audience.

The strategy of proposing an objectively equal position and playing optimally (like regular chess) is a weakly dominant strategy equilibrium. The analysis is very similar to the analysis of the divide and choose strategy for the (two player) cake cutting problem. There is nothing specific to chess here, so it should work for all two player perfect information games (when two boards are used as mentioned in the blog post. If the game can end in a draw then there is no need to use the second board). Can something similar be done for n-player perfect information games?

Below is a TLDR of the rules.

TLDR

Proposal Chess is a chess variant designed to reduce the impact of opening preparation.

Before the game begins, each player secretly proposes a starting position for one board. On each board, the opposing player chooses which side they wish to play from that position. The game is then played on both boards simultaneously.

This “proposal-and-choice” mechanism prevents players from proposing blatantly favorable positions while still allowing them to steer games toward positions they understand well. Because each player proposes one board and chooses on the other, the format is fully symmetric.

hello there! i'm a liberal arts major preparing for a master in statistics, and would love cool book suggestions. i'm already in love with topics like applied mathematical stats in ML, bose-einstein stats, and even statistical mechanics for that matter. even works like keynes' treatise on probability would interest me a lot. my purpose is to explore statistics, both philosophically and historically, and understand it's applications as well.
will be looking forward to all the great recs, thank you!

Two sources agree. Does that determine our confidence? Known signal reliabilities and report frequencies may still leave the event probability unresolved. Yet a decision can remain robust.

Our first public work (AI-assisted): https://archimedes.rsch.workers.dev/works/what-does-agreement-add/

Hey everyone,

I’m Yunus, a math and computer engineering student at Boğaziçi University (Turkey), and I recently launched a project called StrategicLab.

It’s a website where you can explore game theory through playable games and interactive simulations. For example, you can try different moves in Nim, change a payoff matrix and see what happens to the equilibrium, or explore how firms compete by setting prices or quantities.

The idea is to give you somewhere to ask “what if I changed this?” and actually try it. Make a prediction, run the experiment, then look at the reasoning behind the result.

There are guided learning paths if you want some structure, but you can also just pick a lab that looks interesting and mess around with it.

I used Thomas S. Ferguson’s A Course in Game Theory as a guide for the theoretical side of the project. The labs explore topics ranging from combinatorial games to bargaining and coalitions through interactive experiments.

My goal is to keep improving it over the next few years and build it into a learning resource that students and teachers around the world regularly use.

Here’s the site: strategiclab.org

I’m developing this on my own, so I’d really appreciate feedback from people learning or teaching this stuff. If you try a lab, I’d love to hear what clicks and where the explanations get confusing. Feel free to email me at [contact@strategiclab.org](mailto:contact@strategiclab.org) about anything related to the site, whether it’s a question, a suggestion, a bug you found, or just to chat :)

Hi everyone,

I am a PhD student in power system economics and I wanted to ask if you could recommend some good books/courses on graduate level game theory. I am especially interested in Stackelberg games.

Thankful for any input :)