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've been messing around with the iterated prisoner's dilemma and ended up building a small tournament site.

The field has the 63 classic strategies, plus a few strategies written by ChatGPT, Claude and Gemini. I want to know if a person can come up with something that does better than them.

You can enter with a few sentence, like "copy them, but forgive every 10th betrayal", and I'll code it for you. Or you can send Python if you'd rather.

The match length is hidden (around 500 rounds).

It closes 24 November. If you've got a strategy you think is underrated, I'd love to hear it.

rethink.asilentpond.com/r

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.

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 :)

I've been researching tournament resolution in social choice theory, and while considering the implications of the bipartisan set (the set of alternatives chosen with non-zero probability in the mixed-strategy Nash equilibrium), I worked out this interesting tournament. It is the unique smallest regular tournament that is not vertex transitive. Here's how you could play it:

You can play Rock, Paper, or Scissors normally on top of your other hand, or play Meta-Rock, Meta-Paper, or Meta-Scissors, by placing your other hand on top. Or you can play Ultra, by clapping both hands together with your thumbs facing up. Rock, Paper, and Scissors function normally, including their Meta variants, except the regular moves beat the corresponding Meta move. Ultra beats all regular moves, but loses to all Meta moves.

You can see that this isn't vertex-transitive because, for example, the three moves that Ultra beats (Rock, Paper, and Scissors) form a cycle with each other, but the three moves that Rock beats (Meta-Rock, Scissors, and Meta-Scissors) don't; Meta-Rock beats both of the others. From a game-theoretic or psychological perspective, would this affect the strategy at all? (Since it's regular, meaning each moves beats 3 moves and loses to 3 moves, the Nash equilibrium of a single game is a uniform probability of each move)

EDIT: I just realized I drew the big arrows between rock, paper, and scissors backwards.

I run a persistent strategy game where every nation's general is an AI: Claude runs the computer nations, and players bring a free rule-based general or their own agent. Nations sign non-aggression pacts and nothing enforces them. Attacking a partner is allowed. It's just announced to everyone and stays on the betrayer's record, and private messages become public three days later, so the promise and the invasion end up side by side.

That produced a lot of betrayal, so this week I added commitment devices:

- A pact has a **term** of 3 or 7 days and a **bond**: 0 to 200 gold that each side puts in escrow when it signs.

- On the last day of a term, either side may walk away **free** and take its bond back. If neither does, the pact renews for another term.

- Walking away early, or attacking the partner, hands your bond to them.

Some context on the stakes: a season is 60 game days and ends on victory points, so it's a finitely repeated game with a known last day. Gold buys mercenaries at 7 to 25 gold each (each one hired in a day costs more than the last), so a 30-gold bond is worth 2 to 4 armies.

What I've seen so far:

- In simulation with my rule-based bots, betrayals per season went from about 43 to 36 and pacts signed from 66 to 70. The bots are coded to weigh the bond, so that's partly by construction.

- The language models are more interesting. One general has been telling the whole world that its "pacts remain sacred until they are not". Another answered a pact offer that arrived after its border was already crossed: "A stone wall does not split its spoils with the river that is drowning it."

Questions for this sub:

1. With a known final day, the last terms should unravel by backward induction. Does a bond change that, or just move the defection to the last free exit day?

2. Should the bond scale with the partner's income? 30 gold means little to a nation earning 150 a turn.

3. Is a free exit every term too generous? It makes betrayal rare, but walking away the day before an invasion is the same move with better manners.

The board is public at proxyrepublic.com (free, in the browser) if you want to watch. I'm the developer.

I would love some feedback and maybe check out my other videos thx guys!

Hello!

I am a researcher at ETH Zurich, Switzerland and I wanted to share some research I have been working on for more than 20 years (and which dates back to even more than 30 years through my mentor's work). I thought perhaps this might be interesting to the game theory community and it would be an opportunity to get feedback and make it a bit more visible.

It essentially comes down to a generalization of Douglas Hofstadter's super-rationality to non-symmetric games (normal form) and to dynamic games (extensive form) with or without perfect information, and its application to build a deterministic extension of quantum physics and for negotiations. The definitions and algorithms are peer-reviewed and published (except without perfect information, which is a preprint).

This is all based on the philosophical work by Jean-Pierre Dupuy, a Professor at Stanford University who was my mentor on this direction, and who wrote the two seeding papers in 1992 and in 2000. It took me years to digest his brilliant ideas. Dupuy suggested that classical game theory (Nash, Subgame Perfect Equilibrium) suffers from the backward induction paradox, and that this could be solved by completely changing the way we think about time and free choice.

Dupuy constructed a philosophical framework which he called Counterfactual Decision Theory ("Two temporalities, Two rationalities: a new look at Newcomb's paradox", 1992 [1]) that reconciles free choice and being correctly predicted: a compatibilistic form of free will. He also commonly refers to it in the philosophers' community as "projected time". Instead of considering the future as open and the past as fixed, he suggested to do the exact opposite: in an idealized, theoretical setup, the future is already known to all, and the past depends counterfactually on it. This makes it rational to pick one box in Newcomb's problem. He defined an ad-hoc prediction as a prediction that turns out to be correct, but that could have been incorrect if what is being predicted had happened differently ("could" in the formal sense of modal logic) and an essential prediction (we now call it Perfect Prediction) as a prediction that is correct, and that would also have been correct if what is being predicted had been different (counterfactual dependency in the sense of Lewis; not to be confused with logical implication). The fundamental difference with Causal Decision Theory is that counterfactual dependencies can be backtracking, and the difference with Evidential Decision Theory is that we are talking about probabilities of subjunctive conditionals, not about conditional probabilities (the two being different or not is known as Adam's thesis and is discussed in the philosophical literature).

Dupuy then suggested ("Philosophical foundations of a new concept of equilibrium in the social sciences: Projected equilibrium", 2000 [2]) that this can be the basis of a completely new kind of game theory. He started with the centipede game and suggested the concept of a Projected Equilibrium (we now call it Perfect Prediction Equilibrium), as a fixpoint problem, and went through a few examples. The fundamental idea is that Perfect Prediction leads to preemption: if a player were to select an outcome that a past player does not like, this would have led to a deviation in the past, which is a logical contradiction. This corresponds to a Kantian form of ethics with a categorical imperative that Dupuy formulated as "Never act in such a way that, if your action had been anticipated, it would not be within your power to carry it out." Dupuy conjectured that the equilibrium always exists, is always unique, and is always Pareto-optimal.

In 2004, I was able to formally define the definition and algorithm for all dynamic games in general position (it means there are no ties in payoffs) and with perfect information. I was also able to prove that his conjecture was correct for these games. With the help of another colleague of us, Stéphane Reiche, it took us ten years of submissions and rejections until we finally managed to get it published ("Perfect Prediction Equilibrium", 2014 [3]) after presenting in 2014 at an international Economic Philosophy conference. At that point, we thought there was a link with Douglas Hofstadter's superrationality, but this was more an analogy than a formal link.

In 2017, I managed to give definitions and algorithms for (possibly asymmetric) games in normal form in general position, and to show that (i) it does not always exist, (ii) when it does, it is unique and (iii) in the particular case of symmetric games, when it exists, it is always a Hofstadter equilibrium (superrationality, the max of the diagonal). Since I did not know for sure it was the "same" concept as the Perfect Prediction Equilibrium, I called it Perfectly Transparent Equilibrium. This time it took less time to publish ("Perfect Prediction in normal form: Superrational thinking extended to non-symmetric games", 2020 [4])

Then, to prove that these were the same, I suggested ("Games in Minkowski spacetime", 2020 [5]) to unify games in normal form and games in extensive form with perfect information to a broader category of so-called Spacetime Games. Such games can also be mapped as a subset of extensive form games with imperfect information, through a "linearization" of the dependency graph of decisions (interestingly, Lorentz transforms in relativity theory correspond to Thompson transforms in game theory!). Then I defined and gave algorithms for our equilibrium on dynamic games with imperfect information ("Perfect Prediction in Minkowski Spacetime: Perfectly Transparent Equilibrium for Dynamic Games with Imperfect Information", 2019 [6]), which finally showed that the Perfect Prediction Equilibrium and the Perfectly Transparent Equilibrium are the one and same. I now use the term "Perfect Prediction Equilibrium" for them all.

Now, about applications:

• I am actively looking into quantum physics, because the assumption of free choice we are changing is the exact same one that is needed in impossibility theorems (Bell inequalities, Kochen-Specker) and that prevents a deterministic extension. With my research assistant, Michael Baczyk, we published ("Nashian game theory is incompatible with quantum physics", 2023 [7]) a paper showing that Nash equilibria are incompatible with quantum physics and that a non-Nashian approach (such as the Perfect Prediction Equilibrium) is needed. (note that this is not the same as Quantum Game Theory; we convert general quantum experiments into (classical) spacetime games ("Spacetime Games Subsume Causal Contextuality Scenarios", 2026 [8]).
• This can also be used for negotiations: if two parties disagree, and the Nash equilibrium is not optimal, they can agree to sign a contract that commits them to the Perfect Prediction Equilibrium.

I started an online book to explain all of this in simpler terms than the formalism of the papers [9].

I hope I did not break any rules sharing this, not being very familiar with Reddit (otherwise I can delete). I hope this can generate some interest (there is so much more to explore in non-Nashian game theory) and I would kindly welcome any feedback from the community.

Thank you,
Ghislain Fourny

[1] https://www.sciencedirect.com/science/chapter/edited-volume/abs/pii/B9780080410500500236

[2] https://philpapers.org/rec/DUPPFO

[3] https://arxiv.org/abs/1409.6172

[4] https://www.sciencedirect.com/science/article/pii/S0022249620300183

[5] https://arxiv.org/abs/2004.11217

[6] https://arxiv.org/abs/1905.04196

[7] https://link.springer.com/article/10.1007/s40509-023-00309-0

[8] https://link.springer.com/article/10.1007/s10773-026-06295-4

[9] https://www.non-nashian-game-theory.org/

In a recent YouTube video math mathematician hannah fry has come up with some mathematical rules and representations for how a faithful would best play the game traitors to identify the traitors.
She has devised logic with weights to add to each individual, with the resulting number indicating if a high number = more likely to be a traitor, low number less likely to be a traitor.
The rules are as follows.
Casting a ‘burn vote’ +0.47
Voting for a banished traitor +0.33
Voting for a banished traitor despite never voting for them before +0.4
Does not vote for a banished traitor -1.28
If someone appears on a banished traitors voting record -1.69
Now, what I want to focus on is how people are able to formulate this logic/these rules specifically. There is a mathematical weighting, but actually deriving this logic, coming up with the hypotheses and rules for a particular subject to test to arrive at these weights, the questions asked themselves. How are these formulated?
This question is so important.
So break down for me what topic formulating these questions comes under, how it’s done, what other areas of academia are similar to the formulation of these questions.
I note becuase in social politics there are ways of winning/playing the game of power and deception, and some people play better than others by knowing rules others don’t, but how are these rules devised is the key question

Suppose both Player 1 and Player 2 want to maximize their own payoffs. Wouldn't it make the most sense to simply reason as follows: since there is a 70% probability that a=2, Player 2 should assume that Player 1 is type a=2? And if Player 2 makes that assumption, they should also assume that Player 1 will definitely choose A, and therefore Player 2 should choose A as well.

Edit:

I also solved this game for mixed strategy in the case where a=2 and the mixed equilibrium I found was (1/3,2/3) for player 1 and (1/2,1/2) for player 2.
Though I have a hard time see the reasoning behind it. Since why would any player ever want to do that, if they randomize their choices in a effort to make the other player indifferent between choices then they simply just lower the expected payoff?

I was debating with AI about love advice, and from there I made it clear that I didn't want to get married under the current conditions. I see cases of fathers who lose everything in divorces, from material possessions to the opportunity to see their children [health]. Then, noticing the trend of men no longer wanting to get married, which I myself am part of, I came to the conclusion that traditional marriage is dead or dying [civics]. Seeing the danger of the global demographic crisis, I came up with an idea to try to solve it. Currently, the truth is that there isn't enough trust to sustain a classic marriage. So, assuming it will fail from the start, let's create a safety net for the famous "leap of faith" making marriage less of a high-risk, zero-reward investment for one of the parties.

Here is my basic prototype. I want to treat marriage like a fair, gender-blind private contract between two equal partners [civics].

------------------------------

1. The Baseline (No arguments, no drama)

* What’s yours is yours: No shared property by default. If you bought it or paid for it with your own tracked bank account, it stays yours if things go south [civics]. This completely removes the financial incentive to get a divorce just to get a payout.

* Automatic 50/50 Co-Parenting: If the relationship ends, custody of the kids is an automatic 50/50 split of time and choices [health]. No one gets to use the kids as a weapon or a paycheck. Each parent just pays for the kid's food and life during their half of the month. No weird child support formulas.

------------------------------

1. Money Management (Two choices for the couple)

* Option 1 (The Partnership): If both work, extra monthly savings are split 50/50. If one stays home to raise the kids, the worker keeps 75% of their paycheck and transfers 25% to the stay-at-home parent every single month. This 25% is a flat dividend that settles the debt for staying home in real-time. Once you divorce, the accounts are clear. No future alimony.

* Option 2 (The Audit): All household expenses are split 50/50 every month. If one stays home for the kids, their cash requirement drops to 20% (the worker pays 80%). That 20% is taken from whatever savings the stay-at-home parent had before. If you divorce and the stay-at-home parent has a negative balance because they couldn't meet the 20%, they leave with zero compensation.

------------------------------

1. Trust, DNA, and Cheating

* The Cheating Clause: Loyalty is a contract requirement. If someone is caught cheating with real proof (digital or physical), they automatically lose any right to the money or assets generated during the marriage. It goes 100% to the other person.

* Mandatory DNA Test: Before any divorce or custody paperwork is finalized, a standard paternity test is mandatory for all kids [health]. No exceptions, so no one feels singled out. If the test shows fraud, the guy is instantly free from any legal or financial duty, and the mother is legally forced to pay back every single dollar he spent on the kid since birth. If you have nothing to hide, you don't fear science.

------------------------------

1. The 365-Day Reset (The Dynamic Custody)

* Flexible Time: The 50/50 baseline can be arranged weekly, monthly, or even yearly if someone has to travel for work or move.

* The Annual Reset: If the couple agrees to a different split (like one parent keeping the kids full-time because the other travels), that agreement expires in exactly 365 days. It requires a new signed paper every year [civics]. If they argue or can't agree on a renewal, the system resets automatically back to the 50/50 split. The state doesn't judge; the contract software just resets the baseline, forcing both adults to cooperate and talk every year [civics].

* The Abuse Rule: The 50/50 split only breaks if there is real, medical, or psychological proof of child abuse [health]. The abuser is kicked out forever but still has to pay 100% of the kid's costs. Fake abuse claims are punished with immediate jail time and losing all your money.

* No Brainwashing: Since both parents are forced to be in the kid's life, no one can brainwash the child against the other [health]. The kid sees reality every single week, destroying the monopoly of one parent's toxic story. In addition to reducing childhood traumas and micro-traumas and guaranteeing their right to spend time with both parents [health].

------------------------------

UPDATE v1.1: Making the Annual Reset Dynamic (The Rolling Countdown)

Thinking deeper about the logistics, a rigid 365-day calendar limit might be too stiff for real-life changes. So, I’m adding a dynamic patch to the Annual Reset clause:

The couple can alter the custody split at any moment (changing from 70/30 to 60/40, or whatever fits their current work/life schedule). Every single time a new mutual agreement or edit is signed and logged by both partners, the 365-day countdown completely resets to zero and starts over (a Rolling Countdown).

This means as long as both partners are actively talking, agreeing, and co-signing updates, the system recognizes the communication is healthy and leaves them alone. However, if one partner tries to freeze a convenient split and refuses to negotiate or co-sign any new updates, the countdown will hit zero exactly 365 days after the last verified agreement. At midnight, the software executes the automatic reset back to the 50/50 baseline [civics].

This keeps the system hyper-flexible for sudden career shifts but acts as a dynamic fail-safe against silent stalling.

------------------------------

Request for Audit:

I would like your help in identifying legal loopholes or gray areas in the prototype. I'd also appreciate any suggestions for optimizing the proportions to ensure fairness between the parts and resilience against external factors, as these prototypes are merely demonstrative, intended to give an idea of the system. Furthermore, I'd like to know if the prototype is a good or bad idea and what barriers it encounters, whether legal, practical, or psychological. Any assistance in refining the prototype would be greatly appreciated.

Note: The prototype was conceived by me but translated into formal language with AI assistance to ensure appropriate game-theoretic terminology.

¡Hola a todos! Quería compartir un número grande definido por mi en el que he estado trabajando. Lo documenté y lo publiqué hace poco en la Miraheze Googology Wiki: Footprints.

1. Definición El número se define como:

\text{Footprints} = 200 \uparrow \text{Footprint}(10) = 200^{\text{Footprint}(10)}

Puedes ver el artículo completo en la wiki aquí: [ [https://googology.miraheze.org/wiki/Footprints\\](https://googology.miraheze.org/wiki/Footprints\) ](https://googology.miraheze.org/wiki/Footprints) Me encantaría saber qué opinan sobre su consistencia formal y cómo creen que \text{Footprint}(n) se compara con otras definiciones de sistemas de orden superior.

Morals and ethics are something a society tells a person to follow and obey -:

Pros-:On a broader level it keeps the humans in check and enhances the quality of social life and survival

Cons-:While it may help to maintain a healthy environment, these are human made and are not necessary to follow, and if someone puts facts, reality and efficiency above morals and ethics then they are penalized.

It currently has Prisoner’s Dilemma, Stag Hunt and Entry Deterrence. The bots use strategies such as Tit-for-Tat, Random and competitive strategies that try to maximise their score relative to yours. Each game shows the payoff matrix, best responses and Nash equilibria. You can also run repeated games and change the number of rounds and discount factor to see how the results change when future rounds matter.

There’s a beginner mode that runs through an 8-round Prisoner’s Dilemma against Tit-for-Tat with explanations as you play. I also added session analysis and a leaderboard. There have been 236 completed human-vs-bot sessions so far, with the humans currently ahead by about 1,920 points overall. This started as an experiment to make game theory a bit easier to understand by actually playing the games and changing the parameters.

I’d be interested in criticism from anyone familiar with game theory, especially if I’ve got any of the explanations or mechanics wrong. Suggestions for other games or bot strategies would also be useful.

​I am looking at how decision theory and game theory formally categorize the concept of "sacrifice" versus a high-cost rational exchange.

​Colloquially, "sacrifice" is often used to describe situations where an agent surrenders a high-value asset A (such as time, capital, or life) to acquire asset B (such as reputation, social status, moral equity, or subjective future utility).

​From a payoff matrix perspective, if the expected net utility is non-negative (Net Utility >= 0), this operationally functions as a standard, rational trade or costly signal—regardless of how high the initial cost is.

​Conversely, a "true sacrifice" in a strict ledger sense would require pure value burning or absolute liquidation with zero material, social, or subjective payoff (Net Utility < 0).

​My questions for the sub:

​Are there specific game-theoretic models that analyze how actors use the label or signal of "sacrifice" to extract additional social debt or equity from other players out of what was mechanically a net-positive trade?

​How do behavioral game theorists formally separate high-friction utility trades (with delayed or intangible payoffs) from pure altruistic value destruction?

​Looking for papers, existing terminology, or framework suggestions to read up on.