THE NASHIAN VIEW
My choice ↗
your strategy stays fixed.
your strategy stays fixed.
Find a best response to the others’ strategies.
NON-NASHIAN GAME THEORYA NEW PERSPECTIVE ON RATIONAL CHOICE
What if the future
could see you coming?
When our choices can be predicted, the rules of rationality change. Meet the Perfect Prediction Equilibrium.
PREDICTION
DECISION
＋ A FUTURE CONSISTENT WITH ITS OWN PREDICTION
GAME THEORY, REIMAGINED.SCROLL TO EXPLORE ↓
Rationality✳Counterfactuals✳Artificial intelligence✳Quantum foundations
01 / THE PARADIGM SHIFTONE ASSUMPTION. A DIFFERENT WORLD.
Same players.
Same incentives.
Different rationality.
“What if I had chosen differently?”
Every theory of rational choice needs an answer. In a Nash equilibrium, a player tests a different strategy while holding everyone else’s strategies fixed.
Perfect prediction asks a different question: if my choice had been different, would others have predicted that difference—and acted differently, too?
A decision and its anticipation become counterfactually dependent. The outcome must fit the reasoning that predicts it. [1]
THE PERFECT PREDICTION VIEW
My choice ⇄
your anticipation.
your anticipation.
Find an outcome consistent with being predicted.
↳ Counterfactual dependence does not mean sending a signal into the past. It compares different possible histories.
02 / THINK IT THROUGH INTERACTIVE EXPLORATION
A promise.
A prediction.
A different ending.
Alice can give Bob a loaf of bread. Bob can then pay her—or keep it without paying. Both want the best outcome for themselves.
Switch the reasoning. Watch the game change.
Equilibrium pathPayoffs: Alice, Bob · Higher is better
BACKWARD INDUCTIONThe exchange
The exchange
never happens.
- With bread already in hand, Bob prefers not paying: 2 is better than 1.
- Alice anticipates this. Keeping the bread gives her 0 instead of −1.
- She keeps it. Both receive 0.
ALICE0BOB0
The promise / assurance game from the supplied research. Nashian mode shows the subgame-perfect equilibrium; PPE mode assumes perfect prediction and strict preferences. [1] [4]
02B / PERFECT PREDICTION IN NORMAL FORMAN ASYMMETRIC GAME · THREE STRATEGIES EACH
Different choices.
A shared prediction.
Perfect Prediction Equilibrium also applies when players choose without observing each other’s decision. Here, the row player chooses D, E or F; the column player chooses A, B or C.
Switch the reasoning to compare the Nash equilibrium with Perfect Prediction Equilibrium. In perfect-prediction mode, step through the elimination of outcomes that contradict rational payoff guarantees.
| ROW ↓ | A | B | C |
|---|---|---|---|
| D | 6, 8Possible | 1, 2Possible | 4, 4Possible |
| E | 3, 1Possible | 0, 7Possible | 2, 3Possible |
| F | 7, 0Possible | 8, 5Possible | 5, 6Possible · Nash |
Nash equilibriumPayoffs: row player, column player
STEP 0 / 2
THE ORIGINAL GAMENine possibilities.
Nine possibilities.
No symmetry required.
Each cell is one joint outcome. Both players have strict preferences: no two outcomes give either player the same payoff. The reasoning compares guaranteed payoffs without holding the other player’s strategy fixed.
9possible outcomes
NASH EQUILIBRIUMF, C → (5, 6)
Stable against unilateral deviations, but not Pareto optimal: (6, 8) improves both players’ payoffs.
PERFECT PREDICTION EQUILIBRIUMD, A → (6, 8)
Pareto optimal. In this game, both players obtain a higher payoff than at Nash.
Exact game from Figure 11, page 29 (supplied revision v4) of Perfect Prediction in Normal Form by Ghislain Fourny. The paper calls this equilibrium PTE; we use the unified name PPE here. This example improves on Nash for both players; that relationship does not hold for every game.
03 / THE MATHEMATICAL COREMORE THAN A THOUGHT EXPERIMENT
A prediction that
holds itself together.
PPE is a fixed-point solution: predicting the outcome leads rational players to that very outcome. Forward induction eliminates futures that would contradict their own anticipation. [1]
It exists.
For finite extensive-form games with perfect information and strict preferences, a PPE always exists.
It is unique.
Under those assumptions, the reasoning selects a single equilibrium path and outcome.
It is Pareto optimal.
No alternative outcome can improve one player’s payoff without making another worse off.
One equilibrium. Two historical names. ＋
PPE and PTE denote the same unified equilibrium concept. The original sequential-game work used “Perfect Prediction Equilibrium”; the normal-form work used “Perfectly Transparent Equilibrium,” before their unification became clear. In normal form, the reasoning repeatedly eliminates outcomes below players’ current maximin guarantees, under rationality and knowledge of strategies in all possible worlds.
The guarantees depend on the game class. For finite sequential games with perfect information and strict preferences, existence, uniqueness and Pareto optimality are guaranteed. For the finite normal-form games with pure strategies and no payoff ties studied in the paper, the equilibrium is unique and Pareto optimal when it exists; existence is not guaranteed. On symmetric normal-form games, it coincides with Hofstadter’s superrationality. [1] [2]
04 / NEW FRONTIERSFROM MINDS TO MEASUREMENTS
One idea.
Two extraordinary horizons.
What changes when independence is an assumption
we can question?
01 — ARTIFICIAL INTELLIGENCE & AI SAFETYWhen agents can
When agents can
model each other.
Prediction sits at the heart of the connection to AI. The normal-form paper points to machine learning and data analysis as motivation for questioning independent decisions. [2]
For artificial agents that inspect, simulate or predict one another, an intriguing question follows: should a counterfactual change in one policy leave the other policy’s response fixed?
This perspective is relevant to AI safety: how can predictive agents make credible commitments, cooperate with humans, and avoid incentives for harmful behavior?
FROM ASIMOV TO AI SAFETY
PPE can be seen as a proposed game-theoretic implementation of Isaac Asimov’s Three Laws of Robotics, provided their priorities are explicitly encoded in the model: protect humans first, follow human instructions consistent with that protection, and preserve the robot subject to both.
This interpretation connects equilibrium reasoning with safety-oriented design. PPE itself supplies no ethical priorities: the laws must be built into the model, and real-world safety requires more than an equilibrium guarantee.
Perfect prediction is an idealization. Applying this approach to imperfect AI systems remains an open research direction.
Explore the theoretical foundation ↗MUTUAL PREDICTION / CONCEPTUAL ILLUSTRATION
SPACELIKE SEPARATION / CONCEPTUAL ILLUSTRATION
02 — QUANTUM FOUNDATIONSCould nature
Could nature
play a different game?
Spacetime games place choices inside Minkowski spacetime. Light cones determine which information can reach a decision—connecting game structures to relativistic causality. [3]
Baczyk and Fourny model measurement settings and outcomes as decisions by experimentalists and nature. Within this framework, they argue that Nashian free-choice assumptions conflict with Bell-inequality violations, and propose non-Nashian reasoning instead. [4]
AN OPEN QUESTION IN PHYSICS
Read the quantum foundations paper ↗The proposal accommodates counterfactual dependence and contextuality. It leaves nature’s precise payoffs open; it is a research framework, not a completed derivation of quantum predictions.
“If I had chosen differently,
you would have seen it coming.”
05 / GO TO THE SOURCESIX PAPERS. ONE RESEARCH JOURNEY.
Ideas deserve
a closer look.
Explore the original arguments, formal definitions and proofs—from the early work on projected equilibrium and Newcomb’s paradox to the four manuscripts behind this introduction.
THE FOUNDATION · EXTENSIVE FORM
Perfect Prediction Equilibrium
Ghislain Fourny, Stéphane Reiche & Jean-Pierre Dupuy
THE EXTENSION · NORMAL FORM
Perfect Prediction in Normal Form
Superrational Thinking Extended to Non-Symmetric Games · Ghislain Fourny
THE BRIDGE · RELATIVISTIC CAUSALITY
Games in Minkowski Spacetime
Ghislain Fourny
THE FRONTIER · QUANTUM FOUNDATIONS
Nashian game theory is incompatible with quantum physics
Michal Baczyk & Ghislain Fourny
THE PHILOSOPHICAL ROOTS · PROJECTED EQUILIBRIUM
Philosophical foundations of a new concept of equilibrium in the social sciences: Projected equilibrium
Jean-Pierre Dupuy
THE EARLY FOUNDATIONS · NEWCOMB’S PARADOX
Two temporalities, two rationalities: a new look at Newcomb’s paradox
Jean-Pierre Dupuy
TAKE THE NEXT STEP
Explore the research engine ↗From understanding to exploration.
A FEW GOOD QUESTIONS
Stay curious.
Does this mean Nash equilibrium is wrong? ＋
Nash equilibrium answers a well-defined question under a particular counterfactual assumption: hold the others’ strategies fixed. PPE explores a different assumption. Which model is useful depends on the agents and situation being studied.
Is perfect prediction realistic? ＋
It is an idealization, like other strong knowledge assumptions in game theory. The papers establish what follows under that idealization. Partial predictability and imperfect models require further analysis.
Does cooperation require altruism? ＋
No. The model’s players still pursue their own payoffs. Cooperation can emerge because a different choice would have been anticipated differently. The promise-game example is played once and needs no repeated-game reward or punishment.
Does this prove a new theory of quantum mechanics? ＋
The quantum paper proposes a decision-theoretic framework and argues for revisiting free-choice assumptions. It does not specify a complete physical payoff model or derive all quantum statistics. Its connection to quantum foundations is an invitation to further research.