game theory.md

the study of strategic interaction — what happens when the outcome of your choice depends on the choices of others

from first principles

a game has three primitives:

• players — agents who choose. in cyber, these are neurons
• strategies — the available actions. in cyber, which cyberlinks to create, where to stake focus
• payoffs — the consequences. in cyber, karma, focus shifts, delegation rewards

the central question: given that every player reasons about what others will do, what happens? the answer is equilibrium — the state where no player gains by unilaterally changing strategy. Nash (1950) proved every finite game has at least one. in cyber, equilibrium is the fixed point where focus distribution across the cybergraph ceases to shift

the four archetypes

every strategic situation reduces to one of four archetypes:

branches

non-cooperative game theory — each agent optimizes alone. Nash equilibrium, dominant strategies, mixed strategies. the workhorse for modeling consensus, auction, and adversarial behavior in open protocols

cooperative games — agents form coalitions and share joint gains. the Shapley value gives the unique fair attribution satisfying efficiency, symmetry, null player, and additivity. in cyber, distributes focus rewards proportionally to each neuron's causal impact via probabilistic shapley attribution. see also core stability and Nash bargaining

mechanism design — the inverse of game theory: given a desired outcome, design the rules so self-interested agents produce it. Myerson (1981) showed how to build incentive-compatible mechanisms. the cyberlink market protocol, auction formats, token engineering, and governance quadrants are all mechanism design

evolutionary game theory — strategies reproduce proportionally to their fitness. replicator dynamics, evolutionarily stable strategies. explains the emergence of cooperation without rationality: kin selection (Hamilton's rule $r \cdot B > C$), reciprocal altruism (Trivers), indirect reciprocity through reputation (Nowak). in cyber, karma serves as reputation enabling indirect reciprocity at planetary scale

algorithmic game theory — computational complexity of finding equilibrium. some equilibria are PPAD-complete to compute. probabilistic shapley attribution addresses this by reducing $O(n!)$ Shapley computation to $O(k \cdot n)$ via Monte Carlo sampling

information and signaling

games differ fundamentally in who knows what:

• complete information — all players know all payoffs (chess)
• incomplete information — private types, Bayesian reasoning (Harsanyi, 1967)
• imperfect information — simultaneous moves, hidden actions (poker)

information asymmetry creates two pathologies:

adverse selection — the informed party exploits ignorance before contracting. solved by screening and signaling (Spence, 1973)

moral hazard — hidden action after contracting. solved by monitoring, bonding, incentive alignment

in cyber, the costly signal resolves both: a cyberlink costs focus to create, making it an honest indicator of what the neuron values. focus is the cost, cyberlink is the signal, cyberank is the outcome. cheap talk is impossible when the signal burns a scarce resource

information aggregation

aggregating dispersed knowledge across agents:

wisdom of the crowds — Condorcet jury theorem (1785): independent voters with $p > 0.5$ converge to truth as group size grows. fails under correlated errors, herding, conformity bias

prediction markets — prices aggregate private information weighted by stake. LMSR for thin markets, inversely coupled bonding surface for self-scaling liquidity. the cyberlink market protocol makes every cyberlink simultaneously a structural assertion and a market on its own truth

Bayesian Truth Serum — extracts honest beliefs without ground truth. rewards beliefs more popular than predicted. a proper scoring rule applied peer-to-peer. in cyber, implemented via the valence field in cyberlinks

proper scoring rules unify all three: log score, Brier score, and ICBS settlement factors are all instantiations of the same Bregman divergence structure. honesty is enforced because distortion always costs

public goods and externalities

public goods — non-excludable, non-rival. the cybergraph is a public good: anyone can query or extend it. the free rider problem leads to underprovision. solutions: quadratic funding, staking incentives, token engineering

externality — costs or benefits to non-participants. every cyberlink generates positive externalities by enriching the shared knowledge graph. Pigouvian taxes internalize negatives; Coase theorem handles bilateral cases with clear property rights

coordination and stigmergy

coordination asks how agents synchronize. Schelling focal points: convergence without communication through shared salience. coordination graphs model dependencies among agent actions for optimal joint decisions

stigmergy — indirect coordination through a shared environment. ants leave pheromones; neurons leave cyberlinks. the cybergraph is a stigmergic medium at planetary scale

in cyber

the cyber protocol is a game-theoretic construction from the ground up:

the governance quadrant maps the design space:

key figures

John von Neumann — founded the field (1928), minimax theorem, zero-sum games John Nash — Nash equilibrium (1950), existence proof via fixed-point theorem Lloyd Shapley — Shapley value (1953), stable matching (Nobel 2012) William Vickrey — Vickrey auction, revelation principle (Nobel 1996) Condorcet — jury theorem (1785), foundation of wisdom of the crowds Thomas Schelling — focal points, commitment strategies (Nobel 2005) Leonid Hurwicz — mechanism design (Nobel 2007)

see cooperative games for coalition theory. see coordination for synchronization. see cooperation for evolutionary foundations. see cybernomics for token economy design. see token engineering for applied mechanism blueprints