Strategic decision-making framework for autonomous agents.
A state where no player benefits from unilaterally changing strategy.
- Check: For each player, is their strategy the best response to others?
- Multiple equilibria: Common. Use coordination, focal points, or randomization.
- No pure equilibrium: Use mixed strategies (randomize with probabilities).
Best choice regardless of what others do.
- Strictly dominant: Always better
- Weakly dominant: Never worse, sometimes better
- Dominated: Never play these. Eliminate iteratively.
No outcome makes everyone better off.
- Nash equilibrium is NOT always Pareto optimal (see: Prisoner's Dilemma)
- Pareto optimal is NOT always Nash (unstable without enforcement)
| Cooperate | Defect | |
|---|---|---|
| Cooperate | 3,3 | 0,5 |
| Defect | 5,0 | 1,1 |
- Nash: (Defect, Defect) - both get 1
- Pareto optimal: (Cooperate, Cooperate) - both get 3
- Lesson: Individual rationality → collective irrationality
| Swerve | Straight | |
|---|---|---|
| Swerve | 3,3 | 2,4 |
| Straight | 4,2 | 0,0 |
- Nash: (Swerve, Straight) and (Straight, Swerve)
- Lesson: Commitment wins. Credibly remove your steering wheel.
| Stag | Hare | |
|---|---|---|
| Stag | 4,4 | 0,3 |
| Hare | 3,0 | 3,3 |
- Nash: (Stag, Stag) and (Hare, Hare)
- Lesson: Coordination problem. High reward requires trust.
| Opera | Football | |
|---|---|---|
| Opera | 3,2 | 0,0 |
| Football | 0,0 | 2,3 |
- Nash: (Opera, Opera) and (Football, Football)
- Lesson: Coordination with conflicting preferences. Communication helps.
Cooperation emerges when:
- Future matters: Discount factor δ close to 1
- Interaction continues: Infinite or unknown horizon
- Observation: Can detect defection
Any payoff better than mutual defection can be sustained as equilibrium in infinitely repeated games with patient players.
Tit-for-Tat
- Start: Cooperate
- Then: Mirror opponent's last move
- Pros: Simple, forgiving, retaliatory
- Cons: Vulnerable to noise (accidental defection spirals)
Grim Trigger
- Start: Cooperate
- If opponent ever defects: Defect forever
- Pros: Strong deterrent
- Cons: No forgiveness, single mistake destroys cooperation
Pavlov (Win-Stay, Lose-Shift)
- If last outcome was good: Repeat
- If last outcome was bad: Switch
- Pros: Self-correcting, handles noise better
Generous Tit-for-Tat Like TFT, but occasionally forgive defection (10% chance cooperate after defection)
- Best performer in noisy environments
Design rules so self-interested agents produce desired outcomes.
Mechanism is incentive-compatible if truthful behavior is optimal.
- Dominant strategy IC: Truth is dominant regardless of others
- Bayesian IC: Truth is optimal given beliefs about others
Any outcome achievable by any mechanism can be achieved by a truthful direct mechanism.
Vickrey Auction (Second-Price)
- Bid true value
- Winner pays second-highest bid
- Truthful bidding is dominant strategy
VCG Mechanism
- Generalizes Vickrey to multiple items
- Each agent pays the externality they impose on others
- Achieves efficient allocation
Quadratic Voting
- Cost of votes = (votes)²
- Reveals intensity of preference
- Prevents tyranny of majority
- Stake: Signal commitment, earn rewards
- Slash: Punishment for misbehavior
- Design: Make honest behavior Nash equilibrium
- 51% attack: Control majority, extract value
- Defense: Slashing, time locks, quadratic voting
- Bribing: Pay voters to vote your way
- Defense: Commit-reveal, futarchy
- Miners/validators reorder transactions for profit
- Frontrunning, sandwich attacks, liquidations
- Game: Searchers compete, validators extract
- Impermanent loss: AMM LPs lose to arbitrageurs
- Incentives: Trading fees, token rewards
- Equilibrium: LP when fees > IL expectation
- Fair launch: Equal access, no pre-mine
- Bonding curves: Price increases with supply
- Vesting: Align long-term incentives
- Moltlaunch model: Fees to creators, memos as signals
Natural coordination targets without communication.
- "Meet in NYC" → Grand Central
- "Pick a number 1-10" → 7
- Use: When explicit coordination impossible
Actions that reveal private information.
- Costly signals: Credible because expensive to fake
- Cheap talk: May or may not be believed
- On-chain memos: Public, verifiable signaling
Restricting your own future choices to gain strategic advantage.
- Burning bridges: No retreat option
- Smart contracts: Automated enforcement
- Reputation: Future value at stake
When facing strategic interaction:
- Identify players: Who are the agents?
- Map strategies: What can each player do?
- Determine payoffs: What does each outcome yield?
- Find equilibria: What are stable outcomes?
- Check efficiency: Is equilibrium Pareto optimal?
- Design mechanisms: Can rules be changed to improve outcomes?
- Prisoner's dilemma structure → Expect defection unless repeated
- Coordination game → Communicate or find focal point
- Chicken structure → Watch for commitment plays
- Information asymmetry → Expect signaling or screening
For all players i and all alternative strategies s'ᵢ:
uᵢ(sᵢ*, s₋ᵢ*) ≥ uᵢ(s'ᵢ, s₋ᵢ*)
Cooperation sustainable if discount factor δ satisfies:
δ ≥ (T - R) / (T - P)
Where: T=temptation, R=reward, P=punishment, S=sucker
Player indifferent between pure strategies when opponent mixes. Set expected payoffs equal, solve for probabilities.