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Nash Equilibrium

A strategy profile where no player can improve their payoff by unilaterally changing their strategy.

Formal Definition

A Nash equilibrium is a set of strategies (s1*, s2*, ..., sn*) such that for each player i:

ui(si*, s-i*) >= ui(si, s-i*) for all si in Si

Where:

  • ui is player i's payoff function
  • si* is player i's equilibrium strategy
  • s-i* is the equilibrium strategies of all other players
  • Si is player i's strategy set

Types of Nash Equilibrium

Pure Strategy Nash Equilibrium

Each player chooses a single deterministic action.

Example: In a coordination game where both players benefit from choosing the same option, both choosing A or both choosing B are pure strategy equilibria.

Mixed Strategy Nash Equilibrium

Players randomize over actions with specific probabilities.

Example: In matching pennies, each player randomizes 50/50 between heads and tails.

Crypto relevance: MEV searchers often use mixed strategies to avoid predictability.

Symmetric Nash Equilibrium

All players use the same strategy.

Crypto relevance: In many DeFi games, symmetric equilibria are focal points.

Finding Nash Equilibria

Method 1: Best Response Analysis

  1. For each player, find best response to every possible opponent strategy
  2. Nash equilibrium exists where best responses intersect

Example (2x2 game):

                Player 2
              L       R
Player 1  U  (3,3)   (0,5)
          D  (5,0)   (1,1)
  • If P2 plays L: P1's best response is D (5 > 3)
  • If P2 plays R: P1's best response is D (1 > 0)
  • If P1 plays U: P2's best response is R (5 > 3)
  • If P1 plays D: P2's best response is L (0 > 1)

Nash equilibrium: (D, L) with payoffs (5, 0)? No - P2 would deviate. Need to check: at (D,L), P2 gets 0, could get 1 at (D,R). Deviates. At (D,R), P1 gets 1, could get 5 at (D,L). Check P2: gets 1, could get 0 at (D,L). No deviation. (D,R) is Nash equilibrium with payoffs (1,1).

Method 2: Iterated Elimination of Dominated Strategies

Remove strategies that are never best responses, iterate until stable.

Method 3: Support Enumeration (for mixed equilibria)

For each possible support (set of strategies played with positive probability), solve for probabilities that make opponent indifferent.

Properties

Existence

Every finite game has at least one Nash equilibrium (possibly mixed).

Uniqueness

Not guaranteed. Many games have multiple equilibria.

Efficiency

Nash equilibria may not be Pareto efficient (see: Prisoner's Dilemma).

Stability

Players have no incentive to deviate unilaterally, but:

  • Coalition deviations may be profitable
  • Trembling (small mistakes) may destabilize

Refinements

Subgame Perfect Equilibrium

Nash equilibrium in every subgame. Rules out non-credible threats.

Crypto relevance: Evaluating whether protocol threats (slashing) are credible.

Trembling Hand Perfect Equilibrium

Equilibrium survives if players make small mistakes with tiny probability.

Crypto relevance: Robust mechanism design that works despite user errors.

Sequential Equilibrium

For games with imperfect information. Combines strategies and beliefs.

Crypto relevance: Trading games where participants have private information.

Crypto Applications

Validator Economics

Game: Validators choose how much to stake.

Players: n validators Strategies: Stake amount si >= 0 Payoffs: Rewards proportional to stake share, minus opportunity cost

Equilibrium: Staking continues until marginal reward equals opportunity cost for all validators.

Analysis questions:

  • Is the security budget sufficient?
  • Can cartels form to extract more value?
  • What happens if a large player exits?

AMM Liquidity Provision

Game: LPs choose where to provide liquidity.

Players: Liquidity providers Strategies: Amount and price range for liquidity Payoffs: Fees earned minus impermanent loss

Equilibrium: Liquidity distributed such that marginal return equals across positions.

Insight: Concentrated liquidity shifts equilibrium toward more competition at active prices.

Gas Price Bidding

Game: Users compete for block inclusion.

Players: Transaction senders Strategies: Gas price bid Payoffs: Value of inclusion minus gas cost

Equilibrium: Bids rise until marginal bidder is indifferent between inclusion and waiting.

Pathology: During congestion, equilibrium can involve extreme overbidding.

Governance Voting

Game: Token holders vote on proposals.

Players: Token holders with varying stakes Strategies: Vote yes, no, or abstain Payoffs: Depend on proposal outcome and voting costs

Equilibrium: Often rational apathy - small holders don't vote because impact is negligible.

Attack: Voter apathy allows concentrated interests to capture governance.

Limitations

Rationality Assumption

Nash equilibrium assumes perfect rationality. Real actors:

  • Have bounded rationality
  • Make mistakes
  • Have incomplete information about payoffs

Equilibrium Selection

With multiple equilibria, theory doesn't predict which will occur.

Computational Complexity

Finding Nash equilibria is PPAD-complete (computationally hard).

Dynamic Considerations

Static Nash analysis misses:

  • Learning dynamics
  • Reputation effects
  • Time preferences

Best Practices for Analysis

  1. Start simple: Model the core interaction first, add complexity later.

  2. Identify all players: Including passive ones (e.g., retail in MEV games).

  3. Be precise about payoffs: Vague payoffs lead to vague conclusions.

  4. Check for multiple equilibria: Don't assume uniqueness.

  5. Stress test equilibrium: What breaks it? Collusion? New entrants? Parameter changes?

  6. Consider dynamics: Is the equilibrium reachable? Stable to perturbations?