npbuilds Evolutionary Games — The Survivors
Evolutionary game theory replaces the rational player with a population of agents using inherited or imitated strategies. Strategies don't "choose" — they spread or die based on fitness. This framework explains how cooperation emerges, why conflict persists, and how conventions form — all without assuming anyone does any strategic reasoning. Grounded primarily in Maynard Smith (1982), Weibull (1995), and Sandholm (2010).
The Evolutionary Framework
Instead of "players choose strategies," evolutionary GT says:
- A large population of agents is randomly matched to play a game
- Each agent uses a fixed strategy (inherited, learned, or culturally transmitted)
- Payoffs determine fitness — higher payoff → more offspring / more imitators
- Over time, fitter strategies spread and less fit strategies decline
- Equilibrium is a population state that resists invasion by alternative strategies
This removes two key assumptions of classical game theory: (1) agents don't need to know the game structure, and (2) agents don't need to be rational. Evolution is the optimizer.
Evolutionarily Stable Strategy (ESS)
The central concept (Maynard Smith & Price 1973):
A strategy s* is an ESS if, when the entire population plays s*, no small fraction of mutants playing any alternative strategy s can invade.
Formal definition: s* is an ESS if for all s ≠ s*:
- u(s*, s*) > u(s, s*), OR
- u(s*, s*) = u(s, s*) AND u(s*, s) > u(s, s)
Condition 1: s* is a strict best response to itself (a strict Nash equilibrium is always an ESS). Condition 2: If s does equally well against s*, then s* must do better against the mutant s than the mutant does against itself.
Key properties:
- Every ESS is a Nash equilibrium (but not every NE is an ESS)
- ESS may not exist (some games have no ESS, only mixed ESS)
- A strict Nash equilibrium is always an ESS
- A completely mixed Nash equilibrium is an ESS if and only if the game matrix satisfies certain negative-definiteness conditions
Canonical Evolutionary Games
Hawk-Dove (The Fundamental Conflict Game)
Two animals compete for a resource of value V. Hawks fight; Doves share or retreat.
Dove Hawk
Dove (V/2, V/2) (0, V)
Hawk (V, 0) ((V-C)/2, (V-C)/2)
Where C = cost of fighting.
If V > C (low-cost fighting): Hawk is the unique ESS. Aggression pays. If V < C (costly fighting): No pure ESS. The mixed ESS is p* = V/C (proportion of Hawks). This is the canonical example of a frequency-dependent equilibrium — Hawks thrive when rare (easy targets), suffer when common (costly fights).
Biological insight: The hawk-dove model explains why animals in the same species rarely fight to the death — C is usually high, so the mixed ESS involves a lot of display behavior (dove) with occasional escalation (hawk).
Stag Hunt (Evolutionary Coordination)
Stag Hare
Stag (4, 4) (0, 3)
Hare (3, 0) (3, 3)
Two ESS: All-Stag and All-Hare. Both are evolutionarily stable — once established, neither can be invaded. But their basins of attraction under replicator dynamics differ. If initial Stag frequency > 3/4, population converges to All-Stag; otherwise to All-Hare.
Key insight: The risk-dominant equilibrium (Hare) has a larger basin of attraction than the payoff-dominant equilibrium (Stag). Evolution favors safety over optimality when coordination failure is costly. This explains why inferior conventions persist — they're harder to dislodge.
Prisoner's Dilemma (Evolutionary Cooperation)
Cooperate Defect
Cooperate (3, 3) (0, 5)
Defect (5, 0) (1, 1)
Defect is the unique ESS in the one-shot game. Cooperation cannot invade a population of defectors, and defection can always invade a population of cooperators.
How cooperation evolves despite this:
- Kin selection (Hamilton 1964): Copies of your strategy in relatives make cooperation viable when relatedness × benefit > cost
- Direct reciprocity (Trivers 1971): Repeated interaction. Tit-for-tat and other reciprocal strategies are ESS in repeated games
- Indirect reciprocity (Nowak & Sigmund 1998): Reputation — cooperate with cooperators, defect on defectors
- Spatial structure: On networks/lattices, cooperators can cluster and protect each other from exploitation
- Group selection (multi-level): Groups of cooperators outcompete groups of defectors, even if defectors outcompete cooperators within groups
These are the "five mechanisms for the evolution of cooperation" (Nowak 2006).
Rock-Paper-Scissors (Cyclic Dominance)
Rock Paper Scissors
Rock (0,0) (-1,1) (1,-1)
Paper (1,-1) (0,0) (-1,1)
Scissors (-1,1) (1,-1) (0,0)
No pure ESS. The unique Nash equilibrium (1/3, 1/3, 1/3) is NOT an ESS — it fails condition 2. Under replicator dynamics, orbits cycle perpetually around the interior fixed point without converging.
Biological significance: Models cyclic dominance in nature. The side-blotched lizard (Uta stansburiana) has three male morphs — orange (aggressive), blue (mate-guarders), yellow (sneakers) — that cycle in frequency, matching the RPS dynamic (Sinervo & Lively 1996).
ESS and Nash Equilibrium
| Property | Nash Equilibrium | ESS |
|---|---|---|
| Assumes | Rational deliberation | Evolutionary process |
| Stability | No profitable deviation | Resistant to invasion |
| Always exists? | Yes (in mixed strategies) | Not always |
| Multiple? | Often | Can have 0, 1, or multiple |
| Refinement of NE? | N/A | Every ESS is a NE (stronger concept) |
| Dynamic foundation | Requires separate justification | Replicator dynamics provide natural dynamic |
The connection: ESS provides an evolutionary justification for Nash equilibrium. If a population converges to an ESS, it's playing a Nash equilibrium — without anyone knowing what a Nash equilibrium is.
Beyond Two-Player Symmetric Games
Asymmetric games (different roles): In asymmetric contests, ESS is defined for the population of role pairs. The "owner vs. intruder" model shows how property conventions emerge — "owners fight, intruders retreat" is an ESS even when roles are assigned randomly (Maynard Smith 1982).
Multi-player games: ESS generalizes to population games with n-player interactions. Payoffs depend on the population strategy distribution, not just a single opponent.
Continuous strategy spaces: When strategies are real-valued (e.g., investment level, body size), ESS analysis uses calculus. Convergence-stability and evolutionary branching become relevant (adaptive dynamics framework).
Sources
Read references/sources.md for the full bibliography — primary texts (Maynard Smith, Weibull, Sandholm), key papers on ESS, cooperation evolution, and biological applications.
When This Applies
- Analyzing competition or cooperation in biological populations
- Understanding how norms, conventions, or cultural practices emerge and persist
- Evaluating whether a Nash equilibrium is evolutionarily plausible
- Modeling situations where agents adapt rather than optimize
- Understanding frequency-dependent phenomena (strategies that succeed when rare)