GameTheory.jl

Algorithms and data structures for game theory in Julia

Example usage

Create a NormalFormGame:

using GameTheory
player1 = Player([3 3; 2 5; 0 6])
player2 = Player([3 2 3; 2 6 1])
g = NormalFormGame(player1, player2)
println(g)

3×2 NormalFormGame{2, Int64}:
 [3, 3]  [3, 2]
 [2, 2]  [5, 6]
 [0, 3]  [6, 1]

lrsnash calls the Nash equilibrium computation routine in lrslib (through its Julia wrapper LRSLib.jl):

lrsnash(g)

3-element Vector{Tuple{Vector{Rational{BigInt}}, Vector{Rational{BigInt}}}}:
 ([4//5, 1//5, 0//1], [2//3, 1//3])
 ([0//1, 1//3, 2//3], [1//3, 2//3])
 ([1//1, 0//1, 0//1], [1//1, 0//1])

A 2x2x2 NormalFormGame:

g = NormalFormGame((2, 2, 2))
g[1, 1, 1] = [9, 8, 12]
g[2, 2, 1] = [9, 8, 2]
g[1, 2, 2] = [3, 4, 6]
g[2, 1, 2] = [3, 4, 4]
println(g)

2×2×2 NormalFormGame{3, Float64}:
[:, :, 1] =
 [9.0, 8.0, 12.0]  [0.0, 0.0, 0.0]
 [0.0, 0.0, 0.0]   [9.0, 8.0, 2.0]

[:, :, 2] =
 [0.0, 0.0, 0.0]  [3.0, 4.0, 6.0]
 [3.0, 4.0, 4.0]  [0.0, 0.0, 0.0]

hc_solve computes all isolated Nash equilibria of an N-player game by using HomotopyContinuation.jl:

NEs = hc_solve(g)

9-element Vector{Tuple{Vector{Float64}, Vector{Float64}, Vector{Float64}}}:
 ([2.63311e-36, 1.0], [0.333333, 0.666667], [0.333333, 0.666667])
 ([0.25, 0.75], [1.0, 0.0], [0.25, 0.75])
 ([0.0, 1.0], [0.0, 1.0], [1.0, 0.0])
 ([0.25, 0.75], [0.5, 0.5], [0.333333, 0.666667])
 ([0.5, 0.5], [0.5, 0.5], [1.0, 1.37753e-40])
 ([1.0, 0.0], [0.0, 1.0], [0.0, 1.0])
 ([0.5, 0.5], [0.333333, 0.666667], [0.25, 0.75])
 ([1.0, 0.0], [1.0, 9.40395e-38], [1.0, -9.40395e-38])
 ([0.0, 1.0], [1.0, 0.0], [0.0, 1.0])

See the tutorials for further examples.

Implemented algorithms

Nash equilibrium computation

• pure_nash: Find all pure-action Nash equilibria of an N-player game (if any)
• support_enumeration: Find all mixed-action Nash equilibria of a two-player nondegenerate game
• lrsnash: Find all mixed-action Nash equilibria (or equilibrium components) of a two-player game
• hc_solve: Find all isolated mixed-action Nash equilibria of an N-player game

Learning/evolutionary dynamics

• BRD: Best response dynamics
• KMR: Best response with mutations dynamics of Kandori-Mailath-Rob
• SamplingBRD: Sampling best response dynamics
• FictitiousPlay: Fictitious play
• StochasticFictitiousPlay: Stochastic fictitious play
• LocalInteraction: Local interaction dynamics
• LogitDynamics: Logit dynamics

Repeated games

• outerapproximation: Equilibrium payoff computation algorithm by Judd-Yeltekin-Conklin

Tutorials

• Tools for Game Theory in GameTheory.jl
• A Recursive Formulation of Repeated Games

See also the game_theory submodule of QuantEcon.py, the Python counterpart of this package.