A set of functions in pure Julia for analysing strategic generic N-players games using concepts and tools of Game Theory.
While written in Julia, the library is easily accessible directly in Python or R using the PyJulia and JuliaCall packages respectively (Python and R examples).
Check out the companion repository GameTheoryNotes for introductory notes on the Game Theory approach and the documentation at the links below for further information on the package and how to use it.
Other examples are available in the Examples section of the documentation.
julia> # Prisoner's dilemma (N players are also supported in all functions)
payoff = [(-1,-1) (-3, 0);
( 0,-3) (-2, -2)];
julia> # From N-dimensional array of tuples to N+1 arrays of scalars
payoff_array = expand_dimensions(payoff);
julia> # Find all the dominated strategies for the two players
dominated_actions(payoff_array)
2-element Vector{Vector{Int64}}:
[1]
[1]
julia> # Compute one Nash Equilibrium of the Game using complementarity formulation
eq = nash_cp(payoff_array).equilibrium_strategies
2-element Vector{Vector{Float64}}:
[0.0, 0.9999999887780999]
[0.0, 0.9999999887780999]
julia> # Compute all isolated Nash equilibria using support enumeration
eqs = nash_se(payoff_array,max_samples=Inf)
1-element Vector{NamedTuple{(:equilibrium_strategies, :expected_payoffs, :supports), Tuple{Vector{Vector{Float64}}, Vector{Float64}, Vector{Vector{Int64}}}}}:
(equilibrium_strategies = [[0.0, 0.9999999999999999], [0.0, 0.9999999999999999]], expected_payoffs = [-1.9999999999999678, -1.9999999999999678], supports = [[2], [2]])
julia> # Best response for player 2
best_response(payoff_array,[[0.5,0.5],[0.5,0.5]],2).optimal_strategy
2-element Vector{Float64}:
0.0
1.0
julia> # Expected payoffs given a specific strategy profile
expected_payoff(payoff_array,[[1,0],[1,0]])
2-element Vector{Int64}:
-1
-1
julia> # Is this strategy profile a Nash equilibrium ?
is_nash(payoff_array,[[1,0],[1,0]])
falseJulia
- Nash.jl Has several functions to generate games in normal form, determine best response and if a strategy profile is a Nash Equilibrium (NE) but it doesn't provide a functionality to retrieve a NE, except for 2 players simmetric games
- GameTheory.jl Inter alia, compute N-players pure strategy NE, 2-players mixed strategy games (
lrsnash, using exact arithmetics) and N-players mixed strategies NE using a solver of the polynomial equation representation of the complementarity conditions (hc_solve). However this "generic" N-player solver is slow, as it doesn't seem to have a dominance check.
Non-Julia
- Nashpy: two players only
- Gambit: many algorithms require installation of gambit other than pygambit. No decimal/rational payoffs in Python
- Sage Math: two players only
- GtNash
The following benchmarks have been run on a Intel Core 5 laptop on StrategicGames v0.0.5. See benchmarks/benchmarks_other_libraries.jl for details.
Row │ benchmark_name library method time memory alloc neqs notes
│ String String String Float64 Float64? Int64? Int64 String
─────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
1 │ Test rand rand(2,2) 59.4469 96.0 1 1
2 │ small_3x2 StrategicGames nash_cp 1.58605e7 306112.0 6002 1
3 │ small_3x2 StrategicGames nash_se 2.42753e6 348928.0 8022 3
4 │ small_3x2 GameTheory hc_solve 1.54957e7 2.90228e6 38420 3
5 │ small_3x2 GameTheory support_enumeration 1.26744e5 6848.0 103 3
6 │ small_3x2 GameTheory lrsnash 81660.0 25208.0 1064 3
7 │ small_3x2 nashpy vertex_enumeration 2.24663e7 missing missing 3
8 │ small_3x2 nashpy lemke_howson_enumeration 1.70237e6 missing missing 5 repeated results
9 │ small_3x2 nashpy support_enumeration 2.88622e6 missing missing 3
10 │ small_3x2 pygambit lcp_solve 626370.0 missing missing 3
11 │ small_3x2 pygambit ExternalEnumPolySolver 3.04543e6 missing missing 3
12 │ rand_6x7 StrategicGames nash_cp 6.79384e7 832992.0 16774 0
13 │ rand_6x7 StrategicGames nash_se 5.81445e7 8.39402e7 1489215 1
14 │ rand_6x7 GameTheory hc_solve 2.29249e10 2.13365e8 5674406 1
15 │ rand_6x7 GameTheory support_enumeration 1.07393e6 255888.0 3223 1
16 │ rand_6x7 GameTheory lrsnash 1.22114e6 189432.0 11058 1
17 │ rand_6x7 nashpy vertex_enumeration 4.69925e8 missing missing 1
18 │ rand_6x7 nashpy lemke_howson_enumeration 1.12535e7 missing missing 13 repeated results
19 │ rand_6x7 nashpy support_enumeration 1.14592e9 missing missing 0
20 │ rand_6x7 pygambit lcp_solve 9.02179e6 missing missing 1
21 │ rand_6x7 pygambit ExternalEnumPolySolver 5.88408e11 missing missing 1
22 │ rand_dec_6x5 StrategicGames nash_cp 2.62495e7 666384.0 12585 1
23 │ rand_dec_6x5 StrategicGames nash_se 1.91476e7 1.63948e7 301926 3
24 │ rand_dec_6x5 StrategicGames nash_se 2.10698e7 1.63948e7 301928 3
25 │ rand_dec_6x5 GameTheory hc_solve 2.81279e9 1.31916e7 129193 3
26 │ rand_dec_6x5 GameTheory support_enumeration 3.6849e5 82976.0 1112 3
27 │ rand_dec_6x5 GameTheory lrsnash 1.18709e6 194672.0 9179 3
28 │ rand_dec_6x5 nashpy vertex_enumeration 1.22334e8 missing missing 3
29 │ rand_dec_6x5 nashpy lemke_howson_enumeration 5.08469e6 missing missing 11 repeated results
30 │ rand_dec_6x5 nashpy support_enumeration 2.52825e8 missing missing 3
31 │ rand_4x4x2 StrategicGames nash_cp 7.2921e8 2.99718e6 85361 0
32 │ rand_4x4x2 StrategicGames nash_se 3.10279e9 7.36352e7 1353042 7 1 eq repeated
33 │ rand_4x4x2 GameTheory hc_solve 6.81009e9 1.87611e7 168777 4 2 eq missing
34 │ rand_4x4x2 pygambit ExternalEnumPolySolver 6.67698e8 missing missing 5 1 eq missed
35 │ rand_6x7_1st_eq StrategicGames nash_se 7.89602e6 3.82298e6 69988 1
36 │ rand_6x7_1st_eq GameTheory hc_solve 1.17059e10 1.06781e8 2643505 1
37 │ rand_6x7_1st_eq nashpy vertex_enumeration 2.20641e8 missing missing 1
38 │ rand_6x7_1st_eq nashpy lemke_howson_enumeration 8.8455e5 missing missing 1
39 │ rand_6x7_1st_eq nashpy support_enumeration 3977.38 missing missing 0 no eq reported
40 │ degenerated_3x2 StrategicGames nash_cp 1.74314e7 315424.0 6320 1
41 │ degenerated_3x2 StrategicGames nash_se 2.39501e6 430528.0 9999 3
42 │ degenerated_3x2 GameTheory hc_solve 1.58658e7 2.91026e6 38445 1
43 │ degenerated_3x2 GameTheory support_enumeration 128884.0 6544.0 98 2
44 │ degenerated_3x2 GameTheory lrsnash 79857.5 23688.0 1002 3
45 │ degenerated_3x2 nashpy vertex_enumeration 1.8437e7 missing missing 3
46 │ degenerated_3x2 nashpy lemke_howson_enumeration 1.75092e6 missing missing 5 2 repeated results
47 │ degenerated_3x2 nashpy support_enumeration 3.03576e6 missing missing 2
48 │ degenerated_3x2 pygambit lcp_solve 672832.0 missing missing 3
49 │ degenerated_3x2 pygambit ExternalEnumPolySolver 3.39349e6 missing missing 2
The development of this package at the Bureau d'Economie Théorique et Appliquée (BETA, Nancy) was supported by the French National Research Agency through the Laboratory of Excellence ARBRE, a part of the “Investissements d'Avenir” Program (ANR 11 – LABX-0002-01).
