DoubleQuack.jl

Two-player zero-sum continous game solver.

Install

Install the package directly from Github. Open Pkg mode with ], then

add https://github.com/votroto/DoubleQuack.jl.git

Gurobi

This package uses Gurobi. You need a valid license to use it. Supress debugging output by creating gurobi.env in your current folder, and paste into it the settings:

OutputFlag 0
LogToConsole 0

Model 1: Nie 2021, Example 6.1.(i)

Consider the following two-player zero-sum continuous game with $3\times 3$ variables, where Player 1 has the loss function:

$l_1(\mathbf{x}, \mathbf{y}) = x_1x_2 + x_2x_3 + x_3y_1 + x_1y_3 + y_1y_2 + y_2y_3$

and where the strategy spaces are symmetrically defined by

$v_i \ge 0 \quad \forall i \in {1,2,3}$,

$\sum_{i=1}^3 v_i = 1$.

We model it in DoubleQuack as

l(x, y) = x[1]x[2] + x[2]x[3] + x[3]y[1] + x[1]y[3] + y[1]y[2] + y[2]y[3]

g = @game x{3} y{3} begin
    -l(x, y)
end
@strategy_set g (x, y) as v in begin
    v .>= 0
    sum(v) == 1
end

The game has a known saddle point at $x^\star = (0, 1, 0)$ and $y^\star = (0.25, 0.5, 0.25)$.

Model 2: Stein 2008, Example 2.3

Consider the following two-player general-sum continuous game with $1\times 1$ variables, where the utility function are:

$u_1(\mathbf{x}, \mathbf{y}) = 2x_1y_1 + 3y_1^3 - 2x_1^3 - x_1 - 3x_1^2y_1^2$

$u_2(\mathbf{x}, \mathbf{y}) = 2x_1^2y_1^2 - 4y_1^3 - x_1^2 + 4y_1 + x_1^2y_1$

The strategy spaces are: $-1 \leq x_1 \leq 1$ and $-1 \leq y_1 \leq 1$.

We model it in DoubleQuack as

g = @game x{1} y{1} begin
    2 * x[1] * y[1] + 3y[1]^3 - 2x[1]^3 - x[1] - 3x[1]^2 * y[1]^2
    2x[1]^2 * y[1]^2 - 4y[1]^3 - x[1]^2 + 4y[1] + x[1]^2 * y[1]
end
@strategy_set g (x, y) in [-1, 1]

Experiment

Find the $\epsilon$-NE of either game with $\epsilon=0.001$ using:

result = solve(g, eps=1e-3, max_iters=20)
DoubleQuack.clean_print_ne(result)

the result is a mixed strategy $\epsilon$-NE