diff --git a/.gitignore b/.gitignore
index b067edd..ef76bda 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1 +1,31 @@
-/Manifest.toml
+*.mp4
+
+# Files generated by invoking Julia with --code-coverage
+*.jl.cov
+*.jl.*.cov
+
+# Files generated by invoking Julia with --track-allocation
+*.jl.mem
+
+# System-specific files and directories generated by the BinaryProvider and BinDeps packages
+# They contain absolute paths specific to the host computer, and so should not be committed
+deps/deps.jl
+deps/build.log
+deps/downloads/
+deps/usr/
+deps/src/
+
+# Build artifacts for creating documentation generated by the Documenter package
+docs/build/
+docs/site/
+
+# Emacs files.
+#*
+.#*
+*~
+
+# File generated by Pkg, the package manager, based on a corresponding Project.toml
+# It records a fixed state of all packages used by the project. As such, it should not be
+# committed for packages, but should be committed for applications that require a static
+# environment.
+Manifest.toml
diff --git a/Project.toml b/Project.toml
index 728af29..3b2acaa 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,12 +1,10 @@
- name = "ZeroSumGameSolve"
+name = "ZeroSumGameSolve"
 uuid = "acbac2c1-af1f-4c5c-8072-533d2c216f7a"
 authors = ["Anonymized for submission"]
 version = "0.1.0"
 
 [deps]
 Arpack = "7d9fca2a-8960-54d3-9f78-7d1dccf2cb97"
-CUDA = "052768ef-5323-5732-b1bb-66c8b64840ba"
-Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
 DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
 LineSearches = "d3d80556-e9d4-5f37-9878-2ab0fcc64255"
@@ -14,11 +12,8 @@ LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Optim = "429524aa-4258-5aef-a3af-852621145aeb"
 Optimisers = "3bd65402-5787-11e9-1adc-39752487f4e2"
 OrdinaryDiffEq = "1dea7af3-3e70-54e6-95c3-0bf5283fa5ed"
-Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
 ReverseDiff = "37e2e3b7-166d-5795-8a7a-e32c996b4267"
-Revise = "295af30f-e4ad-537b-8983-00126c2a3abe"
 StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
-StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
 Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
 Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 
diff --git a/experiments/GANExample/Project.toml b/experiments/GANExample/Project.toml
index da174a4..ea505c5 100644
--- a/experiments/GANExample/Project.toml
+++ b/experiments/GANExample/Project.toml
@@ -16,3 +16,6 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 ZeroSumGameSolve = "acbac2c1-af1f-4c5c-8072-533d2c216f7a"
 Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"
 cuDNN = "02a925ec-e4fe-4b08-9a7e-0d78e3d38ccd"
+
+[sources]
+ZeroSumGameSolve = {path = "../.."}
\ No newline at end of file
diff --git a/experiments/MinMaxComparison/Manifest.toml b/experiments/MinMaxComparison/Manifest.toml
new file mode 100644
index 0000000..422c5b5
--- /dev/null
+++ b/experiments/MinMaxComparison/Manifest.toml
@@ -0,0 +1,1863 @@
+# This file is machine-generated - editing it directly is not advised
+
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+manifest_format = "2.0"
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+
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+
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+
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+
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+    ArrayInterfaceBlockBandedMatricesExt = "BlockBandedMatrices"
+    ArrayInterfaceCUDAExt = "CUDA"
+    ArrayInterfaceCUDSSExt = "CUDSS"
+    ArrayInterfaceChainRulesCoreExt = "ChainRulesCore"
+    ArrayInterfaceChainRulesExt = "ChainRules"
+    ArrayInterfaceGPUArraysCoreExt = "GPUArraysCore"
+    ArrayInterfaceReverseDiffExt = "ReverseDiff"
+    ArrayInterfaceSparseArraysExt = "SparseArrays"
+    ArrayInterfaceStaticArraysCoreExt = "StaticArraysCore"
+    ArrayInterfaceTrackerExt = "Tracker"
+
+    [deps.ArrayInterface.weakdeps]
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+[[deps.Xorg_xcb_util_jll]]
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+[[deps.Xorg_xcb_util_wm_jll]]
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+[[deps.eudev_jll]]
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diff --git a/experiments/MinMaxComparison/MinMaxComparison.jl b/experiments/MinMaxComparison/MinMaxComparison.jl
new file mode 100644
index 0000000..fa8c6e8
--- /dev/null
+++ b/experiments/MinMaxComparison/MinMaxComparison.jl
@@ -0,0 +1,17 @@
+module MinMaxComparison
+
+using Plots, Symbolics, ForwardDiff, LinearAlgebra
+
+include("algorithm.jl")
+include("utils.jl")
+include("domain.jl")
+include("config.jl")
+
+
+# TODO: @kushy write code here
+function foo()
+    println("hello")
+end
+
+
+end # module
diff --git a/experiments/MinMaxComparison/Project.toml b/experiments/MinMaxComparison/Project.toml
new file mode 100644
index 0000000..7cdd724
--- /dev/null
+++ b/experiments/MinMaxComparison/Project.toml
@@ -0,0 +1,8 @@
+[deps]
+Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
+ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
+Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
diff --git a/experiments/MinMaxComparison/Project.toml~ b/experiments/MinMaxComparison/Project.toml~
new file mode 100644
index 0000000..7ac5162
--- /dev/null
+++ b/experiments/MinMaxComparison/Project.toml~
@@ -0,0 +1,9 @@
+[deps]
+Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
+StayOnTheRidge = "250d2a78-eb8c-4fdb-8c50-61bd39400985"
+
+[sources]
+StayOnTheRidge = {path = "../../external/"}
diff --git a/experiments/MinMaxComparison/algorithm.jl b/experiments/MinMaxComparison/algorithm.jl
new file mode 100644
index 0000000..917185d
--- /dev/null
+++ b/experiments/MinMaxComparison/algorithm.jl
@@ -0,0 +1,258 @@
+using LinearAlgebra
+using ForwardDiff
+using Symbolics
+
+include("domain.jl")
+include("config.jl")
+
+P(x; x_min = 0, x_max = 1) = min.(max.(x, x_min), x_max)
+
+function get_V(conf::Config_sym, point)
+    V = [conf.V[i](point) for i in 1:conf.n]
+    return V
+end
+
+function get_V(conf::Config_FD, point)
+    V = ForwardDiff.gradient(conf.f_obj, point)
+    V[conf.min_coords] .*= -1
+    return V
+end
+
+function get_V_origin(conf::Config_sym, point)
+    V_origin = [conf.V_origin[i](point) for i in 1:conf.n]
+    return V_origin
+end
+
+function get_V_origin(conf::Config_FD, point)
+    V_origin = ForwardDiff.gradient(conf.f, point)
+    V_origin[conf.min_coords] .*= -1
+    return V_origin
+end
+
+function get_Vi(conf::Config_sym, point, i)
+    Vi = conf.V[i](point)
+    return Vi
+end
+
+function get_Vi(conf::Config_FD, point, i)
+    Vi = get_V(conf, point)[i]
+    return Vi
+end
+
+function get_V_tr(conf::Config_sym, new_point, tr_coords)
+    V_tr = [conf.V[tr](new_point) for tr in tr_coords]
+    return V_tr
+end
+
+function get_V_tr(conf::Config_FD, new_point, tr_coords)
+    V_tr = get_V(conf, new_point)[tr_coords]
+    return V_tr
+end
+
+function get_jacobian_Si(conf::Config_sym, point, d_nonzero_idxs, S)
+    jacobian_Si = [conf.J[i,j](point) for i in S, j in d_nonzero_idxs]
+    return jacobian_Si
+end
+
+function get_jacobian_Si(conf::Config_FD, point, d_nonzero_idxs, S)
+    jacobian = ForwardDiff.hessian(conf.f_obj, point)
+    jacobian[conf.min_coords,:] .*= -1
+    jacobian_Si = [jacobian[i,j] for i in S, j in d_nonzero_idxs]
+    return jacobian_Si
+end
+
+function compute_direction(point, i, S, conf::Config)
+    d = zeros(conf.n)
+
+    if isempty(S)
+        d[i] = 1
+        return d
+    end
+
+    d_nonzero_idxs = vcat(S, [i])
+    jacobian_Si = get_jacobian_Si(conf, point, d_nonzero_idxs, S)
+    d_nonzero = nullspace(jacobian_Si)
+    if size(d_nonzero, 2) != 1
+        # direction is not uniquely defined
+        error("Assumption 1 violated at i = $i, S = $S, x = $point, nullspace dimension is $(ndims(d_nonzero)), jacobian_Si_at_point = $jacobian_Si")
+    end
+
+    d_nonzero = vec(d_nonzero)
+    decision_mat = hcat(transpose(jacobian_Si), d_nonzero)
+    decision_det = det(decision_mat)
+    if sign(decision_det) != (-1)^(length(S))
+        d_nonzero .*= -1
+    end
+
+    d[d_nonzero_idxs] = d_nonzero
+    return d
+end
+
+function good_exit(point, i, conf::Config)
+    Vi = get_Vi(conf, point, i)
+    xi = point[i]
+    zs = abs(Vi) <= conf.ϵ
+    return zs || (xi == 0 && Vi < conf.ϵ) || (xi == 1 && Vi > -conf.ϵ), zs
+end
+
+function bad_exit(point, i, S, direction)
+    tr_coords = vcat(S, [i])
+    point_tr = point[tr_coords]
+    direction_tr = direction[tr_coords]
+
+    for (j, (pt, dt)) in enumerate(zip(point_tr, direction_tr))
+        if (pt == 0 && dt < 0) || (pt == 1 && dt > 0)
+            return tr_coords[j]
+        end
+    end
+
+    return 0
+end
+
+function middling_exit(point, i, S, direction, conf::Config)
+    new_point = point + conf.γ*direction
+    tr_coords = setdiff(1:i-1, S)
+    point_tr = point[tr_coords]
+    V_tr = get_V_tr(conf, new_point, tr_coords)
+    for (j, (pt, Vt)) in enumerate(zip(point_tr, V_tr))
+        if (pt == 0 && Vt > 0) || (pt == 1 && Vt < 0)
+            return tr_coords[j]
+        end
+    end
+
+    return 0
+end
+
+function Vx_wrst_case_y(dom::Hyperrectangle, conf::Config, point)
+    Vx = get_V_origin(conf, point)
+    y_worst = []
+    for i in eachindex(Vx)
+        if Vx[i] < 0
+            push!(y_worst, dom.sides[i][1])
+        else
+            push!(y_worst, dom.sides[i][2])
+        end
+    end
+    return Vx, y_worst
+end
+
+function Vx_wrst_case_y(dom::Default, conf::Config, point)
+    Vx = get_V_origin(conf, point)
+    return Vx, ifelse.(Vx .<= 0, 0, 1)
+end
+
+# check if point satisfies the variational inequality
+function is_solution_VI(point, conf::Config; α = 0.1)
+    Vx = get_V(conf, point)
+    y_worst = ifelse.(Vx .<= 0, 0, 1)
+    dot1 = dot(Vx, point)
+    dot2 = dot(Vx, y_worst)
+    VI_val = dot1 - dot2
+    return VI_val >= -α
+end
+
+function is_solution_VI(point, conf::Config, dom::Domain; α = 0.1)
+    Vx, y_worst = Vx_wrst_case_y(dom, conf, point)
+    dot1 = dot(Vx, point)
+    dot2 = dot(Vx, y_worst)
+    VI_val = dot1 - dot2
+    if VI_val >= -α
+        display("$α-VI satisfied at $point with value $VI_val")
+    else
+        @warn "$α-VI not satisfied at $point with value $VI_val"
+    end
+    return VI_val >= -α
+end
+
+function assumption_2_violated(i, S, point)
+    coords = point[vcat(S,[i])]
+    count_zeros = count(x -> x == 0, coords)
+    count_ones = count(x -> x == 1, coords)
+    if (count_ones+count_zeros > 1)
+        @warn "Assumption 2 violated at i = $i, S = $S, x = $point"
+    end
+end
+
+"""
+run_dynamics(conf::Config)
+
+Executes STON'R dynamics with a given configuration conf.
+
+# Arguments
+- `conf::Config` : See `config_FD`, `Config_sym`.
+
+# Output
+- `point`: The min-max critical point.
+- `pts`: A vector of trajectory points explored during the dynamics.
+- `m`: The number of epochs (outer iterations) performed.
+- `k`: The number of iterations within each epoch.
+
+# Example
+
+```julia-repl
+julia> n = 2
+julia> min_coords = [1]
+julia> γ = 1e-3
+julia> ϵ = 1e-2
+julia> f_fd(x) = -2*x[1]*x[2]^2+x[1]^2+x[2]
+julia> domain = Hyperrectangle([[-1,1],[-1,1]])
+julia> conf = Config_FD(f_fd, n, min_coords, γ, ϵ, domain)
+julia> min_max, trajectory, m, k = run_dynamics(conf)
+```
+"""
+function run_dynamics(conf::Config)
+    point = fill(0, conf.n)
+    i = 1
+    S = Vector{Int}()
+    pts = Vector{Vector{Float64}}()
+    m = 0
+    k = 0
+
+    while  i <= conf.n # && !is_solution_VI(point, conf)
+        k = 0
+        println("starting epoch ($i, $S) at $point")
+
+        while true
+            assumption_2_violated(i, S, point)
+            is_good_exit, is_zs = good_exit(point, i, conf)
+            if is_good_exit
+                println("good exit at $point")
+                if is_zs
+                    S = vcat(S,[i])
+                end
+                i += 1
+                break
+            end
+            
+            direction = compute_direction(point, i, S, conf)
+            j = bad_exit(point, i, S, direction)
+            if j != 0
+                println("bad exit at $point for j = $j")
+                if j == i
+                    S = setdiff(S, [i-1])
+                    i -= 1
+                else
+                    S = setdiff(S, [j])
+                end
+                break
+            end
+
+            j = middling_exit(point, i, S, direction, conf)
+            if j != 0
+                println("middling exit at $point for j = $j")
+                S = vcat(S, [j])
+                break
+            end
+            
+            point += conf.γ*direction
+            point = P(point)
+
+            k += 1
+            push!(pts, point)
+        end
+
+        m += 1
+    end
+    return conf.domain isa Default ? (point, pts, m, k) :
+        (H(conf.domain)(point), map(x->H(conf.domain)(x),pts), m, k)
+end
\ No newline at end of file
diff --git a/experiments/MinMaxComparison/config.jl b/experiments/MinMaxComparison/config.jl
new file mode 100644
index 0000000..cc1826c
--- /dev/null
+++ b/experiments/MinMaxComparison/config.jl
@@ -0,0 +1,129 @@
+abstract type Config end
+
+"""
+Config_FD
+
+Configuration structure to store data for STON'R execution.
+Using ForwardDiff.jl for differentiation.
+
+# Arguments
+- `f::Function` : Function on domain [0,1]ⁿ.
+- `n::Int` : Number of variables.
+- `min_coords::Vector{Int}` : Vector of minimizing coordinates.
+- `γ::Real` : Step size.
+- `ϵ::Real` : Precision parameter of min-max critical point.
+- `domain::Domain` : Specifies the domain of the function (see Default or Hyperrectangle).
+# Example
+```julia-repl
+julia> f_fd(x) = -2*x[1]*x[2]^2+x[1]^2+x[2]
+julia> n = 2
+julia> min_coords = [1]
+julia> γ = 1e-2
+julia> ϵ = 1e-2
+julia> domain = Default()
+julia> conf = Config_FD(f_fd, n, min_coords, γ, ϵ, domain);
+```
+"""
+struct Config_FD <: Config
+    f::Function
+    f_obj::Function
+    n::Int
+    min_coords::Vector{Int}
+    γ::Real
+    ϵ::Real
+    domain::Domain
+
+    function Config_FD(f::Function, n::Int, min_coords::Vector{Int},
+        γ::Real, ϵ::Real, domain::Domain)
+        @assert n > 0
+        @assert all(x -> x > 0, min_coords)
+        @assert γ > 0
+        @assert ϵ > 0
+        @assert length(min_coords) <= n
+        if domain isa Default # default [0,1]ⁿ
+            f_obj = f
+        else
+            inner = H(domain) # closure to gain performance
+            f_obj = x -> f(inner(x))
+        end
+        return new(f, f_obj, n, min_coords, γ, ϵ, domain)
+    end
+end
+
+"""
+Config_sym
+
+Configuration structure to store data for STON'R execution.
+Using Symbolics.jl for differentiation.
+
+# Arguments
+- `f_expr::Num` : Symbolic function on domain [0,1]ⁿ.
+- `x::Num` : Vector of symbolic variables.
+- `n::Int` : Number of variables.
+- `min_coords::Vector{Int}` : Vector of minimizing coordinates.
+- `γ::Real` : Step size.
+- `ϵ::Real` : Precision parameter of min-max critical point.
+- `domain::Domain` : Specifies the domain of the function (see Default or Hyperrectangle).
+# Example
+```julia-repl
+julia> n = 2
+julia> x = [Symbolics.variable(:x,i) for i in 1:n]
+julia> f = (x[1]-1/2)*(x[2]-1/2)
+julia> min_coords = [1]
+julia> γ = 1e-2
+julia> ϵ = 1e-2
+julia> domain = Default()
+julia> conf = Config_sym(f, x, n, min_coords, γ, ϵ, domain);
+```
+"""
+struct Config_sym <: Config
+    f::Function
+    V::Vector{Function} # modified gradient of f
+    V_origin::Vector{Function} # modified gradient of f on the [0,1]ⁿ
+    J::Matrix{Function} # modified Jacobian matrix of V
+    x::Vector{Num}
+    n::Int
+    min_coords::Vector{Int}
+    γ::Real
+    ϵ::Real
+    domain::Domain
+
+    # for testing
+    grad_f::Vector{Function}
+    hess_mat_f::Matrix{Function}
+
+    function Config_sym(f_expr::Num, x::Vector{Num}, n::Int, min_coords::Vector{Int},
+        γ::Real, ϵ::Real, domain::Domain)
+        @assert n > 0
+        @assert all(x -> x > 0, min_coords)
+        @assert γ > 0
+        @assert ϵ > 0
+        @assert length(x) == n
+        @assert length(min_coords) <= n
+        f = eval(build_function(f_expr, x))
+        V_expr = prepare_V(f_expr, x, min_coords, domain)
+        V = [eval(build_function(V_expr[i], x)) for i in 1:n]
+        V_origin_expr = prepare_V(f_expr, x, min_coords, Default())
+        V_origin = [eval(build_function(V_origin_expr[i], x)) for i in 1:n]
+        J_expr = Symbolics.jacobian(V_expr, x)
+        J = [eval(build_function(J_expr[i,j], x)) for i in 1:n, j in 1:n]
+        grad_f_expr = Symbolics.gradient(f_expr, x)
+        grad_f = [eval(build_function(grad_f_expr[i], x)) for i in 1:n]
+        hess_mat_f_expr = Symbolics.jacobian(grad_f_expr, x)
+        hess_mat_f = [eval(build_function(hess_mat_f_expr[i,j], x)) for i in 1:n, j in 1:n]
+        return new(f, V, V_origin, J, x, n, min_coords, γ, ϵ, domain, grad_f, hess_mat_f)
+    end
+end
+
+eval_expr(expr, dict) = Symbolics.value.(Symbolics.substitute(expr, dict))
+
+function prepare_V(f, x, min_coords, domain)
+    if domain isa Default
+        f_obj = f
+    else
+        f_obj = eval_expr(f, Dict(x .=> H(domain)(x)))
+    end
+    V = Symbolics.gradient(f_obj, x)
+    V[min_coords] .*= -1
+    return V
+end
\ No newline at end of file
diff --git a/experiments/MinMaxComparison/domain.jl b/experiments/MinMaxComparison/domain.jl
new file mode 100644
index 0000000..04f3edf
--- /dev/null
+++ b/experiments/MinMaxComparison/domain.jl
@@ -0,0 +1,26 @@
+abstract type Domain end
+
+"""
+Default domain [0,1]ⁿ.
+"""
+struct Default <: Domain end # unit hypercube [0,1]ⁿ
+
+"""
+Hyperrectangle
+
+# Arguments
+- `sides::Vector{Vector{Real}}` : Define the hyperrectangle, e.g. sides = [[-1,1],[0,1]] define the rectangle [-1,1]×[0,1].
+"""
+struct Hyperrectangle <: Domain
+    sides::Vector{Vector{T}} where T <: Real
+    function Hyperrectangle(sides::Vector{Vector{T}} where T <: Real)
+        @assert all(subvec -> length(subvec) >= 2 && subvec[1] < subvec[2], sides)
+        return new(sides)
+    end
+end
+
+# general hyperrectangle mapping
+function H(domain::Hyperrectangle)
+    sides = domain.sides
+    return x -> getindex.(sides,1) .+ (getindex.(sides,2) .- getindex.(sides,1)) .* x
+end
\ No newline at end of file
diff --git a/experiments/MinMaxComparison/utils.jl b/experiments/MinMaxComparison/utils.jl
new file mode 100644
index 0000000..9917df8
--- /dev/null
+++ b/experiments/MinMaxComparison/utils.jl
@@ -0,0 +1,78 @@
+using Plots
+
+function get_bounds(domain::Hyperrectangle)
+    p11 = domain.sides[1][1]
+    p12 = domain.sides[1][2]
+    p21 = domain.sides[2][1]
+    p22 = domain.sides[2][2]
+    x1_min_bound = p11 - abs(p11-p12)/20
+    x1_max_bound = p12 + abs(p11-p12)/20
+    x2_min_bound = p21 - abs(p21-p22)/20
+    x2_max_bound = p22 + abs(p21-p22)/20
+    return [x1_min_bound, x1_max_bound, x2_min_bound, x2_max_bound]
+end
+
+function get_bounds(domain::Default)
+    return [-1/20,21/20,-1/20,21/20]
+end
+
+function get_ranges(domain::Hyperrectangle)
+    return [range(domain.sides[1][1], domain.sides[1][2], length=300),
+        range(domain.sides[2][1], domain.sides[2][2], length=300)]
+end
+
+function get_ranges(domain::Default)
+    return [range(0, 1, length=300),
+    range(0, 1, length=300)]
+end
+
+function plot_trajectory2D(min_max, trajectory, domain::Domain)
+    x1_coords = [pt[1] for pt in trajectory]
+    x2_coords = [pt[2] for pt in trajectory]
+    bounds = get_bounds(domain)
+    scatter(x1_coords, x2_coords; color=:blue, legend = false, markersize = 1,
+        markershape=:auto, xlims = [bounds[1], bounds[2]], ylims = [bounds[3], bounds[4]])
+    scatter!([min_max[1]], [min_max[2]], markershape=:star, markersize=10, color=:blue)
+    title!("Trajectory")
+end
+
+function plot_contour2D(min_max, f, domain::Domain)
+    f_plot(x,y) = f([x,y])
+    rs = get_ranges(domain)
+    bounds = get_bounds(domain)
+    contourf(rs[1], rs[2], f_plot, levels=20, xlims = [bounds[1], bounds[2]], ylims = [bounds[3], bounds[4]]) # size=(800, 600), dpi=300
+    scatter!([min_max[1]], [min_max[2]], markershape=:star, markersize=10, legend=false, color=:blue)
+    title!("Contour plot of the function")
+    # xlabel!("θ")
+    # ylabel!("ω")
+end
+
+function plot_surface(min_max, f, domain::Domain)
+    f_plot(x,y) = f([x,y])
+    rs = get_ranges(domain)
+    zs = [f_plot(x,y) for y in rs[2], x in rs[1]]
+    surface(rs[1], rs[2], zs, legend=false) # size=(800, 600), dpi=300
+    # zticks!(0.25:0.25:1)
+    # xlabel!("θ")
+    # ylabel!("ω")
+    scatter!([min_max[1]], [min_max[2]], [f(min_max)], markershape=:star, markersize=10, legend=false, color=:blue)
+    title!("3D plot of the function")
+end
+
+function pretty_print(point, time, m, k)
+    str_point = "($(join(round.(point, digits=4), ", ")))"
+    str_time = format_elapsed(time)
+    display("Min max critical point: $str_point, elapsed time: $str_time, outer cycles: $m, inner cycles: $k")
+end
+
+function format_elapsed(elapsed_seconds)
+    if elapsed_seconds < 1e-6
+        return string(round(elapsed_seconds * 1e9, digits=3), " ns")
+    elseif elapsed_seconds < 1e-3
+        return string(round(elapsed_seconds * 1e6, digits=3), " μs")
+    elseif elapsed_seconds < 1
+        return string(round(elapsed_seconds * 1e3, digits=3), " ms")
+    else
+        return string(round(elapsed_seconds, digits=3), " s")
+    end
+end
\ No newline at end of file
diff --git a/experiments/PE Game/pegame.jl b/experiments/PE Game/pegame.jl
new file mode 100644
index 0000000..c1a6c34
--- /dev/null
+++ b/experiments/PE Game/pegame.jl	
@@ -0,0 +1,24 @@
+using ZeroSumGameSolve, LinearAlgebra, StaticArrays, ReverseDiff, Zygote
+include("pegame_utils.jl")
+
+delta = 1.0
+dt = 1.0
+v_amax=0.8
+T = 5
+tol = 1e-5
+max_iters = Int(1e4)
+α = 0.1
+
+pegame = makePEGame(dt, delta, v_amax, T, tol, max_iters, α)
+
+guess = zeros(pegame.prob_dim)
+guess[1] = 0.5
+guess[2:pegame.n_x] .= 1/(pegame.n_x - 1)
+guess[pegame.n_x+1:end] .= pegame.v_amax-0.1
+println("Initial guess: ", guess)
+func = x -> PEObjective(x, pegame.delta, pegame.dt, pegame.T, pegame.v_amax, pegame.i_iter_start, pegame.num_possible_x0s, pegame.n_x)
+println("function", func)
+sol, func_val, iters, path = solver_PE(guess, func, pegame.n_x, pegame.tol, pegame.max_iters, pegame.α, pegame.v_amax)
+println("Solution: ", sol)
+println("Function Value: ", func_val)
+println("Iterations: ", iters)
diff --git a/experiments/PE Game/pegame_fixed.jl b/experiments/PE Game/pegame_fixed.jl
new file mode 100644
index 0000000..d5b34e3
--- /dev/null
+++ b/experiments/PE Game/pegame_fixed.jl	
@@ -0,0 +1,27 @@
+using ZeroSumGameSolve, LinearAlgebra, StaticArrays, ReverseDiff, Zygote, Plots
+include("pegame_utils.jl")
+
+delta = 1.0
+dt = 1.0
+v_amax=0.8
+T = 5
+tol = 1e-5
+max_iters = Int(1e4)
+α = 0.1
+radius = 1
+
+pegamefixed = makePEGamefixed(dt, delta, v_amax, T, tol, max_iters, α)
+
+guess = zeros(pegamefixed.prob_dim)
+guess[1] = 0.5
+println("Initial guess: ", guess)
+guess[pegamefixed.n_x+1:end] .= 0.75
+println("Initial guess: ", guess)
+func = x -> PEObjectivefixed(x, pegamefixed.delta, pegamefixed.dt, pegamefixed.T, pegamefixed.v_amax, pegamefixed.x0, pegamefixed.n_x, radius)
+println("function", func)
+sol, func_val, iters, path = solver_PE_fixed(guess, func, pegamefixed.n_x, pegamefixed.tol, pegamefixed.max_iters, pegamefixed.α, pegamefixed.v_amax)
+println("Solution: ", sol)
+println("Function Value: ", func_val)
+println("Iterations: ", iters)
+
+plot_traj(sol[2:end], pegamefixed.v_amax, pegamefixed.x0, pegamefixed.delta, sol[1])
diff --git a/experiments/PE Game/pegame_utils.jl b/experiments/PE Game/pegame_utils.jl
new file mode 100644
index 0000000..72d5a3d
--- /dev/null
+++ b/experiments/PE Game/pegame_utils.jl	
@@ -0,0 +1,348 @@
+using ZeroSumGameSolve
+
+struct PEGame
+    n_x::Int
+    dt::Float64 
+    delta::Float64
+    v_amax::Float64
+    i_iter_start::Int
+    prob_dim:: Int
+    T::Int
+    num_possible_x0s::Int
+    tol::Float64
+    α::Float64
+    max_iters::Int
+
+end
+
+function makePEGame(dt::Float64, delta::Float64, v_amax::Float64, T::Int, tol::Float64, max_iters::Int, α::Float64)
+    i_iter_start = Int(round(delta / dt))
+    num_possible_x0s = T - 2*i_iter_start
+    n_x = 1 + num_possible_x0s
+    prob_dim = n_x + T
+    return PEGame(n_x, dt, delta, v_amax, i_iter_start, prob_dim, T, num_possible_x0s, tol, α, max_iters)
+end
+
+
+
+struct PEGamefixed
+    n_x::Int
+    dt::Float64 
+    delta::Float64
+    v_amax::Float64
+    prob_dim:: Int
+    T::Int
+    x0::Int
+    tol::Float64
+    α::Float64
+    max_iters::Int
+end
+
+function makePEGamefixed(dt::Float64, delta::Float64, v_amax::Float64, T::Int, tol::Float64, max_iters::Int, α::Float64)
+    n_x = 1
+    prob_dim = n_x + T
+    x0 = T*dt-delta
+    return PEGamefixed(n_x, dt, delta, v_amax, prob_dim, T, x0, tol, α, max_iters)
+end
+
+
+# export PEgradient
+# function PEgradient(point, func)
+#     a = SVector{size(point)...}(point)
+#     return ReverseDiff.gradient(x->func(x), a)
+# end
+
+
+# ## Gives out
+# ## J = [∇_xx,   ∇_xy]
+# ##     [∇_yx,   ∇_yy]
+# export PEhessian
+# function PEhessian(point, func)
+#     a = SVector{size(point)...}(point)
+#     return ReverseDiff.hessian(x->func(x), a)
+# end
+
+
+# ## Takes the form
+# ## w = [∇_x; -∇_y]
+# export PEzero_sum_gradient
+# function PEzero_sum_gradient(point, func, n_x)
+#     grad = PEgradient(point, func)
+#     update_step_grad = [grad[1:n_x], -1.0*grad[n_x+1:end]]
+#     update_step_grad = SVector{size(update_step_grad)...}(update_step_grad)
+#     return update_step_grad
+# end
+
+
+# ## Fundamental matrix of the paper.
+# ## J = [∇_xx,   ∇_xy]
+# ##     [-∇_yx, -∇_yy]
+# export PEzero_sum_true_hessian
+# function PEzero_sum_true_hessian(point, func, n_x)
+#     hess = PEhessian(point, func)
+#     ∇_xx = hess[1:n_x, 1:n_x]
+#     ∇_xy = hess[1:n_x, n_x+1:end]
+#     ∇_yx = -1.0*hess[n_x+1:end, 1:n_x]
+#     ∇_yy = -1.0*hess[n_x+1:end, n_x+1:end]
+#     update_step_hess = [∇_xx ∇_xy; ∇_yx ∇_yy]
+#     update_step_hess = SMatrix{size(update_step_hess)...}(update_step_hess)
+# end
+## ∇f(x)
+export PEgradient
+function PEgradient(point::AbstractVector, func::Function)
+    grad = Zygote.gradient(func, point)[1]
+    return SVector{length(grad)}(grad)
+end
+
+## ∇²f(x)
+export PEhessian
+function PEhessian(point::AbstractVector, func::Function)
+    hess = Zygote.hessian(func, point)
+    return SMatrix{length(point), length(point)}(hess)
+end
+
+## w = [∇_x; -∇_y]
+export PEzero_sum_gradient
+function PEzero_sum_gradient(point::AbstractVector, func::Function, n_x::Int)
+    grad = PEgradient(point, func)
+    update_step_grad = vcat(grad[1:n_x], -grad[n_x+1:end])
+    return SVector{length(update_step_grad)}(update_step_grad)
+end
+
+## J = [∇_xx,   ∇_xy;
+##      -∇_yx, -∇_yy]
+export PEzero_sum_true_hessian
+function PEzero_sum_true_hessian(point::AbstractVector, func::Function, n_x::Int)
+    hess = PEhessian(point, func)
+    ∇_xx = hess[1:n_x, 1:n_x]
+    ∇_xy = hess[1:n_x, n_x+1:end]
+    ∇_yx = -hess[n_x+1:end, 1:n_x]
+    ∇_yy = -hess[n_x+1:end, n_x+1:end]
+    update_step_hess = [∇_xx ∇_xy; ∇_yx ∇_yy]
+    return SMatrix{size(update_step_hess)...}(update_step_hess)
+end
+
+
+export solver_PE
+function solver_PE(guess, func, n_x, tol, max_iters, α, v_amax)
+    x = guess
+    prob_dim = length(guess)
+    k = 0
+    error = tol+1.0
+    path = [x]
+    while k<max_iters && error>tol
+        # println("k = ", k)
+        update_step_grad = PEzero_sum_gradient(x, func, n_x)
+        println("update_step_grad: ", update_step_grad)
+        update_step_hess = PEzero_sum_true_hessian(x, func, n_x)
+        static_array_update = PE_regularization_g_d(update_step_grad, update_step_hess, n_x, prob_dim)
+        # update = zeros(size(static_array_update)) for i in 1:size(static_array_update)[1]
+        #     update[i] = static_array_update[i][1]
+        # end
+        update = Vector(static_array_update)
+        w = Vector(update_step_grad)
+        m = (transpose(update)*w/norm(w))*w
+        m = m / norm(w)
+        x_new = enforce_PE_constraints(x-α*m, n_x, v_amax)
+        push!(path, x_new)
+        error = norm(x_new - x)
+        # println(x)
+        x = x_new
+        k += 1
+    end
+    println("iters: ", k)
+    if k==max_iters
+        println("constrained g_d did not converge!")
+        println("Error: ", error)
+    end
+    return x, func(x), k, path 
+end
+
+function project_to_simplex(x::AbstractVector{<:Real})
+    n = length(x)
+    u = sort(x, rev=true)
+    cssv = cumsum(u) .- 1
+    rho = findlast(i -> u[i] > cssv[i] / i, 1:n)
+    theta = cssv[rho] / rho
+    return max.(x .- theta, 0)
+end
+
+function enforce_PE_constraints(x, n_x, v_amax)
+    #clip x[1] to be in [0, 1]
+    x[1] = clamp(x[1], 0.0, 1.0)
+    # project x[2:n_x] to simplex
+    x[2:n_x] = project_to_simplex(x[2:n_x])
+    x[n_x+1:end] = clamp.(x[n_x+1:end], -v_amax, v_amax)
+    return x
+end
+
+export PEObjective
+function PEObjective(x, delta, dt, T, v_amax, i_iter_start, num_possible_x0s, n_x)
+    return sum(x[i-i_iter_start+1]*(x[1]*(norm(sum(x[n_x+1:n_x+ i - i_iter_start])*dt - i*dt + delta)^2 + norm(sum(sqrt.(v_amax^2 .- x[n_x+1:n_x + i - i_iter_start].^2))*dt)^2 ) + (1-x[1])*(norm(sum(x[n_x+1:n_x+ i + i_iter_start])*dt - i*dt - delta)^2 + norm(sum(sqrt.(v_amax^2 .- x[n_x+1:n_x + i + i_iter_start].^2))*dt)^2)) for i in 1+i_iter_start:T-i_iter_start)
+    # total = zero(eltype(x))
+    # for i in (1+i_iter_start):(T - i_iter_start)
+    #     xi = x[i - i_iter_start + 1]
+    #     p = x[1]
+        
+    #     # Left side
+    #     left_vels = x[n_x+1 : n_x + i - i_iter_start]
+    #     left_pos = dt * sum(left_vels)
+    #     left_term1 = norm(left_pos - i*dt + delta)^2
+    #     left_term2 = norm(sum(sqrt.(v_amax^2 .- left_vels.^2)) * dt)^2
+    #     left_cost = left_term1 + left_term2
+
+    #     # Right side
+    #     right_vels = x[n_x+1 : n_x + i + i_iter_start]
+    #     right_pos = dt * sum(right_vels)
+    #     right_term1 = norm(right_pos - i*dt - delta)^2
+    #     right_term2 = norm(sum(sqrt.(v_amax^2 .- right_vels.^2)) * dt)^2
+    #     right_cost = right_term1 + right_term2
+
+    #     # Weighted average cost
+    #     total += xi * (p * left_cost + (1 - p) * right_cost)
+    # end
+    # return total
+    # vel = x[n_x+1:end]
+    # weight = x[1]
+    # total = 0.0
+
+    # for i in (1 + i_iter_start):(T - i_iter_start)
+    #     idx = i - i_iter_start
+    #     xi = x[idx + 1]
+        
+    #     # Velocity ranges
+    #     v_range1 = vel[1:idx]
+    #     v_range2 = vel[idx+1:2idx]
+
+    #     # Sum terms
+    #     sum_v1 = sum(v_range1) * dt
+    #     sum_v2 = sum(v_range2) * dt
+
+    #     sqrt_term1 = sqrt.(v_amax^2 .- v_range1.^2) .* dt
+    #     sqrt_term2 = sqrt.(v_amax^2 .- v_range2.^2) .* dt
+
+    #     # Avoid norm^2: use sum of squares
+    #     loss1 = sum((sum_v1 - i*dt + delta).^2) + sum(sqrt_term1.^2)
+    #     loss2 = sum((sum_v2 - i*dt - delta).^2) + sum(sqrt_term2.^2)
+
+    #     total += xi * (weight * loss1 + (1 - weight) * loss2)
+    # end
+
+    # return total
+end
+
+
+function PE_regularization_g_d(update_step_grad, update_step_hess, n_x, problem_dim)
+    beta = zeros(problem_dim, problem_dim)
+    if isposdef(update_step_hess[1:n_x, 1:n_x])
+        beta[1:n_x, 1:n_x] = LinearAlgebra.I(n_x)
+    end
+    if isposdef(update_step_hess[n_x+1:end, n_x+1:end])
+        beta[n_x+1:end, n_x+1:end] = -LinearAlgebra.I(problem_dim - n_x)
+    end
+    return inv(circle_theorem_regularize(transpose(update_step_hess)*update_step_hess*(update_step_hess+transpose(update_step_hess)))+ beta) * transpose(update_step_hess)* update_step_grad
+end
+
+#x is now [p, vel_1, ..., vel_T]
+export PEObjectivefixed
+function PEObjectivefixed(x, delta, dt, T, v_amax, x0, n_x, radius)
+    t1 = Int((x0-delta)/dt)
+    t2 = T
+    return x[1]*(norm(x0-delta + sum(x[2:end-2*Int(delta/dt)]) - dt*t1)^2 + norm(sum(sqrt.(v_amax^2 .- x[2:end-2*Int(delta/dt)].^2))*dt)^2 + sum(norm(x0-delta +sum(x[2:1+i])*dt - i*dt)^2 + norm(sum(sqrt.(v_amax^2 .- x[2:1+i].^2))*dt)^2 - radius^2 for i in 1:T)) + (1-x[1])*(x0+delta + norm(sum(x[2:end]) - dt*t2)^2 + norm(sum(sqrt.(v_amax^2 .- x[2:end].^2))*dt)^2 + sum(norm(x0 + delta + sum(x[2:1+i])*dt - i*dt)^2 + norm(sum(sqrt.(v_amax^2 .- x[2:1+i].^2))*dt)^2 - radius^2 for i in 1:T))
+end
+
+export solver_PE_fixed
+function solver_PE_fixed(guess, func, n_x, tol, max_iters, α, v_amax)
+    x = guess
+    prob_dim = length(guess)
+    k = 0
+    error = tol+1.0
+    path = [x]
+    while k<max_iters && error>tol
+        # println("k = ", k)
+        update_step_grad = PEzero_sum_gradient(x, func, n_x)
+        # println("update_step_grad: ", update_step_grad)
+        update_step_hess = PEzero_sum_true_hessian(x, func, n_x)
+        static_array_update = PE_regularization_g_d(update_step_grad, update_step_hess, n_x, prob_dim)
+        # update = zeros(size(static_array_update)) for i in 1:size(static_array_update)[1]
+        #     update[i] = static_array_update[i][1]
+        # end
+        update = Vector(static_array_update)
+        w = Vector(update_step_grad)
+        if norm(w) > tol
+            if check_if_fixed_active(x, n_x, v_amax, tol)
+                m = (transpose(update)*w/norm(w))*w
+                m = m / norm(w)
+                x_new = enforce_fixed_PE_constraints(x-α*m, n_x, v_amax)
+            else
+                x_new = enforce_fixed_PE_constraints(x-α*update, n_x, v_amax)
+            end
+            push!(path, x_new)
+            error = norm(x_new - x)
+            # println("x_new: ", x_new)
+            x = x_new
+        else
+            println("Gradient norm is below tolerance, stopping.")
+            break
+        end
+        k += 1
+    end
+    println("iters: ", k)
+    if k==max_iters
+        println("constrained g_d did not converge!")
+        println("Error: ", error)
+    end
+    return x, func(x), k, path 
+end
+
+function enforce_fixed_PE_constraints(x, n_x, v_amax)
+    #clip x[1] to be in [0, 1]
+    x[1] = clamp(x[1], 0.0, 1.0)
+    # project x[2:n_x] to simplex
+    # x[2:n_x] = project_to_simplex(x[2:n_x])
+    x[n_x+1:end] = clamp.(x[n_x+1:end], -v_amax, v_amax)
+    return x
+end
+
+function check_if_fixed_active(x, n_x, v_amax, tol_active)
+    # Check if the first element is in [0, 1]
+    result = false
+    if x[1] < 0.0 + tol_active || x[1] > 1.0 - tol_active
+        result = true
+    end
+    # Check if the velocities are within bounds
+    result = result || any(abs.(x[n_x+1:end]) .> v_amax - tol_active)
+    return result
+end
+
+
+function get_traj(x_vels, v_max)
+    T = length(x_vels)
+    traj = zeros(T, 2)
+    for i in 1:T
+        traj[i, 1] = sum(x_vels[1:i])
+        traj[i, 2] = sum(sqrt.(v_max^2 .- x_vels[1:i].^2))
+    end
+    return traj
+end
+
+export plot_traj
+
+function plot_traj(x_vels, v_max, x0, delta, p)
+    traj = get_traj(x_vels, v_max)
+
+    # Base trajectory plot
+    plt = plot(traj[:, 1], traj[:, 2], 
+        xlabel="Position_x", 
+        ylabel="Velocity", 
+        title="Trajectory", 
+        label="Trajectory", 
+        linewidth=3)
+
+    # Barograph: vertical bars at x0 - delta and x0 + delta
+    bar_xs = [x0 - delta, x0 + delta]
+    bar_ys = [p, 1 - p]
+    bar!(bar_xs, bar_ys, label="Barograph", bar_width=1e-3, legend=:topright)
+    
+    return plt
+end
diff --git a/experiments/Rebuttal/GrimmerExample.jl b/experiments/Rebuttal/GrimmerExample.jl
new file mode 100644
index 0000000..eedb41d
--- /dev/null
+++ b/experiments/Rebuttal/GrimmerExample.jl
@@ -0,0 +1,23 @@
+using ZeroSumGameSolve
+using Plots
+guess = [4.0, 4.0]
+tol = 10e-11
+ball_tol = 1e-2
+max_iters = 500000
+# α = 0.001
+α = 0.01
+n_x = 1
+func = grimmer
+sol2, value2, iters_taken2, path2 = g_d(guess, func, n_x, tol, max_iters, α)
+println("g_d done")
+println("Solution: ", sol2)
+println("Value: ", value2)
+println("Iterations: ", iters_taken2)
+xlims = [-4, 4]
+ylims = [-4, 4]
+xs = range(xlims...; length = 1000)
+ys = range(ylims...; length = 1000)
+plt = contourf(xs, ys, grimmer; color = :viridis)
+nashpt = [0, 0]
+plot!([x[1] for x in path2], [x[2] for x in path2], label="\$\\texttt{DND}\$", show=true)
+scatter!([nashpt[1]], [nashpt[2]], color=:green, label="Local Nash Equilibrium", markershape=:star5, markersize=12)
diff --git a/experiments/ToyExample/Manifest.toml b/experiments/ToyExample/Manifest.toml
new file mode 100644
index 0000000..65f1214
--- /dev/null
+++ b/experiments/ToyExample/Manifest.toml
@@ -0,0 +1,3164 @@
+# This file is machine-generated - editing it directly is not advised
+
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+version = "1.0.13+0"
+
+[[deps.Xorg_libXcursor_jll]]
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+uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
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+[[deps.Xorg_libXdmcp_jll]]
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+[[deps.Xorg_libXfixes_jll]]
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+git-tree-sha1 = "a376af5c7ae60d29825164db40787f15c80c7c54"
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+
+[[deps.Xorg_libXinerama_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
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+[[deps.Xorg_libXrandr_jll]]
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+version = "1.5.5+0"
+
+[[deps.Xorg_libXrender_jll]]
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+
+[[deps.Xorg_libxcb_jll]]
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+git-tree-sha1 = "bfcaf7ec088eaba362093393fe11aa141fa15422"
+uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
+version = "1.17.1+0"
+
+[[deps.Xorg_libxkbfile_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
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+version = "1.1.3+0"
+
+[[deps.Xorg_xcb_util_cursor_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
+git-tree-sha1 = "c5bf2dad6a03dfef57ea0a170a1fe493601603f2"
+uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
+version = "0.1.5+0"
+
+[[deps.Xorg_xcb_util_image_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
+uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
+version = "0.4.1+0"
+
+[[deps.Xorg_xcb_util_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
+git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
+uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
+version = "0.4.1+0"
+
+[[deps.Xorg_xcb_util_keysyms_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
+uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
+version = "0.4.1+0"
+
+[[deps.Xorg_xcb_util_renderutil_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
+uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
+version = "0.3.10+0"
+
+[[deps.Xorg_xcb_util_wm_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
+git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
+uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
+version = "0.4.2+0"
+
+[[deps.Xorg_xkbcomp_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
+git-tree-sha1 = "801a858fc9fb90c11ffddee1801bb06a738bda9b"
+uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
+version = "1.4.7+0"
+
+[[deps.Xorg_xkeyboard_config_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
+git-tree-sha1 = "00af7ebdc563c9217ecc67776d1bbf037dbcebf4"
+uuid = "33bec58e-1273-512f-9401-5d533626f822"
+version = "2.44.0+0"
+
+[[deps.Xorg_xtrans_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
+uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
+version = "1.6.0+0"
+
+[[deps.ZeroSumGameSolve]]
+deps = ["Arpack", "CUDA", "Colors", "DifferentialEquations", "ForwardDiff", "LineSearches", "LinearAlgebra", "Optim", "Optimisers", "OrdinaryDiffEq", "Plots", "ReverseDiff", "Revise", "StaticArrays", "StatsPlots", "Symbolics", "Zygote"]
+path = "../.."
+uuid = "acbac2c1-af1f-4c5c-8072-533d2c216f7a"
+version = "0.1.0"
+
+[[deps.Zlib_jll]]
+deps = ["Libdl"]
+uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
+version = "1.2.13+1"
+
+[[deps.Zstd_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "446b23e73536f84e8037f5dce465e92275f6a308"
+uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
+version = "1.5.7+1"
+
+[[deps.Zygote]]
+deps = ["AbstractFFTs", "ChainRules", "ChainRulesCore", "DiffRules", "Distributed", "FillArrays", "ForwardDiff", "GPUArraysCore", "IRTools", "InteractiveUtils", "LinearAlgebra", "LogExpFunctions", "MacroTools", "NaNMath", "PrecompileTools", "Random", "SparseArrays", "SpecialFunctions", "Statistics", "ZygoteRules"]
+git-tree-sha1 = "a29cbf3968d36022198bcc6f23fdfd70f7caf737"
+uuid = "e88e6eb3-aa80-5325-afca-941959d7151f"
+version = "0.7.10"
+
+    [deps.Zygote.extensions]
+    ZygoteAtomExt = "Atom"
+    ZygoteColorsExt = "Colors"
+    ZygoteDistancesExt = "Distances"
+    ZygoteTrackerExt = "Tracker"
+
+    [deps.Zygote.weakdeps]
+    Atom = "c52e3926-4ff0-5f6e-af25-54175e0327b1"
+    Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
+    Distances = "b4f34e82-e78d-54a5-968a-f98e89d6e8f7"
+    Tracker = "9f7883ad-71c0-57eb-9f7f-b5c9e6d3789c"
+
+[[deps.ZygoteRules]]
+deps = ["ChainRulesCore", "MacroTools"]
+git-tree-sha1 = "434b3de333c75fc446aa0d19fc394edafd07ab08"
+uuid = "700de1a5-db45-46bc-99cf-38207098b444"
+version = "0.2.7"
+
+[[deps.demumble_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "6498e3581023f8e530f34760d18f75a69e3a4ea8"
+uuid = "1e29f10c-031c-5a83-9565-69cddfc27673"
+version = "1.3.0+0"
+
+[[deps.eudev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "c3b0e6196d50eab0c5ed34021aaa0bb463489510"
+uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
+version = "3.2.14+0"
+
+[[deps.fzf_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
+uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
+version = "0.61.1+0"
+
+[[deps.libaom_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "4bba74fa59ab0755167ad24f98800fe5d727175b"
+uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
+version = "3.12.1+0"
+
+[[deps.libass_jll]]
+deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "125eedcb0a4a0bba65b657251ce1d27c8714e9d6"
+uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
+version = "0.17.4+0"
+
+[[deps.libblastrampoline_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
+version = "5.11.0+0"
+
+[[deps.libdecor_jll]]
+deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
+git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
+uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
+version = "0.2.2+0"
+
+[[deps.libevdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557"
+uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
+version = "1.13.4+0"
+
+[[deps.libfdk_aac_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
+uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
+version = "2.0.4+0"
+
+[[deps.libinput_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"]
+git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499"
+uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
+version = "1.28.1+0"
+
+[[deps.libpng_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "07b6a107d926093898e82b3b1db657ebe33134ec"
+uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
+version = "1.6.50+0"
+
+[[deps.libvorbis_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
+git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
+uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
+version = "1.3.8+0"
+
+[[deps.mtdev_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1"
+uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
+version = "1.1.7+0"
+
+[[deps.nghttp2_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
+version = "1.59.0+0"
+
+[[deps.oneTBB_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "d5a767a3bb77135a99e433afe0eb14cd7f6914c3"
+uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
+version = "2022.0.0+0"
+
+[[deps.p7zip_jll]]
+deps = ["Artifacts", "Libdl"]
+uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
+version = "17.4.0+2"
+
+[[deps.x264_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
+uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
+version = "10164.0.1+0"
+
+[[deps.x265_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl"]
+git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
+uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
+version = "4.1.0+0"
+
+[[deps.xkbcommon_jll]]
+deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
+git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1"
+uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
+version = "1.9.2+0"
diff --git a/experiments/ToyExample/Project.toml b/experiments/ToyExample/Project.toml
new file mode 100644
index 0000000..fe77d3e
--- /dev/null
+++ b/experiments/ToyExample/Project.toml
@@ -0,0 +1,9 @@
+[deps]
+Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
+ZeroSumGameSolve = "acbac2c1-af1f-4c5c-8072-533d2c216f7a"
+
+[sources]
+ZeroSumGameSolve = {path = "../.."}
diff --git a/src/ObjectiveFunctions.jl b/src/ObjectiveFunctions.jl
index 607ed27..6c16540 100644
--- a/src/ObjectiveFunctions.jl
+++ b/src/ObjectiveFunctions.jl
@@ -8,4 +8,16 @@ end
 export entropy_regularized_bilinear
 function entropy_regularized_bilinear(x, y)
     return x*y + x*log(x) - y*log(y)
+end
+
+export grimmer
+function grimmer(x, y)
+    return x^4 - 10*(x^2) + 10*x*y -y^4 + 10*(y^2)
+end
+
+export norm_grad_grimmer
+function norm_grad_grimmer(x, y)
+    w1 = 4*x^3 - 20*x + 10*y
+    w2 = 4*y^3 - 20*y-10*x
+    return sqrt(w1^2 + w2^2)
 end
\ No newline at end of file
diff --git a/src/Regularization.jl b/src/Regularization.jl
index b7d79b8..db939df 100644
--- a/src/Regularization.jl
+++ b/src/Regularization.jl
@@ -39,7 +39,7 @@ function regularization_g_d(update_step_grad, update_step_hess)
     if update_step_hess[1, 1] > 0
         b1 = 1.0
     end
-    if update_step_hess[2, 2] < 0
+    if update_step_hess[2, 2] > 0
         b2 = -1.0
     end
     beta = [b1 0.0; 0.0 b2]
diff --git a/src/TwoPlayer.jl b/src/TwoPlayer.jl
index 8a30bc4..a805263 100644
--- a/src/TwoPlayer.jl
+++ b/src/TwoPlayer.jl
@@ -266,6 +266,9 @@ function g_d(guess, func, n_x, tol, max_iters, α)
     k = 0
     error = tol+1.0
     path = [x]
+    update_step_grad = zeros(2)
+    update_step_hess = zeros(2, 2)
+    update = zeros(2)
     while k<max_iters && error>tol
         update_step_grad = zero_sum_gradient(x, func, n_x)
         update_step_hess = zero_sum_true_hessian(x, func, n_x)
@@ -285,6 +288,10 @@ function g_d(guess, func, n_x, tol, max_iters, α)
         println("g_d did not converge!")
         println("Error: ", error)
     end
+    println("update_step_grad: ", update_step_grad)
+    println("update_step_hess: ", update_step_hess)
+    println("update: ", update)
+    println("assumption", transpose(update_step_hess)*update_step_grad)
     return x, func(x[1], x[2]), k, path 
 end
 
diff --git a/src/ZeroSumGameSolve.jl b/src/ZeroSumGameSolve.jl
index b9a7894..7754cea 100644
--- a/src/ZeroSumGameSolve.jl
+++ b/src/ZeroSumGameSolve.jl
@@ -1,5 +1,10 @@
 module ZeroSumGameSolve
 
+# TODO: isolate all deps here
+# TODO: move tests into test/
+# TODO: rename all non-module files "file_name.jl" instead of "FileName.jl"
+# TODO: function names are "FunctionName" if constructors or structs, and "function_name" otherwise, or "function_name!" if they modify any arguments (usually the first argument)
+
 # Write your package code here.
 include("LineSearch.jl")
 include("ObjectiveFunctions.jl")