stochastic.jl

### A Pluto.jl notebook ###
# v0.19.14

using Markdown
using InteractiveUtils

# ╔═╡ 26c3d3fe-af66-4d79-8aaa-5f5b0784bab0
using CairoMakie

# ╔═╡ 0bee0d4b-272d-406f-b9fd-b947b901278b
using StatsBase

# ╔═╡ 9b0a5eee-a0d0-4b5e-83cf-7badb1753688
using PlutoUI: Slider

# ╔═╡ 30c0a8df-2340-4f66-a415-9375d0a6c893
using Distributions

# ╔═╡ c708e88a-4771-4144-8765-a78aa730e7f9
using LinearAlgebra

# ╔═╡ 6259db20-c705-4e7c-ba8f-18cc8b733f8b
using Random: AbstractRNG, Xoshiro, default_rng, randexp

# ╔═╡ aa7a9420-f7e1-4a44-9375-882c79f97e1a
using LsqFit

# ╔═╡ 7c145e9e-5d12-11ed-3c4b-796dde8540c7
md"""
# Random Variables

Useful links for this section:
- [Random docs](https://docs.julialang.org/en/v1/stdlib/Random/#Random-Numbers)
- [Mathematical operations docs](https://docs.julialang.org/en/v1/manual/mathematical-operations/#Mathematical-Operations-and-Elementary-Functions)
- [Histogram docs](https://juliastats.org/StatsBase.jl/stable/empirical/#Empirical-Estimation-1)
- [Plotting docs](https://docs.makie.org/stable/)
- [Plotting example gallery](https://beautiful.makie.org/dev/)

In Julia, the [rand](https://docs.julialang.org/en/v1/stdlib/Random/#Base.rand) function samples a number in [0,1). press SHIFT + RETURN with the cursor in the cell below to sample a new number.
"""

# ╔═╡ 9128a75c-bcc4-4131-bdd0-73a6a8490165
rand()

# ╔═╡ f89e1123-2dd8-4c40-b501-909829e8c714
md"## Exercise 1: Bernoulli Trial
Below is a function definition and evaluation. Right now the function only returns *true* no matter what random number rand spits out."

# ╔═╡ 985916d7-8105-48d7-a3c9-9447afa50416
bernoulli(ω, p) = true

# ╔═╡ f3e091d6-5359-40b2-817e-db2848d70f90
bernoulli(rand(), 0.5)

# ╔═╡ f1ff8af1-6954-4771-9981-db2e295360a3
# this counts how often bernoulli(rand(), 0.7) returns true out of 1_000 evaluations
# If you've implemented the function correctly this should return 0.7
count([bernoulli(rand(), 0.7) for _ in 1:1_000]) / 1_000

# ╔═╡ 6382e7e4-2508-4bf6-9b09-41692268089b
md"Here are some examples of how to define functions in Julia."

# ╔═╡ bed0c3f8-0017-4fdd-b520-e4a265677140
# This is a one line function definition
# f adds the inputs x, y and z and subtracts 5
f(x, y, z) = x + y + z - 5 

# ╔═╡ 0642ade6-1869-4848-bc85-f60a85ec15ff
f(0.6, 0.1, 10)

# ╔═╡ 154cd13b-391d-46c2-83d6-241ca23a154e
# Functions often do more complicated things and may require multiple lines
# I show the return staement here, but you can delete return and just keep
# max(a,b) at the end. The last line of a function is automatically returned.
function g(x, y)
	a = 2x
	b = 5y + x
	return max(a, b)  # returns the larger of a and b
end

# ╔═╡ 08548873-ffd7-4f0c-83cd-8ba4c8d536c6
g(5, 7)

# ╔═╡ bbfdd8d8-f27f-4830-b468-df189c778de8
md"To return something else, try changing the bernoulli function above to return *false* instead. To do the first exercise, you may find the less than or equals (<=) comparison operator useful.

To be able to use greek letters and other symbols, type \\omega and press TAB. You can get all kinds of greek letters and mathematical symbols this way."

# ╔═╡ a666d797-ecf3-441f-9406-f616305f2d31
rand() <= 0.9

# ╔═╡ a736afcc-b477-45f9-ae5b-0fdd31b4f4f7
md"One major use of the *true* and *false* values is to conditionally execute code. For example,"

# ╔═╡ ca39bb7a-39e6-4756-810b-20813aede18f
if bernoulli(rand(), 0.8)
	println("80% of the time I'm happy! :)")
else
	println("But 20% of the time I'm sad :(")
end

# ╔═╡ 57ea1a43-33ec-46d9-a26c-9a52d5fb52f1
md"## Exercise 2: Uniform Distribution"

# ╔═╡ e09214e7-82b3-4119-8b97-ff92da6c9f3f
uniform(ω, a, b) = ω

# ╔═╡ e12a61f0-b0bc-4fad-8712-64668dad552b
# Not quite right yet
5 <= uniform(rand(), 5, 9) <= 9

# ╔═╡ f35904f0-a884-4979-a464-2ac04fcd5cf7
md"To generate a uniformly distributed variable, you may find the basic mathematical operations useful. Below are some examples. For more details, click on the docs link at the beginning of this section."

# ╔═╡ 4cf08f8c-b7ee-4700-b556-b61abe166b15
# This is a variable representing the number 2.3
foo = 2.3

# ╔═╡ 04a883ff-f076-47a3-9c7a-b645abb6e2c3
# It can be multiplied
foo * 3

# ╔═╡ b45f6141-c1ee-4ec0-949f-b4fbc7b512e3
# Or divided. Get π by typing \pi and then TAB
foo / π

# ╔═╡ 7d0b767d-61c4-4041-909e-1fb2a177ccf9
# Or added and subtracted
foo + 3 - 20 <= 0

# ╔═╡ 41d77d2a-b38b-46a7-97a1-a3e530df33fb
# and much more...
sqrt(sin(log(foo^3)))

# ╔═╡ 9e61d9c4-1308-46d5-b32d-7e8419a48f67
md"## Making a Histogram
Let's test whether your random variable is doing the right thing by sampling many times and plotting the distribution of the samples we get.

To make a list or vector of samples, you can use a list comprehension as below. I also provide some further examples so you can see how list comprehensions and Vectors work."

# ╔═╡ 08cb36a2-4b14-4003-b2f1-131f87e82db2
[i for i in 1:5]

# ╔═╡ 8d59dc93-cc98-4e58-aaf0-3112758c7cb5
# Note that these a:b objects are not Vectors, they are called ranges.
# However, they behave like a Vector and they can be converted to one with collect
# In fact, anything you can iterate over can be turned into a Vector with collect
5:8, collect(5:8)

# ╔═╡ bca187e3-04c6-4a30-81ba-c639f288db3e
1:10 isa Vector

# ╔═╡ ce084ee6-1b2f-40da-8a70-1005dd861f05
# we don't need the variable, so we use underscore _ to signal this fact
[2 for _ in 1:5]

# ╔═╡ fca887ee-28d2-4c21-9938-b892d87380eb
# cube each number from 0 to 0.5, going in steps of size 0.1
[x^3 for x in 0.0:0.1:0.5]

# ╔═╡ dca96c3e-e674-4124-a794-ad1ea5daecb8
md"vectors can be iterated over and indexed into as follows."

# ╔═╡ 9dd88140-fc85-4896-bd3a-90a79d4c86b3
squares = [n^2 for n in 1:10]

# ╔═╡ 0cc8e35a-412f-4675-b9db-1c79ed305821
for x in squares
	if isodd(x)
		println("forward")
	else
		println("backward")
	end
end

# ╔═╡ 5f8ce592-ab59-49f1-babb-ae7d759bebbf
# get the first entry
squares[1]

# ╔═╡ edb3721e-8fc2-4cf4-b925-bc2c729fc401
# get the last entry
squares[end]

# ╔═╡ db53f578-7050-4831-8c98-8e5e243eb606
# get the third to fifth entry
squares[3:5]

# ╔═╡ 681ddd77-9da9-4777-b0f5-64169fc47e3c
# set the 9th entry
squares[9] = -1

# ╔═╡ f4631d98-dad4-41c9-ac78-af0aee40a119
# and the 2nd to 3rd entries
squares[2:3] = [-2, -3]

# ╔═╡ 81fe6e9a-2af8-40b2-a364-a9512537aeeb
squares

# ╔═╡ 7d29e024-975f-4d61-8c5a-d3c9a6dfb0e2
md"Back to our samples... Let's make a histogram and plot it."

# ╔═╡ 93e93bcb-29cd-4b26-9047-0d7ff9c0ebb1
md"## Exercise 6: Exponential Distribution"

# ╔═╡ fd3754b0-f8cd-428d-9c97-f5d2583eb82f
# not quite right jet. This should be sampled exponentially
myrandexp(ω) = sqrt(ω)

# ╔═╡ b545f815-545e-44bf-be76-25ee69881bf8
samples = [myrandexp(rand()) for _ in 1:100_000]

# ╔═╡ c994b386-e183-471c-b161-5a4b5a3374ff
samples isa Vector

# ╔═╡ 3af8c14b-fa91-457a-87e3-d29866cc3e49
# Make a histogram
hist = fit(Histogram, samples)

# ╔═╡ 53073c14-bf67-4f84-9a56-7d179dc77301
plot(hist)

# ╔═╡ d216cf65-2de7-46c7-8b0c-f1e3a355ab5f
# compare to the built in randexp()
randexp()

# ╔═╡ 9ec38300-de96-4860-9893-1a8a00166004
md"## Exercise 9: Semicircle"

# ╔═╡ 831a9d8f-3eeb-4f1c-8c50-fb09413ee689
function semicircle(rng::AbstractRNG)
	x = rand(rng)
	y = rand(rng)
	# accept some, reject others

	return x
end

# ╔═╡ 4912e73b-d15c-4024-9bb2-e355d0de663d
let fig = Figure(), n = 500
	ax = Axis(fig[1, 1]; aspect=DataAspect(), xlabel="x", ylabel="y")

	pdf(x) = sqrt(1 - x^2)
	
	rxs = 2 .* rand(n) .- 1
	rys = rand(n)
	idx = rys .^2 .+ rxs .^2 .< 1
	notidx = map(!identity, idx)
	scatter!(ax, rxs[idx], rys[idx], color=:red, label="accepted")
	scatter!(ax, rxs[notidx], rys[notidx], color=:black, label="rejected")

	xs = -1:0.01:1
	ys = pdf.(xs)
	lines!(ax, xs, ys; label="pdf")

	axislegend(ax)
	
	fig
end

# ╔═╡ 0391e8fe-0a18-4782-a797-157d253417af
let n = 100_000  # estimating π
	p = 4 * count(_ -> rand()^2 + rand()^2 <= 1, 1:n) / n
	(π = p, error = abs(π - p) / π)
end

# ╔═╡ 01004f3b-fb1e-4da0-801e-786eb12a1e76
md"# Monte Carlo"

# ╔═╡ 4e88f8a9-fe2d-40b8-829d-b5d636127e83
function walk(
	jump,
	rng::AbstractRNG,
	x0s::AbstractVector{T},
	nt::Integer,
	p
) where T
	# initialise memory for trajectories
	xs = map(_ -> Vector{T}(undef, nt), x0s)

	for i in 1:length(x0s)  # for each trajectory
		x = x0s[i]  # init position
		xs[i][1] = x  # # save initial position
		
		for j in 2:nt  # for each time step: jump and save
			x = jump(rng, x, p)
			xs[i][j] = x
		end
	end

	return xs
end

# ╔═╡ f8acef37-ce4a-49f7-81df-745ba26b50a8
function walk(jump, x0s::AbstractVector{T}, nt::Integer, p) where T
	walk(jump, default_rng(), x0s, nt, p)
end

# ╔═╡ c75338b4-6bf0-4ef3-a07f-4736e3bcd4dd
opendiffuse(rng::AbstractRNG, x, p) = x + p.σ * randn(rng)

# ╔═╡ f0a0c065-600a-44e7-afec-9a43c1c02698
discrete(rng::AbstractRNG, x, p) = rand(rng, Bool) ? x + p.σ : x - p.σ

# ╔═╡ 9086bcd0-3fc4-4ba9-8f30-2f78b5ead628
function boxdiffuse(rng::AbstractRNG, x, p)
	y = x + p.σ * randn(rng)
	y = y < 0.0 ? -y : y
	while y > p.L
		y = p.L - (y - p.L)
		y = y < 0.0 ? -y : y
	end
	return y
end

# ╔═╡ 7b2c2c89-ff31-4f15-99b9-3a46eee4d19e
ohrnstein_uhlenbeck(rng, x, p) = x - p.a * x + p.σ * randn(rng)

# ╔═╡ 9286d291-cc9b-450a-aae5-7bf8fa32f185
periodic(rng, x, p) = x - p.a * sin(x) + p.σ * randn(rng)

# ╔═╡ a1a66164-d290-40f9-8f76-0fe3aef90564
pareto(rng, x, p) = x + p.σ * (rand(rng, Pareto(p.α)) - 1) * rand(rng, (-1, 1))

# ╔═╡ d7d7a506-993a-48a7-8215-d0c1a192dcf7
kelly(rng, x, p) = rand(rng) <= p.p ? x * (1.0 + p.f * p.b) : x * (1.0 - p.f)

# ╔═╡ a3bc1aa1-0795-4e7e-a55a-9e76366ce0f2
rw = walk(opendiffuse, 0.1 * randn(100), 100, (σ = 0.1,))

# ╔═╡ f1b1ecb1-4147-48c6-a9bc-b4949372094c
let fig = Figure()
	# add the following after fig[1, 1] to plot with y axis using log scale
	# ; yscale=log10
	ax = Axis(fig[1, 1])

	for i in 1:length(rw)
		lines!(ax, rw[i], color=(:black, 0.2))
	end

	fig
end

# ╔═╡ fa122c23-f3c8-4bdb-a6ed-585bb694da6f
let fig = Figure()
	ax = Axis(fig[1, 1])

	N = length(rw[1])
	μs = [mean([xs[i] for xs in rw]) for i in 1:N]

	lines!(ax, μs)

	fig
end

# ╔═╡ 6ee96f9d-4e8a-4c9a-aa44-c30f14496b97
getvars(rw) = [var([xs[i] for xs in rw]) for i in 1:length(rw[1])]

# ╔═╡ e20309dd-03d0-4003-b383-581fadaf74d0
let fig = Figure()
	ax = Axis(fig[1, 1])

	lines!(ax, getvars(rw))

	fig
end

# ╔═╡ 98179e10-52fa-44b7-9155-7300c27cd5a7
linmodel(x, p) = x .* p[1] .+ p[2]

# ╔═╡ 05873c45-3276-4df6-8cf0-3c97b55aa48d
function getslope(vars)
	fit = curve_fit(linmodel, 1:length(vars), vars, [1.0, 0.0])
	fit.param[1]
end

# ╔═╡ 6fe7dcc5-805a-4d09-8baa-949952762d4c
let fig = Figure()
	ax = Axis(fig[1, 1])

	σs = 0.1:0.1:1.0
	slopes = [getslope(getvars(
		walk(opendiffuse, 0.1 * randn(1_000), 10, (σ = σ,))
	)) for σ in σs]
	
	scatter!(ax, σs .^ 2, slopes)

	fig
end

# ╔═╡ f2f81a76-a2c8-40e8-8282-be7bf869370c
md"# Markov Chains"

# ╔═╡ b62f7c11-07b8-4dd9-b6c7-bc8a13ea7faf
T = [0.2 0.2; 0.8 0.8]

# ╔═╡ 8f001bf1-b456-4fbd-b025-8faee9754768
p0 = [0.5, 0.5]

# ╔═╡ 3e9244ab-1160-4381-820f-4ae01df68383
p = T^1000 * p0

# ╔═╡ 099de66c-9163-4fdc-9f94-52d83ce345cf
sum(p)

# ╔═╡ 854bdeba-0c10-4631-995e-81ca747d8c4f
E = eigen(T)

# ╔═╡ 557f84d7-7526-4978-942c-c2d0fe328dc8
pstat = E.vectors[:, 2]

# ╔═╡ 6ac10e92-8687-4a99-b64c-4979f96826aa
T * pstat ≈ pstat

# ╔═╡ 9ebe8846-ee64-44be-8726-fafcee2488b2
let fig = Figure(), p0 = [1, 0], T = T, nsteps = 10
	ax = Axis(fig[1, 1]; xlabel = "t", ylabel = "p")
	heat = zeros(nsteps, length(p0))
	heat[1, :] .= p0
	for i in 2:nsteps
		mul!(view(heat, i, :), T, view(heat, i-1, :))
	end

	xs = 1:nsteps
	ys = 1:length(p0)
	hmap = heatmap!(ax, xs, ys, heat)
	Colorbar(fig[1, 2], hmap; width = 15, ticksize = 15)

	fig
end

# ╔═╡ ca0fb426-6214-4818-9db7-d7a57b2f9b7e
md"# Poker"

# ╔═╡ f0e60438-584c-476f-a832-3e59bc42010e
struct Game
	die::Int
	b1::Float64
	b2::Float64
end

# ╔═╡ 5983efa0-5cf3-4fb5-abcd-5390a578e95d
game = Game(6, 2.5, 5.0)

# ╔═╡ 87d0ddbb-07d5-4f60-938a-0930478800e7
game.die

# ╔═╡ aa3a4673-f16e-4efc-910a-c94fa7e2939a
game.b1

# ╔═╡ 46d31d61-eb5a-47e5-8bb7-6c06bb31091a


# ╔═╡ 46d9aac7-18d2-4a04-8c15-dc83aa23b86f
function winner(dA, dB, df)

end

# ╔═╡ 49980be2-be66-49c9-a131-21d580dd2813
function play(
	rng::AbstractRNG,
	(; die, b1, b2)::Game,
	pr1::Vector{Float64},
	pc1::Vector{Float64},
	pr2::Vector{Float64},
	pc2::Vector{Float64},
	nrounds::Int
)
	if !(die == length(pr1) == length(pc1) == length(pr2) == length(pc2))
		throw(ArgumentError("die and probability lengths must match"))
	end

	if !all(p -> 0 <= p <= 1, pr1) || !all(p -> 0 <= p <= 1, pc1) ||
		!all(p -> 0 <= p <= 1, pr2) || !all(p -> 0 <= p <= 1, pc2)
		throw(ArgumentError("Probabilities must take values between 0 and 1"))
	end

	# player A's profit == - player B's profit
	pft = 0.0
	
	for _ in 1:nrounds
		dA = rand(rng, 1:die)
		if rand(rng) <= pr1[dA]  # A raises b1
			dB = rand(rng, 1:die)
			if rand(rng) <= pc1[dB]  # B calls b1
				if rand(rng) <= pr2[dB]  # B raises b2
					if rand(rng) <= pc2[dA]  # A calls b2
						pay = sign(dA - dB) * (b1 + b2 + 1.0)
					else  # A folds to b2 raise
						pay = -(1.0 + b1)
					end
				else  # B checks
					pay = sign(dA - dB) * (b1 + 1.0)
				end
			else  # B folds to b1 raise
				pay = 1.0
			end
		else # A folds initially
			pay = -0.5
		end

		pft += pay
	end

	return pft
end

# ╔═╡ d3378855-4ca5-4e82-9785-57ab9747ef8e
function eprofit((; die, b1, b2)::Game, pr1, pc1, pr2, pc2)
	if !(die == length(pr1) == length(pc1) == length(pr2) == length(pc2))
		throw(ArgumentError("die and probability lengths must match"))
	end

	pft = 0.0
	for dA in 1:die
		for dB in 1:die
			s = sign(dA - dB)  # =1 if A wins, -1 if B wins and 0 if tie
			pft -= 0.5 * (1-pr1[dA])  # A folds initially
			pft += 1.0 * pr1[dA] * (1-pc1[dB])  # B folds to b1 raise

			# probability of getting to second betting round
			ps2 = pr1[dA] * pc1[dB]

			# B checks
			pft += s * (b1 + 1.0) * ps2 * (1-pr2[dB])
			# A folds to b2 raise
			pft -= (1.0 + b1) * ps2 * pr2[dB] * (1-pc2[dA])
			# A calls b2 raise
			pft += s * (b1 + b2 + 1.0) * ps2 * pr2[dB] * pc2[dA]
		end
	end

	return pft / die^2
end

# ╔═╡ c4bbf394-0811-4966-8488-7d3864d266f8
# grad points in direction of improving prfoit for A and B respectively,
# so both players want to change their strategies in the direction of grad
function gradprofit!(
	∇pr1, ∇pc1, ∇pr2, ∇pc2,
	(; die, b1, b2)::Game,
	pr1, pc1, pr2, pc2
)
	if !(die == length(pr1) == length(pc1) == length(pr2) == length(pc2)) ||
		!(die == length(∇pr1) == length(∇pc1) == length(∇pr2) == length(∇pc2))
		throw(ArgumentError("die, grad and probability lengths must match"))
	end

	fill!(∇pr1, 0.0)
	fill!(∇pc1, 0.0)
	fill!(∇pr2, 0.0)
	fill!(∇pc2, 0.0)

	for dA in 1:die
		for dB in 1:die
			s = sign(dA - dB)  # =1 if A wins, -1 if B wins and 0 if tie

			# pft -= 0.5 * (1-pr1[dA])  # A folds initially
			∇pr1[dA] += 0.5
			
			# pft += 1.0 * pr1[dA] * (1-pc1[dB])  # B folds to b1 raise
			∇pr1[dA] += 1.0 * (1-pc1[dB])
			∇pc1[dB] += 1.0 * pr1[dA]

			# B checks
			# pft += s * (b1 + 1.0) * pr1[dA] * pc1[dB] * (1-pr2[dB])
			∇pr1[dA] += s * (b1 + 1.0) * pc1[dB] * (1-pr2[dB])
			∇pc1[dB] -= s * (b1 + 1.0) * pr1[dA] * (1-pr2[dB])
			∇pr2[dB] += s * (b1 + 1.0) * pr1[dA] * pc1[dB]
			
			# A folds to b2 raise
			# pft -= (1.0 + b1) * pr1[dA] * pc1[dB] * pr2[dB] * (1-pc2[dA])
			∇pr1[dA] -= (1.0 + b1) * pc1[dB] * pr2[dB] * (1-pc2[dA])
			∇pc1[dB] += (1.0 + b1) * pr1[dA] * pr2[dB] * (1-pc2[dA])
			∇pr2[dB] += (1.0 + b1) * pr1[dA] * pc1[dB] * (1-pc2[dA])
			∇pc2[dA] += (1.0 + b1) * pr1[dA] * pc1[dB] * pr2[dB]
			
			# A calls b2 raise
			# pft += s * (b1 + b2 + 1.0) * pr1[dA] * pc1[dB] * pr2[dB] * pc2[dA]
			∇pr1[dA] += s * (b1 + b2 + 1.0) * pc1[dB] * pr2[dB] * pc2[dA]
			∇pc1[dB] -= s * (b1 + b2 + 1.0) * pr1[dA] * pr2[dB] * pc2[dA]
			∇pr2[dB] -= s * (b1 + b2 + 1.0) * pr1[dA] * pc1[dB] * pc2[dA]
			∇pc2[dA] += s * (b1 + b2 + 1.0) * pr1[dA] * pc1[dB] * pr2[dB]
		end
	end

	return inv(die^2) .* (∇pr1, ∇pc1, ∇pr2, ∇pc2)
end

# ╔═╡ 565575d7-4bf7-41a4-89d4-5d9a9e331f06
function gradprofit(game::Game, pr1, pc1, pr2, pc2)
	∇pr1 = Vector{Float64}(undef, game.die)
	∇pc1 = Vector{Float64}(undef, game.die)
	∇pr2 = Vector{Float64}(undef, game.die)
	∇pc2 = Vector{Float64}(undef, game.die)
	gradprofit!(∇pr1, ∇pc1, ∇pr2, ∇pc2, game, pr1, pc1, pr2, pc2)
end

# ╔═╡ 47f410d9-9fac-437c-a273-72c52ca36236
function gradascent!(praise, pcall, N, bet, niter, rate)
	rgrad = Vector{Float64}(undef, N)
	cgrad = Vector{Float64}(undef, N)
	for _ in niter
		rgrad, ∇cgrad = gradprofit!(rgrad, cgrad, N, bet, praise, pcall)
		praise .= clamp.(praise .+ rate .* rgrad, 0.0, 1.0)
		pcall .= clamp.(pcall .+ rate .* cgrad, 0.0, 1.0)
	end
	praise, pcall
end

# ╔═╡ c5258501-01e5-4f7f-a408-ed54c7d4ce6a
begin
	mypr1 = [0.0, 0.0, 0.0, 0.5, 1.0, 1.0]
	mypc1 = [0.0, 0.0, 0.0, 0.0, 1.0, 1.0]
	mypr2 = [0.0, 0.0, 0.0, 1.0, 0.0, 1.0]
	mypc2 = [0.0, 0.0, 0.0, 0.0, 1.0, 1.0]
end

# ╔═╡ ddedae38-867a-4771-96c6-4d6953805955
let nrounds = 10_000_000
	play(default_rng(), game, mypr1, mypc1, mypr2, mypc2, nrounds) / nrounds
end

# ╔═╡ f1330b02-b271-43fe-8c66-01773c3d4818
@profview play(default_rng(), game, mypr1, mypc1, mypr2, mypc2, 10_000_000)

# ╔═╡ f296db36-7ae6-4f70-a8de-c959daf32254
eprofit(game, mypr1, mypc1, mypr2, mypc2)

# ╔═╡ 315a94c7-2ba5-4ccd-bd24-bab1cfd12bc8
gradprofit(game, mypr1, mypc1, mypr2, mypc2)

# ╔═╡ ec17738d-2f28-42a1-b3f9-b231edb394c4
function optimise!(praise, pcall, N, bet, niter)
	rgrad = Vector{Float64}(undef, N)
	cgrad = Vector{Float64}(undef, N)
	for _ in niter
		∇rpft, ∇cpft = gradprofit(N, bet, praise, pcall)
		for i in 1:N
			∂pr = ∇rpft[i]
			if ∂pr > 0.0
				praise[i] = 1.0
			elseif ∂pr < 0.0
				praise[i] = 0.0
			else
				praise[i] = rand()
			end
			∂pc = ∇cpft[i]
			if ∂pc > 0.0
				pcall[i] = 1.0
			elseif ∂pc < 0.0
				pcall[i] = 0.0
			else
				# since grad is zero, any probability is equally good
				pcall[i] = rand()
			end
		end
	end
	praise, pcall
end

# ╔═╡ 4fe533dc-3ccc-4a0c-b868-ac1b2bb014f9
optimise!(rand(2), rand(2), 2, 2.0, 20)

# ╔═╡ ce97e2ed-3d99-424b-92bd-99224bec7a5b
clamp(-2.0, 0.0, 1.5)

# ╔═╡ 824836d2-9d47-4ce4-ae96-8d3933381bce
gradascent!(rand(2), rand(2), 2, 2.0, 100_000, 10.0)

# ╔═╡ c4d79a40-7173-4fc7-8ea7-d1b95f88bbfd
fuzz(ps::Vector, fz) = ps .* (1 .- fz) .+ (1 .- ps) .* fz

# ╔═╡ bfcb7dd2-3d08-4eac-bd3c-40b4fbd71b82
fuzz(fill(0.4, 10), 0.2)

# ╔═╡ 83862c32-40e1-4bd3-89d6-8882d7b7ea1a
let N = 10, bet = 10.0, nrounds = 1_000_000
	g = Game(N, bet)
	pro, pco = optimise!(rand(N), rand(N), N, bet, 100)
	prg, pcg = gradascent!(rand(N), rand(N), N, bet, 10_000, 100.0)
	∇rpft, ∇cpft = gradprofit(N, bet, prg, pcg)
	praise = prg
	pcall = pco
	gpft = play(g, praise, pcall, nrounds) ./ nrounds
	epft = eprofit(N, bet, praise, pcall)
	(
		(praise, pcall),
		(pro, pco),
		(prg, pcg),
		(∇rpft, ∇cpft),
		gpft, epft 
	)
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0"
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
LsqFit = "2fda8390-95c7-5789-9bda-21331edee243"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

[compat]
CairoMakie = "~0.9.2"
Distributions = "~0.25.76"
LsqFit = "~0.13.0"
PlutoUI = "~0.7.48"
StatsBase = "~0.33.21"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.8.2"
manifest_format = "2.0"
project_hash = "9edc8ebdaf323159a2dd679a78c4ed43778fb7d5"

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version = "1.2.1"

[[deps.AbstractPlutoDingetjes]]
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version = "1.1.4"

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[[deps.ColorSchemes]]
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[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.isoband_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51b5eeb3f98367157a7a12a1fb0aa5328946c03c"
uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4"
version = "0.2.3+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"
"""

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