### A Pluto.jl notebook ###
# v0.19.19

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ ec777da4-8ef0-44b4-8d56-38b31790a5b7
begin
  using Plots # pour le tracé de figures
  using PlutoUI # pour les objets Pluto
  using LinearAlgebra # pour les matrices identité
  using SparseArrays # pour les matrices creuses
  using LaTeXStrings # pour les strings en LaTeX (pour les plots)
  TableOfContents(depth=4)
end

# ╔═╡ dc4eb528-82b2-11ed-3ca2-c321ff8ef647
html"""
<center>
	<strong style="font-size: 2rem;">
		Exercices - Reinforcement learning <br/>
		Laurent Fainsin <br/>
		2021 - 2022
	</strong>
</center>
"""

# ╔═╡ 7fb608fa-da53-47d3-a585-235b4692c4dd
md"""
# Exercice 1 - Finite Horizon MDP

> Revenue management: Littlewood’s model
>
> An airplane has 20 seats available, and the sell closes in 50 days. At every time epoch, the airplane decides the selling price: Either ``p_1`` = 5, and then it will sell a seat with probability ``q_1`` = 0.1, or ``p_2`` = 1, and then it will sell a seat with probability ``q_2`` = 0.8.
"""

# ╔═╡ 58baae0a-ae46-443c-8c08-8743a13eb860
md"""
D'après l'énoncé on a:

$\mathcal{S} = \left\{ s_1, s_2 \right\}$

$\mathcal{A} = \left\{ p_1, p_2 \right\}$

$\mathcal{R} = \left\{ p_1 q_1, p_2 q_2 \right\}$
"""

# ╔═╡ dc2fcab2-7b23-47e3-b53a-0ad966645afd
md"""
D'après l'équation d'optimalité de Bellmann:

$V(s) = r(s) + \gamma \sum_{s'} p(s,s')V(s')$

Dans notre cas on a donc (``\gamma = 1``):

$V_T(s_1) = p_1 q_1 + q_1 V_{T-1}(s_1) + (1-q_1) V_{T-1}(s_2)$
$V_T(s_2) = p_2 q_2 + q_2 V_{T-1}(s_2) + (1-q_2) V_{T-1}(s_1)$
"""

# ╔═╡ a6100c90-9a12-43d8-8ec3-dbf04b4ab5d1
md"""
On peut alors poser:

$P = \begin{pmatrix}
q_1 & 1-q_1 \\
q_2 & 1-q_2
\end{pmatrix}$

$R = \begin{pmatrix}
p_1 q_1 \\
p_2 q_2
\end{pmatrix}$

$V_T = \begin{pmatrix}
V_T(s_1) \\
V_T(s_2)
\end{pmatrix}$

tel que l'on puisse reformuler Bellmann:

$V_T = R + \gamma P V_{T-1}$
"""

# ╔═╡ a5df336b-3db1-441d-845b-3ddb0aa3213a
md"""
En selectionnant la valeur maximale de ``V_T`` on en déduit la politique optimal pour ce jour ``T``. Intuitivement on choisi le prix ``p_2`` si l'on souhaite maximiser notre gain. On peut le vérifier numériquement:
"""

# ╔═╡ a30def9e-50f0-4aa2-8e79-6d10ecee5bee
begin
  p1_slider = @bind p1 Slider(0:1:10, default=5, show_value=true)
  q1_slider = @bind q1 Slider(0:0.1:1, default=0.1, show_value=true)
  
  p2_slider = @bind p2 Slider(0:1:10, default=1, show_value=true)
  q2_slider = @bind q2 Slider(0:0.1:1, default=0.8, show_value=true)

  md"""  
    ``p_1``: $(p1_slider) ``\quad\quad``
    ``q_1``: $(q1_slider)
  
    ``p_2``: $(p2_slider) ``\quad\quad``
    ``q_2``: $(q2_slider)
  """
end

# ╔═╡ 3176c1f0-c1fa-4c74-9133-f4371e584b6c
P = [
	q1 1-q1
	q2 1-q2
]

# ╔═╡ f5566288-5ef4-495c-8442-a59f17f0814b
R = [
	p1 * q1
	p2 * q2
]

# ╔═╡ f01bd868-a180-4327-98bf-de2298673523
γ = 1

# ╔═╡ f8c9b0c8-3f81-441b-b673-f07ed2ae3e5a
begin
	text = "";
	T = 50;
	
	V = [0 ; 0];
	text *= "``V_{50} = $(V)``\n\n";
	
	for t in 1:T
		V = R + γ * P * V;
		choix = argmax(V);
		V_display = round.(V, digits=2);
		text *= "``V_{$(T-t)} = $(V_display) → p_$(choix)``\n\n";
	end

	Markdown.parse(text)
end

# ╔═╡ 73a0d6a0-94a9-45b3-9db9-6cc0c845ab95
md"""
# Exercice 2 - Infinite Horizon MDP

> See figure
>
> What is the optimal policy (for total discounted reward) for various values of ``\gamma`` ?
"""

# ╔═╡ 3b87e4e9-81c9-48d9-9f58-e6271f975c40
md"""
Équation d'optimalité pour nos deux états (``s_0`` and ``s_1``):

``V_\star(s_0) = \max( 10 + \gamma V_\star(s_1), 1 + \gamma V_\star(s_0) )``

``V_\star(s_1) = \max( 0 + \gamma V_\star(s_1), -15 + \gamma V_\star(s_0) )``
"""

# ╔═╡ 045e6ef0-84a8-4ad1-bc7c-0494599fd825
md"""
Si ``\gamma \approx 0``:

``V_\star(s_0) \approx \max(10, 1) = 10``

``V_\star(s_1) \approx \max(0, -15) = 0``
"""

# ╔═╡ 8cd6c607-2094-4fa0-b1d8-445842ac5091
md"""
Si ``\gamma \approx 1``:

``V_\star(s_0) = 1 + \gamma V_\star(s_0) \implies V_\star(s_0) = \displaystyle\frac{1}{1-\gamma}``

``V_\star(s_0) = -15 + V_\star(s_0) \implies V_\star(s_1) = -15 + \displaystyle\frac{1}{1-\gamma}``
"""

# ╔═╡ fe6dc66f-2627-4c13-b32a-9751b2ff3a73
md"""
Si l'on résoud ces deux dernières équations on a:

``\gamma = 0.9``

``\gamma = \displaystyle\frac{15}{16} \approx 0.94``
"""

# ╔═╡ fdd906ab-2224-4c26-9fb9-d9855bfda257
md"""
Par disjonction des cas:
"""

# ╔═╡ 65dcb2b0-1021-45a6-afe7-a95f6e139636
md"""
Si ``\gamma \in [0, 0.9]``:

``V_\star(s_0) = 1 + \gamma V_\star(s_0)``

``V_\star(s_1) = \gamma V_\star(s_1)``
"""

# ╔═╡ 69a4815d-1a84-400c-ae2b-4753e6c96abc
md"""
Si ``\gamma \in [0.9, 0.94]``:

``V_\star(s_0) = 10 + \gamma V_\star(s_1)``

``V_\star(s_1) = \gamma V_\star(s_1)``
"""

# ╔═╡ 0b1a0996-d98d-43a5-8d4e-3faa6ded79a8
md"""
Si ``\gamma \in [0.94, 1]``:

``V_\star(s_0) = 1 + \gamma V_\star(s_0)``

``V_\star(s_1) = -15 + \gamma V_\star(s_0)``
"""

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[[deps.URIs]]
git-tree-sha1 = "ac00576f90d8a259f2c9d823e91d1de3fd44d348"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.1"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[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_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[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.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "868e669ccb12ba16eaf50cb2957ee2ff61261c56"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+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.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"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
# ╟─ec777da4-8ef0-44b4-8d56-38b31790a5b7
# ╟─dc4eb528-82b2-11ed-3ca2-c321ff8ef647
# ╟─7fb608fa-da53-47d3-a585-235b4692c4dd
# ╟─58baae0a-ae46-443c-8c08-8743a13eb860
# ╟─dc2fcab2-7b23-47e3-b53a-0ad966645afd
# ╟─a6100c90-9a12-43d8-8ec3-dbf04b4ab5d1
# ╟─a5df336b-3db1-441d-845b-3ddb0aa3213a
# ╟─a30def9e-50f0-4aa2-8e79-6d10ecee5bee
# ╟─3176c1f0-c1fa-4c74-9133-f4371e584b6c
# ╟─f5566288-5ef4-495c-8442-a59f17f0814b
# ╟─f01bd868-a180-4327-98bf-de2298673523
# ╟─f8c9b0c8-3f81-441b-b673-f07ed2ae3e5a
# ╟─73a0d6a0-94a9-45b3-9db9-6cc0c845ab95
# ╟─3b87e4e9-81c9-48d9-9f58-e6271f975c40
# ╟─045e6ef0-84a8-4ad1-bc7c-0494599fd825
# ╟─8cd6c607-2094-4fa0-b1d8-445842ac5091
# ╟─fe6dc66f-2627-4c13-b32a-9751b2ff3a73
# ╟─fdd906ab-2224-4c26-9fb9-d9855bfda257
# ╟─65dcb2b0-1021-45a6-afe7-a95f6e139636
# ╟─69a4815d-1a84-400c-ae2b-4753e6c96abc
# ╟─0b1a0996-d98d-43a5-8d4e-3faa6ded79a8
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002