{"id":2817,"date":"2020-09-26T10:02:17","date_gmt":"2020-09-26T08:02:17","guid":{"rendered":"https:\/\/blog.atlant.is\/?p=2817"},"modified":"2021-06-13T20:08:10","modified_gmt":"2021-06-13T18:08:10","slug":"infinitely-many-hats-3-3","status":"publish","type":"post","link":"https:\/\/blog.douzeb.is\/?p=2817","title":{"rendered":"Infinitely Many Hats (3\/3)"},"content":{"rendered":"\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/infinite-alpha.webp\" alt=\"Countably infinitely many hats\" class=\"wp-image-2832\" width=\"366\" height=\"2025\"\/><figcaption>Countably infinitely many hats<\/figcaption><\/figure><\/div>\n\n\n\n<p>This is the last in a series of three <em>exercices de style<\/em> to show some interesting aspects of the <em>game of hats<\/em>: a puzzle which was initially proposed by Lionel Levine.<\/p>\n\n\n\n<ul><li><a href=\"https:\/\/blog.atlant.is\/?p=1183\">First exercice<\/a><\/li><li><a href=\"https:\/\/blog.atlant.is\/?p=1893\">Second exercice<\/a><\/li><\/ul>\n\n\n\n<p>Here we look at case where each player has a tower of countably infinitely many hats on their head and try constructing efficient strategies from the results of the first exercice.<\/p>\n\n\n\n<!--more-->\n\n\n\n\n\n<h2>Definitions and notations<\/h2>\n\n\n\n<h3>Towers and strategies<\/h3>\n\n\n\n<p>We use notations inspired by <span class=\"zp-InText-zp-ID--6833562-DB3TDTNR--wp2817 zp-InText-Citation loading\" rel=\"{ 'pages': 'np', 'items': '{6833562:DB3TDTNR}', 'format': '(%a%, %d%, %p%)', 'brackets': '', 'etal': '', 'separator': '', 'and': '' }\"><\/span>.<\/p>\n\n\n\n<p>Let $\\mathbb{T}_{&lt;\\mathbb{N}}$ denote the set of all towers of finite height and $\\mathbb{T}_{\\mathbb{N}}$ the set of towers of countably infinite height.<\/p>\n\n\n\n<p>\\begin{align}<br>\\mathbb{T}_{&lt;\\mathbb{N}}<br>&amp;\\stackrel{\\text{def}}{=}\\bigcup_{n\\in\\mathbb{N}} \\mathbb{T}_n\\\\<br>\\mathbb{T}_{\\mathbb{N}}<br>&amp;\\stackrel{\\text{def}}{=}[2]^{\\mathbb{N}}<br>\\end{align}<\/p>\n\n\n\n<p>$\\mathbb{T}_{&lt;\\mathbb{N}}$ is countable but $\\mathbb{T}_{\\mathbb{N}}$ has the cardinality of the real line. $\\mathbb{T}_{\\mathbb{N}}$ is also known as the <em>Cantor space<\/em>.<\/p>\n\n\n\n<p>Let $\\mathbb{S}_{\\mathbb{N}}$ denote the set of strategies for the game of hats with towers of countably infinite height. A (symmetric) strategy maps a tower to a chosen position:<\/p>\n\n\n\n<p>\\begin{align}<br>\\mathbb{S}_{\\mathbb{N}}<br>&amp;\\stackrel{\\text{def}}{=}{\\mathbb{Z}}^{\\mathbb{T}_{\\mathbb{N}}}<br>\\end{align}<\/p>\n\n\n\n<p>$\\mathbb{S}_{\\mathbb{N}}$ has cardinality strictly greater than the real  line.<\/p>\n\n\n\n<h3>Hit set and win rate<\/h3>\n\n\n\n<p>With infinite towers we can still use $\\eta_{t,u}(s)$ to denote the outcome of applying strategy $s\\in\\mathbb{S}_{\\mathbb{N}}$ to the configuration $(t,u)$, however it is no longer possible to calculate the hit score as a sum and the win rate as a mean value.<\/p>\n\n\n\n<p>We substitute with $\\eta(s)$, the <em>hit set<\/em> of strategy $s$:<\/p>\n\n\n\n<p>\\begin{equation}<br>\\eta(s)\\stackrel{\\text{def}}{=}\\{(t,u)\\in\\mathbb{T}_{\\mathbb{N}}\\times\\mathbb{T}_{\\mathbb{N}}\\mid\\eta_{t,u}(s)=1\\},<br>\\end{equation}<\/p>\n\n\n\n<p>and write $\\mu(s)$ for the <em>win rate<\/em> of strategy $s$, defined as the Baire\/Borel measure of  $\\eta(s)$ in the space $\\mathbb{T}_{\\mathbb{N}}\\times\\mathbb{T}_{\\mathbb{N}}$ (when it is measurable).<\/p>\n\n\n\n<p>\\begin{equation}<br>\\mu(s)\\stackrel{\\text{def}}{=}\\mu(\\eta(s)).<br>\\end{equation}<\/p>\n\n\n\n<h3>Embedding<\/h3>\n\n\n\n<p>For convenience we identify the towers in $\\mathbb{T}_n$ with towers in $\\mathbb{T}_{\\mathbb{N}}$ (by completing with black hats) and the strategies in $s\\in\\mathbb{S}_n$ with strategies in $s\\in\\mathbb{S}_{\\mathbb{N}}$ (by discarding hats at position $n$ and beyond). <\/p>\n\n\n\n<p>Thus we loosely write $s(t)$ and $eta_{t,u}(s)$ even when $s$, $t$ and $u$ are a mix of finite\/infinite strategies and towers.<\/p>\n\n\n\n<p>\\begin{align}<br>\\forall i&lt;j, \\;\\;<br>\\left\\{<br>\\begin{array}{l}<br>\\mathbb{T}_i<br>\\subset \\mathbb{T}_j<br>\\subset \\mathbb{T}_{&lt;\\mathbb{N}}<br>\\subset \\mathbb{T}_{\\mathbb{N}}\\\\<br>\\mathbb{S}_i<br>\\subset \\mathbb{S}_j<br>\\subset \\mathbb{S}_{\\mathbb{N}}\\\\<br>\\end{array}<br>\\right.<br>\\end{align}<\/p>\n\n\n\n<h2>Limits<\/h2>\n\n\n\n<h3>Infinite towers as limits of finite towers<\/h3>\n\n\n\n<p>Every tower $t\\in\\mathbb{T}_{\\mathbb{N}}$ is the limit of a sequence of towers in $\\mathbb{T}_{&lt;\\mathbb{N}}$ (with the topology of the Cantor space).<\/p>\n\n\n\n<p>Proof: if $s\\!\\mid\\!k$ denotes the initial segment of $s$ of size $k\\in\\mathbb{N}$, then:<\/p>\n\n\n\n<p>\\begin{equation}<br>s = \\lim_{n\\to\\infty} s\\!\\mid\\!n<br>\\end{equation}<\/p>\n\n\n\n<p>In fact, we can view $t\\in\\mathbb{T}_{\\mathbb{N}}$ as the Cauchy completion of $t\\in\\mathbb{T}_{&lt;\\mathbb{N}}$.<\/p>\n\n\n\n<h3>Finite strategies are continuous<\/h3>\n\n\n\n<p>A finite strategy $s\\in\\mathbb{S}_n$, embedded into $\\mathbb{S}_{\\mathbb{N}}$ as a mapping $s: \\mathbb{T}_{\\mathbb{N}} \\to \\mathbb{Z}$,  is <em>continuous<\/em>.<\/p>\n\n\n\n<h3>Limits of strategies<\/h3>\n\n\n\n<p>Here we allow strategies as functions $s: \\mathbb{T}_{\\mathbb{N}} \\to \\mathbb{Z}$ with <em>partial domain<\/em>.<\/p>\n\n\n\n<p>Given a sequence $s_n\\in\\mathbb{S}_{\\mathbb{N}}$ we define its <em>pointwise limit<\/em>:<\/p>\n\n\n\n<p>\\begin{align}<br>\\lim (s_n):\\; &amp;\\big\\{t\\mid \\lim_{n\\to\\infty} s_n(t)\\text{ exists}\\big\\} \\to \\mathbb{Z}\\\\<br>&amp;\\;t \\mapsto \\lim_{n\\to\\infty} s_n(t) \\nonumber<br>\\end{align}<\/p>\n\n\n\n<p>It is natural to try constructing efficient infinite strategies as limits of finite strategies.<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th><\/th><th>Type&nbsp;&nbsp;&nbsp;<strong>&nbsp;&nbsp;&nbsp;<\/strong><\/th><th>Win<br>rate<\/th><th>Domain<\/th><th class=\"has-text-align-left\" data-align=\"left\">Computable<\/th><\/tr><\/thead><tbody><tr><td>$0_\\omega$<\/td><td>Constant<\/td><td>$1 \\over 4$<\/td><td>$\\mathbb{T}_{\\mathbb{N}}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">on $\\mathbb{T}_{\\mathbb{N}}$<\/td><\/tr><tr><td>$\\mathbf{W}_\\omega$<\/td><td>Lowest white<\/td><td>$1 \\over 3$<\/td><td>$\\mathbb{T}_{\\mathbb{N}}\\setminus\\{(0,0,\\cdots)\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">on its domain<\/td><\/tr><tr><td>$\\mathbf{B}_\\omega$<\/td><td>Lowest black<\/td><td>$1 \\over 3$<\/td><td>$\\mathbb{T}_{\\mathbb{N}}\\setminus\\{(1,1,\\cdots)\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">on its domain<\/td><\/tr><tr><td>$\\mathbf{C}_\\omega$<\/td><td>Carter<\/td><td>$7 \\over 20$<\/td><td>$\\mathbb{T}_{\\mathbb{N}}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">on $\\mathbb{T}_{\\mathbb{N}}\\setminus$ $\\{(0,0,\\cdots),$ $(1,1,\\cdots)\\}$<\/td><\/tr><tr><td>$\\mathbf{R}_\\omega$<\/td><td>Reyes<\/td><td>$7 \\over 20$<\/td><td>$\\big\\{t\\in\\mathbb{T}_{\\mathbb{N}}\\mid\\exists k,$ $\\text{ card}\\{t_{3k},t_{3k+1},$ $\\:t_{3k+2}\\}\\neq1\\big\\}$<\/td><td class=\"has-text-align-left\" data-align=\"left\">on its domain<\/td><\/tr><\/tbody><\/table><figcaption>Sample of infinite strategies<\/figcaption><\/figure>\n\n\n\n<h2>Infinite lowest-white strategy<\/h2>\n\n\n\n<p>Consider the infinite lowest-white strategy $\\mathbf{W}_\\omega$:<\/p>\n\n\n\n<p>\\begin{equation}<br>\\mathbf{W}_\\omega \\stackrel{\\text{def}}{=} \\lim_{n\\to\\infty} \\mathbf{W}_n<br>\\end{equation}<\/p>\n\n\n\n<p>$\\mathbf{W}_\\omega$ has win rate $1\\over3$.<\/p>\n\n\n\n<h3>Proof<\/h3>\n\n\n\n<p>Considering all towers in $\\mathbb{T}_{\\mathbb{N}}$, let&rsquo;s represent the combined output of the $\\mathbf{W}_n$ strategies as a tree, as shown in the picture below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/white-full.webp\" alt=\"\" class=\"wp-image-2874\"\/><figcaption>Output of the $\\mathbf{W}_n$ strategies<\/figcaption><\/figure>\n\n\n\n<p>Let&rsquo;s denote in gray the points where the pointwise limit is reached:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/white-short.webp\" alt=\"\" class=\"wp-image-2875\" width=\"427\" height=\"190\"\/><figcaption>Pointwise limit of the $\\mathbf{W}_n$ strategies<\/figcaption><\/figure>\n\n\n\n<p>We see that $\\mathbf{W}_\\omega=\\lim(\\mathbf{W}_n)$ is defined almost everywhere<span id='easy-footnote-1-2817' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/blog.douzeb.is\/?p=2817#easy-footnote-bottom-1-2817' title='In fact $\\mathbf{W}_\\omega$ is defined everywhere except at $(0, 0, \\cdots)$ '><sup>1<\/sup><\/a><\/span> with respect to the Baire\/Borel measure, and we can squeeze its hit set the following way:<\/p>\n\n\n\n<p>\\begin{equation}<br>I_k \\subset \\eta(\\mathbf{W}_\\omega) \\subset S_k <br>\\end{equation}<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<p>$$<br>\\left\\{<br>\\begin{array}{l}<br>I_k = \\eta(\\mathbf{W}_k) \\setminus U_k\\\\<br>S_k = \\eta(\\mathbf{W}_k) \\cup U_k\\\\<br>U_k = \\big\\{(t,u)\\mid \\text{ card}\\bigcup_{i\\geq k}\\{\\mathbf{W}_i(t)\\}\\neq1<br>\\text{ or card}\\bigcup_{i\\geq k}\\{\\mathbf{W}_i(u)\\}\\neq1\\big\\}<br>\\end{array}<br>\\right.<br>$$<\/p>\n\n\n\n<p>$I_k$ is an increasing sequence of sets and $S_k$ is a decreasing sequence, with $\\lim_{k\\to\\infty} \\mu(U_k) = 0$, $\\lim_{k\\to\\infty} \\mu(I_k) = 1\/3$, $\\lim_{k\\to\\infty} \\mu(S_k) = 1\/3$, hence $\\mu(\\mathbf{W}_\\omega) = 1\/3$. $\\quad \\Box$<\/p>\n\n\n\n<p>Note (as shown in the picture below) that $\\mathbf{W}_\\omega$ is a computable function and can be written as a finite automaton that processes the <em>stream<\/em> of hats from $t\\in\\mathbb{T}_{\\mathbb{N}}$:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/white-auto.webp\" alt=\"\" class=\"wp-image-2885\" width=\"138\" height=\"206\"\/><figcaption>$\\mathbf{W}_\\omega$ as a finite automaton<\/figcaption><\/figure><\/div>\n\n\n\n<h2>Infinite Carter strategy<\/h2>\n\n\n\n<p>Consider the infinite Carter strategy $\\mathbf{C}_\\omega$:<\/p>\n\n\n\n<p>\\begin{equation}<br>\\mathbf{C}_\\omega \\stackrel{\\text{def}}{=} \\lim_{n\\to\\infty} <br>\\mathbf{X}_n,<br>\\end{equation}<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<p>$$<br>\\left\\{<br>\\begin{array}{l}<br>\\mathbf{X}_{2k-1} = \\mathbf{C}_k\\!\\mid\\!(2k-1)\\\\<br>\\mathbf{X}_{2k} = \\mathbf{C}_k\\!\\mid\\!(2k)<br>\\end{array}<br>\\right.<br>$$<\/p>\n\n\n\n<p>$\\mathbf{W}_\\omega$ has win rate $7\\over20$.<\/p>\n\n\n\n<h3>Proof<\/h3>\n\n\n\n<p>Considering all towers in $\\mathbb{T}_{\\mathbb{N}}$, let&rsquo;s represent the combined output of the $\\mathbf{X}_n$ strategies as a tree, as shown in the picture below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/carter-full.webp\" alt=\"\" class=\"wp-image-2902\"><figcaption>Output of the Carter $\\mathbf{X}_n$ strategies<\/figcaption><\/figure>\n\n\n\n<p>Let&rsquo;s denote in gray the points where the pointwise limit is reached:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/carter-short.webp\" alt=\"\" class=\"wp-image-2903\"><figcaption>Pointwise limit of the Carter $\\mathbf{X}_n$ strategies<\/figcaption><\/figure>\n\n\n\n<p>We see that $\\mathbf{C}_\\omega=\\lim(\\mathbf{X}_n)$ is defined everywhere, and we can squeeze its hit set the following way:<\/p>\n\n\n\n<p>\\begin{equation}<br>I_k \\subset \\eta(\\mathbf{C}_\\omega) \\subset S_k <br>\\end{equation}<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<p>$$<br>\\left\\{<br>\\begin{array}{l}<br>I_k = \\eta(\\mathbf{X}_k) \\setminus U_k\\\\<br>S_k = \\eta(\\mathbf{X}_k) \\cup U_k\\\\<br>U_k = \\big\\{(t,u)\\mid \\text{ card}\\bigcup_{i\\geq k}\\{\\mathbf{X}_i(t)\\}\\neq1<br>\\text{ or card}\\bigcup_{i\\geq k}\\{\\mathbf{X}_i(u)\\}\\neq1\\big\\}<br>\\end{array}<br>\\right.<br>$$<\/p>\n\n\n\n<p>$I_k$ is an increasing sequence of sets and $S_k$ is a decreasing sequence, with $\\lim_{k\\to\\infty} \\mu(U_k) = 0$, $\\lim_{k\\to\\infty} \\mu(I_k) = 7\/20$, $\\lim_{k\\to\\infty} \\mu(S_k) = 7\/20$, hence $\\mu(\\mathbf{C}_\\omega) = 7\/20$. $\\quad \\Box$<\/p>\n\n\n\n<p>Note (as shown in the picture below) that $\\mathbf{C}_\\omega$ is a computable function almost everywhere<span id='easy-footnote-2-2817' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/blog.douzeb.is\/?p=2817#easy-footnote-bottom-2-2817' title='$\\mathbf{W}_\\omega$ is not computable only at points $(0,0,\\cdots)$ and $(1,1,\\cdots)$'><sup>2<\/sup><\/a><\/span> and can be written as a finite automaton that processes the <em>stream<\/em> of hats from $t\\in\\mathbb{T}_{\\mathbb{N}}$:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/carter-auto.webp\" alt=\"\" class=\"wp-image-2906\" width=\"615\" height=\"385\"><figcaption>$\\mathbf{C}_\\omega$ as a finite automaton<\/figcaption><\/figure>\n\n\n\n<h2>Infinite Reyes strategy<\/h2>\n\n\n\n<p>Consider the infinite Reyes strategy $\\mathbf{R}_\\omega$:<\/p>\n\n\n\n<p>\\begin{equation}<br>\\mathbf{R}_\\omega \\stackrel{\\text{def}}{=} \\lim_{n\\to\\infty} <br>\\mathbf{X}_n,<br>\\end{equation}<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<p>$$<br>\\left\\{<br>\\begin{array}{ll}<br>\\mathbf{X}_{3k-2} &amp;= \\mathbf{C}_k\\!\\mid\\!(3k-2)\\\\<br>\\mathbf{X}_{3k-1} &amp;= \\mathbf{C}_k\\!\\mid\\!(3k-1)\\\\<br>\\mathbf{X}_{3k} &amp;= \\mathbf{C}_k\\!\\mid\\!(3k)<br>\\end{array}<br>\\right.<br>$$<\/p>\n\n\n\n<p>$\\mathbf{W}_\\omega$ has win rate $7\\over20$.<\/p>\n\n\n\n<h3>Proof<\/h3>\n\n\n\n<p>Considering all towers in $\\mathbb{T}_{\\mathbb{N}}$, let&rsquo;s represent the combined output of the $\\mathbf{X}_n$ strategies as a tree, as shown in the picture below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/reyes-full.webp\" alt=\"\" class=\"wp-image-2911\"\/><figcaption>Output of the Reyes $\\mathbf{X}_n$ strategies<\/figcaption><\/figure>\n\n\n\n<p>Let&rsquo;s denote in gray the points where the pointwise limit is reached:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/reyes-short.webp\" alt=\"\" class=\"wp-image-2912\"\/><figcaption>Pointwise limit of the Reyes $\\mathbf{X}_n$ strategies<\/figcaption><\/figure>\n\n\n\n<p>We see that $\\mathbf{R}_\\omega=\\lim(\\mathbf{X}_n)$ is defined almost everywhere<span id='easy-footnote-3-2817' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/blog.douzeb.is\/?p=2817#easy-footnote-bottom-3-2817' title='$\\mathbf{R}_\\omega$ is defined for all $t$ such that $\\exists k, \\text{ card}\\{t_{3k},t_{3k+1},t_{3k+2}\\}\\neq1$'><sup>3<\/sup><\/a><\/span>, and we can squeeze its hit set the following way:<\/p>\n\n\n\n<p>\\begin{equation}<br>I_k \\subset \\eta(\\mathbf{C}_\\omega) \\subset S_k <br>\\end{equation}<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<p>$$<br>\\left\\{<br>\\begin{array}{l}<br>I_k = \\eta(\\mathbf{X}_k) \\setminus U_k\\\\<br>S_k = \\eta(\\mathbf{X}_k) \\cup U_k\\\\<br>U_k = \\big\\{(t,u)\\mid \\text{ card}\\bigcup_{i\\geq k}\\{\\mathbf{X}_i(t)\\}\\neq1<br>\\text{ or card}\\bigcup_{i\\geq k}\\{\\mathbf{X}_i(u)\\}\\neq1\\big\\}<br>\\end{array}<br>\\right.<br>$$<\/p>\n\n\n\n<p>$I_k$ is an increasing sequence of sets and $S_k$ is a decreasing sequence, with $\\lim_{k\\to\\infty} \\mu(U_k) = 0$, $\\lim_{k\\to\\infty} \\mu(I_k) = 7\/20$, $\\lim_{k\\to\\infty} \\mu(S_k) = 7\/20$, hence $\\mu(\\mathbf{R}_\\omega) = 7\/20$. $\\quad \\Box$<\/p>\n\n\n\n<p>Note (as shown in the picture below) that $\\mathbf{R}_\\omega$ is a computable function on its definition domain and can be written as a finite automaton that processes the <em>stream<\/em> of hats from $t\\in\\mathbb{T}_{\\mathbb{N}}$:<\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/blog.atlant.is\/wp-content\/uploads\/2020\/09\/reyes-auto.webp\" alt=\"\" class=\"wp-image-2914\" width=\"623\" height=\"385\"\/><figcaption>$\\mathbf{R}_\\omega$ as a finite automaton<\/figcaption><\/figure>\n\n\n\n<h2>References<\/h2>\n\n\n\n\n<div id='zp-InTextBib-zotpress-e391fbbbec8fb7f1bc8715e9d03d87ef' class='zp-Zotpress zp-Zotpress-InTextBib wp-block-group zp-Post-2817'>\r\n\t\t<span class=\"ZP_ITEM_KEY\" style=\"display: none;\">{6833562:DB3TDTNR}<\/span>\r\n\t\t<span class=\"ZP_STYLE\" style=\"display: none;\">apa<\/span>\r\n\t\t<span class=\"ZP_SORTBY\" style=\"display: none;\">default<\/span>\r\n\t\t<span class=\"ZP_ORDER\" style=\"display: none;\">asc<\/span>\r\n\t\t<span class=\"ZP_TITLE\" style=\"display: none;\">no<\/span>\r\n\t\t<span class=\"ZP_SHOWIMAGE\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_SHOWTAGS\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_DOWNLOADABLE\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_NOTES\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_ABSTRACT\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_CITEABLE\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_TARGET\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_URLWRAP\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_FORCENUM\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_HIGHLIGHT\" style=\"display: none;\"><\/span>\r\n\t\t<span class=\"ZP_POSTID\" style=\"display: none;\">2817<\/span><div class='zp-List loading'>\n<div class=\"zp-SEO-Content\"><div id=\"zp-ID-2817-6833562-DB3TDTNR\" class=\"zp-Entry zpSearchResultsItem\"><div class=\"csl-bib-body\" style=\"line-height: 2; padding-left: 1em; text-indent:-1em;\">\n  <div class=\"csl-entry\">Kechris, A. S. (2012). <i>Classical descriptive set theory.<\/i> Springer.<\/div>\n<\/div><\/div><\/div><!-- .zp-zp-SEO-Content -->\n<\/div><!-- .zp-List --><\/div><!--.zp-Zotpress-->\n\n\n","protected":false},"excerpt":{"rendered":"<p>This is the last in a series of three exercices de style to show some interesting aspects of the game of hats: a puzzle which was initially proposed by Lionel Levine. First exercice Second exercice Here we look at case where each player has a tower of countably infinitely many hats on their head and [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":1451,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[20,16,19],"tags":[],"ppma_author":[76],"authors":[{"term_id":76,"user_id":1,"is_guest":0,"slug":"fred","display_name":"Fred","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/ec0326d654fdf9f66e9eb42bb34a9bc4?s=96&d=mm&r=g","description":"","first_name":"","last_name":"","user_url":""}],"_links":{"self":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2817"}],"collection":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2817"}],"version-history":[{"count":107,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2817\/revisions"}],"predecessor-version":[{"id":3687,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/2817\/revisions\/3687"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/media\/1451"}],"wp:attachment":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2817"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2817"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2817"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=2817"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}