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                <p><img src="http://route50.net/data/gallery/submissions/3391_full.png" alt=""></p>
                <p>In a <a href="http://koaning.io/pokemon-recommendations.html">previous</a> document I've explained how to use the CAPM model to create naive ranking for pokemon. Even though the approach had some logic to it, I wanted to explore if a naive <a href="http://koaning.io/mini-trueskill-demo.html">probibalistic implementation</a> of trueskill could do the job as well. This document contains a python implementation my naive approach as well as an application to a pokemon dataset. The first part will explain how it is made and tested. The second part will analyse the ranking of all 750 pokemon.</p>

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                <h2 id="Starting-Data">Starting Data<a class="anchor-link" href="#Starting-Data"></a></h2><p>Before writing any application code, it seems sensible to first think about how to test if the code works. It would be a good idea to simulate some random players with predetermined skill levels such that we can test if our ranking algorithm can at least rank simulated data well enough. The code below generates this data.</p>

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                    <div class=" highlight hl-ipython2"><pre><span class="kn">import</span> <span class="nn">uuid</span>
                        <span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
                        <span class="kn">import</span> <span class="nn">pandas</span> <span class="kn">as</span> <span class="nn">pd</span>
                        <span class="kn">import</span> <span class="nn">json</span> 
                        <span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
                        <span class="o">%</span><span class="k">matplotlib</span> inline  
                        <span class="kn">import</span> <span class="nn">seaborn</span><span class="p">;</span> <span class="n">seaborn</span><span class="o">.</span><span class="n">set</span><span class="p">()</span>

                        <span class="n">SIZE_ARR</span> <span class="o">=</span> <span class="mi">100</span>

                        <span class="k">def</span> <span class="nf">sim</span><span class="p">(</span><span class="n">p1</span><span class="p">,</span> <span class="n">p2</span><span class="p">,</span> <span class="n">entropy</span> <span class="o">=</span> <span class="mi">1</span><span class="p">):</span>
                        <span class="sd">&quot;&quot;&quot; </span>
                        <span class="sd">    Given two players (tuples: &lt;name, skill&gt;) simulate </span>
                        <span class="sd">    a game between them. Returns (winner-name, loser-name)</span>
                        <span class="sd">    &quot;&quot;&quot;</span>
                        <span class="n">r</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="o">-</span><span class="n">entropy</span><span class="p">,</span> <span class="n">entropy</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
                        <span class="k">if</span> <span class="n">p1</span><span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="o">+</span> <span class="n">r</span> <span class="o">&gt;</span> <span class="n">p2</span><span class="p">[</span><span class="mi">1</span><span class="p">]:</span>
                        <span class="k">return</span> <span class="n">p1</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">p2</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
                        <span class="k">else</span><span class="p">:</span>
                        <span class="k">return</span> <span class="n">p2</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">p1</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>

                        <span class="k">def</span> <span class="nf">gen_data</span><span class="p">(</span><span class="n">n_users</span> <span class="o">=</span> <span class="mi">20</span><span class="p">,</span> <span class="n">n_games</span> <span class="o">=</span> <span class="mi">100</span><span class="p">,</span> <span class="n">entropy</span> <span class="o">=</span> <span class="mi">1</span><span class="p">):</span>
                        <span class="sd">&quot;&quot;&quot;</span>
                        <span class="sd">    Generate a lot of games and outputs a list of user tuples</span>
                        <span class="sd">    (&lt;name&gt;, &lt;skill&gt;)as well as a list of games played.</span>
                        <span class="sd">    &quot;&quot;&quot;</span>
                        <span class="n">data</span> <span class="o">=</span> <span class="p">[]</span>
                        <span class="n">users</span> <span class="o">=</span> <span class="p">[(</span><span class="nb">str</span><span class="p">(</span><span class="n">uuid</span><span class="o">.</span><span class="n">uuid4</span><span class="p">())[:</span><span class="mi">6</span><span class="p">],</span> <span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_users</span><span class="p">)]</span>
                        <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_games</span><span class="p">):</span> 
                        <span class="n">i</span><span class="p">,</span> <span class="n">j</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">choice</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">users</span><span class="p">)),</span> <span class="mi">2</span><span class="p">)</span>
                        <span class="n">u1</span><span class="p">,</span> <span class="n">u2</span> <span class="o">=</span> <span class="n">users</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">users</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> 
                        <span class="n">data</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sim</span><span class="p">(</span><span class="n">u1</span><span class="p">,</span> <span class="n">u2</span><span class="p">,</span> <span class="n">entropy</span><span class="p">))</span>
                        <span class="k">return</span> <span class="n">users</span><span class="p">,</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span> <span class="o">=</span> <span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">,</span> <span class="s1">&#39;loser&#39;</span><span class="p">])</span>
                    </pre></div>

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                <h1 id="Code-that-does-the-math">Code that does the math<a class="anchor-link" href="#Code-that-does-the-math"></a></h1><p>We'll now concern ourselves with writing code that updates a system of beliefs. The belief will be captured in a numpy array and can be interpreted as a histogram of likelihood values for a certain level of skill. The algorithm is explained in the <a href="http://koaning.io/pokemon-recommendations.html">previous blogpost</a> as well but schematically, this is what the belief update looks like;</p>
                <p><img src="http://i.imgur.com/Q9T5V8M.png" alt=""></p>
                <p>We start out with two arrays of prior skill beliefs for player 1 and player 2. We then combine them in a two dimensional density. This is our prior belief of the skill levels of the two players. Once a player wins, we can update this belief. One player has a lower skill level. To translate this to our belief system, we cut away all likelihood in the triangle of the losing player. Then we marginalise our belief back to the two players. By repeating this for all the games, we hope to achieve a belief of skill that can be updated in real time.</p>
                <h2 id="Towards-Code">Towards Code<a class="anchor-link" href="#Towards-Code"></a></h2><p>Next we'll define all the functions that do the ranking. These functions are meant to be minimal, as we only want to check if this algorithm can work. It makes sense to think about what our app code might look like and to think about how to test individual components before writing the code though. I'm not particularily religious about testing but I do see some value for writing a test before writing code here. We're going to be munging with numpy matrices and typos in indices will be unforgiving.</p>

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                <pre><code>p1 = np.array([1.,2.,3.])
                    p2 = np.array([3.,2.,1.])
                    p3 = np.array([1.,1.,1.])
                    mat1 = gen_prior(p1, p2)
                    mat2 = gen_prior(p2, p1)
                    mat3 = gen_prior(p3, p3)
                    m1, m2 = gen_marginals(mat1)
                    m3, m4 = gen_marginals(mat2)
                    np1, np2 = gen_marginals(cut_matrix(mat1))

                    assert all(m1 == m2[::-1]) #marginals should still be same shape 
                    assert all(m3 == m4[::-1]) #marginals should still be same shape 
                    assert np.sum(mat1) == np.sum(mat2) #matrices should have equal values
                    assert np.sum(mat3) == 9 #all ones should sum to 9 
                    assert all(np1 == np2[::-1]) #marginals should still be same shape</code></pre>
                    <p>With these tests in place, I could worry about the code.</p>

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                        <div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">get_player_stats</span><span class="p">(</span><span class="n">name</span><span class="p">):</span>
                            <span class="sd">&quot;&quot;&quot;</span>
                            <span class="sd">    Check if a player exists in the dictionary. If not, add it. </span>
                            <span class="sd">    Then returns the player belief. </span>
                            <span class="sd">    &quot;&quot;&quot;</span>
                            <span class="k">if</span> <span class="n">name</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">d</span><span class="o">.</span><span class="n">keys</span><span class="p">():</span> 
                            <span class="n">d</span><span class="p">[</span><span class="n">name</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">SIZE_ARR</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">(</span><span class="n">SIZE_ARR</span><span class="p">))</span>
                            <span class="k">return</span> <span class="n">d</span><span class="p">[</span><span class="n">name</span><span class="p">]</span>

                            <span class="k">def</span> <span class="nf">gen_prior</span><span class="p">(</span><span class="n">arr1</span><span class="p">,</span> <span class="n">arr2</span><span class="p">):</span>
                            <span class="sd">&quot;&quot;&quot;</span>
                            <span class="sd">    Combines two arrays into a matrix.</span>
                            <span class="sd">    &quot;&quot;&quot;</span>
                            <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">arr1</span><span class="p">)</span><span class="o">.</span><span class="n">T</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="n">arr2</span><span class="p">)</span>

                            <span class="k">def</span> <span class="nf">cut_matrix</span><span class="p">(</span><span class="n">mat</span><span class="p">):</span> 
                            <span class="sd">&quot;&quot;&quot;</span>
                            <span class="sd">    Cuts the matrix according to likelihood update rule. </span>
                            <span class="sd">    Also applies normalisation</span>
                            <span class="sd">    &quot;&quot;&quot;</span>
                            <span class="n">posterior</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="n">mat</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.000001</span>
                            <span class="n">posterior</span> <span class="o">=</span> <span class="n">posterior</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">posterior</span><span class="p">)</span>
                            <span class="k">return</span> <span class="n">posterior</span>

                            <span class="k">def</span> <span class="nf">gen_marginals</span><span class="p">(</span><span class="n">posterior_mat</span><span class="p">):</span>
                            <span class="sd">&quot;&quot;&quot;</span>
                            <span class="sd">    From a cut matrix, generate the appropriate marginals back</span>
                            <span class="sd">    &quot;&quot;&quot;</span>
                            <span class="n">winner</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">posterior_mat</span><span class="p">,</span> <span class="mi">0</span><span class="p">)))</span>
                            <span class="n">loser</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">squeeze</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">posterior_mat</span><span class="p">,</span> <span class="mi">1</span><span class="p">)))</span>
                            <span class="k">return</span> <span class="n">winner</span><span class="p">,</span> <span class="n">loser</span>

                            <span class="k">def</span> <span class="nf">update</span><span class="p">(</span><span class="n">winner</span><span class="p">,</span> <span class="n">loser</span><span class="p">):</span>
                            <span class="sd">&quot;&quot;&quot;</span>
                            <span class="sd">    Given a winner and user, update our state. </span>
                            <span class="sd">    &quot;&quot;&quot;</span>
                            <span class="n">winner_arr</span> <span class="o">=</span> <span class="n">get_player_stats</span><span class="p">(</span><span class="n">winner</span><span class="p">)</span>
                            <span class="n">loser_arr</span> <span class="o">=</span> <span class="n">get_player_stats</span><span class="p">(</span><span class="n">loser</span><span class="p">)</span>
                            <span class="n">prior_mat</span> <span class="o">=</span> <span class="n">gen_prior</span><span class="p">(</span><span class="n">loser_arr</span><span class="p">,</span> <span class="n">winner_arr</span><span class="p">)</span>
                            <span class="n">posterior_mat</span> <span class="o">=</span> <span class="n">cut_matrix</span><span class="p">(</span><span class="n">prior_mat</span><span class="p">)</span>
                            <span class="n">new_winner</span><span class="p">,</span> <span class="n">new_loser</span> <span class="o">=</span> <span class="n">gen_marginals</span><span class="p">(</span><span class="n">posterior_mat</span><span class="p">)</span>
                            <span class="n">d</span><span class="p">[</span><span class="n">loser</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_loser</span>
                            <span class="n">d</span><span class="p">[</span><span class="n">winner</span><span class="p">]</span> <span class="o">=</span> <span class="n">new_winner</span>
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                    <p>I wrote the code and the tests passed. For now this is enough. Before running this code against the pokemon dataset, let us first run it against some simulated data.</p>
                    <h2 id="Running-the-code-a-few-times.">Running the code a few times.<a class="anchor-link" href="#Running-the-code-a-few-times."></a></h2>
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                        <div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">plot_scores</span><span class="p">(</span><span class="n">plot_idx</span><span class="p">):</span> 
                            <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="n">plot_idx</span><span class="p">)</span>
                            <span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span>
                                <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">users</span><span class="p">)),</span> 
                                <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">d</span><span class="p">[</span><span class="n">usr</span><span class="p">[</span><span class="mi">0</span><span class="p">]]</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">SIZE_ARR</span><span class="p">))</span> <span class="k">for</span> <span class="n">usr</span> <span class="ow">in</span> <span class="n">users</span><span class="p">]</span>
                                <span class="p">)</span>

                                <span class="k">def</span> <span class="nf">plot_dist</span><span class="p">(</span><span class="n">plot_idx</span><span class="p">):</span>
                                <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="n">plot_idx</span><span class="p">)</span>
                                <span class="k">for</span> <span class="n">usr</span> <span class="ow">in</span> <span class="n">users</span><span class="p">:</span>
                                <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">SIZE_ARR</span><span class="p">),</span> <span class="n">d</span><span class="p">[</span><span class="n">usr</span><span class="p">[</span><span class="mi">0</span><span class="p">]],</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
                            </pre></div>

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                        <h2 id="Simulate-n_players-=-100,-n_games-=-400,-entropy-=-1">Simulate <code>n_players</code> = 100, <code>n_games</code> = 400, <code>entropy</code> = 1<a class="anchor-link" href="#Simulate-n_players-=-100,-n_games-=-400,-entropy-=-1"></a></h2>
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                            <div class=" highlight hl-ipython2"><pre><span class="n">d</span> <span class="o">=</span> <span class="p">{}</span> 
                                <span class="n">users</span><span class="p">,</span> <span class="n">df</span> <span class="o">=</span> <span class="n">gen_data</span><span class="p">(</span><span class="n">n_users</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_games</span><span class="o">=</span><span class="mi">400</span><span class="p">,</span> <span class="n">entropy</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
                                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">json</span><span class="o">.</span><span class="n">loads</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="n">orient</span> <span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)):</span> 
                                <span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">],</span> <span class="n">i</span><span class="p">[</span><span class="s1">&#39;loser&#39;</span><span class="p">])</span>

                                <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
                                <span class="n">plot_scores</span><span class="p">(</span><span class="mi">221</span><span class="p">)</span>
                                <span class="n">plot_dist</span><span class="p">(</span><span class="mi">222</span><span class="p">)</span>

                                <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span> 
                                <span class="n">users</span><span class="p">,</span> <span class="n">df</span> <span class="o">=</span> <span class="n">gen_data</span><span class="p">(</span><span class="n">n_users</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_games</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">entropy</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
                                <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">json</span><span class="o">.</span><span class="n">loads</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="n">orient</span> <span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)):</span> 
                                <span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">],</span> <span class="n">i</span><span class="p">[</span><span class="s1">&#39;loser&#39;</span><span class="p">])</span>

                                <span class="n">plot_scores</span><span class="p">(</span><span class="mi">223</span><span class="p">)</span>
                                <span class="n">plot_dist</span><span class="p">(</span><span class="mi">224</span><span class="p">)</span>
                            </pre></div>

                        </div>
                    </div>
                </div>

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                            >
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                    <h2 id="Simulate-n_players-=-100,-n_games-=-400/10000,-entropy-=-15">Simulate <code>n_players</code> = 100, <code>n_games</code> = 400/10000, <code>entropy</code> = 15<a class="anchor-link" href="#Simulate-n_players-=-100,-n_games-=-400/10000,-entropy-=-15"></a></h2>
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                    <p>Increasing the entropy in the system will cause the ranking algorithm to measure the noise more than the signal, but more games allow for convergence to still occur.</p>

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                        <div class=" highlight hl-ipython2"><pre><span class="n">d</span> <span class="o">=</span> <span class="p">{}</span> 
                            <span class="n">users</span><span class="p">,</span> <span class="n">df</span> <span class="o">=</span> <span class="n">gen_data</span><span class="p">(</span><span class="n">n_users</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_games</span><span class="o">=</span><span class="mi">400</span><span class="p">,</span> <span class="n">entropy</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
                            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">json</span><span class="o">.</span><span class="n">loads</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="n">orient</span> <span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)):</span> 
                            <span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">],</span> <span class="n">i</span><span class="p">[</span><span class="s1">&#39;loser&#39;</span><span class="p">])</span>

                            <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
                            <span class="n">plot_scores</span><span class="p">(</span><span class="mi">221</span><span class="p">)</span>
                            <span class="n">plot_dist</span><span class="p">(</span><span class="mi">222</span><span class="p">)</span>

                            <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span> 
                            <span class="n">users</span><span class="p">,</span> <span class="n">df</span> <span class="o">=</span> <span class="n">gen_data</span><span class="p">(</span><span class="n">n_users</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_games</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">entropy</span><span class="o">=</span><span class="mi">15</span><span class="p">)</span>
                            <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">json</span><span class="o">.</span><span class="n">loads</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="n">orient</span> <span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)):</span> 
                            <span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">],</span> <span class="n">i</span><span class="p">[</span><span class="s1">&#39;loser&#39;</span><span class="p">])</span>

                            <span class="n">plot_scores</span><span class="p">(</span><span class="mi">223</span><span class="p">)</span>
                            <span class="n">plot_dist</span><span class="p">(</span><span class="mi">224</span><span class="p">)</span>
                        </pre></div>

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                <h2 id="Simulate-n_players-=-100,-n_games-=-400/10000,-entropy-=-50">Simulate <code>n_players</code> = 100, <code>n_games</code> = 400/10000, <code>entropy</code> = 50<a class="anchor-link" href="#Simulate-n_players-=-100,-n_games-=-400/10000,-entropy-=-50"></a></h2>
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                <p>Suppose that there is more noise than signal in this sytem? In that case it will correctly indicate this to us.</p>

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                    <div class=" highlight hl-ipython2"><pre><span class="n">d</span> <span class="o">=</span> <span class="p">{}</span> 
                        <span class="n">users</span><span class="p">,</span> <span class="n">df</span> <span class="o">=</span> <span class="n">gen_data</span><span class="p">(</span><span class="n">n_users</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_games</span><span class="o">=</span><span class="mi">400</span><span class="p">,</span> <span class="n">entropy</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">json</span><span class="o">.</span><span class="n">loads</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="n">orient</span> <span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)):</span> 
                        <span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">],</span> <span class="n">i</span><span class="p">[</span><span class="s1">&#39;loser&#39;</span><span class="p">])</span>

                        <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">10</span><span class="p">))</span>
                        <span class="n">plot_scores</span><span class="p">(</span><span class="mi">221</span><span class="p">)</span>
                        <span class="n">plot_dist</span><span class="p">(</span><span class="mi">222</span><span class="p">)</span>

                        <span class="n">d</span> <span class="o">=</span> <span class="p">{}</span> 
                        <span class="n">users</span><span class="p">,</span> <span class="n">df</span> <span class="o">=</span> <span class="n">gen_data</span><span class="p">(</span><span class="n">n_users</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_games</span><span class="o">=</span><span class="mi">10000</span><span class="p">,</span> <span class="n">entropy</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
                        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">json</span><span class="o">.</span><span class="n">loads</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">to_json</span><span class="p">(</span><span class="n">orient</span> <span class="o">=</span><span class="s1">&#39;records&#39;</span><span class="p">)):</span> 
                        <span class="n">update</span><span class="p">(</span><span class="n">i</span><span class="p">[</span><span class="s1">&#39;winner&#39;</span><span class="p">],</span> <span class="n">i</span><span class="p">[</span><span class="s1">&#39;loser&#39;</span><span class="p">])</span>

                        <span class="n">plot_scores</span><span class="p">(</span><span class="mi">223</span><span class="p">)</span>
                        <span class="n">plot_dist</span><span class="p">(</span><span class="mi">224</span><span class="p">)</span>
                    </pre></div>

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