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<!DOCTYPE html>
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<title>Stationary behaviour of EIP 1559 agent-based model</title>
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<meta property="og:description" content="Basefee converges to price out all but a target size of users" />
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Stationary behaviour of EIP 1559 agent-based model
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<h2 id="TL;DR">TL;DR<a class="anchor-link" href="#TL;DR">¶</a></h2><ul>
<li>EIP 1559 is a proposed improvement for the transaction fee market. It sets a variable "base" gasprice to be paid by the user and burned by the protocol, in addition to a "tip" paid by the user to the block producer.</li>
<li>The base price ("basefee") adjusts upwards when demand is high, and downwards otherwise.</li>
<li>We observe in this notebook that in a stationary environmnent, basefee converges to a value that prices out enough users to achieve the target block size.</li>
</ul>
<hr>
<p>We introduce here the building blocks of agent-based simulations of EIP1559. This follows an <a href="https://nbviewer.jupyter.org/github/ethereum/rig/blob/master/eip1559/eip1559.ipynb">earlier notebook</a> that merely looked at the dynamics of the EIP 1559 mechanism. In the present notebook, agents decide on transactions based on the current basefee and form their transactions based on internal evaluations of their values and costs.</p>
<p><a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3025604">Huberman et al., 2019</a> introduced such a model and framework for the Bitcoin payment system. We adapt it here to study the dynamics of the basefee.</p>
<p>All the code is available in <a href="https://github.com/barnabemonnot/abm1559">this repo</a>, with some preliminary documentation <a href="https://barnabemonnot.com/abm1559/build/html/">here</a>. You can also download the <a href="https://pypi.org/project/abm1559/"><code>abm1559</code> package from PyPi</a> and reproduce all the analysis here yourself!</p>
<h2 id="The-broad-lines">The broad lines<a class="anchor-link" href="#The-broad-lines">¶</a></h2><p>We have several entities. <em>Users</em> come in randomly (following a Poisson process) and create and send transactions. The transactions are received by a <em>transaction pool</em>, from which the $x$ best <em>valid</em> transactions are included in a <em>block</em> created at fixed intervals. $x$ depends on how many valid transactions exist in the pool (e.g., how many post a gasprice exceeding the prevailing basefee in 1559 paradigm) and the block gas limit. Once transactions are included in the block, and the block is included in the <em>chain</em>, transactions are removed from the transaction pool.</p>
<p>How do users set their parameters? Users have their own internal ways of evaluating their <em>costs</em>. Users obtain a certain <em>value</em> from having their transaction included, which we call $v$. $v$ is different for every user. This value is fixed but their overall <em>payoff</em> decreases the longer they wait to be included. Some users have higher time preferences than others, and their payoff decreases faster than others the longer they wait. Put together, we have the following:</p>
$$ \texttt{payoff} = \texttt{value} - \texttt{cost from waiting} - \texttt{transaction fee} $$<p>Users expect to wait for a certain amount of time. In this essay, we set this to a fixed value -- somewhat arbitrarily we choose 5. This can be readily understood in the following way. Users estimate what their payoff will be from getting included 5 blocks from now, assuming basefee remains constant. If this payoff is negative, they decide not to send the transaction to the pool (in queuing terminology, they <em>balk</em>). We'll play with this assumption later.</p>
<p>The scenario is set up this way to study <em>stationarity</em>: assuming some demand comes in from a fixed distribution at regular intervals, we must expect basefee to reach some stationary value and stay there. It is then reasonable for users, at this stationary point, to consider that 5 blocks from now basefee will still be at the same level. In the nonstationary case, when for instance a systemic change in the demand happens (e.g., the rate of Poisson arrivals increases), a user may want to hedge their bets by estimating their future payoffs in a different way, taking into account that basefee might increase instead. This strategy would probably be a good idea during the <em>transition</em> phase, when basefee shifts from one stationary point to a new one.</p>
<p>We make the assumption here that users choose their 1559 parameters based on their value alone. We set the transaction <code>max_fee</code> parameter to the value of the user and set the <code>gas_premium</code> parameter to a residual value -- 1 Gwei per unit of gas.</p>
<p>There is no loss of generality in assuming all users send the same transaction in (e.g., a simple transfer) and so all transactions have the same <code>gas_used</code> value (21,000). In 1559 paradigm, with a 20M gas limit per block, this allows at most 952 transactions to be included, although the mechanism will target half of that, around 475 here. The protocol adjusts the basefee to apply economic pressure, towards a target gas usage of 10M per block.</p>
<h2 id="Simulation">Simulation<a class="anchor-link" href="#Simulation">¶</a></h2><p>We import a few classes from our <code>abm1559</code> package.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">config</span> InlineBackend.figure_format = 'svg'
<span class="kn">import</span> <span class="nn">os</span><span class="o">,</span> <span class="nn">sys</span>
<span class="n">sys</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">insert</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">realpath</span><span class="p">(</span><span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">pardir</span><span class="p">))</span>
<span class="c1"># You may remove the two lines above if you have installed abm1559 from pypi</span>
<span class="kn">from</span> <span class="nn">abm1559.utils</span> <span class="kn">import</span> <span class="n">constants</span>
<span class="kn">from</span> <span class="nn">abm1559.txpool</span> <span class="kn">import</span> <span class="n">TxPool</span>
<span class="kn">from</span> <span class="nn">abm1559.users</span> <span class="kn">import</span> <span class="n">User1559</span>
<span class="kn">from</span> <span class="nn">abm1559.userpool</span> <span class="kn">import</span> <span class="n">UserPool</span>
<span class="kn">from</span> <span class="nn">abm1559.chain</span> <span class="kn">import</span> <span class="p">(</span>
<span class="n">Chain</span><span class="p">,</span>
<span class="n">Block1559</span><span class="p">,</span>
<span class="p">)</span>
<span class="kn">from</span> <span class="nn">abm1559.simulator</span> <span class="kn">import</span> <span class="p">(</span>
<span class="n">spawn_poisson_demand</span><span class="p">,</span>
<span class="n">update_basefee</span><span class="p">,</span>
<span class="p">)</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
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<p>And define the main function used to simulate the fee market.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">simulate</span><span class="p">(</span><span class="n">demand_scenario</span><span class="p">,</span> <span class="n">UserClass</span><span class="p">):</span>
<span class="c1"># Instantiate a couple of things</span>
<span class="n">txpool</span> <span class="o">=</span> <span class="n">TxPool</span><span class="p">()</span>
<span class="n">basefee</span> <span class="o">=</span> <span class="n">constants</span><span class="p">[</span><span class="s2">"INITIAL_BASEFEE"</span><span class="p">]</span>
<span class="n">chain</span> <span class="o">=</span> <span class="n">Chain</span><span class="p">()</span>
<span class="n">metrics</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">user_pool</span> <span class="o">=</span> <span class="n">UserPool</span><span class="p">()</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">demand_scenario</span><span class="p">)):</span>
<span class="k">if</span> <span class="n">t</span> <span class="o">%</span> <span class="mi">100</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span> <span class="nb">print</span><span class="p">(</span><span class="n">t</span><span class="p">)</span>
<span class="c1"># `env` is the "environment" of the simulation</span>
<span class="n">env</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"basefee"</span><span class="p">:</span> <span class="n">basefee</span><span class="p">,</span>
<span class="s2">"current_block"</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span>
<span class="p">}</span>
<span class="c1"># We return a demand drawn from a Poisson distribution.</span>
<span class="c1"># The parameter is given by `demand_scenario[t]`, and can vary</span>
<span class="c1"># over time.</span>
<span class="n">users</span> <span class="o">=</span> <span class="n">spawn_poisson_demand</span><span class="p">(</span><span class="n">t</span><span class="p">,</span> <span class="n">demand_scenario</span><span class="p">[</span><span class="n">t</span><span class="p">],</span> <span class="n">UserClass</span><span class="p">)</span>
<span class="c1"># We query each new user with the current basefee value</span>
<span class="c1"># Users either return a transaction or None if they prefer to balk</span>
<span class="n">decided_txs</span> <span class="o">=</span> <span class="n">user_pool</span><span class="o">.</span><span class="n">decide_transactions</span><span class="p">(</span><span class="n">users</span><span class="p">,</span> <span class="n">env</span><span class="p">)</span>
<span class="c1"># New transactions are added to the transaction pool</span>
<span class="n">txpool</span><span class="o">.</span><span class="n">add_txs</span><span class="p">(</span><span class="n">decided_txs</span><span class="p">)</span>
<span class="c1"># The best valid transactions are taken out of the pool for inclusion</span>
<span class="n">selected_txs</span> <span class="o">=</span> <span class="n">txpool</span><span class="o">.</span><span class="n">select_transactions</span><span class="p">(</span><span class="n">env</span><span class="p">)</span>
<span class="n">txpool</span><span class="o">.</span><span class="n">remove_txs</span><span class="p">([</span><span class="n">tx</span><span class="o">.</span><span class="n">tx_hash</span> <span class="k">for</span> <span class="n">tx</span> <span class="ow">in</span> <span class="n">selected_txs</span><span class="p">])</span>
<span class="c1"># We create a block with these transactions</span>
<span class="n">block</span> <span class="o">=</span> <span class="n">Block1559</span><span class="p">(</span><span class="n">txs</span> <span class="o">=</span> <span class="n">selected_txs</span><span class="p">,</span> <span class="n">parent_hash</span> <span class="o">=</span> <span class="n">chain</span><span class="o">.</span><span class="n">current_head</span><span class="p">,</span> <span class="n">height</span> <span class="o">=</span> <span class="n">t</span><span class="p">,</span> <span class="n">basefee</span> <span class="o">=</span> <span class="n">basefee</span><span class="p">)</span>
<span class="c1"># The block is added to the chain</span>
<span class="n">chain</span><span class="o">.</span><span class="n">add_block</span><span class="p">(</span><span class="n">block</span><span class="p">)</span>
<span class="c1"># A couple of metrics we will use to monitor the simulation</span>
<span class="n">row_metrics</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"block"</span><span class="p">:</span> <span class="n">t</span><span class="p">,</span>
<span class="s2">"basefee"</span><span class="p">:</span> <span class="n">basefee</span> <span class="o">/</span> <span class="p">(</span><span class="mi">10</span> <span class="o">**</span> <span class="mi">9</span><span class="p">),</span>
<span class="s2">"users"</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="s2">"decided_txs"</span><span class="p">:</span> <span class="nb">len</span><span class="p">(</span><span class="n">decided_txs</span><span class="p">),</span>
<span class="s2">"included_txs"</span><span class="p">:</span> <span class="nb">len</span><span class="p">(</span><span class="n">selected_txs</span><span class="p">),</span>
<span class="s2">"blk_avg_gas_price"</span><span class="p">:</span> <span class="n">block</span><span class="o">.</span><span class="n">average_gas_price</span><span class="p">(),</span>
<span class="s2">"blk_avg_tip"</span><span class="p">:</span> <span class="n">block</span><span class="o">.</span><span class="n">average_tip</span><span class="p">(),</span>
<span class="s2">"pool_length"</span><span class="p">:</span> <span class="n">txpool</span><span class="o">.</span><span class="n">pool_length</span><span class="p">(),</span>
<span class="p">}</span>
<span class="n">metrics</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">row_metrics</span><span class="p">)</span>
<span class="c1"># Finally, basefee is updated and a new round starts</span>
<span class="n">basefee</span> <span class="o">=</span> <span class="n">update_basefee</span><span class="p">(</span><span class="n">block</span><span class="p">,</span> <span class="n">basefee</span><span class="p">)</span>
<span class="k">return</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">metrics</span><span class="p">),</span> <span class="n">user_pool</span><span class="p">,</span> <span class="n">chain</span><span class="p">)</span>
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<p>As you can see, <code>simulate</code> takes in a <code>demand_scenario</code> array. Earlier we mentioned that each round, we draw the number of users wishing to send transactions from a Poisson distribution. <a href="https://en.wikipedia.org/wiki/Poisson_distribution">This distribution is parameterised by the expected number of arrivals, called <em>lambda</em> $\lambda$</a>. The <code>demand_scenario</code> array contains a sequence of such lambda's. We also provide in <code>UserClass</code> the type of user we would like to model (see the <a href="http://barnabemonnot.com/abm1559/build/html/#users">docs</a> for more details).</p>
<p>Our users draw their <em>value</em> for the transaction (per unit of gas) from a uniform distribution, picking a random number between 0 and 20 (Gwei). Their cost for waiting one extra unit of time is drawn from a uniform distribution too, this time between 0 and 1 (Gwei). The closer their cost is to 1, the more impatient users are.</p>
<p>Say for instance that I value each unit of gas at 15 Gwei, and my cost per round is 0.5 Gwei. If I wait for 6 blocks to be included at a gas price of 10 Gwei, my payoff is $15 - 6 \times 0.5 - 10 = 2$.</p>
<p>The numbers above sound arbitrary, and in a sense they are! They were chosen to respect the scales we are used to (<a href="https://ethereum.github.io/rig/ethdata/notebooks/gas_weather_reports/exploreJuly21.html">although gas prices are closer to 100 Gweis these days...</a>). It also turns out that any distribution (uniform, Pareto, whatever floats your boat) leads to stationarity. The important part is that <em>some</em> users have positive value for transacting in the first place, enough to fill a block to its target size at least. The choice of sample the cost from a uniform distribution, as opposed to having all users experience the same cost per round, allows for <strong>simulating a scenario where some users are more in a hurry than others</strong>.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">demand_scenario</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2000</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="mi">200</span><span class="p">)]</span>
<span class="p">(</span><span class="n">df</span><span class="p">,</span> <span class="n">user_pool</span><span class="p">,</span> <span class="n">chain</span><span class="p">)</span> <span class="o">=</span> <span class="n">simulate</span><span class="p">(</span><span class="n">demand_scenario</span><span class="p">,</span> <span class="n">User1559</span><span class="p">)</span>
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<pre>0
100
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<p>To study the stationary case, we create an array repeating $\lambda$ for as many blocks as we wish to simulate the market for. We set $\lambda$ to spawn on average 2000 users between two blocks.</p>
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<h3 id="Results">Results<a class="anchor-link" href="#Results">¶</a></h3><p>Let's print the head and tail of the data frame holding our metrics. Each row corresponds to one round of our simulation, so one block.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span>
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<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
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vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>block</th>
<th>basefee</th>
<th>users</th>
<th>decided_txs</th>
<th>included_txs</th>
<th>blk_avg_gas_price</th>
<th>blk_avg_tip</th>
<th>pool_length</th>
</tr>
</thead>
<tbody>
<tr>
<th>0</th>
<td>0</td>
<td>1.000000</td>
<td>2031</td>
<td>1562</td>
<td>952</td>
<td>2.000000</td>
<td>1.0</td>
<td>610</td>
</tr>
<tr>
<th>1</th>
<td>1</td>
<td>1.124900</td>
<td>1983</td>
<td>1525</td>
<td>952</td>
<td>2.124900</td>
<td>1.0</td>
<td>1183</td>
</tr>
<tr>
<th>2</th>
<td>2</td>
<td>1.265400</td>
<td>1912</td>
<td>1453</td>
<td>952</td>
<td>2.265400</td>
<td>1.0</td>
<td>1684</td>
</tr>
<tr>
<th>3</th>
<td>3</td>
<td>1.423448</td>
<td>1997</td>
<td>1493</td>
<td>952</td>
<td>2.423448</td>
<td>1.0</td>
<td>2225</td>
</tr>
<tr>
<th>4</th>
<td>4</td>
<td>1.601237</td>
<td>2001</td>
<td>1459</td>
<td>952</td>
<td>2.601237</td>
<td>1.0</td>
<td>2732</td>
</tr>
<tr>
<th>...</th>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
<td>...</td>
</tr>
<tr>
<th>195</th>
<td>195</td>
<td>11.795094</td>
<td>2001</td>
<td>508</td>
<td>508</td>
<td>12.795094</td>
<td>1.0</td>
<td>1924</td>
</tr>
<tr>
<th>196</th>
<td>196</td>
<td>11.893583</td>
<td>2040</td>
<td>459</td>
<td>459</td>
<td>12.893583</td>
<td>1.0</td>
<td>1924</td>
</tr>
<tr>
<th>197</th>
<td>197</td>
<td>11.839914</td>
<td>1979</td>
<td>462</td>
<td>462</td>
<td>12.839914</td>
<td>1.0</td>
<td>1924</td>
</tr>
<tr>
<th>198</th>
<td>198</td>
<td>11.795810</td>
<td>1983</td>
<td>487</td>
<td>487</td>
<td>12.795810</td>
<td>1.0</td>
<td>1924</td>
</tr>
<tr>
<th>199</th>
<td>199</td>
<td>11.829281</td>
<td>1960</td>
<td>444</td>
<td>444</td>
<td>12.829281</td>
<td>1.0</td>
<td>1924</td>
</tr>
</tbody>
</table>
<p>200 rows × 8 columns</p>
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<p>At the start of the simulation we clearly see in column <code>users</code> a demand close to 2000 users per round. Among these 2000 or so, around 1500 decide to send their transaction in (<code>decided_txs</code>). The 500 who don't might have a low value or high per-round costs, meaning it is unprofitable for them to even send their transaction in. Eventually 952 of them are included (<code>included_txs</code>), maxing out the block gas limit. The basefee starts at 1 Gwei but steadily increases from there, reaching around 11.8 Gwei by the end.</p>
<p>By the end of the simulation, we note that <code>decided_txs</code> is always equal to <code>included_txs</code>. By this point, the basefee has risen enough to make it unprofitable for most users to send their transactions. This is exactly what we want! Users balk at the current prices.</p>
<p>In the next chart we show the evolution of basefee and tips. We define <em>tip</em> as the gas price minus the basefee, which is what <em>miners</em> receive from the transaction.</p>
<p>Note that <a href="https://twitter.com/barnabemonnot/status/1284271520311848960">tip is in general <strong>not</strong> equal to the gas premium</a> that users set. This is particularly true when basefee plus gas premium exceeds the max fee of the user. In the graph below, the tip hovers around 1 Gwei (the premium), but is sometimes less than 1 too, especially when users see the prevailing basefee approach their posted max fees.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">df</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="s2">"block"</span><span class="p">,</span> <span class="p">[</span><span class="s2">"basefee"</span><span class="p">,</span> <span class="s2">"blk_avg_tip"</span><span class="p">])</span>
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<pre><AxesSubplot:xlabel='block'></pre>
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