The Fall of Democracy is a Markov Process

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<h1 class=”entry-title”>The Fall of Democracy is a Markov Process</h1><h2 class=’main-sub-title’></h2><div class=’post-summary’>Greek philosophers built theories of political change to explain the rise and fall of their city-states. A modern analysis uses data from 1,035 poleis and Markov simulations to test Polybius’s idea of cyclical government. The findings suggest that stability depends on a balanced mix of monarchy, oligarchy and democracy.</div> <div class=”single-authors”>By <span id=”authorname”></span></div>
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<p>How do democracies die? With thunderous applause? <a href=”https://www.starwars.com/video/so-this-is-how-liberty-dies” target=”_blank” rel=”noreferrer noopener nofollow”>Padme</a> certainly thinks so, but would Polybius agree?</p>

<p>The ancient Greeks were, quite reasonably, concerned by this question, because their democracies died all the time. In fact, this happened so much that the most eminent philosophers and historians of the classical period developed a theory that rationalizes the rise and fall of democracies, oligarchies and tyrannies. In this article, we will investigate whether their theory, called <em>anacyclosis,</em> holds up under scrutiny, and by scrutiny, I mean Monte Carlo simulations of government transition based on historical data from 1,035 Greek city-states. But first, some history.</p>

<h2 class=”wp-block-heading”>The <em>poleis</em> of Ancient Greece</h2>

<p>The peculiarities of classical Greece make empirical theories of political revolution much easier to imagine than in, say, the Persian Empire, which was a hereditary monarchy for pretty much its entire history. The dominant mode of social organization in the archaic and classical Greek periods is the <em>polis, </em>the city-state. Usually, there’s an independent mother city (Athens, Sparta, etc.) that politically, economically and culturally dominates its surrounding hinterland. Each city has its own constitution, or form of government, but shares a common Greek culture and language with its neighboring <em>poleis</em>.</p>

<p>This social structure is as dynamic as it is unstable, and there were <em>many</em> political revolutions. The ancient world’s most sophisticated theories of political evolution grew out of this dynamic — they classify government into a few categories based on which group holds power, and posit that they devolve sequentially from higher to lower forms. Let’s take a quick look.</p>

<h2 class=”wp-block-heading”>Governmental types in Ancient Greek thought</h2>

<p>The <em>Inventory of Archaic and Classical Greek Poleis</em>, about which we will have much more to say later, gives a nice introduction to the types of constitution.</p>

<p style=”padding-left: 50px;”>
In Greek political theory <em>politeiai</em> [political institutions] were divided into types according to how many people constituted and manned the principal organs of government. Basically, there were three constitutional types: the rule of the one, the few and the many. Pindar is the first we know who distinguished between rule by a tyrant, or the wise, or the whole army. About a generation later, Herodotos has a debate about the three basic types of constitution, here described as <em>demos</em>, <em>oligarchia</em> and <em>monarchia</em>. [In the early 4th century BC], Plato called the three forms <em>tyrannis</em>, <em>aristokratia</em> and <em>demokratia</em>.</p>

<h2 class=”wp-block-heading”>Linear evolution in Plato’s <em>Republic</em></h2>

<p>Plato made a finer distinction, dividing government into five categories of constitution in his <a href=”https://en.wikipedia.org/wiki/Republic_(Plato)” target=”_blank” rel=”noreferrer noopener nofollow”><em>Republic</em></a><em>, </em>and additionally giving their sequence of devolution.</p>

<p><strong>Aristocracy</strong> (rule by the best) → <strong>Timocracy</strong> (rule by honor/worth/money) → <strong>Oligarchy</strong> (rule by the few) → <strong>Democracy</strong> (rule by the people/mob) → <strong>Tyranny</strong> (rule by one man).</p>

<p>Plato writes that governments devolve in this order, from best to worst, in a <em>linear</em> fashion, terminating in tyranny. I went back and checked the <em>Republic</em> to see if Plato makes any claims of a cyclical nature, and I don’t <em>think</em> that it does, but the <em>Republic</em> is very hard to read generally, so I’m not 100% sure.</p>

<p>Aristotle, a student of Plato, the tutor of Alexander the Great and a giant of philosophy in his own right, generally agreed with Plato, but distinguished between a good and a bad form of monarchy (<em>basileia</em> versus <em>tyrannis</em>), minority rule (<em>aristokratia</em> versus <em>oligarchia</em>) and majority rule (<em>politeia</em> versus <em>demokratia</em>). His conception, however, was also linear (as best as I understand).</p>

<h2 class=”wp-block-heading”><em>Anacyclosis </em>in Polybius’s <em>Histories</em></h2>

<p>Polybius was a Greek hostage and historian in Rome during Rome’s rise to power, and he improved upon Plato and Aristotle’s framework. Polybius divided government into three categories, each with a virtuous and corrupt form, for a total of six constitutions. They are as follows, from his <a href=”https://en.wikipedia.org/wiki/The_Histories_(Polybius)” target=”_blank” rel=”noreferrer noopener nofollow”><em>Histories</em></a>:</p>

<p>The virtuous aristocracy is corrupted into an oligarchy, which is overthrown by the people as a democracy, which degenerates into mob rule or <em>ochlocracy</em>.<strong><em> </em></strong>A great leader emerges from mob chaos to create a monarchy, which descends into tyranny before being overthrown by the noble aristocracy, beginning the cycle anew. He called this cycle <em>anacyclosis.</em></p>

<figure class=”wp-block-image size-large”><img decoding=”async” width=”1024″ height=”655″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-1-1024×655.jpeg” alt=”” class=”wp-image-159934″ srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-1-1024×655.jpeg 1024w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-1-300×192.jpeg 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-1-150×96.jpeg 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-1.jpeg 1456w” sizes=”(max-width: 1024px) 100vw, 1024px” /><figcaption class=”wp-element-caption”>Anacyclosis as a Diagram.</figcaption></figure>

<p><a href=”https://anacyclosis.org/portfolio/what-is-anacyclosis/” target=”_blank” rel=”noreferrer noopener nofollow”>The Anacyclosis Institute</a> (to whom I must give credit for the genesis of this article), offers the following comments on this process:</p>

<p style=”padding-left: 50px;”>
There is good reason to think that Polybius and his predecessors arrived at this theory empirically. After observing the rise and fall of many hundreds of city-states, most of which cycled through several of the governmental forms mentioned above, Greek political thinkers concluded that these transitions from one form to another were not random. Rather, they seemed to follow simple and recognizable patterns. For example, tyrants were frequently overthrown by groups of aristocrats, while popular revolutions frequently overthrew oligarchies and ushered in democratic rule. Interestingly, the reverse of these trends (aristocracies being overthrown by tyrants or democracies turning into oligarchies) were statistically less likely to occur.</p>

<p style=”padding-left: 50px;”>
Through such observations, Polybius extrapolated the likely complete course of political evolution for an independent state whose lifecycle is not cut short by war or disaster.</p>

<p>Polybius, Plato and Aristotle essentially agree on the pattern — we go from rule-by-few (aristocracy/oligarchy) to rule-by-many (democracy) to rule-by-one (monarchy/tyranny), with an optional cycle back to rule-by-few.</p>

<p>Polybius thinks there’s a way out of this cycle. If one combines all three forms of government into a <em>mixed</em> constitution, a blend of democracy, aristocracy and monarchy, one can create a stable system exempt from <em>anacyclosis. </em>Polybius thought that the Roman Republic was the embodiment of this mixed constitution and the reason for its strength and longevity. The Founding Fathers of the United States of America, and John Adams in particular, were obsessed with Polybius and designed the structure of the United States government to avoid <em>anacyclosis</em>.</p>

<p>So to recap, we have several explicit claims, of which various authors claim subsets:</p>

<ol class=”wp-block-list”>
<li>Political evolution follows a predictable pattern of oligarchy → democracy → monarchy.</li>

<li>This pattern may be linear (Plato) or cyclical (Polybius).</li>

<li>The reverse transitions are unlikely/unnatural.</li>
</ol>

<p>Unfortunately for Polybius, he lacked the tools to quantitatively investigate his theory. Fortunately for us, we are much better than Polybius at linear algebra.</p>

<h2 class=”wp-block-heading”>Political evolution is a Markov process</h2>

<p>Implicit in <em>anacyclosis</em> is actually a fourth claim, the most important claim, that <em>anacyclosis</em> is “memoryless.” In other words, the next type of government <em>depends</em> <em>only on the current type of government</em>: Democracies always devolve to tyrannies, independent of what preceded democracy. In the theory of stochastic processes, this property is called the <a href=”https://en.wikipedia.org/wiki/Markov_property” target=”_blank” rel=”noreferrer noopener nofollow”>Markov property</a>. We can use the Markov property to evaluate the validity of Polybius’ claim.</p>

<p>First, though, we need data. Fortunately, the Copenhagen Polis Centre has done most of the work for us, and <a href=”https://global.oup.com/academic/product/an-inventory-of-archaic-and-classical-poleis-9780198140993?cc=us&lang=en&” target=”_blank” rel=”noreferrer noopener nofollow”>compiled</a> <em>An Inventory of Archaic and Classical Poleis</em>, a monumental work that compiles the existing data/metadata on the 1,035 identifiable Greek city-states of the Archaic and Classical periods (c.650-325 BC). Among the data found in the <em>Inventory </em>is a list of city-states and their known government types, ordered by date.</p>

<figure class=”wp-block-image size-large”><img decoding=”async” width=”1024″ height=”655″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-3-1024×655.jpeg” alt=”” class=”wp-image-159944″ srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-3-1024×655.jpeg 1024w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-3-300×192.jpeg 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-3-150×96.jpeg 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-3.jpeg 1206w” sizes=”(max-width: 1024px) 100vw, 1024px” /><figcaption class=”wp-element-caption”>One of the appendices from the <em>Inventory.</em></figcaption></figure>

<p>This data is actually all we need in order to pretty fairly evaluate the validity of the theory! For each city-state, we can simply extract ordered pairs of government types from this list and count the frequency with which these transitions occur. Because <em>anacyclosis</em> is a Markov chain (remember that means memoryless), these transition frequencies completely define the system! Note that this method completely ignores <em>staying in the same state</em> as a transition of interest (which requires much more sophisticated data parsing). So this method will probe <em>only </em>when governments change between distinct types.</p>

<p>This will make more sense as we actually construct the transition matrix and learn how to analyze Markov processes more generally.</p>

<h2 class=”wp-block-heading”>An introduction to Markov processes through the inventory of Greek <em>poleis</em></h2>

<p>A Markov process, named after the Russian mathematician <a href=”https://en.wikipedia.org/wiki/Andrey_Markov” target=”_blank” rel=”noreferrer noopener nofollow”>Andrey Markov</a>, is a type of random process. It has discrete states and a notion of time. At each time step in the process, each state <em>X </em>has a probability <em>P(X→Y) </em>of transitioning to state <em>Y. </em>The Markov (memoryless) property ensures that this probability is the <em>only</em> relevant characteristic of the system.</p>

<p>Markov processes are often represented by graphs that abstract these transition probabilities, like the one below from Wikipedia showing a two-state Markov process.</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”260″ height=”260″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-17.png” alt=”” class=”wp-image-159946″ style=”width:831px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-17.png 260w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-17-150×150.png 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-17-96×96.png 96w” sizes=”auto, (max-width: 260px) 100vw, 260px” /><figcaption class=”wp-element-caption”>A Markov process with two states, A and E. A transitions to A with probability P(A→A) = 0.6, and to E with probability P(A→E) = 0.4. E transitions to E with probability P(E→E) = 0.3, and to A with probability P(E→A) = 0.7</figcaption></figure>

<p>In essence, a Markov process is the simplest form of a probabilistic state machine that still has interesting behavior. For <em>any</em> process that can be assumed to be stochastic, or perhaps a system complex enough that its behavior approximates a stochastic process, we can model it as a Markov process and immediately extract nontrivial, useful properties (as we shall see later).</p>

<p>Processes that (approximately) have the Markov property show up everywhere. Weather prediction, stock price prediction and population genetics are all examples of approximately-Markov processes. In each of these cases, while the real system may have complex dependencies, a Markov model captures enough of the important behavior to be very useful while still being mathematically tractable.</p>

<p>So how would we construct a Markov model for political evolution in Greek <em>poleis</em>? Like any good scientists, the first thing we have to do is create a good visualization of our data, and stare at it.</p>

<p>Looking at our dataset, we have six distinct types of constitutions listed by the <em>Inventory.</em> I’ll quote briefly from it here:</p>

<p style=”padding-left: 50px;”>
In the <em>Inventory</em>, when we classify the constitution of a polis, we distinguish between <em>basileia</em>, <em>tyrannis</em>, <em>oligarchia</em> and <em>demokratia</em>, but we ignore variants of the latter two types, and all attestations of <em>basileia</em> belong in the Archaic period…</p>

<p style=”padding-left: 50px;”>
In a few cases of serious doubt, we have used <em>Mix.</em> to describe a polis with an unidentifiable mixture of characteristics.</p>

<p>The inventory also has another category in the data not listed above, <em>politeia, </em>which Aristotle defined as the “good” form of democracy, but is also the general term for a “polity” in Greek. Both of these last forms, <em>politeia </em>and “mixed,” are very rare in the <em>Inventory</em> and slightly confusing.</p>

<p>We should also note that the <em>Inventory </em>says that the term <em>basileia </em>might change meaning over time, as it is only attested in the Archaic period and not the Classical period. Second, that “in actual fact, all polis constitutions were mixed,” to one degree or another.</p>

<p>But for a first cut, let’s ignore these complexities and take a look at the data. First, the total frequency of government types:</p>

<figure class=”wp-block-image size-full”><img loading=”lazy” decoding=”async” width=”800″ height=”600″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-21.png” alt=”” class=”wp-image-159950″ srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-21.png 800w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-21-300×225.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-21-150×113.png 150w” sizes=”auto, (max-width: 800px) 100vw, 800px” /><figcaption class=”wp-element-caption”>Total Frequency of Government Types Chart.</figcaption></figure>

<p>The total counts of each type of government in the inventory. I manually removed “klerouchy” and “dynasteia”, additionally (Pol.) in the very first line, because I’m unclear what it means. Each of these occurred only once.</p>

<p>We note that the mixed and <em>politeia</em> types are very rare and not likely to affect our results. Excellent! Let’s ignore them. Second, if we combine <em>basileia </em>and <em>tyrannis</em>, the constitutions are roughly equal in frequency between the autocratic, oligarchic and democratic categories. Interesting! We’ll keep the two types of monarchy separate for now.</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”800″ height=”800″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-20.png” alt=”” class=”wp-image-159949″ style=”width:840px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-20.png 800w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-20-300×300.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-20-150×150.png 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-20-96×96.png 96w” sizes=”auto, (max-width: 800px) 100vw, 800px” /><figcaption class=”wp-element-caption”>Frequency of occurrence for initial and final constitutional types</figcaption></figure>

<p>Next, we can plot the frequency that any constitutional type appears first or last. If Plato is correct, we would expect to see lots of oligarchies initially and lots of tyrannies finally. We … might see some evidence for that? We really see more of a transfer between oligarchy and democracy from this graph, and the number of monarchies slightly decreases, but not a lot. I don’t think Plato gets much help from the data here.</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”800″ height=”800″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-23.png” alt=”” class=”wp-image-159952″ style=”width:840px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-23.png 800w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-23-300×300.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-23-150×150.png 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-23-96×96.png 96w” sizes=”auto, (max-width: 800px) 100vw, 800px” /><figcaption class=”wp-element-caption”>The frequency of occurrence of transitions between constitutional types. Note that there are some self → self transitions, which appear infrequently in the data, likely due to poor or incomplete records recovered of an intervening government type.</figcaption></figure>

<p>Finally, we can plot the frequency of government transition types. We define this naively, taking the sequence of governments in the inventory, and plotting the frequency of each ordered pair, <em>i.e., demokratia, oligarchia, basileia</em> would count as one occurrence each of <em>dem.→ol., ol.→bas..</em></p>

<p>This last plot is pretty much the key to Markov processes. We can simply reinterpret each column of the above heatmap as a <em>probability of transition between states</em>. Thus, by normalizing each column of the transition frequency heatmap, we get a <em>transition matrix</em> <em>T</em> that defines the process. Because our system has the Markov (memoryless) property, the single-step transition matrix <em>entirely</em> defines the process — it is a Markov process.</p>

<p>How does the model work in practice? Let’s enumerate the types of government as the ordered list:</p>

<p>Then we can define a one-hot vector <em>v </em>that corresponds to my state, <em>e.g.,</em> [1, 0, 0, 0, 0, 0] = the system is in the <em>basileia</em> state. The probability of transitioning to any other state is then given by a vector <em>p</em>, equal to the transition matrix <em>T </em>times <em>v.</em></p>

<p>Let’s explicitly construct <em>T </em>for our data. Since “mixed” and <em>politeia</em> are ill-defined and occur so infrequently, I feel fairly justified in simply dropping those columns from the data.</p>

<p>From this matrix, we can then recreate the Markov-process-style node graph.</p>

<h2 class=”wp-block-heading”>Assessing validity</h2>

<p>We’re now in a position to partially assess the validity of <em>anacyclosis </em>as it relates to the data, in a first-order sort of fashion.</p>

<p>The first claim is that political evolution proceeds in the order oligarchy → democracy → monarchy. Our data is fine-grained enough that we can split monarchy into its “virtuous” and “corrupt” forms, <em>basileia</em> and <em>tyrannis</em>, and so let’s look for oligarchy → democracy → <em>basileia </em>→ <em>tyrannis </em>in the data.</p>

<p>Looking first at the <em>basileia</em> → <em>tyrannis</em> transition, we actually find excellent support for this in the data! <em>Basileia</em> to <em>tyrannis</em> transitions happen about 13 times more often than the reverse. However, there’s a confounding variable. Remember that the <em>Inventory</em> says that <em>basileia </em>is attested only in the Archaic period, so any <em>basileia</em> → <em>tyrannis</em> transitions might be the result of redefinition as opposed to transition. Let’s call this one a partial thumbs up, though.</p>

<p>How about the posited oligarchy → democracy transition? Not so much. These two nodes have the tightest connection in the graph, and the transition rates are essentially even, with oligarchy to democracy being ever so slightly more favorable than the reverse.</p>

<p>Finally, what about democracy → monarchy? It seems as though democracies don’t <em>ever</em> go to <em>basileiai</em>, which certainly doesn’t <em>support</em> the theory of <em>anacyclosis, </em>but again, this could be a definition thing — if we started the chain at <em>basileia</em>, we could have passed through the Archaic period before we got back around, when <em>basileiai </em>had turned into <em>tyrannides</em>. Unfortunately, though, the democracy → tyranny transition (29%) is much less common than tyranny → democracy (56%). This gets even worse if we consider <em>basileia</em> the same thing as <em>tyrannis</em>, which has even more asymmetry between the two transition frequencies. So the democracy to monarchy transition doesn’t find much support here; in fact, more the reverse.</p>

<p>In fact, what <em>is</em> the most common cycle? Let’s re-plot the Markov chain where we combine <em>tyrannis</em> and <em>basileia</em> into “monarchy.”</p>

<figure class=”wp-block-image size-large”><img loading=”lazy” decoding=”async” width=”1024″ height=”723″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-4-1024×723.jpeg” alt=”” class=”wp-image-159953″ srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-4-1024×723.jpeg 1024w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-4-300×212.jpeg 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-4-150×106.jpeg 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-4.jpeg 1074w” sizes=”auto, (max-width: 1024px) 100vw, 1024px” /><figcaption class=”wp-element-caption”>Markov chain plot with <em>basileia</em> and <em>tyrannis</em> combined into monarchy.</figcaption></figure>

<p>This doesn’t really help us much. The most plausible cycle by far is simply oscillation between democracy and oligarchy, which does not at <em>all</em> fit into the <em>anacyclosis</em> paradigm. It seems we require a more sophisticated analysis to extract the probable dynamics.</p>

<h2 class=”wp-block-heading”>Markov chain Monte Carlo</h2>

<p>Okay, well then, what <em>does</em> happen to a hypothetical average Greek polis? We can use a Monte-Carlo style simulation to find out.</p>

<p>“Monte Carlo” is a cute name for a very simple technique — if you have the rules of a system and want to understand its behavior, just simulate a whole bunch of random instances of that system and average the results. The simulation method is called “Monte Carlo” because one of the inventors <a href=”https://en.wikipedia.org/wiki/Monte_Carlo_method#History” target=”_blank” rel=”noreferrer noopener nofollow”>had</a> an uncle who gambled too much in real Monte Carlo.</p>

<p>Nevertheless, this simple technique is extremely powerful. To implement, we</p>

<ol class=”wp-block-list”>
<li>Choose an initial one-hot state <em>v.</em></li>

<li>Multiply by our transition matrix to get <em>p = Tv,</em> our vector of state probabilities.</li>

<li>Choose a random state from <em>p</em>, weighted by the probabilities of each state, <em>i.e.,</em> if <em>p = [0.1, 0.2, 0.3, 0.4], </em>we have a 10%, 20%, 30% and 40% probability, respectively, of choosing states 1, 2, 3, 4.</li>

<li>Repeat this for <em>n</em> timesteps.</li>

<li>Repeat steps 1-4 for <em>m</em> simulations.</li>
</ol>

<p>Let’s try this out, keeping the <em>basileia/tyrannis</em> distinction, just for fun.</p>

<figure class=”wp-block-image size-large”><img loading=”lazy” decoding=”async” width=”683″ height=”1024″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-5-683×1024.jpeg” alt=”” class=”wp-image-159955″ srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-5-683×1024.jpeg 683w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-5-200×300.jpeg 200w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-5-150×225.jpeg 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-5.jpeg 800w” sizes=”auto, (max-width: 683px) 100vw, 683px” /><figcaption class=”wp-element-caption”>Markov Chain Monte-Carlo Anacyclosis Analysis. I run 1000 separate simulations with 20 steps each, and plot the fraction of state occupancy at each timestep, along with a 95% confidence interval. The confidence interval is the range in which 95% of averaged occupancy values are expected to fall.</figcaption></figure>

<p>So keeping <em>basileia</em> as a separate category doesn’t do much, its occupancy fraction immediately hits zero (on average) and never recovers. The most distinct feature by far appears to be the oligarchy-democracy oscillation, settling after ten timesteps into an even mixture of democracy and oligarchy (remember this is an <em>average; </em>at each timestep, the system can only be in one state). The <em>tyrannis</em> initialization appears to cause the settling to happen faster, but doesn’t differ in the essential trend. We <em>also</em> appear to stabilize at a steady state for any initialization parameter! Perhaps this is the fabled “mixed” constitution that Polybius thought made the Roman state so powerful and stable? We shall formalize this thought later.</p>

<p>Before we do steady-state analysis, we should check for common cycles. Let’s plot the most common cycles we find, dropping the <em>basileia/tyrannis</em> distinction (both are monarchy) for clarity. 1000 more simulations…</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”780″ height=”483″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-16.png” alt=”” class=”wp-image-159945″ style=”aspect-ratio:1.6149267251140371;width:840px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-16.png 780w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-16-300×186.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-16-150×93.png 150w” sizes=”auto, (max-width: 780px) 100vw, 780px” /><figcaption class=”wp-element-caption”>The most common cycles in 1000 simulations of 20 timesteps, with basileia and tyrannis condensed to a single category, monarchy.</figcaption></figure>

<p>The most common cycle by far is the 2-state democracy → oligarchy → democracy cycle. The next most common state is this same cycle, twice in a row!</p>

<p>But wait, if we look down the list at the fourth most common cycle, it’s <strong>democracy → monarchy → oligarchy → democracy</strong>, that’s <em>anacyclosis</em>! It is the most common three-state cycle! We found it!</p>

<p>Did we just prove <em>anacyclosis</em> is real? Well, uh, it depends on what you mean, I guess.</p>

<p>Instead of answering the above, difficult question, I choose to reinterpret the original theory in light of the data — when Polybius wrote that “<em>anacyclosis </em>consists of predictable, cyclic transitions from democracy → monarchy→ oligarchy<em>,”</em> he clearly must have meant that given the empirical transition probabilities derived from a Markov model of Greek city state constitutional data, a Monte Carlo simulation will show the most common three-state cycle is democracy → monarchy → oligarchy.</p>

<p>I think this is a very reasonable translation of the original Ancient Greek.</p>

<h2 class=”wp-block-heading”>Mixed states and the stable distribution</h2>

<p>Now, Polybius was particularly interested in ways <em>out</em> of this endless cycle. How can we find a stable governmental state? Both our answer and Polybius’ answer are the same, and are already hinted at by the results of the simulations. We noted earlier that the simulations, regardless of initialization state, seemed to settle into a predictable distribution of government types, roughly 40% oligarchy, 40% democracy and 20% tyranny. This was also Polybius’ answer, that a mixed state was a stable point of this Markov process.</p>

<p>Polybius took as his example par excellence the constitution of the Roman Republic, which had popular assemblies (democracy), the Senate (oligarchy), and two consuls with executive power (a dash of monarchy). He felt that this mixture was much more stable than any pure state, and lent Rome its fabulous power.</p>

<p>According to our model, Polybius is absolutely correct.</p>

<p>How can we find the stable state of our Markov process? Well, first, we extend the model to allow for mixed states by not forcing our state <em>v</em> to be one-hot. That’s fairly easy. But how do we find the long-term stable state, if there is one?</p>

<p>Let’s think geometrically about our transition matrix. For a 2 × 2 transition matrix, we can visualize its action by seeing how it transforms a set of vectors arranged along the unit circle in the plane.</p>

<figure class=”wp-block-image size-full”><img loading=”lazy” decoding=”async” width=”977″ height=”389″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-14.png” alt=”” class=”wp-image-159942″ srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-14.png 977w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-14-300×119.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-14-150×60.png 150w” sizes=”auto, (max-width: 977px) 100vw, 977px” /><figcaption class=”wp-element-caption”>The action of a linear transformation on the unit circle</figcaption></figure>

<p>Under this linear transformation, the unit circle becomes an ellipse. The special directions that remain unchanged (up to scaling) by this transformation are called eigenvectors. These are precisely the principal axes of the resulting ellipse.</p>

<p>Mathematically, an eigenvector v with eigenvalue λ is defined by the equation:</p>

<p>Where <em>T </em>is our transition matrix, but this is very helpful for us, because if I now apply <em>T</em> twice, I get:</p>

<p>Thus, when we apply T repeatedly, each application multiplies the magnitude of <em>v</em> by λ while preserving its direction.</p>

<p>This means:</p>

<ul class=”wp-block-list”>
<li>If λ = 1, <em>v</em> maintains its magnitude: it’s a stable state.</li>

<li>If |λ| > 1, <em>v</em> grows without bound.</li>

<li>If |λ| < 1, <em>v</em> shrinks toward zero.</li>
</ul>

<p>For Markov transition matrices, the <a href=”https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem” target=”_blank” rel=”noreferrer noopener nofollow”>Perron-Frobenius theorem</a> guarantees that 1 is <em>always</em> an eigenvalue, and all other eigenvalues have absolute value strictly less than 1, meaning they decay to zero after a long time. When the Markov chain is also <a href=”https://gregorygundersen.com/blog/2019/10/28/ergodic-markov-chains/” target=”_blank” rel=”noreferrer noopener nofollow”>ergodic</a>, meaning that you can visit any state from any other state, and you never get stuck in deterministic cycles, this unit-eigenvalue direction corresponds to a unique stable distribution called the <a href=”https://en.wikipedia.org/wiki/Discrete-time_Markov_chain#Stationary_distributions” target=”_blank” rel=”noreferrer noopener nofollow”><em>stationary distribution</em></a><em>.</em></p>

<p>Any initial distribution will converge to this stationary distribution as we repeatedly apply the transition matrix. We actually saw this in our Monte Carlo simulations — did you notice how, no matter the initial state, we always ended up with the same fraction of oligarchy/democracy/monarchy?</p>

<p>So we can quite quickly find the stationary distribution of our transition matrix by performing an eigendecomposition of our transition matrix. We solve:</p>

<p>Let’s take our transition matrix, where we combine <em>basileia</em> and <em>tyrannis</em> into monarchy.</p>

<p>The eigendecomposition of <em>T</em> yields three eigenvectors.</p>

<p>Which does indeed have a unit-valued eigenvector! To find the stationary distribution, we only need to normalize the eigenvector with eigenvalue 1, by dividing it by its column sum (it’s a probability vector, remember).</p>

<p>We can visualize this final distribution with a bar graph. Bar graphs are the most useless graph type, but they are visually arresting due to large bars of solid primary colors, so I’m making one.</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”727″ height=”579″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-10.png” alt=”” class=”wp-image-159937″ style=”aspect-ratio:1.2556539105089868;width:837px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-10.png 727w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-10-300×239.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-10-150×119.png 150w” sizes=”auto, (max-width: 727px) 100vw, 727px” /><figcaption class=”wp-element-caption”>The stationary distribution of the transition matrix when <em>basileia </em>and <em>tyrannis</em> are combined into monarchy.</figcaption></figure>

<p>This graph shows the <em>anacyclotically stable distribution </em>of the <em>Markov poleis model. </em>I don’t know if the word <em>anacyclotically</em> will ever really catch on in popular discourse, but I think it really rolls off the tongue. Perhaps instead we should call it the <em>Polybian distribution. </em>It does look remarkably similar to the Roman system, which was deeply suspicious of kingship but recognized its utility, and hence had two equal consuls in the place of one tyrant, as well as theoretically balanced popular assemblies and an aristocratic/oligarchic Senate.</p>

<p>So that’s it! This is the final confirmation that Polybius was on the right track, and that if he had only been better at linear algebra, he could have quantitatively estimated the proportion of democracy, monarchy, and oligarchy to inject into a <em>politeia</em> to stabilize it against the inevitable <em>anacyclosis</em>, assuming of course that by <em>anacyclosis</em> he actually meant the stochastic Markov process we’ve been working with this whole time, and not the actual anacyclosis that he wrote down, which is given minimal support by the actual data. Easy!</p>

<p>I should note, for future work, that there is at least one major oversight — for any given year in even the most fractious Greek polis, the probability of government transitioning to an entirely new category is small. In other words, I could have structured this process around a timestep being a single year, instead of an arbitrary “government transition time,” and gotten very different-looking processes, with the same long-run transition probabilities. Oh well, you always have to leave work for the next researcher.</p>

<h2 class=”wp-block-heading”>Addendum: methodological validity</h2>

<p>Mere minutes after posting this article, I had a thought — is the Greek <em>poleis</em> data set even capable of detecting an <em>anacyclosis</em> cycle in principle?</p>

<p>Let’s say I have a sequence A → B → C → A → …, like our <em>poleis</em> dataset. Then, because of spotty recordkeeping, let’s say I decimate this sample by randomly deleting entries, so maybe I’d get A → B → __ → A → … Without knowledge of the original sample, there’s a spurious B → A transition in our data!</p>

<p>So the question I want to ask here is, given a sequence composed of a pure cycle <em>S </em>= A → B → C → A …, if I randomly sample this sequence by throwing away all but a fraction <em>f</em> of the data points, can I still detect my sequence above noise?</p>

<p>Formally, let’s ask the question in the following way: I sample a fraction <em>f</em> of the data points from my sequence <em>S,</em> and construct a Markov transition matrix <em>T </em>by naively measuring transition frequency between neighbors in my sampled sequence. How often will I measure that the probability of the original sequence is greater than that of the reverse sequence? This is a sensible definition of “noise” because if we are solely interested in three-element sequences with unique elements, there are exactly two, A→B→C and C→B→A.</p>

<p>This sounds like an interesting analytical problem, but keeping with the Monte Carlo theme of this article and my own laziness, let’s just try it in code. I’ll construct a sequence of length 100 (A→B→C→…), decimate it, keeping a fraction <em>f</em> of the data, construct my matrix <em>T</em>, and then check whether the original cycle probability exceeds the reverse cycle probability.</p>

<p>The results are as follows.</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”600″ height=”600″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-15.png” alt=”” class=”wp-image-159943″ style=”width:835px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-15.png 600w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-15-300×300.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-15-150×150.png 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-15-96×96.png 96w” sizes=”auto, (max-width: 600px) 100vw, 600px” /><figcaption class=”wp-element-caption”>Probability of detecting A → B → C cycle in the 100-element sequence A → B → C, sampled with fraction <em>f</em> . 1000 simulations.</figcaption></figure>

<p>And for completeness, does this give sensible results when run it on the reverse sequence C → B → A? In other words, what’s the spurious detection rate for a sequence that doesn’t contain the cycle at all?</p>

<figure class=”wp-block-image size-full is-resized”><img loading=”lazy” decoding=”async” width=”600″ height=”600″ src=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-22.png” alt=”” class=”wp-image-159951″ style=”width:839px;height:auto” srcset=”https://www.fairobserver.com/wp-content/uploads/2025/12/image-22.png 600w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-22-300×300.png 300w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-22-150×150.png 150w, https://www.fairobserver.com/wp-content/uploads/2025/12/image-22-96×96.png 96w” sizes=”auto, (max-width: 600px) 100vw, 600px” /><figcaption class=”wp-element-caption”>Probability of detecting A → B → C cycle in the 100-element sequence C → B → A, sampled with fraction <em>f</em> . 1000 simulations.</figcaption></figure>

<p>Yes. This test does indeed give sensible results.</p>

<p>How about a random sequence? What do we expect on average?</p>

<p>Yes, this also makes sense. For a random sequence, we detect that A→B→C is more probable than the reverse about 50% of the time. I think this is a <a href=”https://en.wikipedia.org/wiki/Type_III_error” target=”_blank” rel=”noreferrer noopener nofollow”>“type III” error</a>, where I correctly detect that the probability of my sequence is higher than the reverse sequence (the noise), but this doesn’t mean anything, because the underlying generator is fully random.</p>

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<p>Based on this, I think I can say that the cutoff point where this detection mechanism starts to work is when the probability of detection exceeds ~50% on that curve, so let’s say a sampling fraction of about <em>f = 0.2</em>. That’s not too bad. I think I can consider my method valid enough for a Substack article.</p>

<p><em>[Dylan Black first published this piece on </em><a href=”https://maximumeffort.substack.com/p/the-fall-of-democracy-is-a-markov” target=”_blank” rel=”noreferrer noopener nofollow”><em>Substack</em></a><em>.]</em></p>

<p><em>[</em><a href=”https://www.linkedin.com/in/kaitlyn-diana-aab997278/” target=”_blank” rel=”noreferrer noopener nofollow”><em>Kaitlyn Diana</em></a><em> edited this piece.]</em></p>

<p><strong>The views expressed in this article are the author’s own and do not necessarily reflect Fair Observer’s editorial policy.</strong></p>
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commentmaxcount : 1500,
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commentmincount : 1,
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wc_be_the_first_text: “Be the First to Comment!”,
wc_comment_join_text: “Join the discussion”,
theme: ‘snow’,
debug: ‘error’
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/* .popup_loginmodel .dm-fo-login-box {
max-width: 420px !important
} */
</style>

jQuery(‘.close-popup-btn-signup’).click(function () {
jQuery(‘#myModalPopUpsignup’).hide();

});

jQuery(‘#myBtn , .sign_up_heaer’).click(function () {
jQuery(‘#myModalPopUpsignup’).show();
});

var path = location.pathname;
var popup_not_allowed_in_paths = [“/fo/donate/”, “/fo/support-fo/”, “/fo/membership-account/membership-levels/”,
“/fo/membership-account/membership-checkout/?level=2”, “/fo/membership-account/membership-checkout/?level=1”,
“/fo/contact/”, “payment-success”
];
var isPopupOnPathAllowed = true;
if (popup_not_allowed_in_paths.includes(path)) {
isPopupOnPathAllowed = false;
}

if (jQuery.cookie(‘whenToShowDialog’) != “yesss” && isPopupOnPathAllowed) {
jQuery(window).scroll(function () {
var available, percentage_of_page, half_screen, height;
available = jQuery(document).height();
percentage_of_page = 0.25;
half_screen = available * percentage_of_page;
height = jQuery(window).scrollTop();
if (height > half_screen) {

if (jQuery.cookie(‘whenToShowDialog’) != “yesss”) {
jQuery(“#myModalPopUp”).css(‘display’, ‘flex’);
jQuery.cookie(“whenToShowDialog”, “yesss”);
}
}
});
} else {
jQuery(‘#myModalPopUp’).hide();
}

var dwnldbtn = jQuery(“#download_product_popup_open”);
dwnldbtn.click(() => {
console.log(‘be’);
jQuery(‘#download_product_popup’).show();
});

jQuery(‘#download_product_popup_open’).click(function () {
console.log(‘hellof’);
jQuery(‘#download_product_popup’).fadeIn();
});
jQuery(‘.download_product_popup_close’).click(function () {
jQuery(‘#download_product_popup’).fadeOut();
});
jQuery(‘.download_product_popup_close’).click(function () {
jQuery(‘#download_product_popup’).fadeOut();
});

var container = jQuery(“#download_product_popup”);
container.click((e) => {
if (e.target.id == ‘download_product_popup’) {
jQuery(‘#download_product_popup’).hide();
}
});

jQuery(‘.close-popup-btn’).click(function () {
// $.cookie(‘whenToShowDialog’, “yesss”);
display(“whenToShowDialog”, “yesss”);
jQuery(‘#myModalPopUp’).hide();
});

jQuery(‘.other_donation_amount’).keyup(function (e) {
var $otherRadio = jQuery(‘#other’);
var $otherAmountSection = jQuery(‘.other_donation_amount_section’);

$otherAmountSection.show();
$otherRadio.prop(‘checked’, true);

var value = jQuery(this).val();
jQuery(‘#donation_amount’).html(‘$’ + value);
$otherRadio.val(value);
jQuery(‘#amount’).val(value);
});

jQuery(‘#other’).change(function () {
if (jQuery(this).is(‘:checked’)) {
jQuery(‘.other_donation_amount_section’).show();
} else {
jQuery(‘.other_donation_amount_section’).hide();
}
});
});

<!– –>
jQuery(document).click(function () {

/* login popup */
jQuery(‘#closebtnlogin’).click(function () {
jQuery(“#loginmodal”).hide();
});
var logincontainer = jQuery(“.dm-fo-login-box”);
if (!logincontainer.is(event.target) && !logincontainer.has(event.target).length) {
jQuery(“#loginmodal”).hide();
}
var loginbtn = jQuery(“.fo-login-link”);
if (!loginbtn.is(event.target) && !loginbtn.has(event.target).length) {

} else {
var loginModal = jQuery(“#loginmodal”);
loginModal.find(‘.popup-form input’).val(”); // Clear all input values (optional)
loginModal.find(‘.popup-form’).hide(); // Hide all forms within #loginmodal
loginModal.find(‘.error-message’).remove(); // Remove all error messages within #loginmodal
loginModal.find(‘.input-error’).removeClass(‘input-error’); // Remove error class within #loginmodal
loginModal.find(‘#login-form’).show(); // Show the target form within #loginmodal

loginModal.show(); // Show the login modal
}

/* signupbutton popup */
var signupcontainer = jQuery(“#modal-content-signup”);
if (!signupcontainer.is(event.target) && !signupcontainer.has(event.target).length) {
jQuery(“#myModalPopUpsignup”).hide();
}
var signupcontainerbtn = jQuery(“#myBtn”);
if (!signupcontainerbtn.is(event.target) && !signupcontainerbtn.has(event.target).length) {

} else {
jQuery(“#myModalPopUpsignup”).css(‘display’, ‘flex’);
}

/* signupbutton onscroll popup */
var container = jQuery(“#modal-content-scroll”);
if (!container.is(event.target) && !container.has(event.target).length) {
var displayval = jQuery(‘#myModalPopUp’).css(‘display’);
if (displayval == ‘flex’) {
jQuery(“#myModalPopUp”).hide();
display(“whenToShowDialog”, “yesss”);
}
}

jQuery(‘#other’).click(function () {
jQuery(‘.other_donation_amount_section’).show();
});

jQuery(‘.other_donation_amount’).keyup(function (e) {
var activeElement = document.getElementsByClassName(“donation-amount-box”)[0].getElementsByClassName(“active”)[0];
if (activeElement) {
activeElement.classList.remove(‘active’);
}

var value = $(‘.other_donation_amount’).val();
jQuery(‘#other’).val(value);
jQuery(‘#donation_amount’).html(‘$’ + value);
jQuery(‘#amount’).val(value);
jQuery(‘#paypalamount’).val(value);
jQuery(‘#monthly’).val(value);
})
});

jQuery(document).ready(function () {
jQuery(‘.fo-login-link.btn-popup-footer a’).on(‘click’, function (e) {
e.preventDefault(); // Prevent the default behavior of the link

jQuery(‘.btn-popup-footer-login’).addClass(‘pop-active’); // Add the custom class
});
});

jQuery(document).ready(function () {
jQuery(‘.btn-popup-footer-login-close’).on(‘click’, function () {
jQuery(‘.popup_loginmodel.btn-popup-footer-login’).removeClass(‘pop-active’);
});
});

jQuery(document).ready(function ($) {
jQuery(“.total_counts h1”).click(function () {
jQuery(“.fo-search-popup”).addClass(“active”);
});
});

jQuery(document).ready(function () {
jQuery(‘.tml-pwd-wrap’).append(‘<div id=”show_hide_div” class=”togglepwd” onclick=”showPassword(event)”><i class=”bi bi-eye-slash-fill”></i></div>’);
jQuery(‘meta[property=”og:image:height”]’).attr(“content”, 400);
jQuery(‘meta[property=”og:image:width”]’).attr(“content”, 400);

//Open Bookmark Model
jQuery(‘#screen, #modal’).hide();
jQuery(‘#trigger-modal ,#trigger-modal2’).click(function () {
jQuery(‘#modal’).show();
});
jQuery(‘.x’).click(function () {
jQuery(‘#screen, #modal’).hide();
});

window.addEventListener(‘click’, function (e) {
if (document.getElementById(‘modal’).contains(e.target)) {
jQuery(“#modal”).hide();
} else { }
});

});

function showPassword1() {

var pass = document.getElementById(“upassword”);
if (pass.type === “password”) {
pass.type = “text”;
document.getElementById(“show_hide_div_reg1”).innerHTML = “Hide”;
} else {
pass.type = “password”;
document.getElementById(“show_hide_div_reg1”).innerHTML = “Show”;
}
}

function showPassword2() {

var pass = document.getElementById(“upasswordconfirm”);
if (pass.type === “password”) {
pass.type = “text”;
document.getElementById(“show_hide_div_reg2”).innerHTML = “Hide”;
} else {
pass.type = “password”;
document.getElementById(“show_hide_div_reg2”).innerHTML = “Show”;
}
}

jQuery(“.emaillink”).click(function (e) {
e.preventDefault();
jQuery(“#contact_us_form”).css(‘display’, ‘flex’);
});

jQuery(‘#contact_us_form’).click((e) => {
if (e.target.id === “contact_us_form”) {
jQuery(“#contact_us_form”).css(‘display’, ‘none’);
}
});

</script>

<!– Scroll footer fixed –>

jQuery(function ($) {
$(window).scroll(function () {
var scrollOffset = $(window).scrollTop();
if (scrollOffset > 400) { // change 100 to the desired scroll offset
$(‘.fixed-footer-newsletter’).addClass(‘active’);
} else {
$(‘fixed-footer-newsletter’).removeClass(‘none’);
}
});
$(document).ready(function () {
$(“.up-down-toggle-btn svg”).click(function () {
$(“.fixed-footer-newsletter”).toggleClass(“hide”);
if ($(“.fixed-footer-newsletter”).hasClass(‘hide’)) {
localStorage.setItem(‘hideClass’, ‘true’);
} else {
localStorage.removeItem(‘hideClass’);
}
});

var hideClass = localStorage.getItem(‘hideClass’);
if (hideClass === ‘true’) {
$(‘.fixed-footer-newsletter’).addClass(‘hide’);
}

});
});
</script>

<!– new custom script –>

<!– author slider –>

<!– FO Newsletter –>

jQuery(function($){
$(“.newsletters-click”).on(“click”, function (e) {
$(this).addClass(‘active’);
});

$(‘#amount_50’).click(function () {
$(‘.amount’).removeAttr(‘checked’);
$(‘#other_pay’).val(0);
$(‘#other_amount_pay’).val(0);
$(this).attr(‘checked’, ‘checked’)
});

$(‘#amount_100’).click(function () {
$(‘.amount’).removeAttr(‘checked’);
$(‘#other_pay’).val(0);
$(‘#other_amount_pay’).val(0);
$(this).attr(‘checked’, ‘checked’)
});

$(‘#amount_500’).click(function () {
$(‘.amount’).removeAttr(‘checked’);
$(‘#other_pay’).val(0);
$(‘#other_amount_pay’).val(0);
$(this).attr(‘checked’, ‘checked’)
});

$(‘#amount_1000’).click(function () {
$(‘.amount’).removeAttr(‘checked’);
$(‘#other_pay’).val(0);
$(‘#other_amount_pay’).val(0);
$(this).attr(‘checked’, ‘checked’)
});

$(‘#other_pay’).change(function () {
$(‘.amount’).removeAttr(‘checked’);
$(this).attr(‘checked’, ‘checked’)
});

$(“#other_amount_pay”).on(“input”, function () {
var inputValue = $(this).val();
if (inputValue !== “”) {
$(‘#amount_50’).removeAttr(‘checked’);
$(‘#amount_100’).removeAttr(‘checked’);
$(‘#amount_500’).removeAttr(‘checked’);
$(‘#amount_1000’).removeAttr(‘checked’);
$(‘#other_pay’).val(inputValue);

$(“#other_pay”).prop(“checked”, true);
}
});

});

/* input type number validation */

const numericInput = document.getElementById(‘other_donation_amount’);

if (numericInput) {
numericInput.addEventListener(‘input’, function (e) {
// Get the current input value
const inputValue = e.target.value;

// Remove any non-numeric characters
const numericValue = inputValue.replace(/[^0-9]/g, ”);

// Update the input value with the numeric value
e.target.value = numericValue;
});

}

</script>
<script type=”sortdDeferScript”>
jQuery(document).ready(function () {
jQuery(“.share-btn”).click(function () {
jQuery(“.m-share-events”).toggleClass(“active”);
});
});

jQuery(function($){
// For Login Form
if ($(‘form[name=”login”]’).length) {
const loginForm = $(‘form[name=”login”]’);
// Dynamically remove labels and add placeholders for Login Form
$(‘form[name=”login”] label[for=”user_login”]’).remove();
$(‘form[name=”login”] #user_login’).attr(‘placeholder’, ‘Email or Username’);

$(‘form[name=”login”] label[for=”user_pass”]’).remove();
$(‘form[name=”login”] #user_pass’).attr(‘placeholder’, ‘Password’);

// Form validation and error handling for Login Form
$(‘form[name=”login”]’).on(‘submit’, function (e) {
e.preventDefault(); // Prevent form submission for validation

let isValid = true;

$(this).find(‘input[type=”text”], input[type=”password”]’).each(function () {
const input = $(this);
const value = input.val().trim();

// Remove previous error message and red border
input.closest(‘.tml-field-wrap’).prev(‘.error-message’).remove();
input.removeClass(‘input-error’);

// Determine field name for the error message
const fieldName = input.attr(‘id’) === ‘user_login’ ? ‘Email or Username’ :
input.attr(‘id’) === ‘user_pass’ ? ‘Password’ :
‘This field’;

// Add error if the field is empty
if (!value) {
isValid = false;
input.closest(‘.tml-field-wrap’).before(`<div class=”error-message”>${fieldName} is required.</div>`);
input.addClass(‘input-error’);
}
});

// Submit the form if valid
return isValid;
});

// Remove error message and red border on input for Login Form
$(document).on(‘input’, ‘.input-error’, function () {
$(this).removeClass(‘input-error’);
$(this).closest(‘.tml-field-wrap’).prev(‘.error-message’).remove();
});
}

// For Lost Password Form
if ($(‘form[name=”lostpassword”]’).length) {
// Dynamically remove labels and add placeholders for Lost Password Form
$(‘form[name=”lostpassword”] label[for=”user_login”]’).remove();
$(‘form[name=”lostpassword”] #user_login’).attr(‘placeholder’, ‘Email or Username’);

// Change the submit button text for Lost Password form
$(‘form[name=”lostpassword”] button[type=”submit”]’).text(‘Reset Password’);

// Form validation and error handling for Lost Password Form
$(‘form[name=”lostpassword”]’).on(‘submit’, function (e) {
e.preventDefault(); // Prevent form submission for validation

let isValid = true;

$(this).find(‘input[type=”text”]’).each(function () {
const input = $(this);
const value = input.val().trim();

// Remove previous error message and red border
input.closest(‘.tml-field-wrap’).prev(‘.error-message’).remove();
input.removeClass(‘input-error’);

// Determine field name for the error message
const fieldName = input.attr(‘id’) === ‘user_login’ ? ‘Email or Username’ : ‘This field’;

// Submit the form if valid
return isValid;
});

// Remove error message and red border on input for Lost Password Form
$(document).on(‘input’, ‘.input-error’, function () {
$(this).removeClass(‘input-error’);
$(this).closest(‘.tml-field-wrap’).prev(‘.error-message’).remove();
});
}

$(document).ajaxSuccess(function (event, xhr, settings) {
// Check if the request URL is the one from the lostpassword form (or another condition based on your URL)
if (settings.url.indexOf(‘forgot-the-password’) !== -1) {
var response = JSON.parse(xhr.responseText);

if (response.success) {
// const successMessage = response.data.notice;
const successMessage = “we have sent a link to your email to reset your password.”;
// Toggle visibility of the before and after sections
$(‘#forgot-password-form .forget-password-before’).hide();
$(‘#forgot-password-form .forget-password-after’).fadeIn();

} else {
// const errorMessage = response.data.errors;
// Clear previous error messages
$(‘form[name=”lostpassword”] .error-message’).remove();
$(‘form[name=”lostpassword”] .input-error’).removeClass(‘input-error’);

const errorMessage = ‘Email address or username is not registered’;
// Find the input fields for user_login and user_pass
const userLoginInput = $(‘form[name=”lostpassword”] #user_login’);

userLoginInput.closest(‘.tml-field-wrap’).before(`<div class=”error-message”>${errorMessage}</div>`);
userLoginInput.addClass(‘input-error’);

}
}

if (settings.url.indexOf(‘login’) !== -1) {
var response = JSON.parse(xhr.responseText);

if (response.success) {
// $(‘form[name=”login”]’).after(‘<div class=”success-message”>Login successful! Redirecting…</div>’);
} else {
// Clear previous error messages
$(‘form[name=”login”] .error-message’).remove();
$(‘form[name=”login”] .input-error’).removeClass(‘input-error’);

const errorMessage = response.data.errors; // Assuming this is an HTML string
// Extract plain text error messages from the response
const extractedErrorMessage = $(errorMessage).text().trim();

// Determine where to display the error based on keywords in the message
const userLoginInput = $(‘form[name=”login”] #user_login’);
const userPassInput = $(‘form[name=”login”] #user_pass’);

$(‘form[name=”login”]’).prepend(`<div class=”error-message”>Email address or password is incorrect.</div>`);
userLoginInput.addClass(‘input-error’);
userPassInput.addClass(‘input-error’);
}
}

});

// Switch Between Forms (Login and Forgot Password)
$(‘.switch-form’).on(‘click’, function (e) {
e.preventDefault();
const target = $(this).data(‘target’);
$(‘.popup-form’).hide(); // Hide all forms
$(‘.error-message’).remove(); // Remove all error messages
$(‘.input-error’).removeClass(‘input-error’); // Remove input error classes
$(‘.popup-form input’).val(”); // Clear all input values (optional)

// Show the target form
if (target === ‘forgot-password-form’) {
// Always show forget-password-before when switching to forgot-password-form
$(‘#forgot-password-form .forget-password-after’).hide();
$(‘#forgot-password-form .forget-password-before’).show();
}

$(‘#’ + target).fadeIn(); // Show the target form
});
});
</script>

<!– <script>
document.addEventListener(“DOMContentLoaded”, function () {
// Select the “I Accept” button by its ID or class (adjust selector based on your HTML)
const acceptButton = document.querySelector(“#cookie_action_close_header”); // Replace with the actual button’s class or ID

if (acceptButton) {
acceptButton.addEventListener(“click”, function () {
// Set the cookie with site-wide path and a long expiration date
document.cookie = “viewed_cookie_policy=yes; path=/; expires=Fri, 31 Dec 9999 23:59:59 GMT;”;
console.log(“Cookie consent accepted and cookie set.”);
});
}
});
</script> –>

</script>

class WaveformAudioPlayer {
constructor(playerId) {
this.playerId = playerId;
this.audio = document.getElementById(`audioPlayer_${playerId}`);
this.playBtn = document.getElementById(`playBtn_${playerId}`);
this.playTriangle = document.getElementById(`playTriangle_${playerId}`);
this.pauseBars = document.getElementById(`pauseBars_${playerId}`);
this.currentTime = document.getElementById(`currentTime_${playerId}`);
this.totalTime = document.getElementById(`totalTime_${playerId}`);
this.waveformContainer = document.getElementById(`waveformContainer_${playerId}`);
this.waveformProgress = document.getElementById(`waveformProgress_${playerId}`);
this.waveformBars = document.getElementById(`waveformBars_${playerId}`);
this.menuBtn = document.getElementById(`menuBtn_${playerId}`);

this.isPlaying = false;
this.waveformData = [];
this.totalBars = 200; // Number of waveform bars

this.initWaveform();
this.initEventListeners();
this.updateTotalTime();
}

initWaveform() {
// Generate random waveform data (in real app, you’d analyze the audio)
this.waveformData = Array.from({ length: this.totalBars }, () => {
return Math.random() * 40 + 5; // Random height between 5-45px
});

// Create waveform bars
this.waveformBars.innerHTML = ”;
this.waveformData.forEach((height, index) => {
const bar = document.createElement(‘div’);
bar.className = ‘waveform-bar’;
bar.style.height = `${height}px`;
bar.dataset.index = index;
this.waveformBars.appendChild(bar);
});
}

initEventListeners() {
// Play/Pause button
this.playBtn.addEventListener(‘click’, () => this.togglePlayPause());

// Audio events
this.audio.addEventListener(‘loadedmetadata’, () => this.updateTotalTime());
this.audio.addEventListener(‘timeupdate’, () => this.updateProgress());
this.audio.addEventListener(‘ended’, () => this.audioEnded());

// Waveform container click for seeking
this.waveformContainer.addEventListener(‘click’, (e) => this.seek(e));

// Menu button
this.menuBtn.addEventListener(‘click’, () => this.showMenu());
}

togglePlayPause() {
// Pause all other players first
window.pauseAllPlayersExcept(this.playerId);

if (this.isPlaying) {
this.pause();
} else {
this.play();
}
}

play() {
this.audio.play().then(() => {
this.isPlaying = true;
this.updatePlayButton();
}).catch(error => {
console.error(‘Error playing audio:’, error);
});
}

pause() {
this.audio.pause();
this.isPlaying = false;
this.updatePlayButton();
}

updatePlayButton() {
if (this.isPlaying) {
this.playTriangle.style.display = ‘none’;
this.pauseBars.style.display = ‘block’;
} else {
this.playTriangle.style.display = ‘block’;
this.pauseBars.style.display = ‘none’;
}
}

updateProgress() {
if (this.audio.duration) {
const progress = (this.audio.currentTime / this.audio.duration) * 100;
this.waveformProgress.style.width = `${progress}%`;
this.currentTime.textContent = this.formatTime(this.audio.currentTime);

// Update active bars
const activeBars = Math.floor((progress / 100) * this.totalBars);
const bars = this.waveformBars.querySelectorAll(‘.waveform-bar’);
bars.forEach((bar, index) => {
if (index <= activeBars) {
bar.classList.add(‘active’);
} else {
bar.classList.remove(‘active’);
}
});
}
}

updateTotalTime() {
if (this.audio.duration) {
this.totalTime.textContent = this.formatTime(this.audio.duration);
}
}

seek(e) {
const rect = this.waveformContainer.getBoundingClientRect();
const percent = (e.clientX – rect.left) / rect.width;
const newTime = percent * this.audio.duration;

if (newTime >= 0 && newTime <= this.audio.duration) {
this.audio.currentTime = newTime;
}
}

formatTime(seconds) {
if (!seconds || !isFinite(seconds)) return ’00:00′;

const minutes = Math.floor(seconds / 60);
const remainingSeconds = Math.floor(seconds % 60);
return `${minutes.toString().padStart(2, ‘0’)}:${remainingSeconds.toString().padStart(2, ‘0’)}`;
}

audioEnded() {
this.isPlaying = false;
this.updatePlayButton();
this.waveformProgress.style.width = ‘0%’;
this.currentTime.textContent = ’00:00′;

// Reset all bars
const bars = this.waveformBars.querySelectorAll(‘.waveform-bar’);
bars.forEach(bar => bar.classList.remove(‘active’));
}

showMenu() {
console.log(`Menu clicked for player ${this.playerId} – implement additional features here`);
// Add your menu functionality here
}

// Method to change audio source
changeAudio(src) {
this.pause();
this.audio.src = src;
this.audio.load();
this.waveformProgress.style.width = ‘0%’;
this.currentTime.textContent = ’00:00′;

// Reset bars
const bars = this.waveformBars.querySelectorAll(‘.waveform-bar’);
bars.forEach(bar => bar.classList.remove(‘active’));
}
}

// Global variables to manage multiple players
window.waveformPlayers = {};

// Function to pause all players except the current one
window.pauseAllPlayersExcept = function(currentPlayerId) {
Object.keys(window.waveformPlayers).forEach(playerId => {
if (playerId !== currentPlayerId) {
window.waveformPlayers[playerId].pause();
}
});
};

// Initialize all players when DOM is loaded
document.addEventListener(‘DOMContentLoaded’, () => {
// Find all waveform audio players
const playerElements = document.querySelectorAll(‘.waveform-audio-player’);

playerElements.forEach(playerElement => {
const playerId = playerElement.dataset.playerId;
if (playerId) {
// Extract post ID from player ID (format: player_123)
const postId = playerId.replace(‘player_’, ”);
window.waveformPlayers[postId] = new WaveformAudioPlayer(postId);
}
});

// Global keyboard shortcuts
document.addEventListener(‘keydown’, (e) => {
const isEditable = e.target.isContentEditable;
if (e.code === ‘Space’ && e.target.tagName !== ‘INPUT’ && e.target.tagName !== ‘TEXTAREA’ && !isEditable) {
e.preventDefault();
// This will be handled by individual players
}
});
});
</script>

jQuery(function($){
// Prevent numbers while typing & enforce maxlength for Name
$(document).on(‘input’, ‘#contactFormCustom input[name=”your-name”]’, function(){
// allow letters, spaces, hyphen, apostrophe, limit to 50 chars
this.value = this.value.replace(/[^A-Za-zs’-]/g, ”).slice(0,50);
});

// Email field: limit to 100 characters and block spaces
$(document).on(‘input’, ‘#contactFormCustom input[name=”your-email”]’, function() {
this.value = this.value.replace(/s/g, ”).slice(0, 100);
});

});

</script>

</html>
<style>
/* Hide the view count by default */
.hidden {
display: none;
}

</style>

<p class=”hidden”>Total Views: </p>

<!– <style>
#cliSettingsPopup {display:none}
</style>
<style>
#cookie-law-info-bar,.wt-cli-cookie-bar{box-sizing:border-box;font-size:10pt;margin:0 auto;padding:10px 10px;position:absolute;text-align:center;width:100%;z-index:99999;box-shadow:rgb(0 0 0 / .5) 0 5px 10px;display:none;left:0}#cookie-law-info-again{font-size:10pt;margin:0;padding:5px 10px;text-align:center;z-index:9999;cursor:pointer;box-shadow:#161616 2px 2px 5px 2px}#cookie-law-info-bar span{vertical-align:middle}.cli-plugin-button,.cli-plugin-button:visited{display:inline-block;padding:8px 16px 8px;color:#fff;text-decoration:none;-moz-border-radius:4px;-webkit-border-radius:4px;position:relative;cursor:pointer;text-decoration:none}.cli-accept-button{font-family:Courier;font-variant:small-caps}.cli-plugin-button:hover{background-color:#111;color:#fff;text-decoration:none}.small.cli-plugin-button,.small.cli-plugin-button:visited{font-size:11px}.cli-plugin-button,.cli-plugin-button:visited,.medium.cli-plugin-button,.medium.cli-plugin-button:visited{font-size:13px;font-weight:500;line-height:1}.large.cli-plugin-button,.large.cli-plugin-button:visited{font-size:14px;padding:8px 14px 9px}.super.cli-plugin-button,.super.cli-plugin-button:visited{font-size:34px;padding:8px 14px 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only screen and (max-width:768px){.cli-settings-mobile:hover{box-shadow:none!important}.cli-settings-desktop{display:none}.cli-col-8{max-width:100%!important;width:100%!important;flex:100%!important;-ms-flex:100%!important}.cli-accordion-plusminus{float:right;font-size:22px;line-height:22px}.cli-plugin-button{margin:10px}}@media only screen and (max-width:567px){.cli-switch .cli-slider:after{display:none}.cli-tab-header a.cli-nav-link{font-size:12px}.cli-modal .cli-modal-close{right:-10px;top:-15px}}.wt-cli-iframe-placeholder{background-image:url(../images/cli_placeholder.svg);background-size:80px;max-width:100%;max-height:100%;background-position:center;background-repeat:no-repeat;background-color:#b2b0b059;position:relative;display:flex;align-items:flex-end;justify-content:center}.wt-cli-iframe-placeholder .wt-cli-inner-text{width:100%;text-align:center;padding:1rem 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(min-width:576px){.gdpr-modal .gdpr-modal-dialog{max-width:500px;margin:1.75rem auto;min-height:calc(100% – (1.75rem * 2))}}@media (min-width:992px){.gdpr-modal .gdpr-modal-dialog{max-width:900px}}.gdpr-modal-content{position:relative;display:-ms-flexbox;display:flex;-ms-flex-direction:column;flex-direction:column;width:100%;pointer-events:auto;background-color:#fff;background-clip:padding-box;border:1px solid rgb(0 0 0 / .2);border-radius:.3rem;box-sizing:border-box;outline:0}.gdpr-modal .row{margin:0 -15px}.gdpr-modal .modal-body{padding:0;position:relative;-ms-flex:1 1 auto;flex:1 1 auto}.gdpr-modal .close{position:absolute;right:10px;top:10px;z-index:1;padding:0;background-color:#fff0;border:0;-webkit-appearance:none;font-size:1.5rem;font-weight:700;line-height:1;color:#000;text-shadow:0 1px 0 #fff}.gdpr-modal .close:focus{outline:0}.gdpr-switch{display:inline-block;position:relative;min-height:1px;padding-left:70px;font-size:14px}.gdpr-switch input[type=”checkbox”]{display:none}.gdpr-switch .gdpr-slider{background-color:#e3e1e8;height:24px;width:50px;bottom:0;cursor:pointer;left:0;position:absolute;right:0;top:0;transition:.4s}.gdpr-switch .gdpr-slider:before{background-color:#fff;bottom:2px;content:””;height:20px;left:2px;position:absolute;transition:.4s;width:20px}.gdpr-switch input:checked+.gdpr-slider{background-color:#63b35f}.gdpr-switch input:checked+.gdpr-slider:before{transform:translateX(26px)}.gdpr-switch .gdpr-slider{border-radius:34px}.gdpr-switch .gdpr-slider:before{border-radius:50%}.gdpr-tab-content>.gdpr-tab-pane{display:none}.gdpr-tab-content>.active{display:block}.gdpr-fade{transition:opacity .15s 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.btn-gdpr{background-color:#0070ad;border-color:#0070ad;color:#fff;display:inline-block;font-weight:400;text-align:center;white-space:nowrap;vertical-align:middle;-webkit-user-select:none;-moz-user-select:none;-ms-user-select:none;user-select:none;border:1px solid #fff0;padding:.375rem .75rem;font-size:1rem;line-height:1.5;border-radius:.25rem;transition:color .15s ease-in-out,background-color .15s ease-in-out,border-color .15s ease-in-out,box-shadow .15s ease-in-out;box-sizing:border-box}.gdpr-tab-content p{color:#343438;font-size:14px;margin-top:0}.gdpr-tab-content h4{font-size:20px;margin-bottom:.5rem;margin-top:0;font-family:inherit;font-weight:500;line-height:1.2;color:inherit}.cli-container-fluid{padding-right:15px;padding-left:15px;margin-right:auto;margin-left:auto}.cli-row{display:-ms-flexbox;display:flex;-ms-flex-wrap:wrap;flex-wrap:wrap;margin-right:-15px;margin-left:-15px}.cli-col-4{-ms-flex:0 0 33.333333%;flex:0 0 33.333333%;max-width:33.333333%}.cli-col-8{-ms-flex:0 0 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.cli-modal-dialog{position:relative;width:auto;margin:.5rem;pointer-events:none;font-family:inherit;font-size:1rem;font-weight:400;line-height:1.5;color:#212529;text-align:left;display:-ms-flexbox;display:flex;-ms-flex-align:center;align-items:center;min-height:calc(100% – (.5rem * 2))}@media (min-width:576px){.cli-modal .cli-modal-dialog{max-width:500px;margin:1.75rem auto;min-height:calc(100% – (1.75rem * 2))}}.cli-modal-content{position:relative;display:-ms-flexbox;display:flex;-ms-flex-direction:column;flex-direction:column;width:100%;pointer-events:auto;background-clip:padding-box;border-radius:.2rem;box-sizing:border-box;outline:0}.cli-modal .row{margin:0 -15px}.cli-modal-footer{position:relative}.cli-modal .modal-body{padding:0;position:relative;-ms-flex:1 1 auto;flex:1 1 auto}.cli-modal .cli-modal-close:focus{outline:0}.cli-switch{display:inline-block;position:relative;min-height:1px;padding-left:38px;font-size:14px;width:auto}.cli-switch input[type=”checkbox”]{display:block;margin:0;width:0;position:absolute!important;left:0!important;opacity:0!important}.cli-switch .cli-slider{background-color:#e3e1e8;height:20px;width:38px;bottom:0;cursor:pointer;left:0;position:absolute;right:0;top:0;transition:.4s}.cli-switch .cli-slider:before{bottom:2px;content:””;height:15px;left:3px;position:absolute;transition:.4s;width:15px}.cli-switch input:checked+.cli-slider{background-color:#28a745}.cli-switch input:checked+.cli-slider:before{transform:translateX(18px)}.cli-switch .cli-slider{border-radius:34px;font-size:0}.cli-switch .cli-slider:before{border-radius:50%}.cli-nav-pills{display:-ms-flexbox;display:flex;-ms-flex-wrap:wrap;flex-wrap:wrap;padding-left:0;margin-bottom:0;list-style:none;-ms-flex-direction:column;flex-direction:column}.cli-nav-pills,.cli-tab-content{width:100%;padding:5px 30px 5px 5px;box-sizing:border-box}@media (max-width:767px){.cli-nav-pills,.cli-tab-content{padding:30px 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Source: World Politics - fairobserver.com

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