The label "atheist"

I wrote a similarly-titled post on here about six years ago contemplating whether I could still call myself a Christian. I suppose now is a reasonable time for a follow-up. Although this time around, I think I'm perhaps less uncertain about a different label for me now than I was about the label "Christian" for me then.

It certainly wouldn't feel like a lie if the label "atheist" were applied to me. I don't live my life or make any decisions with regard for the existence of a divine being. God, Jesus, and spirituality are simply irrelevant in my life; I arrived at the point where it doesn't matter whether God exists or not.

However, I think my current hangup is the connotation that the label "atheist" has associated with it. To me, "atheist" suggests a degree of certainty and possibly even closed-mindedness--God does not exist, and that's that. But I think my migration from Christianity to atheism-adjacent or wherever I am now has made it clear to me that you can't go through life closed-mindedly. You need to be open to change if experience and evidence (or lack of evidence) compel you to revise your worldview. I suppose in that sense, applying any label to me could pose a similar challenge.

Perhaps my view of atheism as a closed-minded worldview is due to how atheism was characterized while growing up Christian. Atheists, I was told, assume the premise that atheism is true and make all experiences or evidence conform to that worldview. Atheism was described as requiring as much faith as Christianity to support its worldview. Perhaps to a degree, the latter statement that is true. It cannot be proven that God does not exist, so jumping from, "I have no evidence for God's existence," to, "God does not exist," is indeed a leap of faith.

I've thought before about adopting the label, "agnostic". However, my problem with that label is that it captures an exceedingly broad range of the do-you-believe-in-God spectrum. I think adopting that label would put me under the same umbrella as people who still believe in God but harbor some doubts. I'm far beyond that point in terms of my lack of belief in God.

The image below shows Richard Dawkins' "Spectrum of Theistic Probability" from The God Delusion on top, and how I think about the labels on the bottom. I'm somewhere around "De facto atheist" on top, but I'm not sure where I fall on the bottom (atheist vs. agnostic) since it's kind of at a boundary area.

All that being said, I do find myself in a bit of a similar situation as when I wrote my previous post about the label "Christian". I don't really care, I'm overall happy in life, and identifying a marginally more precise label isn't going to change that. I can definitively say, however, that I am far happier now than I ever was when I was a Christian.

A Child's Prayer

1    Dear God,

2    Thank you for this day

3    Thank you for the food we ate

4    Please watch over our house and protect us while we sleep

5    Please keep us healthy

6    Help us to learn more and more about you

7    Thanks that I had a good day today

8    Help me to have a good day tomorrow

9    Help me not to have any nightmares, just dreams

10    Help our house not to get struck by lightning

11    Help our house not to catch on fire

12    Help our neighbors' houses not to catch on fire and the fire jump over to our house

13    Help someone not to on purpose drop a match on our house

14    Help someone not accidentally drop a match on our house

...

XX    Amen.

This was the prayer that I said--either aloud or to myself--nightly before I went to sleep for many of the formative years of my life, probably up until I was eleven or twelve years old. I said each line in that precise order, and I never omitted any lines. (The "..." refers to additional lines that I said but cannot remember anymore.)

This prayer initially came into existence once my parents started sharing the prayer-before-bedtime responsibility with me when I was probably five or six years old. I borrowed lines from the prayers that my dad used to say with me after tucking me in and from the prayers I overheard my older brother say with my dad as he was tucked in. Lines 3, 7, and 8 came from my older brother's prayers, and Lines 2 and 4-6 came from my dad's prayers with me.

For a few years, my prayers consisted only of Lines 1-8 and XX, but as I grew older, I became more fearful that bad things would happen unless I specifically asked God to prevent them from happening. So I began adding original content to my prayer, starting with Line 9. When I would pray aloud with one of my parents, though, I would say Line 9 to myself and leave them in silence for a couple seconds before I closed with, "Amen". I was embarrassed by Line 9 and didn't want my parents to hear it.

One of the things I was most afraid of as a child was a house fire. I don't know where this fear came from. Perhaps it was stoked when one of my kindergarten classmates came to school sobbing one day, telling the class in tears, "My house burned down." I saw firsthand the chaos and emotional damage that house fires caused. Or perhaps this fear became a permanent fixture of my childhood anxiety repertoire after the incident when my younger brother left an electric charcoal fire starter plugged into an indoor socket while we dropped my older brother off at a friend's house and returned to a house full of smoke. I had never seen my mom so hysterical before as she called the fire department. (The resultant damage was minor and was contained to the interior of the living room, as we had only been gone a few minutes.)

When I was in elementary school, I was also terrified of thunderstorms, and lightning in particular due to its connection with fire. Imagery from films of lightning strikes causing fires pervaded my mind (e.g., The Lion King). Because my fear of lightning causing a house fire was so great, I added Line 10 to my prayers.

Later, I realized that Line 10 only covered a small subset of causes of fire. I feared that God would "catch me", and he could allow our house to catch on fire by some other means. "Well, you didn't ask me not to let a gas main explode," would be an example of a line I imagined God might retort with had I gotten a chance to confront him about why he let our house catch on fire. To mitigate this fear, I added Line 11, which was intended to cover causes of fire originating from within our house.

It wasn't long before I remembered our house could still catch on fire from external causes. Enter Lines 12, 13, and 14. Whenever I said Lines 13 and 14, I imagined one of the villains from Hanna-Barbera's The Greatest Adventure: Stories from the Bible "Noah's Ark" seen below as the perpetrator. I imagined him standing on our porch and walking up to our front door with a lit match in his hand, either holding the match up against the house (Line 13) or turning around and dropping the match from his hand onto our porch (Line 14; hardly an accident in this visualization).

Thus, over the years, my nightly prayer grew lengthier as I accumulated more and more things I needed to ask about to protect myself. All of the original content I added to my prayer--Lines 9 and beyond--consisted entirely of negative requests, in which I asked God for something not to happen as opposed to asking for something. I was afraid that if I did not utter the words, then I would have left myself vulnerable, and the bad thing could have happened. Thus, I had to be sure that I always made every single request in my list.

Looking back, I see that this practice almost perfectly fits the definition of obsessive-compulsive behavior. My mind was full of intrusive, unwanted fears around the possibility of all these negative things happening to me, and I responded to these obsessions by performing a repetitive, ritualistic routine that had to be precisely followed in order to assuage my obsessions...at least until the next night. By then, my prayers would have "worn off" and would no longer apply. I needed to refresh my requests on a nightly basis, as a kind of insurance that horrifying events would not happen to me during that night.

Prayer is hardly a soothing activity if you conduct it from a headspace of perceived risk mitigation--if you're trying to preempt all the ways that God could find a loophole in your requests to allow something bad to happen to you. I feared falling asleep without praying, or falling asleep without completing my nightly list of requests.

So, one night, during a family visit to my grandparents' house when I was probably nine or ten years old, I went to bed so exhausted that I devised a shortcut so I could get to sleep faster: "Dear God, please just say my regular prayer for me." It felt risky to make that ask--would God not honor the requests contained within my regular prayer if I didn't explicitly verbalize them?

But I woke up the next morning, and everything was fine. I realized I didn't necessarily have to say all the words in order for them to come true. Consequently, that shortcut became a technique that I used with increasing frequency when I went to bed and wanted to fall asleep quickly.

Monstro

I recently rewatched Pinocchio, Walt Disney's second feature-length film. One of the film's antagonists, Monstro the whale, got the wheels in my head turning about his role in the story.

Monstro does not really have a character per se. He has no dialogue and is only in the film for the final fifteen minutes or so. We as the viewers do not see the precipitating events that make Monstro an antagonist; there is no point in the story when we decide that we do not like Monstro and we want our hero to defeat him. Instead, upon Monstro's introduction, we already assume his sinister nature. For these reasons, Monstro's function in the story is not so much as a literal being but as a symbol for something.

This assertion is amplified by the fact that although he is described a whale, he seems too extraordinary in size, speed, strength, and emotion to be a literal whale. However, whether Monstro is a literal whale doesn't really matter and misses the point in understanding what Monstro represents within the context of the story.

Monstro represents Geppetto's deep-seated fear of losing someone that you love.

Geppetto was swallowed up by Monstro during his pursuit of Pinocchio after he did not return from his first day of school. Geppetto's fear of losing Pinocchio overwhelmed him until he ended up in a totally helpless place--inside Monstro's belly. His fear disabled him from carrying out his mission of finding Pinocchio. He became mired and trapped within his fear.

In order to become a real boy, two criteria that Pinocchio must meet are that he must be brave and unselfish. He meets both of these criteria during his confrontation with Monstro, whether Monstro is perceived as a literal whale or as a symbol.

The escape from Monstro the literal whale provides an obvious opportunity for Pinocchio to display his bravery and unselfishness. He displays bravery by initiating Geppetto's and his daring escape from Monstro in spite of the whale's extraordinary size and temperamental nature, and by urging Geppetto to keep paddling their raft even though Geppetto protests, "We'll never make it!" as the whale inhales before he sneezes. Pinocchio displays unselfishness after Monstro smashes their raft and Pinocchio pulls Geppetto to shore, against Geppetto's urging to, "Swim for shore; save yourself."

However, assuming Monstro as a symbol, the viewer observes Pinocchio's bravery and unselfishness even before he and Geppetto plot their escape from Monstro's belly. The mere act of seeking out Geppetto where he is--immobilized in the bowels of his deepest fear--is certainly an act of unselfishness. It is also an act of bravery; helping pull someone out of their despair is a daunting and sometimes terrifying task. Yet as soon as Pinocchio learns of Geppetto's state, he immediately marches off to save him, with Jiminy Cricket once again lagging behind and sowing doubt about Pinocchio's decision.

In the novel The Adventures of Pinocchio by Carlo Collodi, upon which the film is based, the equivalent creature is known as The Terrible Dogfish (Il Terribile Pescecane). This character is not named Monstro and is not a whale in the novel. Disney thus made a conscious decision to assign this character in the novel as a whale in the film adaptation. Why should a whale be chosen? Perhaps this is a deliberate allusion to the Biblical story of Jonah, where God sends a great fish (which over the centuries has come to be translated and entered into the cultural lexicon as a whale) to swallow Jonah. In the Biblical story, the whale's purpose is to punish Jonah for his disobedience and to teach him a lesson. If the creative license taken for the film adaptation to have Geppetto swallowed by a whale is indeed an allusion, might it be possible that Geppetto encountered Monstro to learn a lesson?

Perhaps the lesson that Geppetto needed to learn was that sometimes, you as a parent will fuck up when raising your kid. Sometimes, in an abstract sense, you will "lose" your kid. You cannot control your kid's destiny, as much as you may want to or may try to. Instead, what matters the most is that your parental love is constant and unconditional.

This lesson is exemplified soon after Pinocchio comes aboard Geppetto's ship inside Monstro's belly. Geppetto wraps the cold and wet Pinocchio in a blanket, removes Pinocchio's hat, and discovers Pinocchio's donkey ears and tail. At first, Geppetto is shocked and horrified, pleading for Pinocchio to tell him, "what happened?" Pinocchio hesitates and stutters as he thinks about how to respond to his father. Quickly realizing the shame that Pinocchio feels around the subject, Geppetto brushes past the topic, declaring that all that matters is that he and Pinocchio are together. Geppetto has learned the lesson that his love for Pinocchio is most important thing in their relationship.

Only after Geppetto has learned this lesson is he able to fully engage with Pinocchio and participate in the escape plan that Pinocchio devises. His love for Pinocchio has begun the process of pulling himself out of the pits of his deepest fears.

Similarly, in the story of Jonah, he is released from the whale after he learns his lesson and repents to God.

However, the parallelism between the Monstro sub-plot and the Jonah story is certainly not perfect. For instance, the way in which escape is achieved in each story differs significantly. In Pinocchio, Pinocchio comes to save Geppetto, and working together, they escape Monstro from their own efforts. Jonah's escape from the whale comes about from an act of mercy by an external agent, God. Jonah is fairly helpless in his situation.

The Monstro sub-plot of Pinocchio is thus also a story of courage, determination, and decisiveness. As the viewers, we celebrate with the characters as we watch them achieve the goal they set for themselves, overcoming their abysmal situation. By contrast, the whale sub-plot within Jonah's story is a story about God's mercy. Jonah is in fact not the protagonist in this part of his story--God is. Jonah's escape from the whale is not an achievement attributable to Jonah; Jonah goes forth from the whale with an obligation.

This is why the Pinocchio whale sub-plot is a much more compelling story than the Jonah whale story.

Monopoly: A Markov Chain Analysis

I'm sure this analysis is not original.

The question to be answered is the following:

Which squares on a classic Monopoly game board are the most and least likely to be landed on?

I answered this question using a Markov Chain approach.

Markov Chains: Overview

A Markov Chain is a stochastic (i.e., random, or probabilistic) process wherein a system's "state" changes probabilistically over time. A key characteristic of a Markov Chain is that it is a "memoryless" process. That is, it doesn't matter which other states the system has been in before its current state in order to identify the probable state it will be in next; all that matters is the system's current state. The formal definition of a Markov Chain is any process that satisfies the following property:

P(X_n+1 = s | X_n, X_n-1, ... , X₁) = P(X_n+1 = s | X_n)

where X_n is a random variable representing the system's state at time n, and s is a particular state. This property is called the Markov Property.

The game of Monopoly generally satisfies the Markov Property. If you are on a given square, it doesn't matter how you got there in order to determine the probable squares you could travel to next.

How a Markov Chain Model Can be Used for Monopoly

Markov Chains use the concept of "states" to describe a system. The collection of all states in a Markov Chain is known as the "state space". For any stochastic system, a state space can be defined such that the Markov Property holds. In Monopoly, it is fairly easy and intuitive to define the state space simply as the distinct squares of the board. Thus, you could define 41 possible states in a Monopoly Markov Chain. (There are 40 squares on the board, but "Jail (Just Visiting)" and "Jail" are considered two distinct states because they can function differently depending on the player's strategy for exiting Jail.)

A Markov Chain can be represented by a square matrix called the "transition matrix". The entries in such matrices represent the probability of transitioning to state j from state i. The originating states (state i) are the rows of the matrix, and the destination states (state j) are the columns. Because a system in state i must move to some other state j (even if j = i), the sum of each row in a transition matrix is 1. An example transition matrix for a three-state Markov Chain is provided below.

In the above example, if the system is in State 1 (row 1), there is a probability of 0.3 of remaining in State 1 at the next transition; a probability of 0.5 of transitioning to State 2; and a probability of 0.2 of transitioning to State 3. Notice that each row sums to 1.

In Monopoly, if a player is on a given square, the transition probabilities available to the player generally follow the distribution of rolling two six-sided dice to advance between 2 and 12 squares forward. This is not true for every square, such as Chance and Community Chest squares, or when doubles are rolled for the third time in a row, but we'll get back to that.

The Steady-State Distribution

If a transition matrix, denoted as P, has certain properties, then it leads to some interesting outcomes. Specifically, there is a theorem stating that if a Markov Chain is "irreducible" and "aperiodic", then there exists a unique vector 𝞹 such that 𝞹*P = 𝞹 and all the elements of 𝞹 sum to 1. This is called the "steady-state" distribution, in that it represents the convergence of the distribution of the system's state in the long-term. That is, 𝞹 is essentially a vector containing the probabilities (𝞹_i) that the system will be in state i at any given point in time.

For the 3x3 example above, the vector 𝞹 = [0.4113475    0.2695035    0.3191489] (rounded) represents the steady-state distribution of the Markov Chain. The Markov Chain is irreducible and aperiodic (trust me for now; we'll discuss these in a bit), and the vector 𝞹 satisfies the equation 𝞹*P = 𝞹. We thus see that State 1 is the most frequently visited state in the long term and State 2 the least frequently visited.

In the case of Monopoly, 𝞹 represents the probability of a player being on any given square as the number of turns approaches infinity. In other words, this vector tells us the long-term likelihood of landing on all squares--precisely the question we sought to answer.

Before we move on, though, we need to be sure that our matrix P that we establish for a Monopoly board satisfies the necessary properties: irreducibility and aperiodicity.

"Irreducibility" means that the system can go from any given state to any other state after any number of steps. If you have ever played Monopoly, you know that this is true, as every square is landed on multiple times throughout a game. For example, you obviously could not reach Marvin Gardens from Baltic Avenue in one turn, but you could reach it in three turns by rolling 8, 11, 7; or four turns by rolling 6, 7, 9, 4; or seven turns by rolling 12, 12, 5, 10, 9, 11, 7, rounding the board in the process. (A similar exercise could be conducted for every possible pair of squares.) All that matters is that you can eventually reach any other square from a given starting square. Therefore, a Monopoly board is irreducible.

"Aperiodicity" means that the system doesn't get stuck in a fixed-timestep loop between a subset of states. That is, if you are in a given state, you don't know how many transitions you will have to make before you return to that state. An example of periodicity in Monopoly would be if a rule existed such that every time you landed on Park Place, your next six dice rolls were required to total 40, thus landing you on Park Place every sixth turn. Obviously there are no such rules in the game, as advancement around the board is defined by probabilistic dice rolls with no guarantee of the next time you will return to a square. Therefore, a Monopoly board is aperiodic.

So, we see that a Monopoly board Markov Chain satisfies the properties of irreducibility and aperiodicity. Consequently, a steady-state distribution 𝞹 must exist for such a Markov Chain.

For you linear-algebra-minded folk, the theorem alternatively states that for an irreducible and aperiodic Markov Chain with transition matrix P, one of the eigenvalues for the matrix P is 1 with a corresponding eigenvector of 𝞹 such that all the elements of 𝞹 are non-negative and sum to 1.

Defining the Monopoly Markov Chain Transition Matrix

Okay, underlying theory out of the way. Now for application.

A Markov Chain is defined by its state space and transition matrix. For a Monopoly board, we have already identified our state space as the 41 possible squares. Now we must define our transition matrix.

As mentioned before, for most squares on the board, the transition probabilities generally follow the symmetric distribution of the possible outcomes of rolling two dice. This distribution is as follows:

P(roll 2) = 1/36
P(roll 3) = 2/36
P(roll 4) = 3/36
P(roll 5) = 4/36
P(roll 6) = 5/36
P(roll 7) = 6/36
P(roll 8) = 5/36
P(roll 9) = 4/36
P(roll 10) = 3/36
P(roll 11) = 2/36

P(roll 12) = 1/36

As an example, consider what happens when a player is on Virginia Avenue--a typical square that does not have any special movement rules. This is State 16 in the state space. With a transition probability of 1/36, the next state will be State 18 (St. James Place), which represents what happens if a player rolls snake eyes. With a  transition probability of 2/36, the next state will be State 19 (Community Chest), which represents what happens if a player rolls a 3 (and so on). For other squares on the board that are not reachable from Virginia Avenue with a single roll (such as North Carolina Avenue), the transition probability is 0.

Simple enough. But what about special squares from which movement can be irregular?

Wrinkle 1: Community Chest and Chance

There are three Community Chest squares and three Chance squares on a typical board. Upon landing on these squares, a player draws from the respective deck of cards and follows the instructions. Some of these cards instruct the player to move to other squares.

In the Community Chest deck, two of the 16 cards require the player to move to another square. One card sends the player to Go, and the other sends the player to Jail. If any of the other 14 cards is drawn, the player remains on the square and will continue the next turn with a standard roll of the dice.

The following tree illustrates what can happen on a Community Chest square and the corresponding probabilities of each outcome.

In the Chance deck, 10 of the 16 cards require the player to move to another square. These cards can send the player to: Go, Jail, Illinois Avenue, St. Charles Place, Boardwalk, Reading Railroad, the nearest utility, three spaces previous, the nearest railroad, and the nearest railroad (there are two such cards in the deck). If any of the other six cards is drawn, the player remains on the square and will continue the next turn with a standard roll of the dice.

The following tree illustrates what can happen on a Chance square and the corresponding probabilities of each outcome.

Note that four of the Chance cards requiring movement (nearest utility, three spaces previous, nearest railroad [x2]) will bring a player to different squares depending on which Chance square the player is on. For example, the nearest utility for the first and third Chance squares is the Electric Company, whereas the nearest utility for the second Chance square is Water Works.

In this process, we have derived what are known as conditional probabilities. Conditional probabilities answer questions such as: What is the likelihood of advancing to a given square, given that a player drew a certain card?

When a game of Monopoly begins, the Community Chest and Chance card decks are shuffled, and then the order of the cards usually remains fixed for the course of the game. But the way the Markov Chain model is set up assumes that the cards are randomly drawn without regard to a specific, fixed order. Thus, it would be theoretically possible to draw "Go to Jail" twice in a row from the same deck in this model. However, because this model is intended to be inclusive of all possible gameplays, and thus all possible orderings of the decks, this nuance may be insignificant in the limit.

Nevertheless, this is still a simplifying assumption that we have made. If we were to model it more accurately, our state space would need to be much bigger. Our state space would need to be defined to contain the information of the current square, all possible orderings of these card decks, and which cards had already been drawn. Our state space would thus consist of 41*(16!)² states. In addition, defining the Markov Chain in this way (i.e., with a fixed card ordering) would destroy the system's irreducibility, as the system could never reach a state containing a different card ordering.

Wrinkle 2: Triple Doubles

Another rule in Monopoly is that if a player rolls doubles three times in a row, then the player is automatically sent to Jail. In order to fully and accurately include this in the Markov Chain model, the state space would have to be modified to contain the information of the current square and the previous two dice rolls. Similar to the random ordering of the Community Chest and Chance cards, we will make a simplifying assumption to account for this rule as well.

The probability of rolling doubles on any given roll is 1/6. Each doubles pair has a probability of 1/36, and there are six such pairs which count as doubles. Thus, the probability of rolling doubles three times in a row is 1/6 * 1/6 * 1/6, or 1/216. Similar to the conditioning performed for Community Chest and Chance cards, we will condition on whether the roll is a third double roll. The following tree illustrates how the conditional probabilities are calculated.

This addition slightly modifies the preliminary entries in the transition matrix by multiplying each of the dice roll probabilities by 215/216. It also makes Jail accessible from any other square with probability 1/216. The only square that is unaffected by this change is Jail itself. When a player exits Jail, the doubles counter is "reset", and thus it is not possible to be sent to Jail by a third doubles roll from Jail.

Wrinkle 3: Strategy for Exiting Jail

There are two ways that a player can choose to exit Jail: either by paying $50 (or using a "Get out of Jail free" card) and rolling the dice as usual, or waiting to roll doubles, after which the player exits for free and advances the number of spaces returned from the doubles roll. A player can change the strategy at any time. For example, a player may unsuccessfully attempt to get out for free via doubles for three rolls, but then decide to pay the $50 because the player doesn't want to spend any more turns in Jail.

To account for each of these strategies, two transition matrices must be developed. The first assumes that a player always pays $50 for immediate exit and thus may advance anywhere from 2 to 12 squares from Jail following the typical dice roll probability distribution. The second assumes that a player always waits for doubles to exit and thus may advance 2, 4, 6, 8, 10, or 12 squares from Jail, each with equal probability of 1/6. The steady-state distributions for each of these two matrices will be identified, and the effects of each of these strategies on square frequency can be identified.

Sample Entries from the Transition Matrix

All the information needed to create two 41x41 transition matrices is now available. These two matrices are identical except for the row that represents "Jail" as the current state. Examples of entries from these matrices are provided below and are given as probabilities.

Current square: Go

P(Jail | Go) = 1/216 [third doubles rolled]
P(Community Chest 1 | Go) = (215/216) * (1/36) [third doubles was not rolled AND a 2 was rolled]
P(Baltic Avenue | Go) = (215/216) * (2/36) [same logic]
P(Income Tax | Go) = (215/216) * (3/36)
P(Reading Railroad | Go) = (215/216) * (4/36)
P(Oriental Avenue | Go) = (215/216) * (5/36)
P(Chance 1 | Go) = (215/216) * (6/36)
P(Vermont Avenue | Go) = (215/216) * (5/36)
P(Connecticut Avenue | Go) = (215/216) * (4/36)
P(Jail (Visiting) | Go) = (215/216) * (3/36)
P(St. Charles Place | Go) = (215/216) * (2/36)
P(Electric Company | Go) = (215/216) * (1/36)
P(all other squares | Go) = 0

Current square: Community Chest 2 (between St. James Place and Tennessee Avenue)

P(Jail | Community Chest 2) = (1/216) * (14/16) [rolling a third doubles given that a non-movement card was drawn] + (2/16) * (1/2) ["Go to Jail" card drawn]
P(Go | Community Chest 2) = (2/16) * (1/2) ["Advance to Go" card drawn]
P(New York Avenue | Community Chest 2) = (215/216) * (14/16) * (1/36) [third doubles was not rolled AND non-movement card was drawn AND a 2 was rolled]
P(Free Parking | Community Chest 2) = (215/216) * (14/16) * (2/36) [same logic]
P(Kentucky Avenue | Community Chest 2) = (215/216) * (14/16) * (3/36)
P(Chance 2 | Community Chest 2) = (215/216) * (14/16) * (4/36)
P(Indiana Avenue | Community Chest 2) = (215/216) * (14/16) * (5/36)
P(Illinois Avenue | Community Chest 2) = (215/216) * (14/16) * (6/36)
P(B&O Railroad | Community Chest 2) = (215/216) * (14/16) * (5/36)
P(Atlantic Avenue | Community Chest 2) = (215/216) * (14/16) * (4/36)
P(Ventnor Avenue | Community Chest 2) = (215/216) * (14/16) * (3/36)
P(Water Works | Community Chest 2) = (215/216) * (14/16) * (2/36)
P(Marvin Gardens | Community Chest 2) = (215/216) * (14/16) * (1/36)
P(all other squares | Community Chest 2) = 0

Current square: Atlantic Avenue

P(Jail | Atlantic Avenue) = 1/216 [third doubles rolled]
P(Water Works | Atlantic Avenue) = (215/216) * (1/36) [third doubles was not rolled AND a 2 was rolled]
P(Marvin Gardens | Atlantic Avenue) = (215/216) * (2/36) [same logic]
P(Go to Jail | Atlantic Avenue) = (215/216) * (3/36)
P(Pacific Avenue | Atlantic Avenue) = (215/216) * (4/36)
P(North Carolina Avenue | Atlantic Avenue) = (215/216) * (5/36)
P(Community Chest 3 | Atlantic Avenue) = (215/216) * (6/36)
P(Pennsylvania Avenue | Atlantic Avenue) = (215/216) * (5/36)
P(Short Line Railroad | Atlantic Avenue) = (215/216) * (4/36)
P(Chance 3 | Atlantic Avenue) = (215/216) * (3/36)
P(Park Place | Atlantic Avenue) = (215/216) * (2/36)
P(Luxury Tax | Atlantic Avenue) = (215/216) * (1/36)
P(all other squares | Atlantic Avenue) = 0

Current square: Chance 3 (between Short Line Railroad and Park Place)

P(Jail | Chance 3) = (1/216) * (6/16) [rolling a third doubles given that a non-movement card was drawn] + (10/16) * (1/10) ["Go to Jail" card drawn]
P(Go | Chance 3) = (10/16) * (1/10) ["Advance to Go" card drawn] + (215/216) * (6/16) * (3/36) [third doubles was not rolled AND non-movement card was drawn AND a 4 was rolled]
P(Illinois Avenue | Chance 3) = (10/16) * (1/10) ["Advance to Illinois Avenue" card drawn]
P(St. Charles Place | Chance 3) = (10/16) * (1/10) ["Advance to St. Charles Place" card drawn]
P(Boardwalk | Chance 3) = (10/16) * (1/10) ["Advance to Boardwalk" card drawn] + (215/216) * (6/16) * (2/36) [third doubles was not rolled AND non-movement card was drawn AND a 3 was rolled]
P(Reading Railroad | Chance 3) = (10/16) * (1/10) ["Advance to Reading Railroad" card drawn] + (10/16) * (2/10) ["Advance to nearest railroad" card drawn] + (215/216) * (6/16) * (4/36) [third doubles was not rolled AND non-movement card was drawn AND a 9 was rolled]
P(Electric Company | Chance 3) = (10/16) * (1/10) ["Advance to nearest utility" card drawn]

P(Community Chest 3 | Chance 3) = (10/16) * (1/10) ["Go back three spaces" card drawn]
P(Luxury Tax | Chance 3) = (215/216) * (6/16) * (1/36) [third doubles was not rolled AND a 2 was rolled]
P(Mediterranean Avenue | Chance 3) = (215/216) * (6/16) * (4/36) [same logic]
P(Community Chest 1 | Chance 3) = (215/216) * (6/16) * (5/36)
P(Baltic Avenue | Chance 3) = (215/216) * (6/16) * (6/36)
P(Income Tax | Chance 3) = (215/216) * (6/16) * (5/36)
P(Oriental Avenue | Chance 3) = (215/216) * (6/16) * (3/36)
P(Chance 1 | Chance 3) = (215/216) * (6/16) * (2/36)
P(Vermont Avenue | Chance 3) = (215/216) * (6/16) * (1/36)
P(all other squares | Chance 3) = 0

Current square: Go to Jail

P(Jail | Go to Jail) = 1
P(all other squares | Go to Jail) = 0

Current square: Jail (pay $50 to exit)

P(Electric Company | Jail) = 1/36 [A 2 was rolled]
P(States Avenue | Jail) = 2/36 [same logic]
P(Virginia Avenue | Jail) = 3/36
P(Pennsylvania Railroad | Jail) = 4/36
P(St. James Place | Jail) = 5/36
P(Community Chest 2 | Jail) = 6/36
P(Tennessee Avenue | Jail) = 5/36
P(New York Avenue | Jail) = 4/36
P(Free Parking | Jail) = 3/36
P(Kentucky Avenue | Jail) = 2/36
P(Chance 2 | Jail) = 1/36
P(all other squares | Jail) = 0

Current square: Jail (roll doubles to exit)

P(Electric Company | Jail) = 1/6
P(Virginia Avenue | Jail) = 1/6
P(St. James Place | Jail) = 1/6
P(Tennessee Avenue | Jail) = 1/6
P(Free Parking | Jail) = 1/6
P(Chance 2 | Jail) = 1/6
P(all other squares | Jail) = 0

Computing the Steady-State Distribution

Now that the transition matrix (or matrices) P has been defined, and we know that the Markov Chain it represents is irreducible and aperiodic, we can find the steady-state distribution 𝞹 that will reveal which squares have the highest and lowest probabilities of being landed on in the long-term.

I said we were done with theory, but there's actually one more piece that we need.

There is another theorem stating that if a Markov Chain is irreducible and aperiodic, and a steady-state distribution exists, then P^t(i,j) converges to 𝞹_j for all i as t approaches infinity. In English, this means that as P is multiplied with itself over and over again, the entries in each column converge to the same value, and that the value in the jth column is the steady-state probability for state j.

To see this in action, consider the 3x3 matrix at the beginning of this post. If we multiply this matrix by itself, the result is as follows:

Raising the matrix to the third, fourth, fifth, and sixth powers gives:

(All entries have been rounded to three decimals.)

If we continue this matrix multiplication, then eventually the entries in each column will converge to the steady-state distribution, which was 𝞹 = [0.4113475    0.2695035    0.3191489], which we already see beginning to occur as the power to which P is raised increases. This occurs because the Markov Chain represented by the matrix was irreducible and aperiodic, and a steady-state distribution was guaranteed to exist.

In the case of our Monopoly transition matrix (or matrices), we know that a steady-state distribution exists. Therefore, we also know that one row of Pⁿ for a very high value of n will be close to the steady-state distribution of the Markov Chain as we have defined it.

Results

Recall that there are two transition matrices for our Monopoly Markov Chain. Matrix 1 represents Case 1, where the player's strategy to exit Jail is to always pay $50 and take a regular roll. Matrix 2 represents Case 2, where the player's strategy to exit Jail is always to wait for doubles to be rolled. The transition matrices for each of these cases were multiplied by themselves dozens of times, and the first row of the final matrix was taken as the steady-state distribution.

The results for these two cases are provided below.

Case 1: Pay $50 to exit Jail

Most likely to least likely squares:

1. Jail
2. Illinois Avenue
3. Go
4. New York Avenue
5. B & O Railroad
6. Community Chest 2 (between St. James Place and Tennessee Avenue)
7. Reading Railroad
8. Tennessee Avenue
9. Pennsylvania Railroad
10. Free Parking
11. Kentucky Avenue
12. Water Works
13. St. James Place
14. Chance 2 (between Kentucky Avenue and Indiana Avenue)
15. Indiana Avenue
16. Atlantic Avenue
17. Community Chest 3 (between North Carolina Avenue and Pennsylvania Avenue)
18. St. Charles Place
19. Ventnor Avenue
20. Pacific Avenue
21. Go to Jail
22. Boardwalk
23. North Carolina Avenue
24. Electric Company
25. Marvin Gardens
26. Pennsylvania Avenue
27. Virginia Avenue
28. Short Line Railroad
29. States Avenue
30. Income Tax
31. Vermont Avenue
32. Chance 3 (between Short Line Railroad and Park Place)
33. Chance 1 (between Oriental Avenue and Vermont Avenue)
34. Connecticut Avenue
35. Jail (Just Visiting)
36. Oriental Avenue
37. Park Place
38. Luxury Tax
39. Baltic Avenue
40. Community Chest 1 (between Mediterranean Avenue and Baltic Avenue)
41. Mediterranean Avenue

The most likely state (Jail) has a steady-state probability of 3.722%, and the least likely state (Mediterranean Avenue) has a steady-state probability of 1.968%.

Case 2: Must roll doubles to exit Jail

Most likely to least likely squares:

1. Jail
2. Chance 2 (between Kentucky Avenue and Indiana Avenue)
3. Free Parking
4. Electric Company
5. Illinois Avenue
6. Tennessee Avenue
7. Go
8. B & O Railroad
9. Reading Railroad
10. St. James Place
11. Water Works
12. Virginia Avenue
13. New York Avenue
14. St. Charles Place
15. Community Chest 3 (between North Carolina Avenue and Pennsylvania Avenue)
16. Indiana Avenue
17. Pacific Avenue
18. Atlantic Avenue
19. Boardwalk
20. Ventnor Avenue
21. Go to Jail
22. Kentucky Avenue
23. North Carolina Avenue
24. Marvin Gardens
25. Pennsylvania Railroad
26. Pennsylvania Avenue
27. Short Line Railroad
28. Community Chest 2 (between St. James Place and Tennessee Avenue)
29. Income Tax
30. Vermont Avenue
31. Chance 1 (between Oriental Avenue and Vermont Avenue)
32. Chance 3 (between Short Line Railroad and Park Place)
33. Connecticut Avenue
34. Jail (Just Visiting)
35. Oriental Avenue
36. Park Place
37. Luxury Tax
38. Baltic Avenue
39. States Avenue
40. Community Chest 1 (between Mediterranean Avenue and Baltic Avenue)
41. Mediterranean Avenue

The most likely state (Jail) has a steady-state probability of 3.712%, and the least likely state (Mediterranean Avenue) has a steady-state probability of 1.959%.

Case 1 1/2: Average steady-state probabilities of Case 1 and Case 2

Most likely to least likely squares:

1. Jail
2. Illinois Avenue
3. Go
4. Free Parking
5. B & O Railroad
6. Chance 2 (between Kentucky Avenue and Indiana Avenue)
7. Tennessee Avenue
8. Reading Railroad
9. New York Avenue
10. Electric Company
11. St. James Place
12. Water Works
13. Kentucky Avenue
14. Pennsylvania Railroad
15. St. Charles Place
16. Indiana Avenue
17. Community Chest 3 (between North Carolina Avenue and Pennsylvania Avenue)
18. Atlantic Avenue
19. Pacific Avenue
20. Community Chest 2 (between St. James Place and Tennessee Avenue)
21. Ventnor Avenue
22. Virginia Avenue
23. Boardwalk
24. Go to Jail
25. North Carolina Avenue
26. Marvin Gardens
27. Pennsylvania Avenue
28. Short Line Railroad
29. Income Tax
30. Vermont Avenue
31. Chance 3 (between Short Line Railroad and Park Place)
32. Chance 1 (between Oriental Avenue and Vermont Avenue)
33. Connecticut Avenue
34. Jail (Just Visiting)
35. States Avenue
36. Oriental Avenue
37. Park Place
38. Luxury Tax
39. Baltic Avenue
40. Community Chest 1 (between Mediterranean Avenue and Baltic Avenue)
41. Mediterranean Avenue

Findings

Considering that Jail is the most frequently landed on square, it makes you wonder what kind of people Parker Brothers thought landlords were.

Some of the differences between Cases 1 and 2 are quite stark. For instance, States Avenue is #29 in Case 1, but #39 in Case 2. Its low position in Case 2 (relying on doubles to exit Jail) is likely because States Avenue is 3 squares away from Jail, so it cannot be reached upon exit from Jail with one roll. Furthermore, it cannot even be reached within two rolls of exiting Jail in Case 2.

In Case 2, the player's exclusive reliance on doubles to exit Jail results in high rankings for those properties that are within the reach of a doubles roll from Jail: Electric Company (#4), Virginia Avenue (#12), St. James Place (#10), Tennessee Avenue (#6), Free Parking (#3), and Chance 2 (#2). Notice that for this list, as the distance from Jail increases, the ranking increases (except for the Electric Company, which a player can be sent to via a Chance card). This occurs because these squares can be reached by one roll or by two rolls, and the two rolls may have many possible combinations adding up to the square's distance from Jail.

In both Case 1 and Case 2, squares such as Illinois Avenue and B & O Railroad score highly because they are accessible from Jail with 2-3 rolls, and there are many combinations of rolls that can result in one landing on those squares. A player can also be sent to each of these squares via Chance cards.

Mediterranean Avenue and Baltic Avenue, the first properties after Go, are among the least-visited properties (with Mediterranean being the absolute least of all). These are also the lowest-value properties on the board. The combination of those two factors should persuade a player that these properties are not worthwhile investments.

An interesting version of Monopoly would be one based on the principle of "low risk/low reward, and high risk/high reward". In this version of the game, comparatively rarely landed on properties such as Mediterranean and Baltic would be the most expensive properties on the board and would have a high payoff, while the most frequently landed on properties such as Illinois and Tennessee would be the cheapest properties on the board and would have a lower payoff.

Monopoly is primarily about obtaining groups of properties--a monopoly on a certain area of town. So, which monopolies are most likely to be visited? For each Case, the sums of the constituent steady-state probabilities in each monopoly were obtained and ranked.

Case 1

1. Railroads: 10.13% (Reading, Pennsylvania, B & O, Short Line)
2. Orange: 8.13% (St. James, Tennessee, New York)
3. Red: 7.80% (Kentucky, Indiana, Illinois)
4. Yellow: 7.27% (Atlantic, Ventnor, Marvin Gardens)
5. Green: 7.17% (Pacific, North Carolina, Pennsylvania)
6. Magenta: 7.10% (St. Charles, States, Virginia)
7. Light blue: 6.33% (Oriental, Vermont, Connecticut)
8. Utilities: 5.52% (Electric Company, Water Works)
9. Navy: 4.45% (Park Place, Boardwalk)
10. Purple: 3.95% (Mediterranean, Baltic)

Case 2

1. Railroads: 10.51% (Reading, Pennsylvania, B & O, Short Line)
2. Orange: 8.15% (St. James, Tennessee, New York)
3. Red: 8.09% (Kentucky, Indiana, Illinois)
4. Yellow: 7.36% (Atlantic, Ventnor, Marvin Gardens)
5. Green: 7.20% (Pacific, North Carolina, Pennsylvania)
6. Magenta: 6.96% (St. Charles, States, Virginia)
7. Light blue: 6.34% (Oriental, Vermont, Connecticut)
8. Utilities: 4.99% (Electric Company, Water Works)
9. Navy: 4.44% (Park Place, Boardwalk)
10. Purple: 3.96% (Mediterranean, Baltic)

Case 1 1/2

1. Railroads: 10.32% (Reading, Pennsylvania, B & O, Short Line)
2. Orange: 8.14% (St. James, Tennessee, New York)
3. Red: 7.94% (Kentucky, Indiana, Illinois)
4. Yellow: 7.32% (Atlantic, Ventnor, Marvin Gardens)
5. Green: 7.18% (Pacific, North Carolina, Pennsylvania)
6. Magenta: 7.03% (St. Charles, States, Virginia)
7. Light blue: 6.33% (Oriental, Vermont, Connecticut)
8. Utilities: 5.26% (Electric Company, Water Works)
9. Navy: 4.45% (Park Place, Boardwalk)
10. Purple: 3.96% (Mediterranean, Baltic)

The totals do not sum to 100% because of the non-property squares on the board.

Perhaps not surprisingly, the single four-property monopoly ranks the highest in all cases, and the three two-property monopolies rank the lowest. Certainly, the more squares comprising a monopoly, the higher the chance that a square within the monopoly will be visited.

Now, these rankings do not mean that these are the monopolies that should necessarily be sought out first. The expected return on these properties must also be considered. For instance, although the railroads are the monopoly most frequently landed on, the maximum income that can be obtained from them is $200 ($400 if a player is sent to a railroad via a Chance card), which is comparatively low against the $2,000 that can be obtained if a player lands on Boardwalk with a hotel. In addition, the payoff from a colored property (i.e., not railroads or utilities) is fairly minimal unless and until a player has a monopoly involving that property.

Nevertheless, a key takeaway is that the orange and red properties (St. James Place, Tennessee Avenue, New York Avenue, Kentucky Avenue, Indiana Avenue, and Illinois Avenue) are among the most-frequently landed upon squares, a claim which is generally common knowledge.

2020 in flight

Apropos of nothing: Blogger changed their user interface, and it is awful. Formatting is fifteen times harder now.

This post is about two months late, which is silly because I created this post's graphics even before 2020 was over (read: when it was clear that COVID-19 cases were going to keep climbing and I would definitely not be flying again anytime soon). But, better late than never, I suppose.

GOALS FOR 2020

At the end of 2019, I set the following goals for myself with respect to air travel in 2020:

1. Hit my fifth continent--Australia--using the ~$100 credit I had with Qantas.

2. Achieve airline status again.

3. Fly out of my hometown's airport (CCR).

4. Fly in every month of the calendar year.

Here is how I did.

1. [Insert clown emoji] That $100 is almost certainly gone, too. It was originally supposed to expire in October 2020. I haven't even bothered to check.
   

2. I mean, everyone who had status with Alaska at the beginning of 2020 got their status renewed for 2021. So I guess I technically accomplished this goal, although it was through no effort on my part. It feels a bit like cheating.

3. Nope.

4. [Insert clown emoji]

I mean, who could have known what 2020 was going to be, right?

YEAR-OVER-YEAR TRENDS

5,986 miles flown in 2020, measured as great-circle distances in statute miles. 87% decrease from 2019; 90% decrease from 2018.

8 segments flown in 2020. 79% decrease from 2019; 83% decrease from 2018. The total segments I flew in 2018 is equal to the number of segments flown in 2019 and 2020 combined.

$1,168.37 in airfare paid in 2020, which considers the actual amounts charged to the card, inclusive of airline credits, miles used, taxes, and fees. 79% decrease from 2019; 88% decrease from 2018.

MORE ON MILES

Gee, what happened in March?

American unseated Alaska this year, although I did fly more segments with Alaska (3) than with American (2).

Regretfully, I did fly with United in 2020. Although it was for one segment on which I used miles, so I at least have the satisfaction of knowing that I didn't give them any money.

"Others" should really just be "Other", as it consists only of Porter Airlines.

The B737 unseated the A320 this year. The last year in which I flew more miles on the B737 than the A320 was 2016.

"Others" includes the B767, B777, CRJ, and Q400.

It is Wednesday, my dudes.

MORE ON MONEY

Southwest's "Wanna Get Away" fare is the sole element of the "Other or N/A" category. Since I only flew 8 segments, this pie chart comes out much too cleanly for my liking.

My average cost per mile for leisure travel was not quite as low as I would have liked to have achieved (I got it down to $0.09/mile in 2019), but in my defense, I had only 3 leisure reservations to work with in 2020. I also had only 2 business reservations.

The minimum cost per mile for leisure travel is so low because that segment was purchased with miles, so I only had to pay the $5.60 in taxes and fees.

"Lead time" is defined as how far in advance of departure the ticket was purchased.

In a Normal Year^TM, I would show the minimum, 25th percentile, median, average, 75th percentile, and maximum of these distributions. However, you can't really show the 25th or 75th percentiles when your sample sizes are 2 business and 3 leisure reservations.

The small sample size is also why the leisure distribution is so low. The 16-day lead time was a (relatively) short-notice decision to spend a weekend in Los Angeles, and the other two lead times of 1 and 2 days were when I spontaneously decided to fly to San Diego for my sister's college graduation.

Rebecca Black would be so proud.

MORE ON WHERE

Border closures were trending in 2020.

A "visit" is defined as a segment either originating or terminating at a given airport. So connections grant two visits to the connecting airport.

The size of the font is directly proportional to the number of visits.

Similar to the fare class pie chart, it's not terribly interesting this year. This is what happens when you're dealing with a small sample size. I flew four round trips out of SFO to each of the four destinations in the graphic. In fact, the next image shows all the routes I flew in 2020.

Really riveting stuff.

WHILE IN FLIGHT

I don't know if I had ever made a graph before this one that plots natural numbers but has a y-axis that only goes up to 2.

This was one of my greatest accomplishments of 2020: avoiding the middle seat. And, for the record, 6 of my 8 seats were before the airlines began blocking middle seats.

Once again, not a terribly exciting shape with a sample size of 8.

A "pushback pause" is defined as the time when the aircraft is stationary after being pushed back from the gate. Specifically, it begins when the aircraft stops moving backward and the tug begins detaching, and it ends when the aircraft begins moving forward on its own power.

I made up for my lack of in-flight drinking in 2020 by drinking more at home. It's cheaper that way.

Each of the upgrades were on segments between Northern and Southern California. It almost seems a waste to get upgraded on such segments which are only an hour long. Although, my first class upgrade was when I had my one in-flight drink of 2020, so perhaps that made it worthwhile.

HIGHLIGHTS FROM 2020

Typically, this is where I would list of my "firsts", such as the first time flying on a specific aircraft type, to/from a country or U.S. state, on an airline, or some other activity unique to the year.

Unfortunately, there is nothing exciting of this manner to report for 2020. I think the year 2020 was not the time to be taking risks or to pursue ambitious "firsts" while flying. "First time wearing a face mask during a flight" is about as creative as I can get.

GOALS FOR 2021

Even as I write this in March, what air travel will look like even a month from now is very much up in the air (pun intended). As was the case in 2020, it may be foolish to set goals for air travel in 2021 given how much uncertainty still exists in the world.

Nevertheless, goals are supposed to be something that challenge you. So I will set one sole goal for myself in 2021, which is not new to this year: achieve airline status for 2022. This could be very easy if most of the world remains shut down and airlines decide again to automatically renew statuses or lower their eligibility requirements; or it could be very difficult if things reopen given how much airlines have cut back their schedules.

For example, starting in April or May, Alaska will only fly SFO-EWR and EWR-SFO non-stops once a day on Sunday, Monday, Thursday, and Friday. This route used to have 2-3 daily non-stops in both directions. Furthermore, this route is now much more expensive on Alaska than on the obvious candidate for this route, United (hub to hub). So, if the winter continues to thaw and things reopen, I may need to make some tough decisions in order to achieve this goal.

Thoughts After an Attempted Coup

On January 6, 2021, both Houses of Congress were to meet to count the electoral college votes, completing our country's head-scratchingly bizarre electoral process. I say "head-scratchingly bizarre" because there are multiple points during the process in which the public's preference can be distorted or even nullified. Yesterday was one of those points. Twelve Republican Senators went into the meeting vowing to object to the counted electoral votes on the grounds of widespread election irregularities and/or conspiracy theories about widespread voter fraud. If enough Senators had agreed to do the same, then the electoral college votes--already a distortion of the national popular vote--would have been disregarded.

As the proceedings began, American citizens who believed the 2020 presidential election was stolen from trumpet breached the United States Capitol building and stormed the halls of Congress. Representatives and Senators hid in secure locations until the Capitol was secured. Late in the evening, the House and Senate continued with the proceedings. Six of the twelve Senators who had vowed to reject the electoral college results spoke, saying that they would recant their previous positions given What Had Happened in the Capitol That Day.

Now, What Had Happened in the Capitol That Day had nothing to do with what the dissenting Senators were calling into question--the legitimacy of the voting process and/or the electoral process in the 2020 election. Consequently, it had absolutely no bearing on the perception of the legitimacy of the 2020 election. The six Senators who recanted thus effectively admitted that they didn't actually believe the election was illegitimate or sufficiently irregular. It was all a ruse to keep trumpet in power and--though this may have not been the express aim--to kill democracy in the United States.

In a strange way, I actually almost hold a bit more respect for the six Senators who did not change their minds despite What Had Happened in the Capitol That Day. They were not swayed by occurrences which were orthogonal to their main arguments. Of course, their steadfastness does not reveal whether they truly believe their own claims of irregularities and fraudulence.

* * *

 

People wearing "Make America Great Again" hats barrelled past law enforcement while carrying Confederate flags into the United States Capitol building.

Okay. There is so much cognitive dissonance there.

The MAGA crowd and the Blue Lives Matter crowd tend to overlap quite a bit. But in this case, they disregarded law enforcement.

What does "Make America Great Again" mean when you carry a flag representing a distinct sovereign nation that no longer exists?

Explain how a Confederate flag relates to claims about a stolen American--and therefore foreign--election.

What "America" are you fighting for?

That America doesn't exist. I don't think that America ever existed. You cannot make it great "again".

* * *

 

terrorism

noun

the unlawful use of violence and intimidation, especially against civilians, in the pursuit of political aims.

[courtesy of a Google search]

Unlawful: Breaking into the United States Capitol.

Use of violence: Trashing the Capitol.

Use of intimidation: Leaving a note in Nancy Pelosi's office saying, "WE WILL NOT BACK DOWN". Graffiti reading, "MURDER THE MEDIA" on an exterior wall.

Against civilians: Are Representatives and Senators considered "civilians"? Let's say yes, because I think the definition is distinguishing between civilians and soldiers. Either way, this isn't a necessary condition to be considered "terrorism"--it is for emphasis ("especially").

Pursuit of political aims: Objective was to stop the certification of an election to keep trumpet in office.

Yep, it checks out.

Does the fact that Confederate flags were carried in mean it was an attack by a foreign power? Would people be more likely to call it "terrorism" then?