11 March 2019

"Alignments that aren't stupid" in action

I wrote about a new concept to map onto the term alignment some years ago here: https://spellsandsteel.blogspot.com/2015/12/alignments-that-arent-stupid.html. I wanted to write a little about using this system in action in a recent game.

The TL;DR of the above system is using the concept of alignment to denote who the NPC looks to for their opinions, and what is guiding their actions. Not Law or Chaos, or Lawful Good, or anything like that, but rather an alignment of "Reeve" (the character likes and supports the Reeve of the village, and agrees with them about most things), or perhaps "Outlaws".

What I've added to this system recently is the concept of true (or secret) alignment and open alignment - characters can act in public, or to strangers, like they are of one alignment (say, Reeve), but in private, they secretly are aligned with the Outlaws.

The setting is a village in Nottinghamshire in about 1347. The King is away in France, and in his absence, the Sheriff of Nottinghamshire (that perennial villain), is abusing his power. When randomly generating the families in the village, I added a line to the random generator which generated their alignment in the sense described above (with a chance of a second, secret alignment). Children under 18 didn't get opinions, but older children did (with a higher-weighted chance of having one of their parent's opinions).

So, when the players were canvassing their neighbours, looking for support, I would have entries for the village households like this:

Head of Household: William, 37; Sheriff (secretly Outlaws)
Partner: Alice, 34; King (secretly Sheriff)
Children: John, 14; Jane, 7

Right away, we have drama. William and Alice openly support the Sheriff and the King, so there's conflict there, but secretly William supports the Outlaws (who are working against the Sheriff) and Alice secretly supports the Sheriff (the natural enemies of the Outlaws). Maybe William is just sympathetic, perhaps because the Sheriff has wronged him or his friends, or maybe he actively aids the Outlaws with food or weapons. If the players can gain his trust, he could put them in touch with the Outlaws.

But Alice, who in public talks about the King's justice and how things will be put right when he returns, secretly supports the Sheriff. Maybe she was a childhood friend of the Sheriff, or one of his retainers. Or maybe Alice can just see which way the wind is blowing, and wants to throw her lot in with the winning side.

So, depending on who in the household the characters approach, and what tactics they use, I'm set up to roleplay an interesting encounter. If the characters are aligned with the Outlaws, William will be reluctant to help them publicly, but if they can get him alone and convince him of their bona fides, he could possibly put them in touch with the Outlaws or give them assistance. If they approach Alice, she's likely to rebuff them in public or private, talking about how the Outlaws are operating outside the King's justice and how the King will deal harshly with them when he returns - but since she's secretly aligned with the Sheriff, she's likely to inform him (directly or indirectly) of her conversation with the PC's, causing them trouble down the line.

Alexis at Tao of D&D talks about "features" in his book, How to Run. This is a design-mindset approach which asks, how will this thing be used? NPCs are there for the characters to interact with, but the traditional approach to character traits (distinctive physical features, backstory, verbal tics, mannerisms) are not features, and do not offer nothing for the players to interact with. Anything the players can't interact with is, in a real sense, irrelevant to the game.

Alignments, as presented, very much are features of the NPC: they define and colours all interactions with them, define whether they will support or oppose the player, and (with secret alignments) add depth and reward for continued interaction.

I will be generating alignments like this for all NPCs in my games going forward.

12 October 2018

Phased Real-Time Combat - The Solution You Didn't Know You Needed

Phased Real-Time round structure dates back to the 1824 Kriegspiel rules; that is to say, to the very origin of wargames and of roleplaying games. The 1959 board game Diplomacy is a more familiar example. It appears in computer games and simulations, as well, such as TacOps v4 and Combat Mission.

Roleplaying games have largely eschewed this structure in favour of the deeply flawed Sequential round structure. I hope to re-popularize Phased Real-Time rounds.

What is Phased Real-Time Round Structure?

PRT breaks a "round" down into phases - typically an orders phase and one or more resolution phases.

Within each phase, there is no concept of "order" or "initiative" as there would be in a Sequential round structure. All operations are considered to happen in real time, as they do in the fiction.

Orders Phase

Here, each player outlines what they choose to do on their turn. In a competitive game, generally speaking there must be some element of secrecy to this phase. Diplomacy, for example, requires the players to write their orders down on paper using a specific (and strictly enforced) syntax. Kriegspiel keeps the staffs of the two sides in separate rooms, where they write orders on paper and hand them to the umpires.

In a cooperative game like D&D, it is hardly necessary to keep the players in the dark as to each other's intentions. Interesting and tense results could be had from enforcing secrecy and non-communication, but in a friendly cooperative game this is not worth the trade-offs. A degree of honesty is required on the part of the DM; they must not alter the orders of the monsters in light of information gleaned during the turn.

Execution Phase

Here, the accumulated orders are resolved. Generally speaking, there will be obvious groupings of orders which must be resolved together, and which can be resolved independently of any other grouping.

In a game like Diplomacy, resolution is strictly algorithmic, requiring no interpretation on the part of the participants. This is possible because of the extremely regimented nature of a board game. In a roleplaying game, some amount of interpretation may be required. In this case, participants must apply their judgement of the reality of the fictional world to the unique situation at hand. This makes the participants and the fictional reality first-class citizens in order resolution.

What is Sequential Round Structure?

Sequential Turn Structure breaks a round down into one turn per agent. Each agent's actions are declared and resolved in a certain order, one after the other. The order of the turns can be set or changeable. There are sometimes mechanisms for interrupting another agent's turn, which can create highly complex situations.

Each turn is considered completely resolved and completed before the next begins (except for interruption rules), with all other agents frozen in time and space for the duration of the turn.

Freezing of inactive agents either requires tactical decisions informed only by the game rules and not by the fictional reality, or complex rules surrounding interrupting the active agent's turn.

There is a certain amount of fudging of the interaction of the game rules and the fictional reality required. A sequential resolution of actions takes more fictional time than a simultaneous resolution of the same actions, but we assume only the time required for simultaneous resolution has passed. This allows exploits like the infamous Peasant Rail Gun, for which the best answer of a sequential ruleset is to add additional game rules which corral the fictional issue.

Why Phased Real-Time Round Structure?

Sequential round structure works well in wholly abstract games like Chess. Chess is not modeling any kind of reality, and so the "freezing" of the black side during white's turn is not a problem of verisimilitude. In a roleplaying game, however, there is a fictional reality being modeled. Sequential round structure fails when asked to model a fictional reality. Verisimilitude is broken when the artifacts of the sequential round structure defy the obvious realities of the fictional world.

Phased Real-Time has no such issues. Actions are resolved within a finite and defined amount of time. The fictional reality, as interpreted through the judgement of the participants, is paramount. Complex special rules for interruptions are not required.

How to Switch to Phased Real-Time Round Structure?

Switching over to phased real-time round structure is quite easy for most RPGs, requiring little more than abandoning initiative, attacks of opportunity, etc. and substituting PRT. My suggested protocol is as follows:
  • Orders Phase
    • DM Hints at Monster/NPC actions
    • Players Declare Actions
  • Execution Phase
    • Interdependent Groups are Identified
    • Actions are Resolved Group by Group (using your existing ruleset)
Let's examine each step in turn.

DM Hints at Monster/NPC Actions (Orders Phase)

This step gives agency to the players. Having a hint of what their opponents are doing allows them to make an informed choice about their own actions.

Players Declare Actions (Orders Phase)

In whatever order is convenient (clockwise from the DM's left works well), the players declare the action they would like their character to undertake this round. There will inevitably be tabletalk regarding what everyone should do. This is a normal and natural part of the game, and should be allowed.

Interdependent Groups are Identified

Rare is the mass melee, where all combatants are engaged in a single interwined struggle. More typical is several fairly discrete combats happening simultaneously. One of the beautiful parts of the PRT round structure is these groups are identified and resolved together, creating a narrative structure to the round which is lacking in Sequential round structures.

Actions are Resolved Group by Group

Resolve each group entirely before moving on to the next. This best allows for the application of judgement to the resolution, as all the facts can easily be kept in mind.

You can use most any existing RPG ruleset to resolve the actual combat.

Whither Initiative?

If you are enamored of initiative, PRT round structure actually can actually make initiative more meaningful if you choose to incorporate it. Here is one example of hooking in an initiative system.

If the players have the initiative, the DM tells them what the monster actions will be and lets them give their orders in full knowledge of the monsters orders. If the monsters/NPCs have the initiative, the players must declare their actions blind and the DM may assign the monster/NPC orders with the knowledge of the player's orders.

Example of Play

Orders Phase

DM: The Pig-faced Orc is drawing a flaming sword and is moving towards Bill, the Horned Orc is charging Jane, and the Fanged Ogre is preparing to throw a rock. What are your actions?

Bill (fighter): I'm going to draw my sword and attack thePig-faced Orc.
Wilma (cleric): I'm going to help Bill fight the Pig-faced Orc.
Steve (wizard): I'll cast Figface on the Fanged Ogre.
Jane (thief): I'm going to flip the table to block the Horned Orc's advance, and look for my Horn of Fury in my pack.
Phil (fighter): I'll shoot an arrow at the Horned Orc to hopefully distract it from Jane while she looks for the Horn.

Execution Phase

DM: Ok, so Bill, Wilma, and the Pig-Faced Orc are engaged. Jane, Phil, and the Horned Orc are engaged. Steve and the Ogre are engaged, as the Ogre is throwing its rock at Steve.

DM: Bill, Wilma, roll to-hit against AC 7. The Pig-faced Orc rolled a 19, and hits Bill for 5 damage.

Bill: I missed.

Wilma: I hit for 3 damage.

DM: The Pig-Faced Orc makes a morale check for taking damage, and it fails! It turns tail.

DM: Jane, you flip the table well before the Horned Orc reaches you. Roll a Dex check to find the Horn of Fury this round. The Horned Orc's charge is blocked by the table; he's clambering over it, though.

DM: Phil, roll to-hit against AC 5.

Jane: Damn, I failed the roll. I'll have to keep looking next round.

Phil: I miss the Horned Orc.

DM: Cool. The Ogre is throwing a big rock at Steve, and Steve is trying to get off the Figface spell to blind the Ogre. Since your spell will be ruined if you're hit by the rock before you cast it, Steve, roll a tiebreaker with the Ogre - d10+Dex Mod [note that this is basically an initiative check like we would see in a Sequential system, but in a PRT system, it is reserved for actions where order matters and is non-obvious].

Phil: I get a 7.

DM: Bad news, the Ogre got a 10. He's going to get his rock attack off before you blind him.

DM: But, the Ogre rolled a 5 to-hit on his rock attack, and the boulder smashes harmlessly into the wall behind you. Your spell shrivels its face up, blinding him and making it very hard to breathe.


Phased-Real Time is a simple drop-in substitution for your existing Sequential round structure. It is simple to understand, and fluid in practice. It produces combats consisting of integrated small scenes, rather than a monotonous series of disjointed actions. It is fiction-first, and respects the reality of your game world. It puts the participant's judgement in the driver's seat regarding action sequencing, while keeping standard initiative rolls in its back pocket for corner cases.

Phased-Real Time is strictly superior to Sequential. Switch your game over today!

16 August 2018

Healing: The Disabled List and Character Rosters

Major League Baseball has something called a disabled list. I'm sure other sports have something similar, but I only follow baseball. Athletes get hurt. Sometimes badly, and sometimes often. With limited rosters, teams need flexibility to bring in a replacement, usually from the team's AAA affiliate, sometimes elsewhere in the farm system, or more seldom by trade/free agency.

Adventurers also get hurt. Sometimes badly, and generally often. In most games, however, players don't have a roster of characters that they use. What if they did? And what if they could put a character on the disabled list in order to call up a fresh replacement "from the farm"? And what if, like a major league team, the roster size was limited, creating (hopefully) interesting decisions about when to DL a player, when to soldier on, and what to do about the roster crunch created by characters coming off the DL?

I feel a mechanic like this is especially important when using a descriptive damage system as I do. Characters tend to get pretty roughed up, and there needs to be a better way of dealing with it than bogging down the whole party going back to home base.

The Character Roster



Players have an "primary roster" (edit: also called the "active roster") of characters they can choose to adventure with (edit: adventure with as player characters). They also have a "secondary roster" which consists of their active roster and any henchmen they have.

The active roster is 5 characters, and the secondary roster is 10 characters.



Players can't just freely put people on and off the disabled list with no consequences. There's pressure to choose their favourite characters, creating interesting decisions.



At the beginning of an adventure (i.e. when leaving "home base"), the player may choose to play any character from their active roster. They may take along any henchmen from their secondary roster.

The Disabled List


There are 2 disabled lists, the 14-day DL and the 60-day DL. Characters on the 14-day DL are removed from the primary roster. Characters on the 60-day DL are removed from the secondary roster.

Once a character has been put on the DL, they are an NPC and simply rest and heal for the duration. They are safe and will not die. They remain an NPC until their time on the DL is up (either 14-day or 60-day).

When you put a character on the DL, either roll up a fresh 1st-level character, take over a henchman or 1st-level hireling, or bring in a standby character from your roster.



The character is healed of all flesh wounds after 14 days, and all (non-permanent) wounds after 60 days. The character is removed from the active roster, making room for a fresh character.

The player doesn't have to play a gimped, wounded character in the meantime.

It encourages procurement and maintenance of a home base.

It gives the party more stamina to work more than a 15-minute adventuring day.



Declare you are moving your active character onto the disabled list, and specify which one.


  • character is not in immediate danger (in combat, in quicksand, surrounded by fire)
  • character could plausibly make it back to base safely (if they can't walk, someone has to go with them)
    • they're not stuck in a pit; they're not surrounded by enemies; they're not being actively hunted for)
  • must have a "home base" available owned/leased by the party (somewhere they can get food and warm bed)

Roster Moves



The character is removed from the active roster and becomes an NPC henchman, with all that entails (they are an NPC with a morale score, they are nominally controlled by the DM but in practice are largely controlled by the PC, etc.).

Characters get 2 "options". Optioning a character uses up an option.


A henchman becomes a PC, with all that entails.


The character is removed from both the active and secondary roster. They are forever more an NPC, and leave the party to make their own way in the world. The DM is free to have the character resurface as a rival, friend, or acquaintance; or perhaps they will never be heard from again; or maybe they'll turn up dead - in any case, their fate is out of the hands of the player.

12 August 2018

Archery: Variable Range Increments

Following up from yesterday's post. A reminder on methodology.

I've taken men's longbow classifications from the Braintree Bowmen in the UK, ranging from 3rd class to Grandmaster Bowman; analyzed them with a tool I made to convert archery scores into to-hit rates on a d20; and applied those results to a variety of ranges for a 65cm target, which I reckon to be comparable to shooting at a human torso.

Each line on the chart represents a score in a "720 round" of archery, converted to be hit/miss against a 65cm target at the stated range.

Analyzing this chart, a thought occurred to me: instead of fixed range increments and a rising attack bonus, what if we had fixed attack bonuses/penalties and range increments that vary with level?

What we get is a system that seems (to me) much more verisimilitudinous. At short range, there's little to no difference in skill. At long range, shots that are essentially impossible for an unskilled bowmen will be easy for a skilled shooter.

What we give up is some granularity in progression, as we probably don't want the range increments to change in such small steps between levels that it makes no real difference.

65cm target 0 -5 -10 -15 -20
3rd Class 5m 10m 15m 20m 25m
2nd Class 10m 20m 30m 40m 50m
Bowman 15m 30m 45m 50m 65m
Master Bowman 20m 40m 60m 80m 100m
Grandmaster Bowman 25m 50m 75m 100m 125m
World-Class Bowman 30m 60m 90m 120m 150m

Now, how about changing target size? The following charts show the effects of changing the target size by doubles and halves.

What we can notice is that for each doubling of  target size, it's as if the lowest-skilled (31 score) bowman is shooting as if they were a middle-skill bowman (157 score). For the "Bowman", it's like shooting as a "Grandmaster Bowman", and so on.

So, changing target sizes can be fairly easily modeled by shifting the effective skill of the archer - double the target size, increase the effective skill two increments. So a "3rd-class" bowman shooting at a target 1.3m across would use the "Bowman" row on the chart.

This is for considered shots in good conditions. Later we will examine more adverse conditions.

8 August 2018

Archery: Towards a Foundation

I was thinking of archery, and was immediately reminded that Delta had done some good work on the subject some time ago: http://deltasdnd.blogspot.ca/2014/04/a-model-of-archery-for-d.html

There's some good stuff there, and I think the methodology is promising, but I think some upstream errors make it less useful/accurate than it could be. With that in mind, I set out to build on those ideas.

Delta used an article in an old Dragon magazine by Robert Barrow (Dragon #58, pp 47-49) as his starting point, which in turn cites Archery by Longman and Walrond, published New York, 1894. I was not able to find that edition online, but did find Archery by Longman and Walrond, London, 1894, which surely must be different edition (likely the original edition) of the same book. Unfortunately, the page numbering is different, and Barrow's citation of pp 240 leads us to a discussion of the history of  the Royal Company of Archers in Scotland, which, while no doubt interesting in its own way, bears not at all on the matter to hand. There is a helpful chart on pp 264, which I would have surmised is the one used by Barrow, except that when I tabulated the data, it doesn't quite match the figures he quoted.

Barrow quotes 92%, 81%, 54% hits at 60, 80, and 100 yards, respectively. I calculated 96%, 86%, and 65%. Perhaps there was an error in his calculations; it was, after all, 1982, and he likely did not have a spreadsheet to do the calculations quickly and easily. Or perhaps he was simply using different data from a different part of the book and the editions are even more different than seems likely. In any case.

What I've done is write a little commandline tool that allows us to enter in a wide range of archery scores and get back out a hit rate for a given range and target size. The Archery data was useful, as it had a hit rate and a score, whereas most modern sources only list the score, and this was very useful to make sure I was getting sane results. I can set the output and input ranges and target sizes to be the same, and the compare the hit rate my model predicted based on the score with the actual hit rate in the data set. I found results were typically within 10% or so, which I'd say is more than good enough for our purposes.

One thing I can tell you - modern archers are much, much better than the archers in the Archery dataset.

So, with this tool in hand, I set out to find some data on how good archers are. I didn't just want a soup of scores, I wanted some guidance/interpretation of what those scores meant. Eventually, I found this site, which helpfully lists scores for 6 "classes" of archers, from 3rd class to Grandmaster. It didn't go all the way to Grandmaster for the 720 round scores I had calibrated my tool for, but this site helpfully allows you to convert a score from one type of round to another (using a similar approach to the CLI tool I made).

So, this got me scores with meanings attached, and my tool allows the conversion of these scores into hit rates for any target size and any range. Results of my investigations in a subsequent post.

3 March 2018

Response to "Fight Semantics" at Tao of D&D

 Alexis and I were discussing why players always want to don their armour before a fight, and we got drawn into the question of the weapon's role in the defense. Alexis is of the opinion that the weapon is already accounted for in the D&D system by the fact that we do not auto-hit an unarmed combatant. This was originally in comments on this post on Tao of D&D, and then continued here, in a new post by Alexis.

Incidentally, that first post is an excerpt from Alexis' "Masterclass" on DM'ing, available to his patrons on Patreon (of which I am one). I can highly recommend it. Alexis analyzes the DMing techniques he has used in his online campaigns in detail, outlining his thought process, what he was trying to do, what worked and what didn't, and generally giving a fascinating look inside the mind of someone who has run more D&D and thought more about the game than probably anyone around.

Alexis is uniquely suited to running a class like this, having DMed for decades, being one of the most considered thinkers on the game (as demonstrated by his long-running blog and several thoughtful books), and by virtue of having two fairly long-running campaigns run on line using Blogger (and which are therefore preserved in their entirety, and available for extensive analysis and discussion).

You should go sign up! Anyway, back to our regularly scheduled programming. I promise I'm not being paid to say that, haha!

Weapons on the Defensive - not in D&D!

Alexis was making the case that the weapon's role in your defense is already accounted for in the system.

There's only one problem with that idea - dropping your weapon does not change your opponent's chance to hit you whatsoever.

Which I would say is proof-positive that the weapon's defensive role is not, in any way, included in the system. Increasing defensive prowess with level is included (as represented by increasing HP per level). The weapon's role is decidedly not.

Increasing Defensive Skill and Hit Points

As Alexis points out, D&D uses increasing hit points to model increasing defensive ability with level, but not quite in the way Alexis described (at least not in its original incarnation, which I would argue is still the metaphor used in the game today, despite its origins being lost in time).

In Chainmail, one hit was one kill. In OD&D, *on average* one hit was one kill, but they made HP and damage both 1d6 to add a little randomness. The idea was still the Chainmail conceit that one hit killed a normal person. The Chainmail "hero" took 4 hits to kill, essentially giving the hero 3 mulligans on death, and that's the same approach that OD&D took - each additional hit die *on average* is one mulligan against death. A "hit" was always supposed to be a lethal, fight-ending blow. This neatly explains why characters fight at 100% until they're dead. The waters were muddied as time went on and hit dice and weapons started being different dice, but if you clear away the murk, that is the foundation of the current system.

The Issues with Defense in D&D

In analyzing the issues with defense in D&D combat, we need to tease out and treat individually each of the three factors to defense: hard-to-hit, hard-to-hurt, and hard-to-kill. D&D rolls hard-to-hit and hard-to-hurt into one stat (AC), but also puts some of the hard-to-hit into the hit point. It ignores the role of the weapon in hard-to-hit, despite being (possibly) the most important factor. The hit point then involves some hard-to-hit and some hard-to-kill. It's all very muddled.

This muddling causes a variety of issues. For just one, a giant-thrown boulder falling on you doesn't care one whit about your defensive abilities or your armour. It just does damage, and will do damage to Lancelot or his squire equally. But because defensive abilities and life force are rolled into your hit points, and because armour and dexterity are rolled into your hard-to-hit, it's difficult or impossible to tease out how to apply damage in a scenario like that. Theoretically, it would be something like having the giant roll against "touch AC" (which I think only existed as of 3e, being your armour class not counting your armour) and then roll damage multiplied by level (to remove the effect of increasing defensive ability being rolled into the hit points).

Also, because damage doesn't scale up with level, it means that two high-level characters who are equally matched will take a long time to finish a fight, whereas two low-level characters who are equally matched will finish quickly. Which doesn't pass the smell test to me.

Healing is difficult to reconcile into this system, because ostensibly some of your hit points are meat and blood, and some of them are fatigue, and some of them are your increased defensive ability. Well, which ones does a cure spell work on? Which ones get whittled away by walking hours in the cold? It's unclear, because the HP is a mixed metaphor from the get-go.


You can actually fix all this pretty easily in D&D combat by making the following changes.


Weapons and level influence hard-to-hit (i.e. your "AC" is set by a level modifier and a per-weapon modifier). One should actually have three hard-to-hit stats - dodge, block, and parry. Dodge is used against the giant's boulder, a ballista bolt, dragon breath, etc - anything that can't be blocked or parried. Block is used against normal missile attacks and includes the shield. Parry is used against normal melee attacks, and includes the weapon's defensive bonus.


Armour influences hard-to-hurt (i.e. it reduces incoming hit point damage).


And hit points represent only hard-to-kill (i.e. just meat and blood - all hit point loss is of a constant magnitude). This integrates naturally with Alexis' hit points for mass system.


Really, the only difference mechanically from D&D combat is including damage reduction, which is easy enough and better matches our intuitive understanding of what armour does anyway. D&D needs the three categories of hard-to-hit, it just ignores the difference (to its detriment), and it's easy enough to add these to the character sheet.

I don't think any of this is better just because it's more "realistic" (although it is). It's better because it makes each stat clearly model something understandable. This makes it intuitive, easy to understand, and easy to apply to edge cases that weren't necessarily envisioned by the designer. Because the model is explicit and understandable, it's easy for the user to tinker with it, and easy for the DM to make rulings on the fly.

18 December 2017

Railroads vs. Sandboxes

In response to Alexis, here: http://tao-dnd.blogspot.ca/2017/12/everything-is-road.html

This is interesting food for thought, but I don't know that I'm sold on such a broad definition of a railroad - unless I've misunderstood, Alexis is saying any voyage with a destination is now a railroad.

In my mind, the defining features of railroads are the permanence and inflexibility of the route, the scheduled departure and arrival, and the impossibility of changing destination once en route. On a railroad voyage, you are not in control except at stations (where your only choice is to disembark or remain on board until the next station). I cannot board a train and go to my home town. The trains haven't gone there for sixty years. It is impossible. I can go to Montreal, but once I have boarded, I can get off at Kingston or a few other points, but I cannot decide to go to Barrie without disembarking and purchasing a new fare.

Contrast this with a road and a car. I get in, I can pull off the road at any point, stop at a rest stop or just on the shoulder, change destinations at will (along the road network, of course, but at any time). I am in control of the route, the schedule, the destination, and even the moment-to-moment course of the  vehicle, whereas with a train I am not.

The word railroad - emphasis on the "rail" - describes the most restrictive form of travel. Even airlines and cruises, which are superficially similar (set schedules and destinations), have more freedom than railroads. The captain can choose to change destinations at any time to anywhere within range with an airport/harbour, say, in the event of a medical or mechanical emergency, or a hijacking. 

I don't think it's coincidental that we use the most restrictive kind of travel as a metaphor for the most restrictive kind of games. An rpg railroad doesn't merely have a destination. It is controlled by an engineer. It is running on rails. Your options are to get off (and find another game group), or continue to the destination. You will visit the scheduled adventure sites in the scheduled order. Anything not on the itinerary is just so many telephone poles whipping by at sixty miles an hour.

So, at one end we have literally the most restricted form of travel. At the other, we have the sandbox - the ultimate example of self-directed, structureless play. Even more so than on the road network, I'm in control in the sandbox. I can make a hill and raze it, build a road network and have my toys run along it until I'm bored and I decide it's a river and we're rafting along it, then build a town and a forest and play at bandits.

In a game context, we can farm mustard until we get bored, then wander the roads helping travellers until we tire of that and decide to open a river freight business, or go be bandits in the woods, or anything else.

The distinction isn't between whether there is a direction/destination/goal/series of obstacles, it's who is in charge. In a railroad, the DM is the engineer, the conductor, the train, the rails, the stations, and the tycoon who planned the route, while the player is the Sunday tourist seeing the sights by rail.

In a sandbox, the player is the child and the DM is the sand.

From Alexis' conclusion:

"Choosing to set themselves on the path to find a McGuffin and obtain some clues is not how a sandbox works; it only shows a willingness for players to buy their own tickets for the destination they choose.  That's nicer than being told which train we're going to board, but it is STILL a railroad."

I don't know about that. If it's a railroad, then they can't get off, visit the stations in any order, or decide to fuck it and just set up as bandits because it turned out this MacGuffin was boring. If it's a railroad, they'll find the DM-as-conductor putting obstacles in any other path to dissuade them, or sweetening the pot, or otherwise coercing/outright forcing them back onto the rails, because in a railroad, the DM is in charge.

If it's a sandbox, they can decide they don't care about the MacGuffin at any point and make the game about being bandits in the woods, and the DM will roll with it, because the DM is the sand, and there to provide the medium of play.