The behavior of “random walkers” is not so random.  You can predict it:  not every square of travel from the first step a roamer takes out the front door of his building until he reaches his walk-finish square if there are stub roads and other branches all along his route, but more than enough to be useful.  Roamers are locked onto a fixed path for several squares at the beginning of most walks, and this little bit of certainty often provides all we need (given the seldom-branched roads in our cities) to direct them where we want.  So, if you want to know in advance where the roaming walkers in your next city will go, this article is for you.  With a little crafty tweaking of the roads, you can even make them go where you want.
   Visitors to Pharaoh Heaven who have suffered through some of my earlier articles on roaming walker behavior may be in for a pleasant surprise.  Brugle helped me a lot with Ambulomancy through his critical review of the manuscript.  In fairness to Brugle, however, I must note that I tore up and redrafted the original text that he saw.  So, if any idiocies or inaccuracies have crept back into the paper when Brugle was not looking, he is not too be blamed; I am.  Comparing the two versions, I have to say Brugle’s suggestions have worked wonders.  He unerringly zeroed in on several areas of ambiguity (forcing me back into the walker labs to iron out details), gently steered me toward a more fact-dense and less chatty style, and expunged at least one flagrant error.  Any readers who make it to the end of the article without becoming hopelessly befuddled owe a great debt of thanks to Brugle for his efforts.  I know I do.
   Actually, reaching the end of the article might not be such a problem.  I wrote it while keeping DawnOfWere’s request for a short-ish summary in mind, so the real meat of roaming walk prediction is packed into the first four pages.  Some illustrations of the four types of roaming walks follow, but they come with glyphy and should not present much of an obstacle.  The article ends with a bunch of addenda providing more fine print about issues like ferries, but few readers would be likely to need this stuff right away to get the essential idea of how the roamers work.  One other constraint imposed on the writing of this article should also increase its reader friendliness: no complex coordinates!  So, with these assurances I hope you will read, and enjoy, this little guide for Pharaoh players to the “divination of walks”:  Ambulomancy.

StephAmon

[This message has been edited by StephAmon (edited 03-18-2005 @ 10:25 AM).]

StephAmon,

Reading through it now and it looks good. You and Brugle did an excellent job with it. Don't know if I'll use any of it soon, but would be an excellent tool for "debugging" an errant walker. Always so much more to learn about this game.

GodOfRandomDeath:

I’m happy you noticed this little guide.  If you ever get a “busted building”, it’ll be here, and it might help.

StephAmon

Hello StephAmon,

finally I had the time to study this excellent work - Thanks to you and Brugle.

When I built my latest city (Itjtawy with almost 24000 people), I had troubles with 3 types of walkers, which behaved unlike the described walkers.
The architect, the tax collector and the senet player vanished or wandered off road or through buildings.
So I experimented with the architect.
I built two parallel roads with the length of 70 and 60 Tiles and 10 Tiles in between - like this:

<- 39 Tiles ->

<- 39 Tiles ->

The roadblocked connection (rbc) to the upper road is at tile 60 of the lower road.
The architect wandered to the northern walk target - 133 tiles away - and then he disappeared.
In the following table I'll use the notation 59R 11U 63L for this trip, which means the architect left his office then he moved 59 tiles to the right, 11 tiles up and 63 tiles left.
Now I moved the rbc step by step to the left and got these results:

rbc at tile	walk-1	walk-2	walk-3	walk-4	distance of walk-4	remarks
60	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	59R 11U 63L	133	very long
44	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 11U 47L	101	very long
43	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 10U 46L	99	off road
42	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 9U 45L	97	off road
35	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 2U 38L	83	off road
34	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 1U 37L	81	off road
33	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 36L	79
16	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R 2L	45
15	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R	43
1	50R 50L	1R 1L 41R 41L	1R 1L 41R 41L	43R	43

I suppose that the algorithm has some problems, if the roadblock is within the default walk distance d.

[This message has been edited by Max (edited 05-09-2005 @ 04:39 AM).]

Max:

I am very glad you got a chance to look over the random walk guide, since you have zeroed right in on an important area.  I fear I am going to have to sit down with your data and probably reconstruct your roads in my own lab before I can get back to you with anything like an intelligent reply to your post.  Before I do so, however, I wanted to offer two general observations and to raise two questions for you.

First, the observations.  You are absolutely right that the architect, tax collector, and senet player do not strictly adhere to the rules laid out in the Ambulomancy guide.  I think of them plus teleporting dancers, musicians and bazaar sellers collectively as “supernatural walkers”.  Awarding them this title at least helps to console me for the fact that I do not understand them at all well.  Even though these walkers do things that most roaming walkers cannot (like vanish, teleport, or go off road), both you and I seem to have found that these supernaturals still somehow use or reference the walk-target squares of ordinary walkers.  I saw dancers and musicians from pavilions appearing on otherwise inaccessible walk targets; you saw architects make walks (of lengths that would be utterly impossible for other long walkers) that terminated by vanishing on a walk-target square.  Like teleporters, I really think that the vanishing and off-roading walkers deserve more study, and I am delighted to find that you feel the same.  I really should have said something more explicit in Ambulomancy to warn readers about architects and tax collectors not playing by the rules.  I hope that your post and this reply will provide the necessary fair warning.

Second, Brugle reminded me in his review of the manuscript that walkers from buildings at dead ends (like your architect) frequently make “false starts” (exactly as you document for walks 1 and 2 in your table) and that my model of roaming walk generation does not account for false starts.  Arrgh!  Too true.  I simply cannot handle dead ends yet.  I tried to be honest about this (without absolutely and painfully trumpeting my ignorance) on p. 7 under “Default Details”.

Now the questions:

#1 - Are you sure you are listing the walks in the right order?   I ask because the walks appear to be progressing in counterclockwise order, i.e., in the reverse of the usual sequence.  Before I saw your data, I had only seen counterclockwise walk order with teleporting walkers.  Very exciting. Thank you!  I will have to reproduce this effect in a lab and play with it some more.

#2 - All the walks in the "walk-3" column in your table look like grounded walks beginning with a destination walk to the (plaza-marked) SE walk target seven squares to the right of the architect's walk-start square.  Do these really end with the disappearance of the architect 50 squares from home, or does he actually return - visibly - to his walk-finish square?  In other words, could you have listed those walk 3's as "50R 50L"?  I ask because I have never before seen a case in which an architect vanished during a grounded walk (unless he wandered off the road first by failing to turn the final corner in a loop).

I will return with a more fleshed out response once I get a chance to reproduce the effects you describe in my lab.  I hope you will not hold it against me if this takes a couple of days, because we are entering final exam period at UMass at the end of this week  (Many exams to write!), and I am still recovering from a bout with asthma that sent me to the emergency room.

Until then,  thank you for some wonderful clues to supernatural roamer behavior.

StephAmon

[This message has been edited by StephAmon (edited 05-09-2005 @ 00:54 AM).]

Hello StephAmon,
you are right - sorry for the inaccuracies - I corrected the table in replay 4.
I guess you noticed, that the formula for the travel distance (td) of walk-4 with rbc between 15 and 60 is always: td = 2 * rbc + 13.
I wish you all the best - no need to hurry..

Max:

Many thanks for the edits. We can now see pretty clear evidence of clockwise rotation, so we may be able to figure some of this out.

Hey! Your architect does not vanish after default walks. I never checked, but I honestly thought he would.

Thanks also for the formula. I was not thinking of the pattern that way (I was working in glyphy), so I missed it. That has got to help.

StephAmon

Most of StephAmon's observations also apply to Caesar III roaming walkers. This comment applies only to C3, but I thought that it belongs here. (I'll link to this thread from the C3 Game Help forum.)

In either a "blockage-shorted" or "grounded" walk, the walker switching from destination to random mode "does not reverse direction". On an intersection-free loop road, that means that a Pharaoh walker will continue around the loop (at least until switching back to destination mode for the return to its building), but a C3 walker might not.

In C3 (unlike Pharaoh), destination walkers who are following a road will cut corners. (Note to StephAmon: using C3, your research might have been easier. ) If a walker was cutting a corner when it switches from destination to random mode, it may go in any road direction (since none of them are the reverse of cutting the corner). An example, in the next paragraph, may make this clearer.

In my latest C3 city (a try at an interesting challenge: the "career" Valentia at Very Hard difficulty without trade, debt, the "rescue" gift, or initial personal funds), I started a housing block with a 46-tile intersection-free rectangular loop road, where I expected all roaming walkers to complete the loop. Most did. However, on 1 walk out of 4, the trader from one market near the east corner had a "grounded" walk which began with a counterclockwise destination walk to the tile just SW of the loop's north tile, and since she cut corners getting there, she arrived travelling toward the west (cutting from just SE of the north tile to just SW of the north tile). Switching to random mode, she chose to travel toward the northeast (not the reverse of toward the west) and continued clockwise. When switching from random back to destination mode near the loop's south tile, she reversed back to counterclockwise to return to her market, and therefore missed some houses on that walk. (Fortunately, I expect that traders from the 2 other markets, who go by all houses on every walk, will be sufficient to prevent devolution.)

Thanks, StephAmon, for explaining what otherwise would have been a puzzling (and annoying) mystery!

[This message has been edited by Brugle (edited 05-17-2005 @ 04:25 PM).]

The March 2005 edition of Ambulomancy contains an error in the description of the pattern used by the walker algorithm to search for road squares around routing centers.  This error is reflected in Table 1 and in text that refers to it.  The March 2005 edition states that routing centers are not searched by the algorithm for roads.  This is incorrect.  Recent research results indicate that a routing center is examined after the square to its NW and before the square to its southeast.  An easy way of seeing this would be as follows:


A single disconnected road square occupying a routing center now appears capable of defaulting out the walk controlled by that center.  Moreover, if a connected road square on a routing center has the highest priority of all the connected roads near the routing center, the algorithm places a walk target on the routing center that will be used for calculating either shorted or grounded walks.  Finally, a road square on a routing center can serve as a teleportation-landing square for an entertainer teleporting from a pavilion.  Naturally, future editions of Ambulomancy will include a completely renumbered version of Table 1, and the text will be appropriately revised.

I apologize abjectly to the community of Pharaoh builders for misinforming them.   I also offer deeply felt thanks to Jimhotep, who brought the use of routing centers as teleportation-landing squares to my attention, and thus alerted me to the potential error.

StephAmon

EDIT 10Sep2007: This error is corrected in the 2nd edition of Ambulomancy. StephAmon

[This message has been edited by StephAmon (edited 09-10-2007 @ 00:38 AM).]

I have taken so long to craft an adequate response to the challenge posed by Max’s architect to ambulomancy that I think I had better offer at least the partial response of which I am presently capable.

In the table Max presents in reply 4, walk #1 must have a formal direction of southeast.  In the accompanying figure, Max clearly shows the walk target (plaza square) associated with a SE formal direction lying seven squares to the right of the architect’s walk start.  Walk #1 takes the architect from his starting point 50 squares to the right (in the figure, or in the SE direction), which corresponds exactly to the sum of the architect’s predicted seven-square destination-mode walk to the plaza (his SE walk target) plus his 43-square random-mode walk to the go-home point.  The expected clockwise rotation in the assignment of formal directions to walks would, therefore, make walks #2 and #3 the SW and NW walks, respectively, and walk #4 (the really interesting one) has a formal direction of NE.

The SW and NW routing centers lie outside Max’s figure, but if the only roads present are the ones in the glyphy, the algorithm’s search for walk targets around either of those two routing centers will fail to find a road square in the 13x13 areas searched.  Consequently, both the SW and NW walks in the architect’s quadramble should default to 43-square long, pure random-mode walks.  With both of these walks, however, a potential ambiguity exists in trying to assign a default direction.  Version 1 of Ambulomancy identified the default directions for cases in which a roaming walker’s start square falls on “a straight stretch of road” or on a corner.  Default directions for walkers starting at the center of three-way and four-way intersections also appear to be variable and untrustworthy.  Max’s example starts default walks (#2 and #3) on a starting square that is not unambiguously within a “straight stretch of road”, since the start square is adjoined only by a single road square rather than a pair of road square on opposite sides of the walk-start point.  Evidently, the architect is not regarded by the algorithm as starting in a “straight stretch of road”; dead ends do not seem to be the same as straight pieces of road.

In Max’s table, the two default walks (#2 and #3) beginning from dead ends make “false starts” in which the total distance traveled from walk-start to go-home point still has the usual length for a default walk (43 squares for a long walker like an architect), but the walk bounces back to this walk start square after his first square of travel (before making the remaining 41 squares in his default walk).  I can offer no explanation based on routing-centers or walk-targets for why this occurs.  A little preliminary checking shows that false starts always occur when 1x1 buildings start a walker on a dead end (not one square away from a dead end) when that dead end terminates a NW-to-SE road.  False starts also occur at NE dead ends (i.e., the end of a road extending away to the SW).  Curiously, I have not been able to produce false starts by roamers from 1x1 buildings at SW dead ends.  The picture gets more complicated when buildings of larger sizes are considered.  I suspect we will need a big Dead End Table in which one cross indexes building size and position with dead end compass direction to find out which combinations will reliably produce or avoid false starts.  I plan to develop such a table in the not-too-distant future, since a few default walks from dead ends are hard to avoid in mining country.

For the present, however, it seems prudent always to assume that a roamer on a default walk will make or will fail to make (whichever is "worse") a false start from a dead end, unless he comes from a 1x1 building and starts on a SW dead end.  Fortunately, default walks tend to be very rare in actual cities, except at map edges or near unbuildable areas like mountains and dunes.

Max’s table makes it evident that 43-square long blockage-shorted walks (e.g., walk #4 in the last two rows of the table) do not begin with false starts, even though they have the same length to the go-home point as a default walk.

Apart from the first two and last two rows in the table, all the different variations of walk #4 shown by Max are examples of “confused walks” and I have discussed the little that I presently know about walks of this kind in a different thread (Walkers and teleporters from pavilions").  I have earnestly tried to develop some ability to predict what a roamer of this sort will do when it reaches the end of the random phase of its roaming walk (when a decent gods-fearing walker would turn around and go home), but so far I have failed.  My only consolation is that once a roamer is in confused mode (e.g., once he has left the road) he will vanish and return instantly to his building ready to begin another walk if he blunders into a non-garden structure.  In densely packed housing blocks, this means the confused walker hits a building immediately when he tries to leave the loop road and soon will begin a new walk.

[This message has been edited by StephAmon (edited 11-06-2005 @ 04:37 PM).]