"Mathematical vs Non-Mathematical Models of Combat" Topic

Richard Taylor, writing in Slingshot magazine, says that modern wargaming is "…fundamentally a mathematical model of conflict… with an emphasis on position, manoeuvre and factorised combat."

He wonders if non-mathematical models might have something to offer future wargame designers.

Do you prefer mathematical models or non-mathematical models?

If you can give me an example of a non-mathematical rules set to compare to the mathematical models, it would be helpful. Even chess breaks down to mathematics at the end, with the firmal grid playing area. Anything involving probability or measuring becomes mathematical. Cards, dice, rulers, hexes… math. It's everywhere.

A matrix game would be non-mathematical. Conflict is resolved through discussion and persuasion.

So, like a group storytelling game?

A matrix game is still technically mathematical. It's just a different subset of MATH than we're used to.

I think what Taylor really means is a probabilistic model of combat.

What about David Weber's battle accounts? He goes into painfully intense descriptions of exact manuvers, shots fired, warheads launched, etc. In the Honor Harington series, we can just blur past it… his tech is fantasy. In the stronghold series he is dealing with real historical tech (five centuries compressed into seven years) where he can say this gun fires this weight of shot at this velocity, this ship design can manuver just this way. I assume he's actually worked out the math on all of this. I definately would not want to be crew on a wooden man'o' war.

This will be a semantics discussion, unless the OP clearly says what he means by a "non-mathematical model".

At some level, game theory always plays a role, whether you model the underlying factors in numbers or descriptors or words or whatever.

He means that the process/resolution is a matter of adding up .. potentially then adding/subtracting a random factor and comparing totals.

As an example of a non-mathematical process, in Andy Gittins' phenomenally successful David and Goliath/Gladiolus, the combat is resolved by players secretly picking where to attack and where to defend – if the attack is on an area not defended the attack succeeds. No mathematical model involved.

As a more bland example I think I would call the outcomes you get in, say, Chess, non-mathematical.

Although the matrix process seems non-mathematical, these days the outcome is most commonly resolved by a die-roll modified according to the plausibility of the arguments made.

Phil

Mathematical

Matrix gaming makes my skin crawl. So mathematical for me.

I must confess, I struggle to think of any games which don't have mathematical models of combat. Some are just more complex than others.

Chess is clearly mathematical, in that you need to manouvre into the correct position to conduct a successful attack. There aren't any random factors though.

One of the few examples I can think of is the old 'Warlord' game where the attacker picked a number (up to the maximum number of attacking armies or 6, whichever was lower) and the defender had ot guess the number. Even that is a classic two party zero sum game though and you can apply decision theory models to predicating the probable answer. I was quite good at Warlord….

Does the OP mean games which don't have any random factors? I can think of more examples of those, although there are usually some sort of strength/posture considerations e.g. Diplomacy.

the combat is resolved by players secretly picking where to attack and where to defend – if the attack is on an area not defended the attack succeeds. No mathematical model involved.

This is discrete set theory mathematics, the math that underlies number math. That's about as math as you can get.

As a more bland example I think I would call the outcomes you get in, say, Chess, non-mathematical.

Chess, as mentioned above is highly mathematical. If it weren't computers couldn't be programmed to walk through its state space.

Which leads to a good point, computers are finite state machines, and mathematical formal systems. If would submit that if a computer can adjudicate a move in a system in accordance with its rules, the system is by its nature mathematical.

Properly, mathematically, you can't just say if a computer isn't capable of adjudicating a move in the system, then it is not mathematical (a->b ~-> ~a->~b). But I can say that as a person. And I will.

A move in a matrix game, even though it may use numbers and dice, cannot be adjudicated in accordance with its rules by a computer, so I would say it is non mathematical.

A bit differently, there are games where their state adjudication is entirely mathematical or requires something beyond mathematics to work.

Pretty much every pulp game I play is non-mathematical in that sense. It is incumbent in players in a pulp game to apply a world-view on the outcomes and adjust accordingly.

Example: DOM had an obscenely low-probability roll (and so did I to counter it) early on in a pulp game that would have crippled my ham-fisted hero resulting in him limping through the remainder of the game with obscenely low odds of success while his allies got picked off with no help from him. Her adjudication was, "That sucks. It isn't going to be any fun. How about instead of taking the hits and falling on the crate of gunpowder that explodes, you dodge, but fall out the warehouse window, take some falling damage and we throw some outdoor terrain on the side of the board? Oh, also I just get to capture your other dudes and hide them for you to find."

Perhaps we are over analysing here, many situations where an outcome needs to be determined can be modelled mathematically, it doesnt mean we actually use set theory, linear programing, quantum mechanics etc in resolved game outcomes.

Perhaps the author simply means that modern rules determine combat outcomes using combat factors, modifiers and dice? and therefore doesn't model manouvre warfare very well. We all understand attritional models of combat, manouvre warfare is much harder to model – defeating the enemy without fighting and all that.

Otherwise I am baffled.

We are using a non-mathematical model in our latest set of rules.

By the way, IMHO, matrix systems are very mathematical. I have spent hours revising and tweaking results on a matrix in order to reflect mathematical chances.

Mathematics is nothing more and nothing less than a means of communication; an ever-growing, ever-evolving family of languages with symbols, syntax and definitions to describe phenomena. So 'non-mathematical' is a non-starter for me.

I suspect the original poster is referring to rules that are 'numbers crunching' or 'fixed in terms of physical variables' versus rules that are more 'set theory' or 'inclusive of non-physical variables.'

In the end, rules authors (whether they are conscious or unconscious of the fact) always select a set of variables to include or exclude in their game. What those variables are and how they are resolved are mechanics.

As a data modeler once told me, building a model is not science. It's more art about what to include and what not to include.

combat resolution via Rock/Paper/Scissors(Lizard/Spock)
might be an example.

The best hope for improving games design is to improve the knowledge of systems modelling and design techniques amongst the games design community.

I'm not saying that they need to design games using professional techniques, but just become more aware of the way warfare works as a system.

There is space for hard (mathematical) and soft (non-mathematical) mechanisms in the same game. It's not a one or the other choice.

Kriegspiel used soft techniques over 100 years ago in the form of umpires using 'seat of the pants' methods to assess situations. One benefit is the uncertainty that a soft model produces in the mind of the player. The player can't assess the odds, but just respond to inputs from the umpire.

The Perfect Captain's WOTR game design immerses the player in an historical environment via it's use of historical terminology and concepts. This is another soft (non-mathematical) technique.

I don't play many commercial products but if anything, I think wargaming is moving more towards toy gaming than an adult hobby. It should be going the other way, for example by looking at warfare using modern design techniques.

Of course everything can pretty much be said to be mathematical but that isn't the point that was being addressed, etotheipi.

The actual combat method of e.g. Chess is not mathematical – the game is, the combat per se is not. Nor is it in the option picking process nor in rock/scissors/paper resolution.

That is the distinction Taylor was trying to address I believe …

Phil

You do the math, I'll play the game.

Resolution a la Diplomacy?

Non-Mathematical Combat = Setting up opposing armies of toy soldiers 6 feet apart and letting fly with rubberbands/marbles to knock them down (obviously not the game for those who like to assemble/paint their minis. ;-)

Do the electric signals in our brains operate mathematically? Of they do, EVERYTHING is mathematical.

All points about "Mathematical" game systems/Matrices/etc are perfectly valid, and useful statements of their nature, but, actually, I think all but one of the previous posters to this thread appear not to understand what alternative methods "Non-mathematical combat systems" means. More troubling to me as a life time war gamer and sometime game designer is that, so far, so few seem to think any alternatives even exist.

It may be that Mr. Taylor of the "Slingshot" article is thinking about ANALOG combat systems. That is, instead of determining, for instance, there's a 10% chance of a shot falling on a given unit, a simulacrum of an ACTUAL shell is physically launched across the gaming area from feet--even yards--away and if it lands on a target, the effects are determined.

This is what the founder of the modern hobby, HG Wells, practiced in his original "Little Wars" using spring loaded Britain's cannon models firing actual lead shot. The chances of a gun inflicting damage on an enemy target were limited to the player's ability to aim and range his fire on a visible target within the maximum range of the "gun."

No rules had to be devised to factor in visibility, drift, and range determination--it was simply let fly with the round in the toy cannon, and a hit was a hit!

Of course the other limitations of the original "Little Wars" included the absence of small arms fire, a simple coin toss (50-50) close combat resolution, and no morale rules per se, meaning the game was pretty strictly about the cannon, and the use of cover to get troops into position with the minimum of loss in order to rush objectives from as close as possible.

Clearly, for anything more sophisticated, this simple "combat system" has serious limitations, but no time was ever lost trying to determine how many of which kind of dice to roll on what table with what modifiers to know if a shot hurt the enemy.

The latest incarnation of Well's work is Padre Paul Wright's "Funny Little Wars" (FLW) which keeps the artillery fire system, but allows for the use of the same spring loaded cannon models (there are several types/sizes) to represent small arms fire, though a "mathematical" system is provided for those who prefer it. The "shells" in this system are 1" matchsticks which have appropriate burst radii associated with them for determining fire effects.

FLW provides rules for Snipers/Stalkers (we are talking pre-WW I here) which call either for use of a spring loaded cannon to actually pick off desired targets, again using matchsticks to prevent damage to any figures struck. Alternatively, there are old Britain's sniper figures (and new ones from TVAG) which are themselves spring loaded and can fire a matchstick out to about 6 feet.

But there are other "analog" methods of determining if a given shot strikes a target.

Fred T. Janes' original Naval War Game rules called for the use of a simple swinging pencil device--a "striker"--that when drawn back and let fly at a variously sized silhouettes of a ship (conveniently available through any of his "Janes Fighting Ships") randomly marked the exact spot of the fall of shot, including being wide or long of the target and missing.

A variant of this system has been developed by this poster to determine wound location for small arms fire, hand-to-hand, and even sword/knife combat on individual human targets in the rules "Mad Dogs And Englishmen" (but by extension, any RPG). Here, instead of a swinging pencil, players choose from among a number of "strikers" (paddle like tools with pins variously sited on their faces), then physically strike a provided silhouette (size appropriate for the distance from the firing character) of a figure once for each "shot" or blow, allowing for misses and/or precise wound location.

This has proved to be much faster than using "mathematical" systems, and is providing accuracy ratios consistent with actual marksmanship at various ranges.

By placing the pins in one of five points on the Striker's face, but not letting the striking player know which one he has chosen, he can try to "aim" at the silhouette, but being required to make all his blows in rapid succession, he can't guarantee making all shots hit. If he only has one shot to make, the player can actually attempt to aim, even choosing the part of the target he wishes to hit.

A simple expansion of this system works as well for games involving formed units in larger scales (1:20, 1:60, etc), and allows for hitting not just targets, but the chances of bagging Officers, Musicians, Color Bearers, etc.

In short, rolling dice/checking matrices are not the only way of determining shot or other weapons effects, and these need not be limited to the scale of the miniatures in use, either.

TVAG

(Anyone interested in play testing TVAG's "Striker Combat System" are welcome to write TVAG@att.net and request a PDF with full instructions.)

Yes, analogue modelling can be very powerful – a huge advantage of HG Wells/FLW approach is that actual matchstick firing cannon mean you immediately model beaten zones, the significance of flanking, dispersion, cover etc. Without a lot of pratting around with dice and modifiers.

"It should be going the other way, for example by looking at warfare using modern design techniques."

These sorts of things are market driven. I have designed plenty of games which model actual weapons effects – which for twentieth century warfare render frontal assaults utterly futile and bunching results in heaps of casualties. The players tend to become baffled and frustrated however. Who really wants to model a 14 hour long firefight where no-one actually gets hit?

(Well I do, but I'm a bit odd like that).

I would commend Phil Sabins 'Simulating War' to anyone interested in wargames design with realistic outcomes using modern gaming design methods.

It would indeed be useful to make a distinction between the gaming engine and combat resolution (which might be based on mathematics, throwing things, pure random outcomes, word games, etc.); and on the other hand the decision process by the players about what to do next (and which will always be firmly based in game theory, using any of the above factors as its input).

Any game, even a pure random one such as rock/papers/scissors has a component in game theory when played over a series of games. In the lobby of my department (I work at a computer science research department) is a setup that consistently beats you in rock/papers/scissors based on pattern analysis of all previous 'battles' you have played so far.

I should have clarified that I meant by modern design techniques was the design of "human systems" aka "interactive systems design".

TMP link

In an earlier post (link above) mention is made of how command and control and human factors are still ignored in preference of the mathematical/positional aspects. I think this is mostly true.

The reason is that few tabletop designers have any real knowledge of how to interpret what occurs on the battlefield as a human system. Consequently they find it difficult to produce a game that looks at a battle relatively realistically.

An example is the command radius model that a lot of rules use. It fails as a realistic model as it ignores command structures and messaging systems, or the fact that a unit might be up to armpits doing something other than waiting for your next activation roll. Modelling a realistic command and control system is not difficult. Making an interesting game of it could be tricky.

Of course everything can pretty much be said to be mathematical but that isn't the point that was being addressed, etotheipi.

Do the electric signals in our brains operate mathematically? Of they do, EVERYTHING is mathematical.

No, everything is not mathematical. Human thought has been mathematically demonstrated to be greater than mathematics (and more generally, formal systems at large).

I am not sure from the quote in the OP what was being addressed as non-mathematical, so I offered a definition. And that definition did not say that everything is mathematical.

That is the distinction Taylor was trying to address I believe …

Don't know the person, so out of context, I couldn't say. I can only describe what I mean when I use the terms.

By the way, IMHO, matrix systems are very mathematical.

We may be talking about different kinds of matrix systems, then. Ultimately, the matrix system I was addressing is adjudicated by a referee (or cadre of referees), which leads to selecting rows and columns to determine either outcomes or odds of outcomes. (More fluid non-mathematical models don't use matrices, but allow the referee(s) to both adjudicate the validity of the propositions and the range of outcomes, with odds if appropriate.) So that even if there is a mathematical element to them, they cannot be adjudicated by math.

The actual combat method of e.g. Chess is not mathematical – the game is, the combat per se is not. Nor is it in the option picking process nor in rock/scissors/paper resolution.

I made a distinction between the game and the player experience. No human decision process is mathematical. People here pretty much agree that the "meat" of the game is in the dynamics, so I posited about adjudication process as the core of the game.

No, everything is not mathematical. Human thought has been mathematically demonstrated to be greater than mathematics (and more generally, formal systems at large).

Really? Can you give a reference?

Because if this is true, this would be HUGE breakthrough in e.g. theoretical computer science. The question whether the human brain is equivalent to a Turing machine (or not), and hence has a different definition of computability is one of the large unanswered questions that would bring you immediate fame and money.

Disclaimer: I teach computer science, including stuff such as modern algebra, Turing machines, Godel's incompleteness theorem, computability, the p=np question, algorithm design, time complexity etc. So perhaps I am biased ;-)

No, everything is not mathematical. Human thought has been mathematically demonstrated to be greater than mathematics (and more generally, formal systems at large).

Really? Can you give a reference?

Because if this is true, this would be HUGE breakthrough in e.g. theoretical computer science. The question whether the human brain is equivalent to a Turing machine (or not), and hence has a different definition of computability is one of the large unanswered questions that would bring you immediate fame and money.

Disclaimer: I teach computer science, including stuff such as modern algebra, Turing machines, Godel's incompleteness theorem, computability, the p=np question, algorithm design, time complexity etc. So perhaps I am biased ;-)

Well, if you teach Godel, you should know that the completeness proof, which comes from human logic is capable of going beyond the power of a formal system. Sir Roger Penrose does a good job at explaining it. Human thought may, in fact, be completely deterministic, but we would need something more powerful than the subset of mathematics that can be implemented on a computer, to prove it. And something more powerful than what we currently consider to be mathematical proof.

Penrose's argument as popularized in The Emperor's New Mind, has been objected to in various articles as well.

It's not because we as humans came up with Godel's theorem that we are above any mathematical system.

The phrase "Human thinking can do more than computational procedures as carried out by computers" – often a misunderstanding of Godel's theorem – has never been proven true.

If human thinking is merely an artefact of physical and chemical processes in the brain, then there is no reason it cannot be modeled by a mathematical process, and hence be carried out by a computer. Penrose has stated he believes quantum uncertainty might play a role in human thinking, and hence it is not determinsitic, but that is more a believe issue at this point and never be shown.

Btw, if you follow the arguments made by Ray Kurzweil about the technological singularity, we might have computers capable of human-like thought within the next 20 years …

Although interesting stuff, this is far removed from wargaming … ;-)

The phrase "Human thinking can do more than computational procedures as carried out by computers" – often a misunderstanding of Godel's theorem – has never been proven true.

I never said that was a part of Godel's theorem. I said it said that human thought (or, more exactly, specific subset of it, mathematical proof) is more powerful than a formal mathematical system. Those are very different things. And, yes, subject for very different types of discussion than posting back and forth on a board.