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Yes, I understand that. You aren't proposing the model I was criticising. This sentence is what seems to sum it up:

"Take movement, for example. While the number of outcome-states for a moving infantry unit is very large, it can be crunched down into three things: Its relation to other units, its relation to terrain, and its relation to itself."

So, if I understand correctly, the decisions to move aren't based on the utility of individual discrete positions, but the position as expressed as a function of these elements, which allows for mathematical operation and/or ranking.

doctorludo:

Yup. The notion being to capitalize on the game theoretic approach of thinking in sets of outcomes rather than getting bogged down in particulars and counter-factuals.

As a friendly reminder, this thread is about a specific approach to Math Hammer and NOT about the validity of Math Hammer in general. Off-topic posts will be deleted.

I'm very interestd in this thread and have been passively reading it.
I'd like to point out that it is possible for a game of 40K to end before turn 5 and that possibility should be included in the moel.
As the extreme end of aggresive play is what can lead to tableing the other player before turn 5. The risks in doing so will be high but ending the game early is the maximum payoff.

Tread'Ead:

That's a good point and one that I hadn't thought of. Thanks for bringing it up!

To follow up on Tread'Ead's point, I think it's an interesting point that in every game of 40k there's two ways to win. The first way to win will be either capturing more objectives than the other person, or scoring more kill points, and the second way to win will be annihilating your opponent.

A player only has to accomplish one of these goals, tournaments that score depending on weird combinations notwithstandning, and stuff we'll be able to adapt anyways. That means that there may be points in the game where the strategy of annihilating your opponent will be a more reliable tactic for winning the game than either scoring objectives or kill points.

Likewise in the Movement, Shooting, and Assault phases of a player turn there are going to be actions that will become 'live' as the turn procedes. I expect it'll be important that an effective strategy will account for these foreshortenings and twists of the game.

If anyone is wondering what I'm talking about, consider the situation of a unit's shooting target removing itself from charge range of that unit via selective casualty removal. Similarly if one has killed all of the priority targets for that turn, it may be useful for units that haven't shot anything yet to engage in running or engaging Star Engines.

Now we're getting into the Warhammer territory that people are more familiar with, the problems of target priority, the issue of wasting a unit's firepower against certain targets, and the choice of using Rapid Fire weapons instead of charging.

Okay, so simple problem for the sake of an example. Suppose you have two Imperial Guard Infantry squads facing off on a 2'x2' board with an objective at the exact centre. Is there a dominant strategy? What is it?

Random game length?

Nurglitch wrote:Okay, so simple problem for the sake of an example. Suppose you have two Imperial Guard Infantry squads facing off on a 2'x2' board with an objective at the exact centre. Is there a dominant strategy? What is it?

Stand and shoot him until he is dead, then worry about the objective. If you move on the objective before he does, he gets a free round to shoot at you. If both of you move on the objective, you contest it and the winner is down to the dice (an irrational risk). If he stands and shoots, then you are guaranteed at least the tie.

Hesperus:

Good point: Random game length, miniaturized Spearhead deployment to fit the 2'x2' board.

Let's see...

I guess we assume you start deployed 12" from the objective.

Regardless of where your opponent is, it's best for you to be within 3" of the objective when the game ends. So you need to be there at the end of turn 5.

To be certain of that, you need to be <10" away at the end of turn 4 (thanks to run). You can be a little further away if you're willing to gamble. This gets a little complicated, though: If you're 9" or more away, you won't be in position to rapid fire your opponent in turn 5, assuming he's close enough. For player 2, the opponent will definitely be in (move and) rapid fire range on turn 5. However, if Player 1 doesn't have to run on turn 5 and Player 2 is within 9", Player 1 can rapid fire Player 2 on turn 5. That means it's best for Player 1 to be within 9" at the end of turn 3, but best for Player 2 to be just outside 9".

That means it's best for Player 1 to move just inside 9" away in turn 4, and best for Player 2 to move to just outside 9" away on turn 4.

Obviously, neither side wants to get shot first, so neither should move forward in turns 1-3.

The game will tie unless it goes to a sixth turn, in which case Player 1 will probably win (he can charge Player 2 and probably break him).

And this causes a problem. If Player 2 has a 50% shot at a tie, he may try a different strategy: don't move at all and take 2+ turns of shooting at Player 1. He has some non-zero chance of breaking him and winning that way. I don't feel like doing the math, and I don't know how much we're valuing a win compared to a tie, but if the numbers come out that way (and Player 2 is risk-neutral), that's his best strategy.

If it is, we'll have to calculate Player 1's payoffs, given his chances of losing/winning if he moves forward and Player 2 stands and shoots. It may be that the best strategy for both players is to not move at all and take the tie. Depends on the payoffs.

Let's call a loss "0", a tie "1", and a win "2" for the sake of reckoning end-game values. These are zero-sum, so there's two points to go around.

Working backwards:
Game Turn 7 ends automatically
Game Turn 6 ends on 3+
Game Turn 5 ends on 4
Game Turn 4
Game Turn 3
Game Turn 2
Game Turn 1
Deployment (5/6 Player 1 deploys first, acts first, 1/6 Player 1 deploys first, acts second)

Deployment-wise, taking a Spearhead-style of deployment at least 12" from the Objective they can deploy within 17" of each other, and beyond 24". So they can either deploy so that the player acting first gets the first shot, or they can deploy so that no-one can shoot first.

So the value of deploying and shooting is [Enemy Squad, 0, 1.68, 10][5/6] or [Enemy Squad, 0, 1.34, 8][1/6] and the value of deploying out of range is 0, and the value of deploying in range and moving is some negative value or -[Enemy Squad, 0, 1.68, 10].

I'll continue this in a bit...

I read over this entire thread last night, and I am watching with excited interest. I really think this model has the potential to become useful to those that know the model, know how to comprehend the model, and know how to utilize the model in actual game play.

What I'd like to do is summarize the thread thus far, for people who might be getting lost (there are quite a few), or people who want the easy way out (also quite a few) . I could very well be wrong in my assumptions this far, but I'd like to try and make it more simple (there's no real simple way of understanding this in enough detail to be useful, so let's at least make it clear what it does). Again, Nurglitch, please correct me on anything that I might be wrong about. No sense in making more confusion.

The Dove-Hawk (Or Prisoner's Dilemma) Game: This is used primarily to illustrate a simple model - that, with a two options being chosen simultaneous, the option that would benefit both players the most (either $2 or 6 months a piece) is virtually never chosen, because no matter what the opponent does, it is better to go "hawk" or "squeal" . While you end up with less than if you have both co-operated together (two hawks get $1 a piece while two squeals get 10 years), it's the only logical move, but you end up with less.

I believe this applies to Warhammer in the sense that the potential outcomes for actions are what need to be applied to most things when considering what to do.

The Node and Branch System: A system by which we can break the game down into reasonable parts in order to analyze possible endings to the game, possible board positions, possible unit differences, and possible advantages we can gleam from our current position. The key perhaps means that they are dealing in "possibilities". It's possible that 3 railguns will explode 3 Leman Russ tanks in a single phase, but I probably shouldn't expect that to happen. However, this would cover what I should do if I do happen to get into that type of situation, or if the railguns do nothing, or if I get something in-between (most likely to happen).

What this is supposed to do: This is a system by which you can look at the game and see what we can except to happen in the future by working backwards from the expected end of the game. By working backwards from the ending we want from our current position (be it draw or win), we can follow the expected sets of moves for each player back to our current position and see how to get to that ending.

Let's assume for a given game of Warhammer there are 30 possible reasonable endings to a given game turn. That means the total number of possible turn endings are between 30^5 and 30^7. That's between 24,300,000 and 21,870,000,000 possible board endings for a standard game of Warhammer. While that seems daunting, (and in reality it is much more), I think the system Nurglitch is attempting to lay out here has the potential to simply that to the point where it can still be accurate, useful, and detailed (enough).

Again, Nurglitch, please correct me if I am misinforming people.

TheRedArmy:

Looks accurate to me. Thanks!

Speaking of which, I should apologize for not continuing where I left off earlier, but I'm trying to fight the need to procrastinate as the end of term hoves into view with its due dates and finals and other stuff. I'll get back to it unless someone else completes it first.

Nurglitch wrote:TheRedArmy:

Looks accurate to me. Thanks!

Speaking of which, I should apologize for not continuing where I left off earlier, but I'm trying to fight the need to procrastinate as the end of term hoves into view with its due dates and finals and other stuff. I'll get back to it unless someone else completes it first.

Glad to see that I actually understand the material. And no one can blame you for end of term tests/paper/bs. Do what you need to do.

Nurglitch wrote:Let's call a loss "0", a tie "1", and a win "2" for the sake of reckoning end-game values. These are zero-sum, so there's two points to go around.

Working backwards:
Game Turn 7 ends automatically
Game Turn 6 ends on 3+
Game Turn 5 ends on 4
Game Turn 4
Game Turn 3
Game Turn 2
Game Turn 1
Deployment (5/6 Player 1 deploys first, acts first, 1/6 Player 1 deploys first, acts second)

Deployment-wise, taking a Spearhead-style of deployment at least 12" from the Objective they can deploy within 17" of each other, and beyond 24". So they can either deploy so that the player acting first gets the first shot, or they can deploy so that no-one can shoot first.

So the value of deploying and shooting is [Enemy Squad, 0, 1.68, 10][5/6] or [Enemy Squad, 0, 1.34, 8][1/6] and the value of deploying out of range is 0, and the value of deploying in range and moving is some negative value or -[Enemy Squad, 0, 1.68, 10].

I'll continue this in a bit...

Almost. The game ends on turn 5 on a 1 or 2, continues on 3+
Turn 6, Game ends on turn 6 on a 1, 2, or 3, continues on a 4+
There is also the weirdness that the game ends when one side completely obliterates the other, which in theory can happen on any of the turns between 1 and 7.
Regardless of the what I just said, since in the main scheme such little things are trivial (although Nurglitch doesn't actively state so and specificly state his assumptions, which I think is bad), the premise works.

However, there are 3 scenarios in the main rulebook, so when weighing values they change depending on the scenario, but all you need to do is adjust and change the weight <- So I guess this is where Nurglitch is going with this.

"Okay, so simple problem for the sake of an example. Suppose you have two Imperial Guard Infantry squads facing off on a 2'x2' board with an objective at the exact centre. Is there a dominant strategy? What is it? Random game length, spearhead deployment" I'll assume for simplicity, that I must deploy at least 6" away from the objective.

It's a good simple example, so if I'm thinking about it, the strategy begins with the roll of the die to see who deploys first. If I make the assumption that I have 12" deployment and am allowed to deploy on the objective,
If I am going first then I will deploy within scoring of the objective (I can move 6 to score) while being as far back as possible. My thinking is that my range essentially covers most of the board, I'm already scoring the moment I move (thus I'm winning), and I'm shooting first. Since I begin with both a positional and numerical advantage (since I shoot first probably), I want to buy as much time for myself to shoot and hold the objective and maintain the status quo.
If I am going 2nd, I recognize that my opponent should do the above, so the first thing I check is to see if he has done so. In the event he has not, I would deploy accordingly. If my opponent has done the above, he intends to get into a shooting standoff with me. I know at this point he has numerical shooting advantage if things go his way, so I look for some other way to gain an advantage (i.e. hth, cover, etc) If I can deploy within assault range, it is very possible for me to win if I steal the initiative or he has a bad run of the dice on the first turn since charging gives a numerical advantage in hth.

Why wouldn't I stand off and get into a shoot war when going 2nd? Two reasons.
1. The most likely "other" outcome to this strategy is a draw. If my opponent doesn't win, odds are I get a draw. If we're assuming a competitive tournament, I don't value a draw nearly as much as I do a win, so it is to my favor to maximize chance of winning, not drawing.
2. Pressing forward gives me both the advantage of a shot at disrupting his strategy, and me having a huge numerical advantage in the event I steal the initiative. It also has the highest chance of losing (i.e., most of the time), but in this particular case of going 2nd, it also has the highest chance of winning since this strategy almost never becomes a draw.

[Enemy Squad, 0, 1.68, 10][5/6] <- Please notate what you're writing.
I assume 0 is models you lose, 1.68 is models enemy lose (although the number should be 1.67, not 1.68), 10 is the number of models the enemy has, and 5/6 is the probability your opponent does not steal the initiative.

NO NO dont stop!!!!

All of this is pretty mind bending but this has so far been a HUGE help in me explaing game theory to people.

While the thought that Hawk-Dove is like 40k in an aspect is correct but then again at another look it is not is most interesting the thoughs of nodes apply a sense of reasoing behind the maddness and then the over all hints of randomness of the dice (which control it all some may argue) is just mind blowing and a major headache at the same time.

So please do not stop this thread I need more input!

Thread is being locked due to thread necromancy.