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Plenty has been said in this forum about the application of Statistics to Warhammer 40,000, and for good reason. Where dice are concerned, Statistics is the go-to mathematical tool for assessing and managing one's expectations of which way Tzeentch may be blowing that day. However, not all of the elements are random, and often the outcome of a particular action will depend on your opponent's reaction to

One interesting application of Game Theory has been instituted in the move from 4th edition to 5th edition. In the 4th edition rational (meaning in this context players that prefer $2 to $1 at the end of a game, all else being equal) players realized several things:

(1) Getting the first turn was an advantage
(2) Preserving units for the last turn was an advantage

This can be demonstrated by a toy-game called Hawk-Dove (or the Prisoner's Dilemma if you swing that way...): In Hawk-Dove the players each have two strategies, named Hawk and Dove. The Hawk tactic is to take. The Dove tactic is to give. The potential outcomes can be represented on a 2x2 table with the following results:

1. Hawk vs Hawk: Each player takes a $1 from the other (1/1)
2. Hawk vs Dove: The Hawk player takes $1 from the Dove, and the Dove gives the Hawk $2 (0/3)
3. Dove vs Dove: Each player gives $2 to the other (2/2)

If it is common knowledge to both players that the game will only be played once, and according to the rules, then it is rational for both players to play Hawk. Though the outcome will be worse for both than if both players played Dove, it will not be as bad for one player if one played Dove and the other played Hawk. In other words, by playing Dove your worst outcomes as a player will be either $2 or $0, while by playing Hawk your worst outcomes as a player will be $3 or $1.

Suppose we change the game to be a set of Hawk-Dove games, called it the Definitely Iterated Hawk-Dove game. If it is common knowledge to both players that this Definitely Iterated game will be played 10 times, and according to the rules, then it is likewise rational for both players to play Hawk each time the game is repeated, or each turn. The Hawk strategy is thus said to dominate the Dove strategy.

Incidentally modification for multiple iterations turns the choice of Hawk and Dove from strategies, or goals, into tactics, or means of achieving said goals. The Hawk strategy generalizes from playing Hawk, to playing Hawks.

Just as in the 4th edition of Warhammer 40,000, knowing when the game will end on a known turn means that certain strategies will dominate other available strategies.

Likewise consider a game more like Warhammer 40,000, where one player gets to play a turn, and then the other player gets to play a turn. Let's call this game the Sequential Iterated Hawk-Dove. If a player has the first turn, again Hawk dominates Dove, because playing Dove depends on the subsequent player preferring $2 to $3, which is irrational.

In 40k terms this meant that a player with access to a 40k equivalent of the Hawk strategy would always prefer that strategy, because it was always the best strategy if they got the first turn, or the second, and whether their opponent decided to take the Hawk strategy, or the 40k equivalent to the Dove strategy.

Now there is a version of the Iterated Hawk-Dove game where Dove dominates Hawk. In this version, the turns are indefinitely iterated, so that players don't know which turn is the last turn, and thus cannot work backwards to figure out the dominating strategy on the first turn, and must play it from turn to turn. Playing from turn to turn, iteration to iteration, means that players can benefit from the goodwill generated by playing Dove, and retaliate against an opponent who played Hawk last turn. In other words, playing Hawk in one turn means that one risks the other player playing Hawk in the next turn, and doing worse than playing Dove.

To illustrate this, consider an Indefinitely Iterated Hawk-Dove game where two turns have passed with both players playing Dove, meaning they each have $4. If player A decides to play a Hawk tactic in the next turn, in order to take advantage of player B continuing to play Dove, then they will have $7 and $4 respectively. This is good if the game ends, but only lucky because either the likelihood of the game ending was common knowledge and thus both would have chosen Hawk, or it was unknown and player A risked retaliation.

But so what if player B relatiated? Suppose the game ends on turn 5, with both players continuing to defect after turn 3? Player A would win with $9 while player B would lose with $6. Not bad, but player A is rational, and prefers $2 to $1. Had player A continued to co-operate, then he would have finished the game with $10. The same goes for player B. That tactical decision would have been irrational.

So why was it irrational if player A played Hawk on turn 3, when it won him the game, which otherwise would have been a tie? Firstly, it is taken as an assumption in Hawk-Dove and its game that player A is rational and prefers the monetary result to the game result, which was established by Hawk-Dove when it was better for players to choose to tie with Hawk rather than risk losing. Similarly trying to play Hawk in the Indefinitely Iterated Hawk-Dove risked losing because it could well have been player B's strategy to play Hawk in turn 2, meaning the result of the game would have been a loss for player A at $5 and a win for player B at $8.

This symmetry between strategies is important because it means that either player trying to outdo the other by playing Hawk first either risks losing, or doing worse in the game than if they had held off playing Hawk and played Dove instead. Just as in the Hawk-Dove game, the players in the Indefinitely Iterated Hawk-Dove game better serve their rationality if they avoid doing worse than otherwise.

Furthermore there may be a greater game afoot. Imagine players in an Indefinitely Iterated Hawk-Dove tournament, where the risk of doing worse than otherwise could scuttle their chances of winning the tournament, or simply knock them out with what little they managed to squeeze out of their last opponent. Again, whether the goal is to win the iteration, the game, or the tournament, the player does better than otherwise by avoiding loss rather than chasing uncertain gain. And where certainty is available, as in the Hawk-Dove game, the strategy is clear.

I'll see if I can look it up, but DashofPepper had a tournament experience exactly like this where he essentially got locked by multiple ties because instead of playing to win, his opponents wisely decided not to risk it and instead avoided losing.

The application to playing games of 40k is clear: If you want to stop losing, stop trying to win and start trying not to lose. Because if both players chose to try and win, and they don't make mistakes, then the game will be settled by the dice, and as mentioned that's an irrational risk if you want to do better than you would otherwise.

In this thread I'm going to use game theory to build on the groundwork laid out here, and introduce additional concepts to the ones of rationality, common knowlege, domination, the Nash Equilibrium (not doing worse than otherwise), and reciprocity (good-will and retaliation) that I've mentioned here. I would also like to further modify the Sequential Iterated Hawk-Dove game to build a model of Warhammer 40,000, and to eventually show where the traditional view of Math Hammer as Statistics fits into the mathematical treatment of Warhammer 40,000 as a game.

My favorite part about people that try to hammer it out, is they don't see the big picture.

I commonly lure them into losing a game by presenting a mathmatically attractive option that drags units out of place or works towards the greater plan.

tactics beat mathammer, people that don't have a concept of the math generally do poorly but too much emphasis is put on it.

I wrote a whole lot initially but my browser killed it. I would be interested to see how you abstract WH40k to match some of these Game Theoretic principles. More importantly, how you can they be transformed to identify winning strategies for these games.

I originally expanded on some of these concepts, but I do not want to step on your toes. Also, I assume you will be building this model into your explanation and I would hate to rehash things.

One interesting read about rationality though (it is short) is by Hey "Do Rational People Make Mistakes"


Automatically Appended Next Post:

Grundz wrote:
tactics beat mathammer, people that don't have a concept of the math generally do poorly but too much emphasis is put on it.

This is the 'Hannibal' syndrome, you can win every battle (tactics) while losing the war (strategy). Sometimes it is okay to lose individual encounters when it advances your aggregate strategy.

Grundz:

Yes, and in presenting your opponents with a suitably sweetened outcome that they find irresistable, you are using game theoretic mathematics better than they are.

Part of this discussion is to situate

calypso2ts:

The plan is not to abstract Warhammer 40,000 to fit game theoretic principles, but to build a model of Warhammer 40,000 using game theoretic principles, while stressing the basics such as the difference between tactics and strategies, and analyzing strategically preferable options.

I might also recommend writing in Notepad and then copy-pasting into your browser. I've been dealing with fishy internet connections for years and it's saved me lots of headache for long posts.

Incidentally, by 'Hannibal', aren't you referring Pyrrhus? Hannibal didn't win all his battles to lose the war, he just won most of them and then was recalled to North Africa when the Romans went on the offensive. Pyrrhus, on the other hand, gave his name to the Pyrrhic Victory.

calypso2ts wrote:This is the 'Hannibal' syndrome, you can win every battle (tactics) while losing the war (strategy). Sometimes it is okay to lose individual encounters when it advances your aggregate strategy.

My most recent game is direct annecdotal evidence of this. I was playing a capture and control game wherin my opponent was orks. By turn 3, I had crippled his mobility but he had a huge central position. Therefore, I sacrificed every unit in my army to prevent him from being able to take my objective, while sending a mobile unit to claim his. By doing this, my army had less points left on the board at the end, but I knew that I could simply make him take too long to go through my army to make it to my objective. I lost many battles in that war, but overall won the game 2 objectives to none.

edit: But its not quite as simply as that either: Had he chosen to play for the draw I would have been hard pressed to remove him from his own objective. Playing for a draw is often a good way to play as it often has the lowest risk.

Excellent post, which definetely makes for interesting reading. Whether I would personally analyse a game of 40k this deeply is up to the jury to decide, but I appreciate that other people take the time to write up an intriguing a read such as this.

L. Wrex

As with any kind of competitive analysis, it's probably better to go by gut instinct than to attempt to crunch the numbers under stress. That said, training yourself to do the math naturally means training your intuitions to match the math, so your intuitions are more accurate and reliable.

I should reiterate that something I'm going to be reiterating is that trying to win is not a sound strategy, and that it's better to avoid losing.

Interesting to note that the Hawk-Dove game is a great analogy. Trying to not lose will help you to win against someone who is trying to win, but is a horribly drawish way to play if you opponent also takes the same strategy.

However, if you examine the strategy as part of a tournament strategy, its actually not as helpful to try to avoid losing. Since avoiding losing results in more drawish games, you are less likely to come out on top at a tournament.

Dracos:

I think it would be helpful here to distinguish between losing the game and losing more in terms of acting irrationally. When I say that you should play to avoid losing, I mean you should follow the Nash Equilibrium and act rationally, not that you should attempt to play to a draw.

Remember, in the Hawk-Dove game, playing the Dove strategy is worse than playing the Hawk strategy. While both players playing the Dove strategy would get more by both playing Dove than both playing Hawk, they would get less by playing Dove if the other played Hawk. Playing Hawk is therefore playing to avoid losing, not playing to win.

As a placehold, therefore, it is useful to consider playing to win as substituting forthe Dove strategy and playing not to lose is the Hawk strategy.

Which brings me back to my point about not losing. If playing not to lose really is the equivalent of the Hawk strategy, then playing not to lose shouldn't result in more tied games, particularly if you're playing against someone playing to win.

It's an interesting point about the Hawk-Dove game that empirically speaking beginners play Dove at a far higher rate than experienced players, and the more experienced they are, the more likely they are to simply play Hawk.

Warhammer 40,000 has a far longer feedback loop between the strategy the player employed (if at all...) and the results of that strategy such that many players don't adapt a losing strategy. Indeed, that's why I raised DashofPepper's tournament report as an example of a good player by all accounts being flummoxed by an opponent that didn't also try to win.

Heh, I'm going to follow this up with something more substantive, but I thought I might mention that one of the Soul Drinker novels has the Adeptus Mechanicus (or an Explorator Fleet thereof) solving fleet-action problems using abaci because they can

To be honest an abacus is handy. I'm currently using custom Litko pieces and dice to calculate the value of stuff like the expected value of Venerable Dreadnoughts. So much easier than calculating in a linear fashion...

I like game theory, and I like that you know game theory, but I don't think the game theory is doing any work here at all. Consider this post an attempt to figure out exactly what you're saying.

First, a technical point. The prisoner's dilemma is a simultaneous game. 40k is a sequential game. The solution happens to be the same either way in this case, but simultaneous games in which there's no dominant strategy come out different if you make them sequential games.

Second, I don't really see where you're getting your payoffs. Where do you get these? I guess you're saying that, if the other player plays to win and you play to not lose, you'll beat them. And here your terminology is confusing. Shouldn't the player who plays to win have a better chance of winning? Otherwise he's not actually playing to win; he's just playing like an idiot. Maybe I simply don't understand your terms.

Maybe what you mean by 'play to win' is 'play riskily,' and by 'play to not lose' you mean 'play conservatively?' That makes your argument at least coherent, but it also makes it wrong.

It's wrong because risky plays don't necessarily have lower expected payoffs than conservative ones. Risk-loving people are just as rational as risk-neutral or -averse people. They just have a different preference when it comes to risk.

Now, you could be making an empirical claim, that in general risky plays in 40k do have lower expected payoffs than their conservative alternatives. That's very interesting if it's true, but (a) it doesn't require any game theory analysis, and (b) you haven't offered much evidence to prove it. Indeed, most tournament winners run pretty risky lists and play pretty risky games. On your payoff scheme, they should be losing to conservative players, but they don't.

Please correct me if I've gotten something wrong. I like the idea of applying game theory to 40k; I just don't think it works in this instance. Applying it to list-building has some potential, though. Maybe you can get us past the inane rock-paper-scissors analogy.

This is an interesting take on playing "not to lose", which is an effective idea, but I am a bit too bold to do that.

I am what you could call an "attack and recieve" tactician, I'll either let my opponent overextend themselves and crush their salient (much like the USSR at Kursk) or I advance on an even and broad front (terrain permitting) and gradually grind their force down. Not allowing them units to kill piecemeal, and making them suffer for ever loss they do enflict. It's a bit of a generalistic approach, but it's very effective.

Good thread Nurglitch!

Nurglitch wrote:1. Hawk vs Hawk: Each player takes a $1 from the other (1/1)
2. Hawk vs Dove: The Hawk player takes $1 from the Dove, and the Dove gives the Hawk $2 (0/3)
3. Dove vs Dove: Each player gives $2 to the other (2/2)

Nurglitch wrote:But so what if player B relatiated? Suppose the game ends on turn 5, with both players continuing to defect after turn 3? Player A would win with $9 while player B would lose with $6. Not bad, but player A is rational, and prefers $2 to $1. Had player A continued to co-operate, then he would have finished the game with $10. The same goes for player B. That tactical decision would have been irrational.

So why was it irrational if player A played Hawk on turn 3, when it won him the game, which otherwise would have been a tie? Firstly, it is taken as an assumption in Hawk-Dove and its game that player A is rational and prefers the monetary result to the game result

That doesn't make any sense. If I win $10 off you, and you win $10 off me, then neither of us win any money. If I play Hawk on turn 3 then I win $3

Firstly, it is taken as an assumption in Hawk-Dove and its game that player A is rational and prefers the monetary result to the game result

If that were the case why would he choose $0 over $3? doesn't make sense.

Nice comparison!
Admittedly I had to read it 2-3 times to understand it, but very thought-provoking none the less.

Nice post

i can see how this is a good strategy.


"Try not to lose"

Playing not to lose has been a core ideal I apply in pretty much every game I've played. 40k, A&A, RTS and TBS video games, etc. For 40k I fell into it naturally playing Necrons and have extended it to every other army I've dabbled in (Nids, IG, Tau). Form a strong defensive core first and formost. Add in glass cannon offensive elements that are designed to quickly cripple the opponent's own offensive capability, then let them make the mistakes and seize an opening if one appears, but don't risk the core to create an opening. Overall I'd say I draw a LOT, win occasionally and rarely ever lose. I also don't have the "I must win to have fun" perspective that I find with many people.

First let me start by saying I love this thread.

It's well thought out and well written. In my opinion the core concept is a relatively simple one but has the ability to fundamentally change the way people play their games. I am extremely interested to see what you do with this and will be watching this closely.

Second, I have been playing Tau since 2002 and learned the "don't play to win, play not to lose" philosphy fairly early on. I have always loved firewarriors and do my best to field 2-3 full squads in any given game. At first it was because from a fluff standpoint the Firewarriors are supposed to be the backbone of the Tau Military and more seen on the battlefield than any other type of unit. But over time I found I was losing many games because my oppnents just weren't failing saves and for all the pulse rifle's strength and range it won't kill anyone if they don't fail that armor save. I wrote posts on other forums and talked to everyone I could find at hobby stores. Eventually I came across a post on warseer where a guy began talking about how Tau players really shouldn't be trying to table their opponents.

"The name of the game in warhammer isn't who killed the most. It's who dies the least."

Since then I have always kept that in mind during my games and in doing so my ability to win while playing the troop heavy army I enjoy has increased dramatically. I no longer attempt to simply wipe out my opponent but try to weaken him so that my defensive strengths play through. (good armor, good sv's, good mobility) All the while thinking the more of my army that is alive than his increases my ability to take points, further damage his force, and control the scale in which statistical abnormalities effect my position.

While I have applied this theory to the Tau it can work for everyone and I'm just glad to see people talking about it... FOR THE GREATER GOOD! lol ;o)

Hesperus:

I do believe I spent some time discussing where sequential games fit into this, specifically the Sequentially Iterated Hawk-Dove game, the fact that Warhammer is a sequential game, and that I would be building a model once I'd covered enough material that the conceptual machinery would be available.

In other words, that first post was just about introducing basic concepts and encouraging people to use them in juxtaposition with Warhammer 40,000. If you feel that I haven't managed to show how game theory can be used to analyze Warhammer 40,000, you'd be right, because I haven't gotten to that point yet.

So instead, as an abstract exercise to get people thinking about the two, I linked playing to win with Dove, and playing not to lose with Hawk. The payoffs follow accordingly to the Hawk-Dove games. Risk here is in terms of not acting rationally and stating a revealed preference for less rather than more. I suppose the problem with introductions is that they don't also contain a body and a conclusion.

The important thing to take home, so to speak, is that one can think about Warhammer 40,000 in game theoretic terms, rather than any specific conclusions about strategy. I think I might go back and tag out that post to clarify.

SmackCakes:

The Hawk-Dove game and its descendents are not zero-sum games. Giving the other player $10 does not proportionately decrease your own score.

You may have noticed that I made no mention of the resources available to each player, and incidentally a concept of resources is one of the things that will distinguish the Hawk-Dove toy game from the game theoretic model I will be building.

Regarding the reason for a player rationally playing Dove in any single iteration of an Indefinitely Iterated Hawk-Dove game, I had explained that while a player may appear to benefit from the Hawk strategy in the Hawk-Dove game, the conditions of the Indefinitely Iterated Hawk-Dove game makes that benefit short-term and near-sighted.

There are three things that make playing Dove, and risking $0, instead of playing Hawk, and risking $3.
First there is the symmetry between the players such that they are both rational, and if doing something is a good idea for one player, it will be a good idea for another player. This means that player need to consider sets of outcomes, rather than particular outcomes.

Secondly there is something called backwards induction which is how one works back from a goal to establish how best to achieve that goal.

Thirdly, and tying backwards induction to symmetry, the player prefer more at the end of the game rather than any momentary gain at the end of any turn or iteration.

So [1, 3] > [0, 2] in the Hawk-Dove game because it is a single iteration and a player playing Hawk will never do as badly as when they play Dove.

In the Indefinitely Iterated Hawk-Dove game the players don't know when the game will end, and so while they will do better than their opponent if they play Hawk earlier, their opponent knows the same thing thanks to common knowledge and symmetry, but they will not do as well as when they continuously play Dove, since $2 > $1. Over five games, a pair of reliable Dove-players will rack up $10 whereas a pair of reliable Hawk-players will rack up $5. The Hawk-players are therefore acting irrationally because they prefer $10 to $5.

Notice that the symmetry in the Hawk-Dove game means that the actual result of play will be $1 for each player, which is preferable to $0 if either played Dove.

Let me return to the notion of backwards induction and the general problem of circularity. Lots of people have difficulty with game theory because it uses cyclical reasons, something that people sometimes mistake for the circular reasoning fallacy. Indeed, many people get caught in a viscious loop the first time they are introduced to the Hawk-Dove game because they want to think of it like "If I know that he knows that I know that he knows, etc"

As mentioned revealed preference circumvents all that blather about minds and thoughts, as it fixes the outcomes as an indexed table, which is why so many toy games are of the 2x2 grid variety with two strategies/tactics, because it's easier to demonstrate concepts with a parsimonious number of elements.

So my earlier post was a little rambling. I'll say things succinctly here.

I disagree.

As far as I can tell, your thesis is: Playing conservatively beats playing riskily.

That's fine, but the parts of game theory you introduce don't support it. They don't hurt it, either. They don't do anything either way.

Evidence suggests your thesis is wrong, at least at higher levels of play. If playing conservatively beat playing riskily most of the time, most tournament winners would be people who played conservatively (assuming those are the only two ways to play). But most tournament winners don't play conservatively.

Here's my opposing position: Playing riskily gives you a higher chance to either win big or lose big, and playing conservatively gives you a chance to either win small or tie. In tournament setting you have to win big, so it makes more sense to play riskily. In non-tournament settings it's a matter or personal preference.

I do think it would be interesting to apply game theory to list writing, ie the metagame, because that is a simultaneous game. It'll be complicated, though.

Hesperus:

I'm not sure how to make this clear: I arbitrarily connected playinng to win and playing not to lose with the Hawk-Dove game to encourage people to think about Warhammer 40,000 in game theoretic terms. I apologize for misleading you into thinking I was making anything more than an abstract claim for the sake of promoting discussion.

If you want to argue with any concrete claims I will actually be making, please stay tuned forwhen I actually make them.

Also I don't see any point to discussing list writing in the Tactics forum: that's Army List forum stuff. If you'd like to start a thread there about applying game theory to army list writing, please feel free to do so.

If i'm reading you correctly, it sounds like youre saying that the best game theory approach is to simply make choices that result in the "best" average or worst case scenario, as opposed to those strategies that result in the "best"best case scenarios.

it seems that in many games, the ability to gain positive value through skill is relatively slight, while the ability to lose value through mistakes is huge. Is that where you're heading?

It would seem that actual game play in 40k (aggressive vs. conservative) is not entirely related: there are times that playing aggressively is actually the way to avoid the largest loss. Playing horde orks against tau gunline is not the time to be coy.

It seems the other concept being alluded to here is the difference between "winning" and "acheiving the best result." the rational player wants to have the best possible value at the end of the day, instead of worrying about winning any given games.

Yeah, sorry, you posted while I was writing my last post and I didn't notice. If your're not actually trying to model anything yet, then cool. It's gonna be really hard to model a game as complex as 40k, but you seem to have the background to do it. I'm interested to see what you can do.

Lol thought it said 'Methhammer' that is all.
Post +1... lol

Hm, maybe I'm misreading Nurglitch's post, but my understanding is that using the "model" that he describes in the first post isn't really an abstraction that should be applied to 40k, either at a single game, or tournament level. It is far too simplistic, and does not accurately describe the game, or a series of games.

His point, I believe, was to give a framework to show how game theory can provide "answers" for rational-acting players in situations. For example, backwards induction is something that can be utilized to assess tournament play. That said, taking the same tourney scenario and simply shoving it into the prisoner's dilemma (or hawk-dove, because that's how I roll) game doesn't really provide the information you'd want.

So... Essentially it's all situational, and we should try not to lose?

Withering hail of lasguns > Game Theory.

My men will advance no matter what, with their bayonets. Strategically smart or not, I will take the entire damn table.

If your dichotomy is risky v. disciplined, then you're using risky in a way I'm not. Your way is something more like "reckless," which is definitely a bad way to play. Of course you always want to play the odds, because then (shockingly) the odds are in your favor.

I was using risk in the way typically used by economists, and especially game theorists. A 'riskier' choice in that sense is something with a lower probability of success, but a higher payoff.

Say, for example, you have a scoring unit on an objective and your opponent has a scoring unit on another objective (an that's it). It's the bottom of the 7th turn. You can sit on your objective and tie, or assault him on his objective, with a 25% chance of wiping him out, a 25% chance of being wiped out, and a 50% chance of both units surviving and contesting. Assume a win's worth 2, a tie's worth and a loss is worth 0.

Which should you do? Well, you could rationally pick either one, because they have the same expected value (of 1). A risk-loving person would go for it, while a risk-averse person would take the guaranteed tie. Neither choice is right or better.

Now, you're saying that disciplined play usually works better. That's probably true: you can't lose the forest for the trees. But a very risky player can be just as disciplined as a very conservative one. He simply has a different kind of battle plan. He can still stick to it, taking calculated risks, though he'll probably have to plan for more contingencies than a conservative player.

Seems like you have two real mottos.

1. Stick to the plan! That's good advice.

2. Do as well as you can! Also good advice. If you think you can only get a draw, then play for it. If you think you can pull a massacre, play for that. But if you think you can pull a massacre, you don't play to not lose.

I think I'm going to continue with the purpose of this thread now...

As mentioned, I wanted to discuss more game theoretic machinery in order to reinforce the principles discussed in relation to Hawk-Dove, and perhaps learn from my mistake in proposing abstracted applications over the specifics of Warhammer 40,000.

But first we need to go back to a fundamental aspect of Warhammer 40,000, and that is its sequential structure. Warhammer 40,000 alternates turns between players rather than simultaneously combining them, as well as ordering movement, shooting, and close combat movement as sequences. Close combat itself is simultaneous in that the results don't sequence into separate combat, and that models may attack at the same Initiative step, but also a sequence defined by the Initiative steps.

Both players pieces interact in close combat, making it a complex piece of simultaneity and sequence nested in the turn sequence. This complexity will be addressed after I've reviewed some implications using the Sequentially Iterated Hawk-Dove game.

In the Sequentially Iterated Hawk-Dove game, the players retain common knowledge of potential outcomes despite the choices made by the players happening in sequence. As noted, backwards induction tells us that players will rationally prefer to play Hawk if they know when the game will end, and prefer Dove if they do not.

Although the second player will appear to have the advantage of knowing the first player's move, both players will know the potential reactions to their movies via backwards induction, so that they will have the information necessary to anticipate the potential reciprocity and hence the dominating strategies.

Now it may seem that the players of Warhammer 40,000 do not have several of the elements available to Sequentially Iterated Hawk-Dove players. They do not appear to have perfect information, for instance, but that is to confuse uncertainty about any particular outcome with all of the potential outcomes in any given turn that form parameters of action. They seem to have complete information of the game, just as players of the Hawk-Dove games do.

However, the Hawk-Dove games lack an element in Warhammer 40,000, the random element of dice. This means that while players of the Sequentially Iterated Hawk-Dove games can assess the potential outcomes of either player's choices and hence the expected utility of the strategies and tactics available to them. For every choice, there is one of four outcomes, according to the payoff table of the Hawk-Dove game.

Players of Warhammer 40,000 are swamped with information about the expected value of their choices as well as the expected utility. The expected utility of an action in Warhammer 40,000 has ceilings ((except in the case of Blood Talons...) and floors with the floor typically set at 0 (though sometimes negative in the case of Gets Hot), and the space in between is divided by all the potentials. For example a squad of five Tactical Space Marines shooting with their Boltguns at a target 12" or less away have 11 outcomes, ranging from 0-10 results (damage/wounds), with the expected value usually closer to 0 than 10.

In a practical sense then the players often lack information about the outcomes of their own actions and those of their opponent, but in game theoretic terms they should be able (given enough time and calculating machinery) to determine the payoff table associated with each turn of the game. This complete pay-off table would suggest complete information is it was not the case that Warhammer 40,000 is concerned with scarcity of resources and space, as well as time (time being the number of turns or iterations).

So let's return to a perceptive 'mistake' made by a poster in this thread where they wondered why the players of the Iterated Hawk-Dove might prefer to exchange $10 in a zero-sum for $3 and a win. In the Hawk-Dove game the players can play Hawk or Dove without regard for whether they have the resources to so, or the space to make it happen. They just have to worry about the dimension of time limiting the dominating strategies.

Played using Victory Points as the payoff, Warhammer 40,000 becomes a zero-sum game, and Kill Points and random numbers of Objectives can be thus be regarded as a feature that broadens the number of strategies available from those of purely zero-sum games.

Regardless, this scarcity of resources and space means that the while players can use backwards induction and the payoff table to think one game turn ahead, they will not have complete information for all the potential game turns ahead. In the Sequentially Iterated Hawk-Dove game, playing Hawk does not prevent one's opponent from playing Hawk in retaliation the next turn.

By contrast destroying a Land Raider on the first turn prevents it from acting in subsequent turn and limits the Land Raider from synergizing with other units. Therefore the ultimate payoff is more than 2x that of destroying the Land Raider in the second turn. It is more than 2x the payoff because as well as potentially limiting the Land Raider from acting twice, it also limits the scope and options for retaliation.

A first-approximation of this situation can be demonstrated by rebuilding the Sequentially Iterated Hawk-Dove game so that both players have access to a limited pot of $10, five turns each and players receive their payoff on the next turn after playing a tactic. This means that the player with the first turn gets no payoff until their opponent chooses a tactic, and so on.

Indefinite iterations and variable first turn will be added to the model later.

I'll iterate that, as usual, rationality is measured by how much money the players accrue over the course of the game rather than whether they win.

See, this is what I was concerned about to start. People who haven't taken game theory are gonna have a heck of a time understanding what Nurglitch is talking about.

The prisoner's dilemma (which is identical to the hawk and dove game, incidentally) is not directly applicable to 40k. It's a simultaneous game and 40k is a sequential game (more specifically, an indefinitely iterated (repeated) sequential game.

In the prisoner's dilemma you don't pick between which player you're going to be, because each player has the same dominant strategy: squeal. There's never a time in the prisoner's dilemma when you shouldn't squeal.

You might keep quiet in one instance of an indefinitely iterated series of prisoner's dilemmas to build some goodwill with your opponent, so maybe you can get to the cooperative outcome. But I have no idea how to connect that to decisions in 40k.

So starsdawn, good try. It's confusing stuff. But it's not Sun Tzu, and it's not...whatever you were doing there.

And Nurglitch: As I said before, I think this is an interesting idea, but it's gonna be crazy hard to actually use. I'm having trouble following you and I have a background in econ. Use small words, and don't use examples unless you're actually modeling something. I think it's just confusing people.

And if I can help in any way, I'd love to. It's a huge project.

Deleted in Compliance with the Moderators

Some of the concepts explored in the Hawk-Dove game.

Rational - Players are rational when they consistently prefer more to less.

Players - Games have a minimum of two.

Iterations - Number of times a game procedure it repeated.

Tactics - Options in a game.

Strategies - Sets of tactics ordered over iterations. Synonymous with tactics when considering single-iteration simultaneous games such as Hawk-Dove and the Prisoner's Dilemma.

Payoff table - A table cross-referencing the utility of simultaneous tactics. It will have a minimum of four cells.

Revealed Preference - The utility of each intersection of each player tactics. In Hawk-Dove the revealed preferences are &[$0, $1, $2, $3] for each player.

The Nash Equilibrium - The strategy that best replies to the set of potential opposing strategies.

Domination - One strategy dominates other strategies if it is rational regardless of any opposing strategy.

Game tree - A flow-chart describing the relation of players to strategies and payoffs in sequential games.

Reciprocity - The property of a strategy reacting to the payoffs of previous iterations.

Common knowledge - Information about the game and its states shared by the players.