View Full Version : What is the DBA Utility Function?

The Black Rider
02-24-2005, 07:08 PM
Reckless Or Risk-Averse? Deriving a DBA Utility Function

Barker’s combat mechanism is fascinating, as much for its elegant simplicity as for its counterintuitive results. Try this: ask any DBA player “What’s the chance of a Knight killing a fellow Knight in open combat?” Try guessing without working it out mathematically!

The actual answer is three percent [just one chance in thirty-six] but new players, and even veterans, will often give much higher answers based on their psychological impression of a Knight’s power. The doubling rules, quick-kill rules, support rules, and dice probabilities all combine to produce a muddle that’s often confusing to new players who want to know the exact odds and preferred matchups of their combat units.

So I drew up a chart which clearly shows the chances out of thirty-six of any unit killing any other in close combat in good going [taking into account support and quick-kills]. Much tidier for new players.

The next step in preparing a guide for newbies was to rank the matchups ordinally for each unit. At this point, some relationships were very clear. A player likes matchups in which he has an advantage in chances-to-kill [Blade vs Pike, 6/0]. Less favorable are those in which he is evenly matched [Blade vs. Warband, 6/6]. Still worse are those in which the enemy is unkillable [Blade vs Psiloi, 0/0], and finally come the matchups in which the attacker is at a distinct disadvantage [Blade vs Knight, 2/15].

But within these categories, is there a way to mathematically determine which matchups are better or worse? If DBA were only a game of matchups, then clearly the greater the margin between the attacker’s chance of victory and the defender’s, the better the matchup. For example, a 4/0 and a 24/20 matchup would have equal value, since in both cases the margin is 4. But is this true? Will most players see the two matchups as having equal value? Or is the 24/20 matchup LESS valuable? Or perhaps, even more valuable?

What about a matchup between Blades and Knights or Scythed Chariots? Suppose you could charge the Blades with either Lancelot [odds: 15/2] or Boudicca [21/10]. Which would you pick?

To understand what I mean about DBA not being a game of “only matchups,” try the following example:

The score is tied at 3-3. Looking across the battlefield, you see that your opponent will surely get a fourth kill within the next two bounds - perhaps he’s trapped a unit against the edge of the board. You also spot a lone enemy Blade, which you can charge with either a Knight [15/2] or a Scythed Chariot [21/10]. Which do you pick?

A smart player reasons as follows: “The chance of my Knight or Scythed Chariot dying is irrelevant since the opponent will win anyway in two turns unless I win first. All that matters is the chance of “not-killing” the enemy [that is, a push, recoil, or destruction]. Whatever form this nonkill comes in, I lose. The Knight has a 34% chance of not killing the Blade over the next two bounds of combat, whereas the Scythed Chariot has an 18% chance. Therefore, Boudicca should charge the Blade.”

Since all DBA games end at some point, one could perhaps argue, as the general in the example did, that all that matters is the mortality rate. Although not played under the contrived constraints of the example, a normal DBA game is still as much about time and PIP management as it as about the margins of matchups.

But looking at only the mortality rate also goes against common sense. For example, all players will prefer a 34-0 matchup over a 35-34, although the second one has a higher chance of attacker success.

Clearly, A, D, and the margin (A-D) all play a role.

We’re no closer to a mathematical model, the utility function we’re looking for that will objectively rank all possible matchups.

But maybe by polling the DBA community and discussing the issue we can come up with a rough approximation. Try these questions:

1. Which is better, 20/10 or 15/5?

2. Which is better, 10/5 or 6/0?

3. You have a matchup in which you have an advantage of 5 chances-to-kill [say, 9/4]. You are given the option of increasing both your chance and the opponent’s chance by 1, to 10/5. Do you agree? Would you continue to do so indefinitely? 11/6, 12/7..... 30/25?

4. Would your answer to question 3 change if the margin was different? Would you stop adding sooner if the margin was smaller? Larger?

02-24-2005, 07:43 PM
Why isn't a SCh vs a Bd 21/15?

02-24-2005, 08:44 PM
Clearly the best match up gives you the best chance of killing your opponent combined with the lowest chance of being killed. Therefore 1/0 is infinitely better then 16/15. So perhaps a ratio of the chance to kill to the chance to be killed would be relevant.

However, these combats do not occur in a vacumn and it may not be the chance to kill an opponent as to kill or recoil them compared to the chance of being killed or recoiled as these two events will effect adjacent combats by improving or decreasing the chances of success.

The Black Rider
02-24-2005, 11:50 PM
Originally posted by Bob.:
Why isn't a SCh vs a Bd 21/15? Because I forget they are killed on a push smile.gif

Anyway, I figured out the answer. All you have to do is set up a game with two matchups and analyze that. In the example, one player would play with a Blade and Knight, and the other with a Scythed Chariot and Blade. Then you just calculate the probabilities for which player wins first. Of course you have to weed out the results in which neither player wins or both players get a kill.

The generale rule: suppose you have two matchups A/B and C/D and don't know which is better.

Plug the chances to kill:

A + D - B - C + BC - AD

And divide the whole thing by 36 when you're done.

If the value is positive, A/B > C/D. If negative, the second matchup is better. The equation gives you a value between -100 and 100 for how the matchups compare.

The Knight/Scythed Chariot matchup versus Blade has the advantage to the Knight, with a value of 4.8.

EDIT: Oh, and 1/0 is actually worse than 16/15! That is, the general playing the 16 and the 0 is more likely to win: a scalar value of 1.15 out of 100. Not much, but still an advantage.

[ February 24, 2005, 20:58: Message edited by: The Black Rider ]

02-26-2005, 04:55 PM
I'm no mathematician, but it seems to me that you will struggle to come up with a satisfactory answer. How do you account for the tactical intention of the player? If my purpose in a certain part of the battlefield is to delay you whilst I win elsewhere, then I will probably look mostly for chances of not-being-killed. I may also initiate combat in order to pull knights or warband out of formation, or to prevent bow or artillery from firing at a more vulnerable target or for any number of reasons which will be hard to quantify.

Nor do you need me to tell you that the flank advantage counts a lot with the low-factor QK types like Wb and Kn, so I may initiate combat with a low chance of killing (say Bd vs Ps) in order to win a flank advantage or deny a flank advantage to the next element down.

I don't consider myself a brilliant player, but generally I only look over the board searching for the best chance of a kill when things are going very badly and my plans have failed. I suspect better players are the same (only their plans fail less often).


p.s. actually the Chariot is better than the knight, because if the chariot dies you don't lose the game and can try again next bound. Even better, send both!