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?

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?