Wikipedia:Reference desk/Archives/Mathematics/2012 September 20

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September 20

Lanchester's Law and Game Theory

Let us consider Lanchester's square law and three groups A,B,C which want to go to war. Making up numbers, let's say A,B,C have armies of sizes 45, 40, and 35 respectively. I am concerned with the optimal strategy for group C. From what I understand, B and C should form an alliance. They'll have a vastly superior force than A. They will beat A and both B and C will suffer casualties in proportions. After A has been beaten, then B and C can fight each other. It seems to be that if C has the smallest army, then it will always lose in the end. A,B,C can randomly all shoot at each other or B and C can form an alliance, wipe out A, and then turn against each other. Is there ever an case (with army sizes being A < B < C...notice strict inequalities) where Lanchester's Law and Multiparty Game Theory predict that C will win? A numerical counterexample is what I am looking for of course. If not then, C will always lose, right? So what is C to do? What incentive is there for C to form an alliance with B? It knows it will be killed in the end anyway so why help B (over A) survive? Is this the best strategy? If yes, then why because C's destruction seems inevitable so why should C care what it does and who it helps and what strategy it chooses? - Looking for Wisdom and Insight! (talk) 04:59, 20 September 2012 (UTC)

I assume you meant "A > B > C" Dbfirs 07:54, 20 September 2012 (UTC)

I can see three possible winning strategies for C:

1) The obvious one is to make peace with both A and B.

2) Form an alliance with one and hope that it holds after the other is defeated.

3) The most Machiavellian way for C to win is to secretly instigate a war between A and B, then, after one is destroyed and the other is weakened, attack. StuRat (talk) 05:18, 20 September 2012 (UTC)

See truel for something very similar. Since peace treaties as unnecessary between friends I view them as declarations of war but not yet. Option 3 certainly seems the best option for C, the problem is what happens if A or B discover it. C could also send an army of size 5 to 'help' B. It would be interesting to try and figure out the strategies to stay alive the longest if they are all fighting each other. Say army A devoted a(t) of its effort to destroying B and 1-a(t) to destroying C, so C was being destroyed by a force of A(1-a(t))+Bb(t), B by a force of C(1-c(t))+Aa(t) and A by B(1-b(t))+Cc(t). Dmcq (talk) 12:40, 20 September 2012 (UTC)

Peace should break out all over in ABC world, for no one can afford to attack alone, nor be the weaker ally in an attack. Rich Farmbrough, 21:10, 20 September 2012 (UTC).

Yep as far as I can see if they have a three way war then all three will be completely destroyed at the same time if the two smaller ones could defeat the largest. What'll happen is that the smaller ones will gang up on the largest until it is no longer the largest, then the two larger will reduce each other till all three are the same size as the smallest. Dmcq (talk) 22:31, 20 September 2012 (UTC)

League problem

Resolved

For a game I'm trying to come up with a particular league system, and I'm unsure how to go about doing it. I tried to do it manually but I couldn't come up with an algorithm that led me to a solution. Basically I have 16 teams in a league, but for logistics' sake I need to break it up into four 4-team series. I figure if I can break it up right, then I can each team have faced every other team in one of the series once, and only once. So in the first leg I have four series:

Series 1	Series 2	Series 3	Series 4
Team A	Team E	Team I	Team M
Team B	Team F	Team J	Team N
Team C	Team G	Team K	Team O
Team D	Team H	Team L	Team P

Each team has to face 15 other teams, and in each leg it faces 3 other teams, so then in 5 legs any given team will have faced all 15 other teams. I can come up with two other legs: one by simply flipping it around the diagonal (so Series 1 would be teams A, E, I, and M), and the other by taking diagonals for each series (so Series 1 would be teams A, F, K, and P), but beyond that I start running into problems. Maybe I'm backing myself into a corner, I don't know. And I calculate that there are 2386 possible ways to sort the 16 teams into 4 series, so I'm not about to check all of those, to find five "orthogonal" legs.

Is there a better way of going about this, in a more mathematical way? Thanks —Akrabbim^talk 16:47, 20 September 2012 (UTC)

I think your approach is reasonable. Your 4th and 5th groups will be "what's left":

Series 1	Series 2	Series 3	Series 4
Team A	Team E	Team I	Team M
Team B	Team F	Team J	Team N
Team C	Team G	Team K	Team O
Team D	Team H	Team L	Team P

Series 1	Series 2	Series 3	Series 4
Team A	Team B	Team C	Team D
Team E	Team F	Team G	Team H
Team I	Team J	Team K	Team L
Team M	Team N	Team O	Team P

Series 1	Series 2	Series 3	Series 4
Team A	Team E	Team I	Team M
Team F	Team J	Team N	Team B
Team K	Team O	Team C	Team G
Team P	Team D	Team H	Team L

 .
 .
 .

Also note that there are many possible solutions. StuRat (talk) 17:12, 20 September 2012 (UTC)

But the other diagonal will intersect. One direction will be A,F,K,P, and the other direction will be A,N,K,H. A and K are in both. —Akrabbim^talk 17:29, 20 September 2012 (UTC)

Yes, you're right, I've revised my advice above. I'm going to write a program to solve this and post it tomorrow. StuRat (talk) 17:38, 20 September 2012 (UTC)

Thanks, I'll be interested to see what you come up with and your algorithm to solve it. I was trying to work it out with Matlab, but it was just turning into a clumsy brute force algorithm, and I wasn't getting anywhere. But I charted it out, and three legs that we have here are definitely not part of the solution. For example, for A and G to meet, the only teams that haven't met A and G would be J and N, and they have both played each other already, so I right when I thought I had painted myself into a corner. —Akrabbim^talk 17:56, 20 September 2012 (UTC)

(Teams have numbers rather than letters, obviously)

Series 1: 1, 2, 3, 4
Series 2: 5, 6, 7, 8
Series 3: 9, 10, 11, 12
Series 4: 13, 14, 15, 16

Series 1: 1, 5, 9, 13
Series 2: 2, 6, 10, 14
Series 3: 3, 7, 11, 15
Series 4: 4, 8, 12, 16

Series 1: 1, 6, 11, 16
Series 2: 2, 5, 12, 15
Series 3: 3, 8, 9, 14
Series 4: 4, 7, 10, 13

Series 1: 1, 7, 12, 14
Series 2: 2, 8, 11, 13
Series 3: 3, 5, 10, 16
Series 4: 4, 6, 9, 15

Series 1: 1, 8, 10, 15
Series 2: 2, 7, 9, 16
Series 3: 3, 6, 12, 13
Series 4: 4, 5, 11, 14

81.159.104.182 (talk) 03:41, 21 September 2012 (UTC)

You beat me too it, 81. Yes, that's a valid solution. 81's version of diagonals in the third grouping seems to work, while mine does not. (Running my program, I found one solution given his first 3 groupings, and no solution given my first 3 groupings.) StuRat (talk) 06:17, 21 September 2012 (UTC)

BTW, Akrabbim, where did the 2386 number come from ? My program seemed to come up with 2,627,625 possible groupings. Here's the formula:

16!
---
4!5

StuRat (talk) 06:36, 21 September 2012 (UTC)

(formerly 81.159) I agree: 16!/(4!^5). I don't understand where that 2386 number comes from either. 86.160.216.189 (talk) 11:28, 21 September 2012 (UTC)

Hey, that's great! Thanks! Can I ask how you figured it out? And about the 2386: I was thinking it would be choose 4 for the first group, then 4 for the next group from the remaining, and so on. So (16/choose/4) * (12/choose/4) * (8/choose/4) * (4/choose/4) = 1820 * 495 * 70 * 1 = 63,063,000 (when I first punched it in accidentally did + instead of *, which is where I got 2386). I haven't thought about probability at all in years though, so I'm not surprised I'm wrong. —Akrabbim^talk 14:34, 21 September 2012 (UTC)

Arrangements in which the split into groups is the same but groups are listed in a different order are counted separately in your count, but only count as one arrangement in StuRat's count. So your count is 4! = 24 times as large as StuRat's count. Gandalf61 (talk) 14:40, 21 September 2012 (UTC)

Agreed, there are the 24 ways to arrange the series, but those are all equivalent. For example, these two are equivalent:

Series 1	Series 2	Series 3	Series 4
Team A	Team E	Team I	Team M
Team B	Team F	Team J	Team N
Team C	Team G	Team K	Team O
Team D	Team H	Team L	Team P

Series 4	Series 3	Series 2	Series 1
Team A	Team E	Team I	Team M
Team B	Team F	Team J	Team N
Team C	Team G	Team K	Team O
Team D	Team H	Team L	Team P

As to the method I used, it wasn't very elegant, I just used brute force. Can we mark this Q resolved ? StuRat (talk) 16:31, 21 September 2012 (UTC)

"when I first punched it in accidentally did + instead of *" ... Ooops!! 86.160.216.189 (talk) 17:28, 21 September 2012 (UTC)

Thanks everybody, yes it is resolved. —Akrabbim^talk 01:09, 25 September 2012 (UTC)

x/y in IV quadrant, is the given ratio negative or positive?

question: "Suppose that the point (x,y) is in the indicated quadrant (IV). Decide whether the given ratio is positive or negative." Is it not positive, simply by sketching? How would it be negative? thanks.24.139.14.254 (talk) 20:08, 20 September 2012 (UTC)

Just look at the signs of x and y. The ratio of two numbers with the same sign is positive; the ratio of two numbers with opposite signs is negative. Widener (talk) 20:28, 20 September 2012 (UTC)