Add Lanchester attrition handicap function

[ajul]

The payoff would be the fraction of remaining force after a Lanchester attrition model is applied to completion.

Like the other existing handicap functions it would take an initial matrix encoding the relative effectiveness of the strategies against each other. This matrix should be log-skew-symmetric. Another parameter would be the Lanchester exponent. Handicaps could be interpreted as unit costs.

This is symmetric. Since this depends only on the ratio of the handicaps it is also one-parameter.

One example appeared previously: Hazard, C. J. 2010. What every game designer should know about game theory. Triangle Game Conference. Raleigh, North Carolina.

• [x] Basic implementation
• [ ] At least one example

[ajul]

Hmm actually as originally stated Hazard's example is not quite the same as the Lanchester model as stated above---the Lanchester model assumes both sides buy up to the same "budget", while Hazard's example has each side having one unit regardless of cost.

Though given that scaling all handicaps (costs) only scales the payoff matrix by the same, I would still expect a unique solution matrix, even if this is not strictly one-parameter as defined in the paper?

[ajul]

Implemented both Lanchester and Hazard separately. The old convergence issues (#1) reared their head when I used an exponential rectifier, with the same solution.

One thing of note is that in the non-symmetric case Hazard is sensitive to global scaling.

[ajul]

Misremembered the Lanchester attrition model---there is actually a non-trivial effect of exponent on surviving proportion. Will need to fix this.

[ajul]

Implemented a corrected version. Unfortunately it appears to have a vertical derivative wherever both sides are evenly matched.

Would be good to have an example also.