kelly criterion with more than two outcomes

Kelly.js

var _ = require('lodash');

var a  = [0.2,0.1];
var b  = [10,30];
var a0 = 1 - _.reduce(a, function(s, n) { return s + n; });

function concave(x) {
  var result = a0 * Math.log(1 - x);
  for (var i = 0; i < a.length; ++i)
    result += a[i] * Math.log(1 + b[i] * x);
  return result;
}

function firstDerivative(x) {
  var result = a0 / (x - 1);
  for (var i = 0; i < a.length; ++i)
    result += a[i] * b[i] / (1 + b[i] * x);
  return result;
}

function sq(x) { return x * x; }

function secondDerivative(x) {
  var result = - a0 / sq(x - 1);
  for (var i = 0; i < a.length; ++i)
    result -= a[i] * sq(b[i]) / sq(1 + b[i] * x);
  return result;
}

function newton() {
  var x = 0.5;
  for (;;) {
    var y = x - firstDerivative(x) / secondDerivative(x);
    if (Math.abs(x - y) < 1e-8)
      return y;
    x = y;
  }
}

console.log(a0);
console.log(a);
console.log(b);
console.log(newton());