Rotating array members
You should use shift and push

function rotate( array , times ){
  while( times-- ){
    var temp = array.shift();
    array.push( temp )
  }
}

//Test
var players = ['Bob','John','Mack','Malachi'];
rotate( players ,2 )
console.log( players );


shift removes the first element, push adds an element at the end.
I am not sure whether you are using players_new because you do not know how to modify the original array or because you do not want to modify the original. If you do not want to modify the original array you could:

function rotate( array , times ){
  array = array.slice();
  while( times-- ){
    var temp = array.shift();
    array.push( temp )
  }
  return array;
}

//Test
var players = ['Bob','John','Mack','Malachi'];
console.log( rotate( players ,2 ) );


Finally, if you meant to declare an array, you should have var players_new = [];, not var players_new = {};. Plus players_new is an unfortunate name.

Golfic edition:

while(times--)array.push(array.shift());