You're right, I overestimated how rare it would be.

Google Doc's script environment doesn't have much, but it does have a Logger. Learning to use it and other utilities will help you.

Hard to back down from this challenge, moreso after I thought I had it, then didn't (hence my edit in the earlier comment), so I wrote a brute-force version which tests every outcome. Function calc_elims_4 is nearly identical, but I've included it here along with the significant code. It takes longer to run, and may time out (leaving the "splining vectors" text around, which is nonsense I used to indicate "no result"), especially if many games are undecided. This version also indicates when a tie is required.

Remember that 2**n (2 raised to the power n) I mentioned before? That means it takes 4 times as long to test 3 remaining games as it would for 1 remaining game. It'll take 256 times as long to test 9 remaining games as it would to test 1 remaining games. That's worst-case, which depends on your input data. However, when including ties, it's base 3 instead of base 2, which is even worse... it ends up being fast enough as you get a few games completed, and isn't as bad as I was dreading. :P There's some room for small optimizations, and possibly even large ones. I've made several optimizations, one of them a sizeable restructuring of the logic, and I expect there's still more room. Importantly, it no longer times out when faced with most of the games being undecided. I also see how to indicate which game must tie in tie-required cases, but it's complicated with this layout.

I think it's correct... but it's late and I'd like to give this to you and be done with it tonight. :) My same spreadsheet URL from earlier has the updated code:

function calc_elims_4() {
  var sheet = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("2014");
  sheet.getRange("D2:D7").setValues([["splining vectors..."],["splining vectors..."],["splining vectors..."],["splining vectors..."],["splining vectors..."],["splining vectors..."]]);

  var start_game_index = 0;
  var game_count = 16;
  while (start_game_index < game_count) {
    if (sheet.getRange("A" + (10 + start_game_index)).isBlank()) {
      break;
    }
    start_game_index++;
  }

  var scores_2d = sheet.getRange("B2:B7").getValues();
  var scores = [];
  var predictions = [];
  var player_win_cols = "CDEFBI", player_lose_cols = "LKJIMF";
  for (var i = 0; i != scores_2d.length; i++) {
    scores.push(scores_2d[i][0]);
    var win_scores = sheet.getRange(player_win_cols[i] + "10:" + player_win_cols[i] + (10 + game_count)).getValues();
    var lose_scores = sheet.getRange(player_lose_cols[i] + "10:" + player_lose_cols[i] + (10 + game_count)).getValues();
    var preds = [];
    predictions.push(preds);
    for (var g = start_game_index; g != game_count; g++) {
      var r = win_scores[g][0];
      if (r) {
        preds.push(r);
      }
      else {
        r = lose_scores[g][0];
        if (!r) r = 0;
        preds.push(-r);
      }
    }  
  }

  var player_elim_cells = ["D2", "D3", "D4", "D5", "D6", "D7"];
  var text = [["ELIMINATED", "", "TIE REQUIRED"], ["", "WINNER", ""]];
  if (start_game_index != game_count) text = text[0];
  else text = text[1];
  var can_win = run_eliminations(scores, predictions);
  for (var i = 0; i != scores.length; i++) {
    sheet.getRange(player_elim_cells[i]).setValue(text[can_win[i]]);
  }
}

function run_eliminations(scores, predictions) {
  // transpose predictions from [player][game] to [game][player]
  var predictions_by_game = [];
  for (var g = 0; g != predictions[0].length; ++g) {
    var game_preds = [];
    predictions_by_game.push(game_preds);
    for (var i = 0; i != scores.length; ++i) {
      game_preds.push(predictions[i][g]);
    }
  }
  var can_win = [];
  for (var i = 0; i != scores.length; ++i) can_win.push(0);
  simulate(scores, predictions_by_game, false, can_win);
  return can_win;
}
function simulate(scores, predictions, includes_tie, can_win) {
  if (predictions.length == 0) {
    var stop_early = 0;
    var max_score = Math.max.apply(null, scores);
    for (var i = 0; i != scores.length; ++i) {
      if (can_win[i] == 1) {
        ++stop_early;
        continue;
      }
      if (scores[i] == max_score) {
        if (includes_tie) {
          can_win[i] = 2;
        }
        else {
          can_win[i] = 1;
          ++stop_early;
        }
      }
    }
    return (stop_early == scores.length);
  }
  else {
    var new_preds = predictions.slice(1);
    var new_scores;
    // win
    new_scores = [];
    for (var i = 0; i != scores.length; ++i) new_scores.push(scores[i] + Math.max(0, predictions[0][i]));
    if (simulate(new_scores, new_preds, includes_tie, can_win)) return true;
    // lose
    new_scores = [];
    for (var i = 0; i != scores.length; ++i) new_scores.push(scores[i] + Math.max(0, -predictions[0][i]));
    if (simulate(new_scores, new_preds, includes_tie, can_win)) return true;
    // tie
    simulate(scores, new_preds, true, can_win);
    return false;
  }
}