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Subject: When will I win the campaign? Analysis. [No spoilers] rss

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Nick Hare
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This thread was prompted by one of my fellow players asking how many (more) victories he would need to outright win the 15-game campaign (i.e. to end up with the most wins by the end of game 15).

When looking into it I was surprised that the answer was not entirely straightforward. The reason is the multiplayer element - with two players the answer is easy to find because any win to one player must equal a loss to the other. With several players (and a Risk: Legacy campaign could involve between 3 and 75 players) it's not quite as simple...

...but it is pretty simple nontheless. Without bothering with a proof (anyone who could interpret it would also be able to derive it) I present the results: a simple set of rules to determine which games are 'crucial' (defined as: could result in a player losing any chance of winning by the end of the tournament, or could result in a player having enough wins to have a certain victory) and for whom.

The first rule gives a 'Personal Minimum', which is calculated as follows:

(Current score of leading player) - (Games remaining)

If your score is higher than this, don't worry - YOU don't HAVE to win the next game to stay in contention (but see below). If your score is EQUAL to this amount, you WILL HAVE TO WIN the next game to stand a chance of winning the campaign. If your score is below this amount, I'm afraid you cannot win the campaign and should start making deals with other players to gain favours etc.

The second rule gives a 'General Minimum', which is very similar:

(Current score of leading player) - (Games remaining) + 1

If your score is equal to this amount, then you will need to make sure that anyone EXCEPT the current leader wins the game, in order to stay in. Again, if your score is above this then the game will not be a crucial one (for you).

These rules have important consequences for the current leader. If you are the leader and the SECOND-HIGHEST-SCORING PLAYER has a score equal to the PERSONAL minimum then preventing him from winning in the next game - e.g. by helping one of the trailing players (the 'scumbag' method) or by winning yourself - will mean you win the campaign. If the second-highest-scoring player has a score equal to the GENERAL minimum then you will need to WIN the next game yourself to win the campaign.

Example: there have been 10 games, and the scores (4-player) are 5-4-1-0. The 'personal minimum' is 5 (leading score) - 5 (games remaining) = 0. This is the score that player 4 has - so he will NEED to win the next game in order to stand a (albeit remote) chance of winning the campaign. The 'general minimum' is 5 - 5 + 1 = 1. This means that player 3 (with 1 win) must stop the current leader from winning in order to stay in the game. Because player 2 (with 4 wins) is above both these scores, there is no chance of the campaign ending because player 1 (the leader) cannot win outright in the next game.

Summary:
If second-highest player score = the personal minimum, he MUST win the next game to prevent the leader from winning the campaign;
If second-highest player score = the general minimum, anyone EXCEPT the leader must win the next game to prevent him winning the campaign;
If your score = the personal minimum, you MUST win the next game to stand a chance of winning the campaign;
If your score = the general minimum, you MUST stop the leading player winning the next game to stand a chance of winning the campaign;
If your score is above the general minimum, then this game is not crucial (for you at least) so relax.

One final consideration - when do you need your next win by to stay in the game? This actually depends on a number of things, including number and composition of players, as-yet-unknown outcome of subsequent games etc., and amounts to asking 'when will the personal minimum equal my score?' An easier-to-answer, and potentially more useful, question is 'when do I need to win a game by to GUARANTEE I won't be knocked out by then?'.

If you are the leading player, you need to work out the following:

T = (Number of games left + size of lead over next player) / 2

Round up. Win at least once in the next T games to GUARANTEE you'll still be in contention for the campaign at that point.

If you are not the leading player, you need to work out the following:

T = (Number of games left - size of lead of winning player over you) / 2

Round up. Win at least once in the next T games to GUARANTEE you'll still be in contention for the campaign at that point.

Example: in my actual Risk:Legacy campaign the scores are 3-1-0-0-0 (yeah, I know, I know). The leader (with 3 wins) will need to win in the next (11 + 2) / 2, rounded up = 7 games to guarantee he'll still be in contention. The runner-up (with 1 win) will need to win in the next (11 - 2) / 2, rounded up = 5 games to guarantee he'll still be in contention. The rest of us will need to win in the next (11 - 3) / 2 = 4 games to guarantee remaining in contention.

However, NOT winning in that time does not mean you'll definitely be knocked out, and the number will actually change from game to game depending on who wins. But right now you know that IF you win in the next time T, you will DEFINITELY still be in contention for the campaign at that point. I like this statistic because it gives you a sense of how much air is left in your tank before you really need to start worrying.

Hopefully this will be useful for constructing Doomsday Clocks describing your personal Risk: Legacy world.
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