With the impending release of Bioshock Infinite: Siege of Columbia and those who’ve pre-ordered starting to receive their copies, it’s time to start evaluating certain strategies.  A key aspect of the game is riding the Sky-Line, a dangerous prospect that can lead to extremely fast troop movements but also result in destruction of either the Unit moving or a number of cards in your hand.  I’ll use a page from the rulebook to illustrate the concept and explain the rules:

 Sky-Line Rules

 

When moving along the Sky-Lines, you roll the three Sky-Line Dice after each movement.  The dice have two “Thumbs Up,” two 0s, and two 1s.  As long as you get at least 1 Thumbs Up, nothing bad happens.  If you fail to roll at least 1 Thumbs Up, you are given the choice of discarding a number of cards from your hand equal to the sum of the numbers rolled, or destroying the Unit you’re trying to move.  Therefore, let’s look at the odds of each outcome to determine the likelihood of bad things happening:

             Rolling three non-Thumbs: (2/3)^3 = 29.6%
 

Thus 70.4% of the time you get at least one Thumbs Up and you’re fine.  If you don’t get any Thumbs Up, there’s still a chance of rolling three 0s.  If that’s the case, you can choose to discard the sum, which is 0, and thus you don’t actually lose anything.  The odds of that happening are:

           Rolling three 0s: (1/3)^3 = 3.7%
  

So effectively you make a safe travel 74.1% of the time, or roughly 3/4 of your trips.  Now let’s look at the odds of losing various quantities of cards.  To do this, we can see that to get a mix of 0s and 1s, we have to roll those two numbers in a two/one split, and any of the 3 dice can be the singleton.  Thus:

         Rolling two 0s and one 1: 3*(1/3)^3 = 11.1%
         Rolling two 1s and one 0: 3*(1/3)^3 = 11.1%
         Rolling three 1s: (1/3)^3 = 3.7%
 

So putting it all together, we can see the total chances of loss:

        “Safe” passage = 74.1%
        Discard 1 card or lose Unit = 11.1%
        Discard 2 cards or lose Unit = 11.1%
        Discard 3 cards or lose Unit = 3.7%
 

Since movement happens after Voting and Production but before Combat, typically you’ll discard cards rather than lose the Unit unless you need those cards for Combat this Round, if you can afford them.  The expected value of losing a card for each space of the Sky-Line can be computed:

        0 Cards * .741 + 1 Card * .111 + 2 Cards *.111 + 3 Cards * .037 = 0.444 Cards
 

So for each space you take on the Sky-Lines, you can expect to lose 0.444 Cards, although ¾ of the time you’ll be safe.  If you are going to take a heavy loss (2-3 Cards) you may be wiser to opt to destroy the Unit and preserve the cards.  Obviously if the Unit is a more expensive Handyman or Leader Unit you may decide to take the heavier Card loss instead.

Let’s look at trips longerthan just a single space.  We just figured out that you lose 0.444 Cards per space travelled on average, so by travelling n spaces on the Sky-Lines you can expect to lose 0.444 * n cards on average.  This is true whether you’re moving a single Unit n spaces or n Units 1 space each, or some combination.  So moving 4 spaces on the Sky-Lines, e.g., will cost you on average 1.776 cards.  

If you feel like pushing your luck by travelling multiple spaces, we can take that 74.1% chance of safe passage on any 1 leg and multiply it by itself n times.  Thus:

Odds of n sequential Safe Passages = (74.1%)^n
 
1 Safe Passage = 74.1%
2 Safe Passages = 54.9%
3 safe Passages = 40.6%
4 Safe Passages = 30.1%
5 Safe Passages = 22.3%
6 Safe Passages = 16.5%
7 Safe Passages = 12.2%
8 Safe Passages = 9.1%
 

How many cards you'll actually lose can be pretty variable, but on average you'll have lost 3.555 cards (or some Units) during those 8 trips.  It's quite possible that early losses mean you won't ever have the chance to even attempt that long of a trip.

Finally, let's wrap things up by looking at the odds of having lost <= x cards after n trips.  This is a little more difficult, as we have to track the odds of losing between 0 and x cards in a certain number of trips.  I’ll spare you the math, but it works out as follows (odds are expressed in decimal form here, meaning 1 = 100%):

 

Lost 0 Cards

Lost <=1 Cards

Lost <=2 Cards

1 Trip

    0.7407

    0.8519

    0.9630

2 Trips

    0.5487

    0.7133

    0.8903

3 Trips

    0.4064

    0.5893

    0.7997

4 Trips

    0.3011

    0.4817

    0.7030

5 Trips

    0.2230

    0.3903

    0.6077

6 Trips

    0.1652

    0.3139

    0.5183

7 Trips

    0.1224

    0.2509

    0.4372

8 Trips

    0.0906

    0.1994

    0.3653

 

So, for instance, if you have 2 cards in hand that you don’t mind discarding you know that taking 6 trips on the Sky-Line will see you make those trips without exceeding 2 cards lost 60.8% of the time.  Since you move 1 space at a time, you can readjust your odds for each space travelled or decide when to cut your losses if one roll is particularly bad. 

At the end of the day, knowing the odds can only take you so far, and ultimately your fate comes down to when you start chucking those dice!  Good luck in your quest for control of Columbia, and watch out for that loose cannon named Booker!


Comments


Adam commented:

Never tell me the odds.

Posted on 2013-08-22.

thenobleknave commented:

Sir, the possibility of successfully navigating an asteroid field is approximately 3,720 to 1.

Posted on 2013-08-22.

BTBAM commented:

I don't want to read this. I just want to ride the skyline.

Posted on 2013-08-22.

Adam commented:

I read it just to confirm my suspicions that math truly is dark magick.

...and because I keep falling off.

Posted on 2013-08-23.

Matthew_Bester commented:

I hate maths!

Posted on 2013-09-15.

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