How I got 3rd place in Battle for Riddler Nation, Round 2

I recently participated in the “The Battle for Riddler Nation, Round 2“, eventually taking 3rd place out of a field of 931 participants.  Here’s how I did it.

HOW TO PLAY

Like most challenges, this game’s concept is easy to grasp but features a great amount of depth. Each participant is a king of a nation of 10 Castles, worth 10,9,8,7… 1 points each but only has 100 soldiers to defend them.  Each monarch uses his authority to specify a specific number of troops to each Castle, the battle begins. Each Castle faces off  with the corresponding Castle of the opponents, and the Castle with the most soldiers wins the points. The monarch with the most points after all 10 Castles have fought is the winner!  The war is actually a round robin tournament between all 931 participants.

The trick to the game is to secure the minimum amount of points to win (28/55) with any combination of Castles possible.  Any points secured over 28 is superfluous and thus wasted resources.  Seems simple, but the here’s the rub: everyone else is doing the same thing you are, and the game is just as much against them as it is to win the points.

The first battle for riddler nation happened in February, and everyone was just shooting from the hip without knowing what others would do. The difference in this battle the data from that tussle was made available to all, as well as the winners strategy.   Here’s the data (both round 1 and 2).

MAXIMUM SOLDIERS PER POINT DISTRIBUTION

The first thing that I did was calculate the soldiers dedicated per point in order to reach 28 points, (i.e, 100/28=3.5_7, and then multiplied them by points per castle to determine the garrison needed in each castle to secure the minimum of 28 points.

Even Distribution of Soldiers with maximum soldiers per point. (rounded)

											Total
Castles	1	2	3	4	5	6	7	8	9	10	55
Soldiers	4	7	11	14	18	21	25	29	32	36	196

Obviously, there are way too many soldiers in defense of the castles; 196 soldiers is 96 more than allotted. Theoretically, any combination of soldiers adding up to 28 points in the above distribution should get close to winning. The trick is which castles to fight over.

 Previous winners distribution

Castles	1	2	3	4	5	6	7	8	9	10	55
Soldiers	3	5	8	10	13	1	26	30	2	2	100

Looking at the previous winners data, he went for castles 1-7, but to one up those doing the same, abandoned 6 and went for 8 instead.  Comparing to the max soldiers per point distribution, the winner undershoots it on every castle except for 7/8.  Everyone competing sees this distribution and every viable strategy has to beat it at a minimum.

USING THE DATA FROM THE LAST BATTLE

Here’s where the previous data came in: I inserted the troop distributions into excel and sorted by from greatest to least placed at each castle.   The previous winner won 84% of the battles, so I wanted to reach that goal.  In order to do so, I found the amount of soldiers that would beat 95% of the castles from the previous battle.

95% win rate compared to Max soldiers per point. 

Castles	1	2	3	4	5	6	7	8	9	10	Total
Soldiers at 95% win	11	11	12	16	20	24	28	34	35	36	227
Soldiers- max per point	4	7	11	14	18	21	25	29	32	36	196

The total for 95% win rate was obviously too high to be reasonable (227 total and more than the maximum allot-able on nearly every Castle) so I lowered my standards to 90%.

90% win rate compared to Max soldiers per point.

Castles	1	2	3	4	5	6	7	8	9	10	Total
Soldiers at 90% win rate	6	9	11	14	18	22	26	31	32	31	200
Soldiers- max per point	4	7	11	14	18	21	25	29	32	36	196

That looked much more reasonable, in fact, it follows the max soldiers per point quite precisely.  In total, it only has 4 more than the max troops per Castle. The question remains, where do you put your soldiers? To solve this question I subtracted the 90% win rate from the Max per point ratio.

Max soldiers per point minus 90% win rate

Castles	1	2	3	4	5	6	7	8	9	10
(Max per point) – (90% win )	-2	-2	0	0	0	-1	-1	-2	0	5

To help with interpretation, a positive score means value, and a negative score means a sub optimal alignment.  This showed that people placed too many soldiers at castles 1,2,6,7,8.  The masses placed the max soldiers allowable at 3,4,5,9.  Castle 10 was a superb value, as 90% win rate was 5 soldiers less the optimal amount.  The best combination of castles and value to reach 28 points was to pursue 4,5,9 and 10.

STRATEGY AIMING FOR 90% WIN AT CASTLES 4,5,9 AND 10

Castles	1	2	3	4	5	6	7	8	9	10	55
90% for 28 points	0	0	0	14	18	0	0	0	32	31	95
First Plan	0	1	1	14	18	1	1	1	32	31	100
Final Plan	0	0	0	15	19	1	1	1	32	31	100

This left me with 5 soldiers to distribute as I saw fit. I experimented briefly but settled on reinforcing the castles 4,5 and placing a solitary soldier on 6,7,8 to steal some points.

I ended up with a record of 737 wins, 7 ties, and 187 losses and a 79.1 % win rate, good enough for 3rd place and one tie from second.

ACTUAL WIN RATE AT CASTLES 4,5,9 AND 10

I started with a goal of 95% wins with each Castle I attacked, but lowered my standards to 90% win.  Here’s my my rank and percent win rate (counting a tie as a loss) at Castles 4,5, 9, 10.

Castle	4	5	9	10
Soldiers	15	19	32	31
Rank	110	78	61	64
%ile	88%	92%	93%	93%

Success! I won just over 90% at 5/9/10 and just under at 4. Interestingly if you assume each loss was independent, multiplying the percentages would have given me a win rate of 70%.  Fortunately for me, about 9% of the time when I lost one of the 4 Castles, I also lost another, giving me a total win percentage of 79%.

And that’s how I got 3rd place in the Battle for Riddler Nation,  Round 2.  Thanks to Oliver Roeder and fivethirtyeight.com for allowing for the battle royale to take place.

Derek Shafer

@dshafe55

3rd Place, Battle for Riddler Nation, Round 2