Colonel Blotto Games!

The (Colonel) Blotto Games!

The Blotto Games (aka the "Divide Dollar" games) is a type of two person, zero sum game. The players involved are supposed to distribute a said amount of limited resources over several object, in this case, battlefields. The player that devote assigns more resources to a specific battlefield wins that battle and the payoff is the number of battlefields that are won.

Sadly, Colonel Blotto is not real. He is just a fictional character in a paper, BUT, in said paper, he was given the task to find the ideal distribution of his soldiers over N battlefields with these conditions:

1. Each Battlefield victory is given to the side that gives more soldiers to that specific field.
2. However, the two parties do not know how many soliders the other side will distribute to that specific battlefield
3. In order to win, you want to capture more battlefield sites than your opponent

 Example Example:

For example, lets say this game is between two players who have to choose 3 positive integers in acending order such that they add up to a specified number S. Once agian, the players must choose these numbers (without knowing what other player will do) and show their distribution at the same time. 

For S = 6, the possible arrangements are (1,1,4), (1,2,3), or (2,2,2). SO:

1. Anything against itself is a draw
2. (1,1,4) draws with (1,2,3) ---> 1 < 2, but 4 > 3
3. (1,2,3) draws with (2,2,2) ---> 1 < 2, but 3 > 2 
4. (2,2,2) beats (1,1,4)

Its clear to see that the optimal strategy is to choose the (2,2,2) as it either draws or beats any other pair with S = 6. If both players decide to choose either (2,2,2) or (1,2,3), then none of them can beat each other so these pairs just so happen to be the Nash Equillibrium. 

Of course, as S goes up, the game becomes harder to analyze. With S = 12, we can come to the conclusion that (2,4,6) is the optimal strategy. However, with S > 12 (lets say 13) , we can see that either choosing (3,5,5) , (3, 3, 7) or (1, 5, 7) each have a .333 probability to be the ideal strategy (as each either lose or win against the other two pairs. 

Colonel Blotto vs Cononel Lotso! 

Lets say Blotto wants to attack and capture 2 different locations. Lucky for him, he has 4 resources he can distribute across as he may. On the other hand, his emeny Lotso wishes to capture those same 2 locations, but he sadly only had 3 resources to choose from to distribute.

SO lets talk Payouts and Utility.

The payout for each "battle" will be 1 + however many troops the less Colonel sent over to that site. So for example:

If each Colonel sends all their troops to one site, then Blotto (having sent more troops)

will earn 1 point for securing the location, and 3 points for capturing Lotso's troops. 

Thus, Blotto will earn a payout of 4 total points. 

On the other hand, if each Colonel were to send all of their troops to different locations, then they would both sure their respected locations. However, they would also not gain any advantage against their enemy. Therefore, the net payout will be 0 for both players. 

What should each Colonel do?

First, lets look at the possible options for both Colonels:

1. Blotto: (4,0), (0,4), (3,1), (1,3) and (2,2) 
2. Lotso: (3,0), (0,3), (2,1) and (1,2)

Where the X coordinate is location 1 and 

the Y coordinate is location 2

Comparing all of the possible values for both Colonel against each other, we get the following payout matrix in terms of Blotto's earnings only:

**Note that this matrix shows a net payout. So Blotto

may send (4,0) to Lotso's (2,1) and Blotto would earn 3 points for 

one site (2 + 1 secure point). However he also loses a point because
Lotso is beating him at the other site earning (0 + 1 secure point).

With this, we can say Blotto's net payout is 3 - 1 = 2.

Optimally, Blotto would want to play either (4,0) or (0,4) most of the time, and (2,2) sometimes. He would never want to play (1,3) or (3,1).
Lotso on the other hand would want to play (2,1) or (1,2) a majority of the time and very rarely will he want to play (3,0) or (0,3).

This happens to be the case because with both players playing ideally, Blotto would want to concentrate all of his troops and resources to one destination knowing that Lotso will never be able to out play him in numbers. Occasionally, he will actually split his troops to ensure that Lotso will not be able to capture a location easily.

Lotso will want to almost always play by splitting up his troops because he knows he is out matched by Blotto by the numbers with hopes that he can secure a lesser location. He will also want to deploy all of his troops to one destination occasionally because he doesn't want Blotto to always send his troops to one place all the time.

Why? Find out the complicated and highly confusing reason and explanation here or here!

Conclusion/Application:

The Blotto games is a simple and easy to follow game that requires the players to simultaneously distribute resources over a certain amount of territories. All you need to play is a friend, a piece of paper, and a pen!

Of course, this game relates well to real life war where an Army would want to have well distributed troops over the battlefield such that the enemy never gets any solid advantage. But ever hear of the game of Risk? 

Risk is a game where each player starts off as a specific country that has its troops randomly distributed across the world. The objective is to use your resources and strategize to capture the enemy troops as well as territories. 

Thats all I got! Thank you!

Jon 

Resources:

http://static3.businessinsider.com/image/51e6e696ecad04fc3c000028/how-to-use-math-to-crush-your-friends-at-risk-like-youve-never-done-before.jpg

https://en.wikipedia.org/wiki/Blotto_games

http://econstor.eu/bitstream/10419/51118/1/563391278.pdf

http://mindyourdecisions.com/blog/2012/01/24/the-colonel-blotto-game/#.Vkke3GSrSCT

http://digitalassets.lib.berkeley.edu/techreports/ucb/text/EECS-2014-19.pdf