Tournaments

As a fan of competitive (and cooperative) games and mathematics, tournaments are a natural place to let both of those passions flourish. Game theory has a wealth of excellent and interesting games that can be tweaked to allow for tournaments, and the age of the internet allows for tournaments on large scales. The following are a list of tournaments that have happened or are happening, and if you are interested in any of the details or want to run a tournament on your own, feel free to send me an email.

The Colonel Blotto Game Tournament

This is my version of the Colonel Blotto Game.

Each player is a colonel with an army of 100 soldiers.

There are ten towns, labelled Town 1, Town 2, ..., Town 10, that each colonel wants to occupy.

Town 1 is worth 1 point, Town 2 is worth 2 points, and so on until Town 10 is worth 10 points.

Each player decides beforehand how many soldiers they want to send to each town, and they must use all of their soldiers. A given strategy is a list of 10 numbers, the first number being how many soldiers you want to send to Town 1, the second number being the number of soldiers you want to send to Town 2, and so on.

Once each player has decided on their strategy, they compare strategies and determine which general occupies which town.

If a player 1 sends more soldiers to Town 1 than player 2 does, player 1 gets 1 point.

If a player 2 sends more soldiers to Town 1 than player 1 does, player 2 gets 1 point.

If both players send the same number of soldiers to Town X, each player receives zero points for that Town.

If a player 1 sends more soldiers to Town X than player 2 does, player 1 gets X points.

If a player 2 sends more soldiers to Town X than player 1 does, player 2 gets X points.

The player that has the most points is the winning colonel of that game. It is possible to tie.

For example, if player 1 uses the strategy 0,0,0,0,0,0,0,0,0,100 and player 2 uses the strategy 10,10,10,10,10,10,10,10,10, then player 1 gets 10 points for occupying Town 10, and player 2 gets 1+2+3+4+5+6+7+8+9 = 45 points for occupying Towns 1 through 9. Player 2 had a superior strategy relative to player 1, and thus is the better colonel.

This game is lots of fun because there is no one perfect strategy, every strategy can be beaten! Even more fun is a round robin style tournament, where multiple teams compete against one another. I've created a macro enabled excel file and linked it here, feel free to download it and try it out.

Submit your strategy here.

The Sunburn Game Tournament

Originally found in Evolutionary Computation for Modeling and Optimization, Sunburn is a model for designing a video-game spacecraft, which was proposed by John Walker. This game is an adaptation of the Sunburn Model described in Chapter 4 of the book. The game involves designing a spacecraft, and pitting that spacecraft against others in one on one combat.

A player designs their spacecraft with the following in mind: each spacecraft has five types of systems, guns, lasers, missiles, drives, and shields and a preferred range for combat.

Each spacecraft has 20 system slots. A system slot can be occupied by a gun (G), laser (L), missile (M), drive (M), or a shield system (S).

Combat range choices are integer values between 1 and 20, inclusive.

A spacecraft is completely described by a list of 20 letters, and an integer between 1 and 20, inclusive. We call this the Sunburn gene, which counts as a complete strategy for the game. For example, GLMDSSMGLSDSMLGMLSDS16 is the gene of a completely described spacecraft.

To derive a spaceship from a Sunburn gene, you count the number of times an S appears and then put 3 times that many shields on the front of the ship. The remaining systems are placed behind the shields in the order in which they appear in the gene. The example gene given above would become SSSSSSSSSSSSSSSSSSGLMDMGLDMLGMLD16 with the front of the ship being to the left.

Combat between two ships is conducted as follows. Combat is initiated at a normally distributed starting range, with a mean of 16.2 and a standard deviation of 2, rounded normally.

Once combat has started, the ships iteratively go through turns consisting of shooting and then moving. This loop continues until a winner and loser are found or until a draw occurs.

The two ships shoot simultaneously using each gun, missile, or laser system once each turn. Each of the three types of weapons has an effectiveness curve that specifies its probability of scoring a hit at each possible range, defined later.

Each hit scored removes one system from the front of a ship. A ship that loses a system can no longer use that system in the combat. After shooting, the ships move. Each drive system enables a ship to move a distance of one. The ships take turns moving a distance of one towards their preferred ranges, stopping when they run out of drives.

In order to win a combat, a ship must destroy all the systems on the other ship and have at least one drive left itself. In the event that neither ship has remaining drives or neither ship has remaining weapons systems, the fight is a draw. If two ships have not fought to a draw or victory in 100 combat turns, their combat is a draw.

At a range r the probability of hitting for guns is P_G(r) = 0.9 - (r-1)/38, lasers is P_L(r) = 0.35, and missiles is P_M(r) = 0.1 + (r-1)/38

This tournament is usually automated, and consists of a triple round robin, where each spacecraft faces off against the others three times each, and the winner is the spacecraft with the most victories.