Game Theory, Colonel Blotto and Gerrymandering

All men can see these tactics whereby I conquer, but what none can see is the strategy out of which victory is evolved.

-Sun Tzu

In game theory there is a classical war game called Colonel Blotto. In this game there are two players and the player who devotes most resources to a battlefield wins that battle, and the gain (or payoff) is then equal to the total number of battlefields won.

There are certain rules of the game

  1. In battlefield the player that has allocated the most soldiers will win.
  2. Both players do not know how many soldiers the opponent will allocate to each battlefield
  3. Both players seek to maximize the number of battlefields they expect to win.

Let us take an example. There are two players A & B. Each has 100 soldiers and they have to capture 10 forts (or 10 battle fields), now each player decides how many soldiers to send for each fort ex. suppose A send 10 soldiers to capture fort # 1 while B sends 9 soldiers, then A wins the fort. If both send 10 each then there is a tie. Idea is to win any many forts as possible and one who wins maximum number of forts (or battles) wins the war.

war strategy

Since you don’t know what opposing player will do, there can be lot of combinations. I am not going statistics part of it, there are lots of books on game theory/operations research which will tell you how to come up with optimal solution/Nash equilibrium.

One strategy could be, since each player has 100 soldiers and 10 battles to be won, so each send 10 soldiers per battlefield, but none wins, since it will end in tie.

Other strategy could be A continues with earlier strategy of – “10 10 10 10 10 10 10 10 10 10”, while B comes up with another strategy of “1 11 11 11 11 11 11 11 11 11”. B will lose one battle (1:10) but win other 9 battles (11:10), therefore win the war.

war

This theory has lot of applications esp. in politics. Each party has limited resources, so sends most resources in areas where chances of winning are high. But sometimes politicians alter the boundaries of district/state in such a manner that altered district has maximum number of supporters and victory is assured. It is called Gerrymandering.

640px-The_Gerry-Mander_Edit[1]

There is interesting story on origin of this word. The word was created in reaction to a redrawing of Massachusetts Congressional election districts under the then-governor Elbridge Gerry. In 1812, Governor Gerry signed a bill that redistricted Massachusetts to benefit his Democratic-Republican Party. When mapped, one of the contorted districts in the Boston area was said to resemble the shape of a salamander. Hence the word is combination of Gerry+ Salamander= Gerrymander.

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