Blotto games

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Blotto games (or Colonel Blotto games) constitute a class of two-person zero-sum games in which the players are tasked to simultaneously distribute limited resources over several objects, with the gain (or payoff) being equal to the sum of the gains on the individual objects.

The game is named after the fictional colonel Blotto who was supposed to be tasked to find the optimum distribution of his soldiers over N battlefields knowing that: 1) within each battlefield the party that has allocated most soldiers will win, but 2) both parties do not know how many soldiers the opposing party will allocate to each battlefield, and: 3) the side that wins the majority of the N battles is overall winner.

Contents 1 Example 2 Real life example 3 External links 4 References

[edit] Example

As an example Blotto game, consider the game in which two players each write down three positive integers in non-decreasing order and such that they add up to a pre-specified number S. Subsequently, the two players show each other their writings, and compare corresponding numbers. The player who has two numbers higher than the corresponding ones of the opponent wins the game.

For S = 6 only three choices of numbers are possible: (2, 2, 2), (1, 2, 3) and (1, 1, 4). It is easy to see that:

(1, 1, 4) against (1, 2, 3) is a draw

(1, 2, 3) against (2, 2, 2) is a draw

(2, 2, 2) beats (1, 1, 4)

It follows that the optimum strategy (Nash equilibrium) is (2, 2, 2) as it does not do worse than breaking even against any other strategy whilst beating one other strategy.

For larger S the game becomes progressively more difficult to analyse. For S = 12, it can be shown that (2, 4, 6) represents the optimal strategy, whilst for S > 12, deterministic strategies fail to be optimal. For S = 13, choosing (3, 5, 5), (3, 3, 7) and (1, 5, 7) with probability 1/3 each can be shown to be the optimal probabilistic strategy.

[edit] Real life example

In a recent paper^[1], the 2000 Presidential Elections, one of the closest races in recent history, have been modelled as a colonel Blotto game. It is argued that Gore could have utilized a strategy that would have won the election, but that such a strategy was not identifiable ex ante.

[edit] External links

• Jonathan Partington's Colonel Blotto page

[edit] References

1. ^ [http://www.socsci.duke.edu/ssri/federalism/papers/tofiasmunger.pdf Lotto, Blotto, or Frontrunner: An Analysis of Spending Patterns by the National Party Committees in the 2000 Presidential Election ]

2. Roberson, B. (2006), “The Colonel Blotto Game,” Economic Theory 29, 1–24.

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