Blotto game

Type of game in which the players distribute limited resources over several objects out battlefields
A Colonel Blotto game is a type of two person constant sum game in which the players (officers) are tasked to simultaneously distribute limited resources over several objects (battlefields). In the classic version of the game, the player devoting the most resources to a battlefield wins that battlefield, and the gain (or payoff) is equal to the total number of battlefields won. The game was first proposed by Émile Borel in 1921. In 1938 Borel and Ville published a particular optimal strategy (the "disk" solution). The game was studied after the Second World War by scholars in Operation Research, and became a classic in game theory. Gross and Wagner's 1950 research memorandum states Borel's optimal strategy, and coined the fictitious Colonel Blotto and Enemy names. For three battlefields or more, the space of pure strategies is multi dimensional (two dimensions for three battlefields) and a mixed strategy is thus a probability distribution over a continuous set. The game is a rare example of a non trivial game of that kind where optimal strategies can be explicitly found. In addition to military strategy applications, the Colonel Blotto game has applications to political strategy (resource allocations across political battlefields), network defense, R&D patent races, and strategic hiring decisions. Consider two sports teams with must spend budget caps (or two Economics departments with use or lose grants) are pursuing the same set of candidates, and must decide between many modest offers or aggressive pursuit of a subset of candidates.

In the simpler case of two battlefields, Macdonell and Mastronardi 2015 provide the first complete characterization of all Nash equilibria to the canonical simplest version of the Colonel Blotto game. This solution, which includes a graphical algorithm for characterizing all the Nash equilibrium strategies, includes previously unidentified Nash equilibrium strategies as well as helps identify what behaviors should never be expected by rational players. Nash equilibrium strategies in this version of the game are a set of bivariate probability distributions: distributions over a set of possible resource allocations for each player, often referred to as Mixed Nash Equilibria (such as can be found in Paper Rock Scissors or Matching Pennies as much simpler examples). Macdonell and Mastronardi 2015 solution, proof, and graphical algorithm for identifying Nash equilibria strategies also pertains to generalized versions of the game such as when Colonel Blotto have differing valuations of the battlefields, when their resources have differing effectiveness on the two battlefields (e. g. one battlefield includes a water landing and Colonel Blotto's resources are Marines instead of Soldiers), and provides insights into versions of the game with three or more battlefields. Consider two players (Colonel Blotto and Enemy), two battlefields both of equal value, both players know each other's total level of resources prior to allocation, and they then must make a simultaneous allocation decision. It is often assumed Colonel Blotto is the more resourced officer (his level of resource can be defined to be 1), and Enemy has a fraction of resources less than 1. The Nash equilibrium allocation strategies and payoffs depend on that resource level relationship.

This game is commonly used as a metaphor for electoral competition, with two political parties devoting money or resources to attract the support of a fixed number of voters. Each voter is a "battlefield" that can be won by one or the other party. The same game also finds application in auction theory where bidders must make simultaneous bids. Several variations on the original game have been solved by Jean François Laslier, Brian Roberson, and Dmitriy Kvasov.