Blotto games

Blotto games

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 (or 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 then equal to the total number of battlefields won.

Though the Colonel Blotto game was first proposed by Borel[1] in 1921, most variations of the classic game remained unsolved for 85 years. In 2006, Roberson described the equilibrium payoffs to the classic game for any number of battlefields, and any level of relative resources, as well as characterizing the set of equilibrium to most versions of the classic game.[2]

The game is named after the fictional Colonel Blotto from Gross and Wagner's 1950[3] paper. The Colonel was tasked with finding the optimum distribution of his soldiers over N battlefields knowing that:

  1. on each battlefield the party that has allocated the most soldiers will win, but
  2. both parties do not know how many soldiers the opposing party will allocate to each battlefield, and:
  3. both parties seek to maximize the number of battlefields they expect to win.

Contents

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

It follows that the optimum strategy is (2, 2, 2) as it does not do worse than breaking even against any other strategy while beating one other strategy. There are however several Nash equilibria. If both players choose the strategy (2, 2, 2) or (1, 2, 3), then none of them can beat the other one by changing strategies, so every such strategy pair is a Nash equilibrium.

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, while 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.

Application

The 2000 US presidential election, one of the closest races in recent history, has been modeled as a Colonel Blotto game.[4] 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.

External links

References


Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Blotto — may refer to:* Blotto, a reagent used in immunological assays * Blotto, a colloquial term meaning drunkenness * Blotto a 1930 Laurel and Hardy short comedy film * Blotto, an Albany, NY, rock band in the late 1970s and early 1980s. * Blotto games …   Wikipedia

  • Cooperative game — This article is about a part of game theory. For video gaming, see Cooperative gameplay. For the similar feature in some board games, see cooperative board game In game theory, a cooperative game is a game where groups of players ( coalitions )… …   Wikipedia

  • Rock-paper-scissors — Roshambo redirects here. For the phonetically similar name and terms derived from it, see Rochambeau (disambiguation). For the bullying practice, see sack tapping. Rock paper scissors Rock paper scissors chart Years active Chinese Han Dynasty to… …   Wikipedia

  • Nash equilibrium — A solution concept in game theory Relationships Subset of Rationalizability, Epsilon equilibrium, Correlated equilibrium Superset of Evolutionarily stable strategy …   Wikipedia

  • Best response — In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players strategies as given (Fudenberg Tirole 1991, p. 29; Gibbons 1992, pp. 33–49). The concept of a …   Wikipedia

  • Coordination game — In game theory, coordination games are a class of games with multiple pure strategy Nash equilibria in which players choose the same or corresponding strategies. Coordination games are a formalization of the idea of a coordination problem, which… …   Wikipedia

  • Minimax — This article is about the decision theory concept. For other uses, see Minimax (disambiguation). Minimax (sometimes minmax) is a decision rule used in decision theory, game theory, statistics and philosophy for minimizing the possible loss for a… …   Wikipedia

  • Prisoner's dilemma — This article is about game theory. For the 1988 novel, see Prisoner s Dilemma (novel). For the Doctor Who audiobook, see The Prisoner s Dilemma. For the 2001 play, see The Prisoner s Dilemma (play). The prisoner’s dilemma is a canonical example… …   Wikipedia

  • Chicken (game) — For other uses, see Chicken (disambiguation). The game of chicken, also known as the hawk dove or snowdrift[1] game, is an influential model of conflict for two players in game theory. The principle of the game is that while each player prefers… …   Wikipedia

  • Symmetric game — In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”