Game Theory: Superadditivity & Imputation Explained

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Discussion Overview

The discussion revolves around the concepts of superadditivity and imputation within the context of cooperative game theory. Participants seek clarification on these terms and their implications in game scenarios, particularly in relation to the Shapley value and the core of n-person games. The conversation also touches on the differences between cooperative and non-cooperative games, using the Prisoner's Dilemma as an example.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant requests a simple explanation of superadditivity and imputation in game theory.
  • Another participant explains that a cooperative game is superadditive if the value of a coalition is greater than the sum of individual values, providing a mathematical expression for clarity.
  • A participant confirms that the discussion involves cooperative games and mentions the Shapley value and the core.
  • There is a question regarding the Prisoner's Dilemma, where a participant expresses confusion about the equilibrium strategy and the implications of changing strategies in a non-cooperative game.
  • A later reply emphasizes the importance of distinguishing between cooperative and non-cooperative assumptions in game theory, particularly in the context of the Prisoner's Dilemma.

Areas of Agreement / Disagreement

Participants generally agree on the definitions and context of superadditivity and imputation, but there is some confusion and debate regarding the implications of strategies in non-cooperative games, particularly in the Prisoner's Dilemma.

Contextual Notes

The discussion includes references to specific mathematical expressions and concepts, but some assumptions and definitions may be missing or unclear, particularly regarding the implications of strategy changes in non-cooperative scenarios.

bigjoe5263
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Hi,

In a n-person game theory.. I have encountered these terms, superadditivity and imputation, however i do not understand much their definition. Anyone have a simple explanation to this terms?

Somebody here knows where can I find a dictionary of mathematical expressions/equations in game theory?

thanks in advance.
 
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Can you explain the context? Does it involve cooperative games (e.g. Shapley)?

http://www.britannica.com/ebc/article-22625

For example, a cooperative game may be called superadditive if the value (payoff) of a coalition that consists of two players is greater than the sum of the individual values (payoffs) for each of the two players: V(a + b) > V(a) + V(b). See also http://en.wikipedia.org/wiki/Superadditive

In the same context, an imputation is a payoff vector associated with a possible outcome. In a cooperative game, an outcome is a coalition; more than one imputations may correspond to a coalition. The Britannica article has some examples.

A good starting point for game theory is http://en.wikipedia.org/wiki/Game_theory
 
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yes it involves cooperative games, the shapely value and the core of an n-person game theory...

Just one question on the prisoners dilemma non constant sum game and non-cooperative...the equilibrium is both of them confess but it confuses me because i think when one of them change the strategy the other one will benefit...??
 
That's why you should be careful whether you are operating under "cooperative" assumptions or "non-cooperative" (competitive) assumptions.

Prisoner's Dilemma (P.D.) is a classic example in non-coop. theory. Each prisoner is assumed to care about his/her utility (payoff) only; they are not allowed to interact; there are no side payments; and there are no "future payoffs." Each prisoner will first ask: "if the other prisoner confesses, am I better off confessing or not confessing?" He will find out that he is better off confessing. Then the same prisoner will ask: "if the other one does not confess, am I better off confessing or not confessing?" Again he will find out that he is better off confessing. He will conclude that he is better off confessing regardless of what the other one does.
 
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