I've been looking for a definition of game in game theory. I'd like to know if there is a definition shorter than that of Neumann and Morgenstern in Theory of Games and Economic Behavior and not so vague like "interactive decision problem" or "situation of conflict, or any other kind of interaction". I've started a study of the proof of the existence of Nash equilibria using Brouwer's fixed-point theorem and I think of finding a definition that allows me to understand concepts as normal-form game and mixed strategy without excessive complexity. I'd appreciate some bibliographic suggestion. Thank you!
This definition is from Osborne and Rubinstein, "A Course in Game Theory", section 1.1:
Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue well-defined exogenous objectives (they are rational ) and take into account their knowledge or expectations of other decision-makers’ behavior (they reason strategically).
A game is a description of strategic interaction that includes the con- straints on the actions that the players can take and the players’ interests, but does not specify the actions that the players do take. A solution is a systematic description of the outcomes that may emerge in a family of games. Game theory suggests reasonable solutions for classes of games and examines their properties.
In Fundenberg and Tirole, "Game Theory", subsection 1.1.1, there is a more formal definition of normal-form games.
I know this question already has an accepted answer, but games are usually defined depending on their form and their information structure. Therefore, the definition of a normal form game is different from that one of extensive form of incomplete information (for example). I usually define a game in normal form (its simplest form possible) as:
Notice that normal forms are usually represented by matrices, as you probably know already. Also be aware that a Mixed Strategy can be defined as a probability distribution over pure strategies of a given Player, if that helps. Finally, let me tell you that proving Nash Theorem might be easier using Kakutani's Fixed Point Theorem.
Bibliographical references there are many out there, but you may choose the one you prefer depending on your needs. To introductory but yet precise books are "A Primer on Game Theory", by Gibbons; or "An Introduction to Game Theory", by Osborne. You may also like "A Course on Game Theory", by Osborne and Rubinstein, which is more advanced (I have only read the second one; I use them occasionally as references, so I just share my very personal and uninformed opinion).