In game theory a winning strategy for a player A is a set of rules which, if followed by player A, will result in that player winning, no matter what choices are made by the other players. Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...

To make this more formal requires a little attention to precise formulation. The concept differs significantly from what, intuitively speaking, might be considered 'mastery'.

For example, the set of rules may be infinite. It may simply be a (possibly infinite) lookup table. The rules to follow need not be computable in any way. This is a particularly important distinction in the case of infinite games. Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In computer science, a lookup table is a data structure, usually an array or associative array, used to replace a runtime computation with a simpler lookup operation. ... Computability theory is that part of the theory of computation dealing with which problems are solvable by algorithms (equivalently, by Turing machines), with various restrictions and extensions. ...

All that is required of the winning strategy is the knowledge of at least one way to 'navigate' from a position to another winning position, in which (loosely speaking) the victory is closer. The information on how to play is partial, in the sense that it might give no information at all about the bulk of possible game positions. A standard definition used in mathematical logic gets around this: a strategy gives a rule for playing in any position, but in defining what a winning strategy is, not all that information is necessarily relevant (because in properly following the strategy, many game positions may not ever be reached). Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...

Clearly winning strategy as a concept then unpacks as winning strategy from the initial position(s); that is, from each initial position with which A may be presented at first play. For finite games this means that winning within 50 plays is the same as this play is able to force a position that is winning within 49, next time A plays. For infinite games, though, matters are much less simple.

A two-player game is called determined if either of the players has a winning strategy. In set theory, determinacy is the study of under what circumstances one or the other player of a game must have a winning strategy, and the consequences of the existence of such strategies. ...

If a game has a winning strategy for the Ath player it is called an 'Ath player win'. If both players can force a draw then the game is called a draw. Draw has the following meanings: Drawing is one way of making an image by making marks on a surface with a pen, pencil or other line tool. ...