I have a very basic doubt in Combinatorial Game theory. Whenever I am asked to find a strategy for somebody to win a game or to get the maximum sum or anything as such, what am I exactly supposed to do?

I am asking this because in any question there exist a lot of possibilities for the game to proceed. Moreover, there might exist circumstances in the game when even when the prescribed player does not follow the suggested strategy, he wins.

So what exactly am I supposed to do when I am asked to find a strategy?

PS: This might sound as a very basic doubt, but I am having a lot of difficulty with such questions where I need to find the strategies

  • $\begingroup$ I really think this is too broad. There are a lot of combinatorial games and it can be extremely difficult to sort out strategies (forget Chess, even Checkers is very, very complicated). Best is to go through a lot of the basic cases, like Nim or its many variants. Work by analogy where possible. $\endgroup$ – lulu Dec 28 '18 at 11:43
  • $\begingroup$ my most basic question is basically define 'strategy' $\endgroup$ – saisanjeev Dec 28 '18 at 11:45
  • $\begingroup$ An algorithm that dictates a player's move in any possible situation. $\endgroup$ – lulu Dec 28 '18 at 11:48

A winning strategy is simply that - it is a rule or set of procedures which enable you to win the game if you start from a winning position. There may be other rules available, or quicker ways to win - that is irrelevant. You just need to know what to do in response to any legal move by the opponent.

It can be hard to identify winning positions, and that is another part of combinatorial game theory.

At least two other issues arise:

If you are in a drawn or lost position (often draws cannot occur) a good strategy will give your opponent maximum chance to go wrong. That is not part of choosing a winning strategy, but it is part of identifying good play.

Then a winning strategy need not be complete - it isn't necessary to identify a winning move from every possible winning position, because there may be winning positions you never encounter if you follow the strategy. But it can sometimes be useful to identify winning moves from all winning positions.

For this last, consider chess. A perfect player might never play $1. e4$ as white, and may never respond $1. ... e5$ as black to this move by white, but a human being using a computer to analyse positions, may well want to analyse positions arising after these moves have been played, even though the computer does not need to play such positions itself.


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