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I use the word "game" here both in the sense of "structured playing" and also in the broader sense of "game theory" i.e. "interactive decision theory".
repeated vs. non-repeated: for example, iterated prisoner's dilemma is a repeated game. In repeated games, cooperative behavior is more likely to be optimal, because an action which seems purely altruistic within one game may lead to the reward of more cooperative behavior by other players in future repetitions.
finite vs infinite: when one is using the framework of games and strategy to talk about decision-making outside of the context of a game with a definite end-point after a finite number of moves at which winning/losing is established with finality for all players, one calls the an 'infinite game'. Other games are 'finite games', for example, Chess is finite. Just as we call computers Turing machines even though they have finite memory, if we conceptualize a game in a manner similar to an infinite game, I'll call it an infinite game even if it is technically finite, for example a romantic relationship (one or the other person will eventually die).
zero-sum vs. non-zero-sum: in a non-zero-sum games, utility is not conserved. This leads to the potential for cooperation because there are outcomes which are preferred over other outcomes by multiple players. For example, Chess is a zero-sum game. Prisoner's dilemma is a non-zero-sum game.
public vs. games with private/hidden information: in public information games, the state of the world is known by all players. in games with private/hidden information, some or all players have information about the state of the world which others do not know. Chess is an example of a public information game. Poker is an example of a private information game. An important part of games with private information is trying to infer the hidden state based on the moves of the player who knows it. This is theoretically tricky because it depends on assumptions about the other player's strategy. Also, sometimes players might play an otherwise suboptimal move for the purpose of misleading their opponents about the hidden state. If the players are humans playing in person, body language and human psychology comes into play.
deterministic vs. stochastic
Tactics are usually presented in opposition to strategy. Both are sorts of topics in guidelines for decision-making. The distinction is scale; tactics are details (small-scale in time and possibly in space) and strategy is big-picture (large-scale in time and possibly space).
An important consequence is that often tactics lend themselves to more rigorous analysis. Sometimes tactics are small enough to lend themselves to exhaustive formal analysis.
Call a 'policy' an algorithm that, for any situation, specifies what you should do in that situation (or a set of informal guidelines which attempts to approach this, although it may not be precise enough to say exactly what to do in every situation). If we are on the tactical level, then often we find an optimal policy for some scenarios.
Strategy sometimes takes the form of an informal conceptual partial organization (partial model) of the world, coupled with a narrative. The intent is NOT to formally model the strategy, but rather to stay within the domain of open-ended informal thinking.
However, in many situations strategic analysis takes the form defining equivalence classes and then taking the homomorphism w/r/t those (a quotient), reducing situations to a simple model that is then amenable to exhaustive formal analysis. One form of this is a reduction of everything to 'points', a fungible quantity which attempts to represent how well you are doing, e.g. to choose a single goal metric (in a company, this may be profit or it may be some metric such as revenue per employee; in a game like Chess, it could be the points value of pieces; in a Civ-like game it could be the money value of things, or the time-value of things).
Ideally a good 'points' has these properties:
In other types of strategic analysis, we do not reduce all the way to a single quantity, but rather, to a few quantities (metrics), called KPIs ("key performance indicators"). Sometimes there are specific heuristics regarding these (e.g. "if KPI XY gets below 0.3, you are facing a situation of type A and need to do B immediately.."). Sometimes strategies can be classified by regions of KPI space (strategy type A is when KPI X is high at the expense of KPI Y being low).
Another way to look at strategy is as sets of qualitatively similar policies or equivalently as regions of 'policy space'.
Sometimes the entire game can be 'solved' (e.g. an optimal policy can be found that covers all possible situations; in this case there is no longer any distinction between tactics and strategy because an exhaustive formal analysis can be performed at the single move level). Other times it can be informally solved (e.g. no one bothered to make a formal proof but the general feeling is that there is a known right move in every situation). Sometimes it has not been informally solved, yet there are a set of known heuristics which are undisputed (e.g. there is no case in which move X is correct in a situation where conditions B hold). Other times there are heuristics, but their universal validity is uncertain.
In some cases there do exist groupings of policies into qualitatively coherent types of strategies, but the number of qualitatively different types of strategy is very large and has not been characterized.
This might be called a game that admits 'creative' strategy, in that a player might discover some interesting new niche of strategy space.
I'm actually not very good at most of these..
Puyo puyo is a good game for learning about offence vs. defense, and how this changes the balance of when you should be careful.
Chess is a classic game that is thought by many to be great for learning.
Go is another classic game that is thought by many to be great for learning. Compared to chess, the units are homogeneous and the number of possible moves at each move is larger, which makes brute force lookahead less useful.
Poker is known for being a good game for learning how to deal with reading opponents in person for the purpose of deducing private information, and for hiding one's own private information.
Axis and Allies is a great game but I'm terrible at it and I don't know what it teaches.
Civilization, a computer game, is a good game to learn about resource management, tech trees, military units in the presence of maintenance, the short-term/long-term tradeoff, and the (related) explore/exploit tradeoff, and the tradeoff between safety and improvement and action.
Robotek is an interesting mobile phone game in which you get 3 submoves each move, but you have only partial control over what they are. You choose a move class (out of 3 choices), and a submove within that class, and then the computer semi-randomly chooses 2 other submoves from that class. So it is a place to learn about stochasticity about your own 'actions'. It also has elements of offense vs. defence.
