Description
In the context of this definition, a "phase" refers to a time in the game during which a specific set of actions are available to the player. Games with this mechanic are therefore games in which "the actions that are available to players during their turn are not necessarily the same actions that were available during the previous turn or during the next turn", as the order of the game phases is variable.
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Example: Twilight Imperium
In Twilight Imperium's Strategy Phase, players take turns selecting one of eight strategy cards. Each strategy card corresponds to a specific "phase" in the game, during which players around the table will be allowed to take actions that are unique to that phase. For example, the "Trade" strategy card allows all players to generate income, the "Construction" strategy card allows all players to build structures, etc.
On their turns, players are given the choice to either activate a strategy card (and the corresponding phase and actions) or do something else. (Their other options are not relevant to this example.) As a result, the order in which the phases of the game (and the actions that correspond to them) will become available to the players is variable. For example, in one round the leadership phase might go before the construction phase, in the next round it may be the other way around, and in another round there might be other phases in between them. It is all up to specific players to decide when these phases and actions become available to all the players at the table.
On their turns, players are given the choice to either activate a strategy card (and the corresponding phase and actions) or do something else. As a result, the order in which the phases of the game (and the actions that correspond to them) will become available to the players is variable. For example, in one round the trade phase might go before the construction phase, in the next round it may be the other way around, and in another round there might be other phases in between them. It is all up to specific players to decide when these phases and actions become available to all the players at the table, and it can be a lot of fun to time these phases in a way that they lead to a strategic advantage.
Please note that many games (including in Twilight Imperium) have nested game phases, some of which might be referred to as phases while others might be referred to as something else. For example, in Twilight Imperium, players choose their strategy cards during the "Strategy Phase" (sic) and they activate them during the "Action Phase" (sic). These phases correspond to a set of actions that players can take so it makes sense to call them that. However, the actual activation of the set of rules (aka the "phase") that comes with each strategy card is referred to as a "Strategic Action" (sic) in Twilight Imperium, even though such a strategic action does trigger what would be referred to as a "phase" in most games. As it is much to learn how to play games that do not reuse terminology for nested phases, this is a reasonable decision for the game designer to make, even if it can be confusing when analyzing game mechanics.
Watch a video of people playing Twilight Imperium.