solving a game refers to winning it or reaching its end. Ian Schreiber calls a game solvable if, for every situation, there is a recognizable best action. Generally, it is undesirable if a game can easily be solved, since this makes decisions meaningless, and games become boring faster. There are multiple tiers of solvability: A game might be trivial to solve, but it might also be solvable only in theory with a lot of computing effort. Even games with random elements are solvable since a best action can be found using expected values. Besides high complexity, hidden information and the influence of other human players are what makes it impossible for a human to completely solve a game. Symmetry and asymmetry Symmetric games offer all players identical starting conditions and are therefore automatically fair in the above stated sense. While they are easier to balance, they still must be balanced, e.g. regarding their game elements. Most modern games are asymmetric though, while the grade of asymmetry can vary greatly. Fairness becomes even more important for those. Giving each player identical resources is the simplest game balancing technique. Most competitive games feature some level of symmetry; some (such as Pong) are completely symmetric, but those in which players alternate turns (such as chess) can never achieve total symmetry as one player will always have a first-move advantage or disadvantage. Symmetry is unappealing in games because both sides can and will use any effective strategy simultaneously, or success depends on a very small advantage such as one pawn in chess. An alternative is to offer symmetry with restrictions. Players in Wizard's Quest and Catan have the same number of territories, but choose them in alternating order; the differing combination of territories causes asymmetry. Symmetry can be undone by human psychology; the advantage of players wearing red over players wearing blue is a well-documented example of this. Systems and subsystems In general, games can be viewed as systems of numbers and relations that typically consist of multiple subsystems. All numbers within a game only have a meaning in their given context. Subsystems can be dealt with separately and they might even have different balancing goals, but they also influence each other more or less. It is therefore crucial to consider how changes can affect the balance as a whole. Transitivity and intransitivity (In-)transitivity is a term used for logical relations. In games, this