Succinct game
In algorithmic game theory, a succinct game or a succinctly representable game is a game which may be represented in a size much smaller than its normal form representation. Without placing constraints on player utilities, describing a game of players, each facing strategies, requires listing utility values. Even trivial algorithms are capable of finding a Nash equilibrium in a time polynomial in the length of such a large input. A succinct game is of polynomial type if in a game represented by a string of length n the number of players, as well as the number of strategies of each player, is bounded by a polynomial in n (a formal definition, describing succinct games as a computational problem, is given by Papadimitriou & Roughgarden 2008).
| Consider a game of three players, I,II and III, facing, respectively, the strategies {T,B}, {L,R}, and {l,r}. Without further constraints, 3*23=24 utility values would be required to describe such a game. | ||||
| L, l | L, r | R, l | R, r | |
|---|---|---|---|---|
| T | 4, 6, 2 | 5, 5, 5 | 8, 1, 7 | 1, 4, 9 |
| B | 8, 6, 6 | 7, 4, 7 | 9, 6, 5 | 0, 3, 0 |
| For each strategy profile, the utility of the first player is listed first (red), and is followed by the utilities of the second player (green) and the third player (blue). | ||||