Succinct game
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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*2^{3}=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). 
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^{[1]} (a formal definition, describing succinct games as a computational problem, is given by Papadimitriou & Roughgarden 2008^{[2]}).
Types of succinct games[edit]
Graphical games[edit]
Say that each player's utility depends only on his own action and the action of one other player  for instance, I depends on II, II on III and III on I. Representing such a game would require only three 2x2 utility tables, containing in all only 12 utility values.

Graphical games are games in which the utilities of each player depends on the actions of very few other players. If is the greatest number of players by whose actions any single player is affected (that is, it is the indegree of the game graph), the number of utility values needed to describe the game is , which, for a small is a considerable improvement.
It has been shown that any normal form game is reducible to a graphical game with all degrees bounded by three and with two strategies for each player.^{[3]} Unlike normal form games, the problem of finding a pure Nash equilibrium in graphical games (if one exists) is NPcomplete.^{[4]} The problem of finding a (possibly mixed) Nash equilibrium in a graphical game is PPADcomplete.^{[5]} Finding a correlated equilibrium of a graphical game can be done in polynomial time, and for a graph with a bounded treewidth, this is also true for finding an optimal correlated equilibrium.^{[2]}
Sparse games[edit]
When most of the utilities are 0, as below, it is easy to come up with a succinct representation.  
L, l  L, r  R, l  R, r  

T  0, 0, 0  2, 0, 1  0, 0, 0  0, 7, 0 
B  0, 0, 0  0, 0, 0  2, 0, 3  0, 0, 0 
Sparse games are those where most of the utilities are zero. Graphical games may be seen as a special case of sparse games.
For a two player game, a sparse game may be defined as a game in which each row and column of the two payoff (utility) matrices has at most a constant number of nonzero entries. It has been shown that finding a Nash equilibrium in such a sparse game is PPADhard, and that there does not exist a fully polynomialtime approximation scheme unless PPAD is in P.^{[6]}
Symmetric games[edit]
Suppose all three players are identical (we'll color them all purple), and face the strategy set {T,B}. Let #TP and #BP be the number of a player's peers who've chosen T and B, respectively. Describing this game requires only 6 utility values.

In symmetric games all players are identical, so in evaluating the utility of a combination of strategies, all that matters is how many of the players play each of the strategies. Thus, describing such a game requires giving only utility values.
In a symmetric game with 2 strategies there always exists a pure Nash equilibrium – although a symmetric pure Nash equilibrium may not exist.^{[7]} The problem of finding a pure Nash equilibrium in a symmetric game (with possibly more than two players) with a constant number of actions is in AC^{0}; however, when the number of actions grows with the number of players (even linearly) the problem is NPcomplete.^{[8]} In any symmetric game there exists a symmetric equilibrium. Given a symmetric game of n players facing k strategies, a symmetric equilibrium may be found in polynomial time if k=.^{[9]} Finding a correlated equilibrium in symmetric games may be done in polynomial time.^{[2]}
Anonymous games[edit]
If players were different but did not distinguish between other players we would need to list 18 utility values to represent the game  one table such as that given for "symmetric games" above for each player.

In anonymous games, players have different utilities but do not distinguish between other players (for instance, having to choose between "go to cinema" and "go to bar" while caring only how crowded will each place be, not who'll they meet there). In such a game a player's utility again depends on how many of his peers choose which strategy, and his own, so utility values are required.
If the number of actions grows with the number of players, finding a pure Nash equilibrium in an anonymous game is NPhard.^{[8]} An optimal correlated equilibrium of an anonymous game may be found in polynomial time.^{[2]} When the number of strategies is 2, there is a known PTAS for finding an εapproximate Nash equilibrium.^{[10]}
Polymatrix games[edit]
If the game in question was a polymatrix game, describing it would require 24 utility values. For simplicity, let us examine only the utilities of player I (we would need two more such tables for each of the other players).
If strategy profile (B,R,l) was chosen, player I's utility would be 9+8=17, player II's utility would be 1+2=3, and player III's utility would be 6+4=10. 
In a polymatrix game (also known as a multimatrix game), there is a utility matrix for every pair of players (i,j), denoting a component of player i's utility. Player i's final utility is the sum of all such components. The number of utilities values required to represent such a game is .
Polymatrix games always have at least one mixed Nash equilibrium.^{[11]} The problem of finding a Nash equilibrium in a polymatrix game is PPADcomplete.^{[5]} Moreover, the problem of finding a constant approximate Nash equilibrium in a polymatrix game is also PPADcomplete.^{[12]} Finding a correlated equilibrium of a polymatrix game can be done in polynomial time.^{[2]} Note that even if pairwise games played between players have pure Nash equilibria, the global interaction does not necessarily admit a pure Nash equilibrium (although a mixed Nash equilibrium must exist).
Competitive polymatrix games with only zerosum interactions between players are a generalization of twoplayer zerosum games. The Minimax theorem originally formulated for twoplayer games by von Neumann generalizes to zerosum polymatrix games.^{[13]} Same as twoplayer zerosum games, polymatrix zerosum games have mixed Nash equilibria that can be computed in polynomial time and those equilibria coincide with correlated equilibria. But some other properties of twoplayer zerosum games do not generalize. Notably, players need not have a unique value of the game and equilibrium strategies are not maxmin strategies in a sense that worstcase payoffs of players are not maximized when using an equilibrium strategy. There exists an open source Python library^{[14]} for simulating competitive polymatrix games.
Polymatrix games which have coordination games on their edges are potential games ^{[15]} and can be solved using a potential function method.
Circuit games[edit]
Let us now equate the players' various strategies with the Boolean values "0" and "1", and let X stand for player I's choice, Y for player II's choice and Z for player III's choice. Let us assign each player a circuit: Player I: X ∧ (Y ∨ Z) These describe the utility table below.  
0, 0  0, 1  1, 0  1, 1  

0  0, 0, 0  0, 1, 0  0, 1, 1  0, 0, 1 
1  0, 1, 1  1, 0, 1  1, 0, 1  1, 1, 1 
The most flexible of way of representing a succinct game is by representing each player by a polynomialtime bounded Turing machine, which takes as its input the actions of all players and outputs the player's utility. Such a Turing machine is equivalent to a Boolean circuit, and it is this representation, known as circuit games, that we will consider.
Computing the value of a 2player zerosum circuit game is an EXPcomplete problem,^{[16]} and approximating the value of such a game up to a multiplicative factor is known to be in PSPACE.^{[17]} Determining whether a pure Nash equilibrium exists is a complete problem (see Polynomial hierarchy).^{[18]}
Other representations[edit]
Many other types of succinct game exist (many having to do with allocation of resources). Examples include congestion games, network congestion games, scheduling games, local effect games, facility location games, actiongraph games, hypergraphical games and more.
Summary of complexities of finding equilibria[edit]
Below is a table of some known complexity results for finding certain classes of equilibria in several game representations. "NE" stands for "Nash equilibrium", and "CE" for "correlated equilibrium". n is the number of players and s is the number of strategies each player faces (we're assuming all players face the same number of strategies). In graphical games, d is the maximum indegree of the game graph. For references, see main article text.
Representation  Size (O(...))  Pure NE  Mixed NE  CE  Optimal CE 

Normal form game  NPcomplete  PPADcomplete  P  P  
Graphical game  NPcomplete  PPADcomplete  P  NPhard  
Symmetric game  NPcomplete  The computation of symmetric Nash equilibrium is PPADhard for two players. The computation of nonsymmetric Nash equilibrium for two players is NPcomplete.  P  P  
Anonymous game  NPhard  P  P  
Polymatrix game  PPADcomplete (polynomial for zerosum polymatrix)  P  NPhard  
Circuit game  complete  
Congestion game  PLScomplete  P  NPhard 
Notes[edit]
 ^ Papadimitriou, Christos H. (2007). "The Complexity of Finding Nash Equilibria". In Nisan, Noam; Roughgarden, Tim; Tardos, Éva; et al. (eds.). Algorithmic Game Theory. Cambridge University Press. pp. 29–52. ISBN 9780521872829.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} Papadimitriou, Christos H.; Roughgarden, Tim (2008). "Computing Correlated Equilibria in MultiPlayer Games". J. ACM. 55 (3): 1–29. CiteSeerX 10.1.1.335.2634. doi:10.1145/1379759.1379762.
 ^ Goldberg, Paul W.; Papadimitriou, Christos H. (2006). "Reducibility Among Equilibrium Problems". Proceedings of the thirtyeighth annual ACM symposium on Theory of computing. Seattle, WA, USA: ACM. pp. 61–70. doi:10.1145/1132516.1132526. ISBN 1595931341. Retrieved 20100125.
 ^ Gottlob, G.; Greco, G.; Scarcello, F. (2005). "Pure Nash Equilibria: Hard and Easy Games". Journal of Artificial Intelligence Research. 24 (195–220): 26–37. arXiv:1109.2152. doi:10.1613/jair.1683.
 ^ ^{a} ^{b} Daskalakis, Constantinos; Fabrikant, Alex; Papadimitriou, Christos H. (2006). "The Game World Is Flat: The Complexity of Nash Equilibria in Succinct Games". Automata, Languages and Programming. Lecture Notes in Computer Science. 4051. pp. 513–524. CiteSeerX 10.1.1.111.8075. doi:10.1007/11786986_45. ISBN 9783540359043.
 ^ Chen, Xi; Deng, Xiaotie; Teng, ShangHua (2006). "Sparse Games Are Hard". Internet and Network Economics. pp. 262–273. doi:10.1007/11944874_24. ISBN 9783540681380.
 ^ Cheng, ShihFen; Reeves, Daniel M.; Vorobeychik, Yevgeniy; Wellman, Michael P. (2004). Notes on Equilibria in Symmetric Games. AAMAS04 Workshop on Game Theory and Decision Theory.
 ^ ^{a} ^{b} Brandt, Felix; Fischer, Felix; Holzer, Markus (2009). "Symmetries and the Complexity of Pure Nash Equilibrium". J. Comput. Syst. Sci. 75 (3): 163–177. doi:10.1016/j.jcss.2008.09.001. Retrieved 20100131.
 ^ Papadimitriou, Christos H.; Roughgarden, Tim (2005). "Computing equilibria in multiplayer games". Proceedings of the sixteenth annual ACMSIAM symposium on Discrete algorithms. Vancouver, British Columbia: Society for Industrial and Applied Mathematics. pp. 82–91. ISBN 0898715857. Retrieved 20100125.
 ^ Daskalakis, Constantinos; Papadimitriou, Christos H. (2007). "Computing Equilibria in Anonymous Games". arXiv:0710.5582v1 [cs].
 ^ Howson, Joseph T. (January 1972). "Equilibria of Polymatrix Games". Management Science. 18 (5): 312–318. doi:10.1287/mnsc.18.5.312. ISSN 00251909. JSTOR 2634798.
 ^ Rubinstein, Aviad (20150101). Inapproximability of Nash Equilibrium. Proceedings of the Fortyseventh Annual ACM Symposium on Theory of Computing. STOC '15. New York, NY, USA: ACM. pp. 409–418. arXiv:1405.3322. doi:10.1145/2746539.2746578. ISBN 9781450335362.
 ^ Cai, Y., Candogan, O., Daskalakis, C., & Papadimitriou, C. (2016). Zerosum Polymatrix Games: A generalization of Minimax.https://people.csail.mit.edu/costis/zerosum_final3.pdf
 ^ O. Person https://pypi.org/project/polymatrix/
 ^ Rahn, Mona and Schafer, Guido (2015) Efficient Equilibria in Polymatrix Coordination Games https://arxiv.org/pdf/1504.07518.pdf
 ^ Feigenbaum, Joan; Koller, Daphne; Shor, Peter (1995). A GameTheoretic Classification of Interactive Complexity Classes. Certer for Discrete Mathematics \& Theoretical Computer Science. Retrieved 20100125.
 ^ Fortnow, Lance; Impagliazzo, Russell; Kabanets, Valentine; Umans, Christopher (2005). "On the Complexity of Succinct ZeroSum Games". Proceedings of the 20th Annual IEEE Conference on Computational Complexity. IEEE Computer Society. pp. 323–332. ISBN 0769523641. Retrieved 20100123.
 ^ Schoenebeck, Grant; Vadhan, Salil (2006). "The computational complexity of nash equilibria in concisely represented games". Proceedings of the 7th ACM conference on Electronic commerce. Ann Arbor, Michigan, USA: ACM. pp. 270–279. doi:10.1145/1134707.1134737. ISBN 1595932364. Retrieved 20100125.