1 Introduction
Applications of conventional game theory have played an important role in many modern strategic decision making processes including diplomacy, economics, national security, and business
[Carfì et al.2011, Rabin1993, Roy et al.2010]. Game theory is a mathematical paradigm in which such domainspecific decision situations are modeled [Fudenberg and Tirole1991]. Multiple players interact with each other to collectively complete a task or to enforce their interests. In the classical model of game theory [Von Neumann and Morgenstern1944] all players choose an action simultaneously and obtain a certain payoff (utility), which depends on the actions of the other players. The most common solution concept for such a decision problem is called Nash equilibrium (NE) [Nash1951], in which no player is able to unilaterally improve his payoff by changing his chosen action.There exist many different representations for such simultaneous games. The most popular is the strategic or standard normalform game representation, which is often used for 2player games, like the prisoner dilemma or battle of the sexes. However, due to its exponential growth of the representational size w.r.t the number of players [Kearns et al.2013], a more compact version, called graphical game, is increasingly used to model multiplayer scenarios [La Mura2000, Koller and Milch2003, Littman et al.2002, Palmieri and Lallouet2017]. Here a player’s action only depends on a certain number of other players’ actions (a so called player’s neighborhood). These neighborhoods are visualized by an underlying graph with players as vertices and the dependencies as edges.
While pure strategies, where each player unambiguously decides on a particular action, are conceptually simpler than mixed strategies, the associated computational problems appear to be harder [Gottlob et al.2005]. This also applies to the compact representation of graphical games, for which the complexity of finding pure strategy Nash equilibria (PNEGG) was proven to be NPcomplete, even in the restricted case of neighborhoods of maximum size 3 with at most 3 actions per player [Gottlob et al.2005].
Although there are many algorithms that find mixed or approximated NE in graphical games [Kearns et al.2013, Ortiz and Kearns2003, Bhat and LeytonBrown2004, Soni et al.2007], there are only a couple of algorithms that deal with pure Nash equilibria (PNE). In this paper, we focus on finding the hard computational Nash equilibria in the case of pure strategies, where each player chooses to play an action in a deterministic, nonrandom manner.
With DWave Systems releasing the first commercially available quantum annealer in 2011^{1}^{1}1https://www.dwavesys.com/news/dwavesystemssellsitsfirstquantumcomputingsystemlockheedmartincorporation, there is now the possibility to find solutions for such complex problems in a completely different way compared to classical computation. To use DWave’s quantum annealer, the problem has to be formulated as a quadratic unconstrained binary optimization (QUBO) problem [Boros et al.2007], which is one possible input type for the annealer. In doing so, the metaheuristic quantum annealing seeks to find the minimum of an objective function, i.e., the best solution of the defined configuration space [McGeoch2014].
In this paper, we propose the first quantum annealing algorithm for finding PNEGG, called QNash. The algorithm consists of two phases. The first phase determines all combinations of best response strategies for each player using classical computation. The second phase finds pure Nash Equilibria using a quantum annealing device by mapping the computed combinations to a QUBO formulation based on the Set Cover problem. We empirically evaluate QNash on DWave’s Quantum Annealer 2000Q using different graphical game topologies. The results with respect to solution quality and computing time are compared to a Brute Force algorithm and a Random Search heuristic.
2 Background
2.1 Graphical Games and Pure Nash Equilibria
In an player game, each player , has a finite set of strategies or actions, , with . Such a game can be visualized by a set of matrices . The entry specifies the payoff to player when the joint action (also, strategy profile) of the players is , with being the set of combined strategy profiles. In order to specify a game with players and strategies each, the representational size is , an amount of information exponential with respect to the number of players. However, players often interact only with a limited number of other players, which allows for a much more succinct representation. In [Kearns et al.2013] such a compact representation, called graphical game, is defined as follows:
Definition 2.1.
(Graphical Game)
An player graphical game is a pair , where is an undirected graph with vertices and is a set of matrices with , called the local game matrices. Player is represented by a vertex labeled in . We use to denote the set of neighbors of player in – i.e., those vertices such that the undirected edge appears in . By convention, always includes himself. The interpretation is that each player is in a game with only his neighbors in . Thus, the size of the graphical game representation is only exponential in the maximal node degree of the graph, . If and , denotes the payoff to when his neighbors (including himself) play .
Consider a game with players and strategy sets . For every strategy profile , the strategy of player is denoted by and corresponds to the tuple of strategies of all players but . For every and we denote by the strategy profile in which player plays and all the other players play according to . One has to mention that a strategy profile is called global, if all players contribute to it, i.e., a global combined strategy consists of every player playing one of his actions [Gottlob et al.2005].
Definition 2.2.
(Pure Nash Equilibrium)
A global strategy profile is a PNE, if for every player and strategy we have . That is, no player can improve his expected payoff by deviating unilaterally from a Nash equilibrium.
[Daskalakis and Papadimitriou2006] define a best response strategy as follows:
Definition 2.3.
(Best Response Strategy)
A best response strategy of player p is defined by:
Intuitively, is the set of strategies in that maximize ’s payoff if the other players play according to . Thus, a strategy profile is a pure Nash equilibrium if for every player , .
2.2 Quantum Annealing
Quantum annealing is a metaheuristic for solving complex optimization and decision problems [Kadowaki and Nishimori1998]. DWave’s quantum annealing heuristic is implemented in hardware, designed to find the lowest energy state of a spin glass system, described by an Ising Hamiltonian,
(1) 
where
is the onsite energy of qubit
, are the interaction energies of two qubits and , and represents the spin of theth qubit. The basic process of quantum annealing is to physically interpolate between an initial Hamiltonian
with an easy to prepare minimal energy configuration (or ground state), and a problem Hamiltonian , whose minimal energy configuration is sought that corresponds to the best solution of the defined problem (see Eq. 2). This transition is described by an adiabatic evolution path which is mathematically represented as function and decreases from 1 to 0 [McGeoch2014].(2) 
If this transition is executed sufficiently slow, the probability to find the ground state of the problem Hamiltonian is close to 1
[Albash and Lidar2018]. Thus, by mapping the Nash equilibrium decision problem onto a spin glass system, quantum annealing is able to find the solution of it.For completeness, we map our NE descision problem to an alternative formulation of the Ising spin glass system. The so called QUBO problem [Boros et al.2007] is mathematically equivalent and uses 0 and 1 for the spin variables [Su et al.2016]. The quantum annealer is as well designed to minimize the functional form of the QUBO:
(3) 
with
being a vector of binary variables of size
, and being an realvalued matrix describing the relationship between the variables. Given the matrix , the annealing process tries to find binary variable assignments to minimize the objective function in Eq. 3.2.3 Set Cover Problem
Since the QUBO formulation of QNash resembles the well known Set Cover (SC) problem, it is introduced here. Within the SC problem, one has to find the smallest possible number of subsets from a given collection of subsets with , such that the union of them is equal to a global superset of size . This problem was proven to be NPhard [Karp1972]. In [Lucas2014] the QUBO formulation for the Set Cover problem is given by:
(4) 
with being a binary variable which is , if set is included within the chosen sets, and otherwise. denotes a binary variable which is if the number of chosen subsets which include element is , and otherwise. The first energy term imposes the constraints that for any given exactly one must be , since each element of must be included a fixed number of times, and that the number of times that we claimed was included is in fact equal to the number of subsets we have included with as an element. is a penalty value, which is added on top of the solution energy, described by , if a constraint was not satisfied, i.e. one of the two terms (quadratic differences) are unequal to . Therefore adding a penalty value states a solution as invalid. Additionally, the SC problem minimizes over the number of chosen subsets . We skip discussing the term due to the fact that it has no impact on our QNash QUBO problem later on.
3 Related Work
Most known algorithms focus on finding mixed or approximated NE [Kearns et al.2013, Ortiz and Kearns2003, Bhat and LeytonBrown2004, Soni et al.2007]. However, some investigations in determining PNE in graphical and similar variations of games were made.
Daskalakis and Papadimitriou present a reduction from graphical games to Markov random fields such that PNE can be found by statistical inference. They use known statistical inference algorithms like belief propagation, junction tree algorithm and Markov Chain Monte Carlo to determine PNE in graphical games
[Daskalakis and Papadimitriou2006].In [Jiang and LeytonBrown2007], the authors analyze the problem of computing pure Nash equilibria in action graph games (AGGs), another compact game theoretic representation, which is similar to graphical games. They propose a dynamicprogramming approach that constructs equilibria of the game from equilibria of restricted games played on subgraphs of the action graph. In particular, under the premise that the game is symmetric and the action graph has bounded treewidth, their algorithm determines the existence of a PNE in polynomial time.
Palmieri and Lallouet deal with constraint games, for which constraint programming is used to express players preferences. They rethink their solving technique in terms of constraint propagation by considering players preferences as global constraints. Their approach is able to find all pure Nash equilibria for some problems with 200 players and also shows that performance can be improved for graphical games [Palmieri and Lallouet2017].
With quantum computing gaining more and more attention^{2}^{2}2https://www.gartner.com/smarterwithgartner/theciosguidetoquantumcomputing/ and none of the related work making use of quantum annealing in order to find PNE, we propose a solution approach using a quantum annealer.
4 QNash
In the following sections we present the concept of QNash. QNash consists of two phases, which are described below.
4.1 Determining best response strategies
In the first phase, we identify each player’s best response to what the other players might do. That is, for every strategy profile , we search player ‘s strategy (or strategies) with the maximum payoff . This involves iterating through each player in turn and determining their optimal strategies. An example is given in 4.1. This can be feasibly done in polynomial time, since one can easily explore each player’s matrix representing the utility function, i.e., payoffs [Gottlob et al.2005]. Therefore the first part, which we see as preliminary step for our QNash algorithm, is executed on a classical computer. After doing this, one gets a set of combined strategies with each being a best response to the other players’ played strategies. This set is denoted by and has the cardinality .
Example 4.1.
In Tab. 1 the local payoff matrix of player is visualized. For instance, assume player plays action and player chooses action . In this case, a best response strategy for player is action , due to the fact, that he gets the most payoff in this situation, i.e. 4. This leads to a best response strategy combination for player , denoted by a pointed set with player being the base point of it.
A  B0 C0  B0 C1  B1 C0  B1 C1  B2 C0  B2 C1 

0  4  1  2  2  1  4 
1  1  3  2  1  2  2 
4.2 Finding PNE using quantum annealing
In the second phase, a PNE is identified, when all players are playing one of their best response strategies simultaneously. With the computed set of the classical phase, the following question arises:
“Is there a union of the combined best response strategies of which results in a global strategy profile, under the premise that every player plays one of his best response actions?” — As stated in Definition 2.3 this would lead to a PNE.
This question resembles the Set Cover (SC) problem, stated in Sec. 2.3. It asks for the smallest possible number of subsets to cover the elements of a given global set (in our case, this global set would be a feasible global strategy profile and the subsets correspond to our best response strategy profiles of ). However, for our purposes we have to modify the given formulation in Eq. 4 as follows:
(5) 
Eq. 5 is quite similar to Eq. 4. However, an element of the superset of Eq. 4 corresponds to a player and his chosen action of our global strategy profile. Nevertheless, the intention of those first two energy terms complies with the intention of Eq. 4, stated in Sec. 2.3. Further, another energy term must be added to our QUBO problem as constraint. This last energy term, for which an instance is given in Example 4.2, states that exactly sets of should be included to form the global strategy profile. This constraint implicitly ensures that every player is playing one of his best response strategies.
is called penalty value, which is added on top of the solution energy, if a constraint was not satisfied, i.e. one of the three terms (quadratic differences) are unequal to . Thus, adding a penalty value states a solution as invalid. Only if the total energy described by the corresponding solution is a valid global best response strategy profile and thus a PNE.
All these energy terms are specified within the QUBO problem matrix , which the quantum annealer takes as an input, goes through the annealing process and responds with a binary vector as a solution, see Sec. 2.2. This vector indicates which best response strategy of each player should be chosen to form a PNE.
Example 4.2.
To demonstrate the function of the last energy term (constraint) of Eq. 5, an excerpt of best response strategy combinations of (in form of pointed sets, with the bold playeractioncombination being the base point) for an arbitrary 4player game are visualized in Fig. 1. The green union of four best response sets leads to a PNE, in which every player is playing one of his best response strategies. Although the red union of three sets also leads to a global combined strategy set, in which every player is playing one of his actions, it is not a PNE, due to player not playing a best response strategy.
5 Experiments
5.1 Evaluation GraphTopologies
For evaluating QNash, we implemented a game generator. It creates graphical game instances of three different popular graphical structures, which were often used in literature [Koller and Milch2003, Vickrey and Koller2002, Jiang and LeytonBrown2007, Palmieri and Lallouet2017]. These graphical structures are shown in Fig. 2. As an input for our game generator, one can choose the number of players, the graphtopology and thus the dependencies between the players and the number of actions for each player individually. The corresponding payoffs are sampled randomly from [0, 15]. For our experiments we considered games with three actions per player. The classic theorem of Nash [Nash1951] states that for any game, there exists a Nash equilibrium in the space of joint mixed strategies. However, in this work we only consider pure strategies and therefore there might be (graphical) games without any PNE (see, for instance, [Osborne and Rubinstein1994]). Additionally we want to emphasize, that QNash is able to work on every graphical game structure, even if the dependency graph is not connected, for instance a set of trees (called forest).
5.2 Methods
5.2.1 QBSolv
Due to the fact, that quantum computing is still in its infancy, and corresponding hardware is limited in the number of qubits and their connectivity, we need to fall back to a hybrid method (QBSolv^{3}^{3}3https://github.com/dwavesystems/qbsolv), in order to solve large problem instances. QBSolv is a software that automatically splits instances up into subproblems submitted to DWave’s quantum annealer, and an extensive tabu search is applied to postprocess all DWave solutions. Additionally, QBSolv embeds the QUBO problem to the quantum annealing hardware chip. QBSolv further allows to specify certain parameters such as the number of individual solution attempts (num_repeats), the subproblem size used to split up instances which do not fit completely onto the DWave hardware and many more. For detailed information, see [Booth et al.2017].
5.2.2 Brute Force
For evaluating the effectiveness of QNash we implemented a Brute Force (BF) algorithm to compare with. It determines the best response strategy sets of all players in the same way as QNash does and afterwards tries out every possible combination of those sets to form a valid global strategy set which corresponds to a PNE. The number of combinations is exponential with respect to the number of players of the game, .
5.2.3 Random Search
Additionally, a Random Search (RS) algorithm was implemented, which acts like the BF algorithm, but tries to randomly find combinations of best response sets, which correspond to a Nash equilibrium. Furthermore it takes a timespan as an input parameter and terminates after a timeout occured.
5.3 Computational times of QNash
With respect to the computational results stated in Tab. 6 we first introduce the time components of QNash’s total computation time. For a better understanding a general overview of QNash is given in Algorithm 1.
The find embedding time (3) states the time DWave’s classical embedding heuristic takes to find a valid subproblem hardware embedding. Determining best responses (4) describes the time QNash classical computing phase takes to identify the best response strategies of every player and additionally build up the QUBO matrix. The QBSolv time can be divided into the classical time (5) and the quantum annealing time (6). The classical QBSolv time (5) comprises of not only a tabu search, which iteratively processes all subproblem solutions, it also contains the latency and job queuing time to DWave’s quantum hardware. The quantum annealing time (6) comprises of the number of subproblems times DWave’s qpuaccesstime.
6 Results & Discussion
We investigated the solution quality and computational time of QNash. Fig. 3 shows the ability of finding PNE of the three proposed methods (QNash with QBSolv, BF and RS) in differently structured graphical games. For every graph topology (Tree, Circle and Road) we used games with players ranging from 6 to 10, due to the fact that BF took too long to solve larger games. The results show that QNash found the same amount of PNE as the exact BF algorithm for those instances. We ran RS as long as QNash (total time) took to solve the instances. One can see that RS was only able to find one or two PNE in the smallest game instances with 6 players.
Because of QNash performing well on these instances we tested it for larger game instances, see Fig. 4 and 5
. For the Road and CircleTopology, the solution quality of QNash was evaluated with different numbers of players ranging from 20 to 40. Running QNash 20 times on every instance, one can see that QNash was not always able to find the same amount of PNE per run. Although it seems that for larger scaled game instances the variance of finding PNE decreases, one has to take into account that these game instances might contain less PNE, due to the fact that all the players have to play one of their best response strategies to form a PNE. Furthermore, RS never found a PNE on those game instances, which highlights the huge combinatorial action space on which QNash performs comparatively well with respect to the effectiveness.
Because of quantum annealing being a heuristic, one can set the number of annealing attempts with the num_repeats parameter of QBSolv, to improve the accuracy. In Fig. 6
the influence of this parameter is shown. We exemplary used a 25 player road game and ran QNash 20 times per parameter setting to show its impact on the effectiveness. As expected one can see, that with increasing number of annealing attempts the interquartile range and its median in regard to the number of PNE found, increase. Although an annealing process takes only 20
, it adds up with the num_repeats parameter and therefore leads to a tradeoff between computational time and accuracy.7 Conclusion
We proposed QNash, to our knowledge the first algorithm that finds PNEGG using quantum annealing hardware.
Regarding the effectiveness of QNash, we showed that for small game instances (ranging from 610 players) the algorithm was always able to find all PNE in differently structured graphical games. Anyway we have to mention, that with increasing number of players the variance w.r.t the number of found PNE increased, too.
Due to the fact that quantum computing is still in its infancy and recent hardware is limited in the number of qubits and their connectivity, we had to fall back on a quantumclassical hybrid solver, called QBSolv, which involves additional overhead time. That makes it difficult to draw a fair comparison of QNash and classical stateoftheart solution methods regarding the computational time. We therefore decomposed the total time into its main components to show their impact. According to the experimental results, the only unpredictable time component is QBSolv’s classical tabu search along with latency and job queuing times at DWave’s cloud computing frontend. However with DWave announcing an immense rise of the number of qubits and their connectivity on DWave’s quantum processors in the next years^{4}^{4}4https://www.dwavesys.com/sites/default/files/mwj_dwave_
qubits2018.pdf, it might be possible to embed larger game instances directly onto the chip and therefore omit hybrid solvers like QBSolv. Another possibility is using Fujitsu’s Digital Annealing Unit (DAU) which also takes a QUBO matrix as input. With DAU being able to solve larger fully connected QUBO problems [Aramon et al.2019], a shorter total computation time could be achieved.
References
 [Albash and Lidar2018] Tameem Albash and Daniel A Lidar. Adiabatic quantum computation. Reviews of Modern Physics, 90(1):015002, 2018.
 [Aramon et al.2019] Maliheh Aramon, Gili Rosenberg, Elisabetta Valiante, Toshiyuki Miyazawa, Hirotaka Tamura, and Helmut Katzgraber. Physicsinspired optimization for quadratic unconstrained problems using a digital annealer. Bulletin of the American Physical Society, 2019.

[Bhat and
LeytonBrown2004]
Navin AR Bhat and Kevin LeytonBrown.
Computing nash equilibria of actiongraph games.
In
Proc. of the 20th conference on Uncertainty in artificial intelligence
, pages 35–42. AUAI Press, 2004.  [Booth et al.2017] Michael Booth, Steven P Reinhardt, and Aidan Roy. Partitioning optimization problems for hybrid classical. quantum execution. Technical Report, pages 01–09, 2017.
 [Boros et al.2007] Endre Boros, Peter L Hammer, and Gabriel Tavares. Local search heuristics for quadratic unconstrained binary optimization (qubo). Journal of Heuristics, 13(2):99–132, 2007.
 [Carfì et al.2011] David Carfì, Francesco Musolino, et al. Fair redistribution in financial markets: a game theory complete analysis. Journal of Advanced Studies in Finance, 2(2):4, 2011.
 [Daskalakis and Papadimitriou2006] Constantinos Daskalakis and Christos H Papadimitriou. Computing pure nash equilibria in graphical games via markov random fields. In Proc. of the 7th ACM conference on Electronic commerce, pages 91–99. ACM, 2006.
 [Fudenberg and Tirole1991] Drew Fudenberg and Jean Tirole. Game theory, 1991. Cambridge, Massachusetts, 393(12):80, 1991.
 [Gottlob et al.2005] Georg Gottlob, Gianluigi Greco, and Francesco Scarcello. Pure nash equilibria: Hard and easy games. Journal of Artificial Intelligence Research, 24:357–406, 2005.
 [Jiang and LeytonBrown2007] Albert Xin Jiang and Kevin LeytonBrown. Computing pure nash equilibria in symmetric action graph games. In AAAI, volume 1, pages 79–85, 2007.
 [Kadowaki and Nishimori1998] Tadashi Kadowaki and Hidetoshi Nishimori. Quantum annealing in the transverse ising model. Physical Review E, 58(5):5355, 1998.
 [Karp1972] Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85–103. Springer, 1972.
 [Kearns et al.2013] Michael Kearns, Michael L Littman, and Satinder Singh. Graphical models for game theory. arXiv preprint arXiv:1301.2281, 2013.
 [Koller and Milch2003] Daphne Koller and Brian Milch. Multiagent influence diagrams for representing and solving games. Games and economic behavior, 45(1):181–221, 2003.
 [La Mura2000] Pierfrancesco La Mura. Game networks. In Proc. of the 16th conference on Uncertainty in artificial intelligence, pages 335–342, 2000.
 [Littman et al.2002] Michael L Littman, Michael J Kearns, and Satinder P Singh. An efficient, exact algorithm for solving treestructured graphical games. In Advances in Neural Information Processing Systems, pages 817–823, 2002.
 [Lucas2014] Andrew Lucas. Ising formulations of many np problems. Frontiers in Physics, 2:5, 2014.
 [McGeoch2014] Catherine C McGeoch. Adiabatic quantum computation and quantum annealing: Theory and practice. Synthesis Lectures on Quantum Computing, 5(2):1–93, 2014.
 [Nash1951] John Nash. Noncooperative games. Annals of mathematics, pages 286–295, 1951.
 [Ortiz and Kearns2003] Luis E Ortiz and Michael Kearns. Nash propagation for loopy graphical games. In Advances in Neural Information Processing Systems, pages 817–824, 2003.
 [Osborne and Rubinstein1994] Martin J Osborne and Ariel Rubinstein. A course in game theory. MIT press, 1994.
 [Palmieri and Lallouet2017] Anthony Palmieri and Arnaud Lallouet. Constraint games revisited. In International Joint Conference on Artificial Intelligence, IJCAI 2017, pages 729–735, 2017.
 [Rabin1993] Matthew Rabin. Incorporating fairness into game theory and economics. The American economic review, pages 1281–1302, 1993.
 [Roy et al.2010] Sankardas Roy, Charles Ellis, Sajjan Shiva, Dipankar Dasgupta, Vivek Shandilya, and Qishi Wu. A survey of game theory as applied to network security. In System Sciences (HICSS), 2010 43rd Hawaii International Conference on, pages 1–10. IEEE, 2010.
 [Soni et al.2007] Vishal Soni, Satinder Singh, and Michael P Wellman. Constraint satisfaction algorithms for graphical games. In Proc. of the 6th international joint conference on Autonomous agents and multiagent systems, page 67. ACM, 2007.
 [Su et al.2016] Juexiao Su, Tianheng Tu, and Lei He. A quantum annealing approach for boolean satisfiability problem. In Proc. of the 53rd Annual Design Automation Conference, page 148. ACM, 2016.
 [Vickrey and Koller2002] David Vickrey and Daphne Koller. Multiagent algorithms for solving graphical games. In AAAI/IAAI, pages 345–351, 2002.
 [Von Neumann and Morgenstern1944] John Von Neumann and Oskar Morgenstern. Theory of games and economic behavior. 1944.
References
 [Albash and Lidar2018] Tameem Albash and Daniel A Lidar. Adiabatic quantum computation. Reviews of Modern Physics, 90(1):015002, 2018.
 [Aramon et al.2019] Maliheh Aramon, Gili Rosenberg, Elisabetta Valiante, Toshiyuki Miyazawa, Hirotaka Tamura, and Helmut Katzgraber. Physicsinspired optimization for quadratic unconstrained problems using a digital annealer. Bulletin of the American Physical Society, 2019.

[Bhat and
LeytonBrown2004]
Navin AR Bhat and Kevin LeytonBrown.
Computing nash equilibria of actiongraph games.
In
Proc. of the 20th conference on Uncertainty in artificial intelligence
, pages 35–42. AUAI Press, 2004.  [Booth et al.2017] Michael Booth, Steven P Reinhardt, and Aidan Roy. Partitioning optimization problems for hybrid classical. quantum execution. Technical Report, pages 01–09, 2017.
 [Boros et al.2007] Endre Boros, Peter L Hammer, and Gabriel Tavares. Local search heuristics for quadratic unconstrained binary optimization (qubo). Journal of Heuristics, 13(2):99–132, 2007.
 [Carfì et al.2011] David Carfì, Francesco Musolino, et al. Fair redistribution in financial markets: a game theory complete analysis. Journal of Advanced Studies in Finance, 2(2):4, 2011.
 [Daskalakis and Papadimitriou2006] Constantinos Daskalakis and Christos H Papadimitriou. Computing pure nash equilibria in graphical games via markov random fields. In Proc. of the 7th ACM conference on Electronic commerce, pages 91–99. ACM, 2006.
 [Fudenberg and Tirole1991] Drew Fudenberg and Jean Tirole. Game theory, 1991. Cambridge, Massachusetts, 393(12):80, 1991.
 [Gottlob et al.2005] Georg Gottlob, Gianluigi Greco, and Francesco Scarcello. Pure nash equilibria: Hard and easy games. Journal of Artificial Intelligence Research, 24:357–406, 2005.
 [Jiang and LeytonBrown2007] Albert Xin Jiang and Kevin LeytonBrown. Computing pure nash equilibria in symmetric action graph games. In AAAI, volume 1, pages 79–85, 2007.
 [Kadowaki and Nishimori1998] Tadashi Kadowaki and Hidetoshi Nishimori. Quantum annealing in the transverse ising model. Physical Review E, 58(5):5355, 1998.
 [Karp1972] Richard M Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85–103. Springer, 1972.
 [Kearns et al.2013] Michael Kearns, Michael L Littman, and Satinder Singh. Graphical models for game theory. arXiv preprint arXiv:1301.2281, 2013.
 [Koller and Milch2003] Daphne Koller and Brian Milch. Multiagent influence diagrams for representing and solving games. Games and economic behavior, 45(1):181–221, 2003.
 [La Mura2000] Pierfrancesco La Mura. Game networks. In Proc. of the 16th conference on Uncertainty in artificial intelligence, pages 335–342, 2000.
 [Littman et al.2002] Michael L Littman, Michael J Kearns, and Satinder P Singh. An efficient, exact algorithm for solving treestructured graphical games. In Advances in Neural Information Processing Systems, pages 817–823, 2002.
 [Lucas2014] Andrew Lucas. Ising formulations of many np problems. Frontiers in Physics, 2:5, 2014.
 [McGeoch2014] Catherine C McGeoch. Adiabatic quantum computation and quantum annealing: Theory and practice. Synthesis Lectures on Quantum Computing, 5(2):1–93, 2014.
 [Nash1951] John Nash. Noncooperative games. Annals of mathematics, pages 286–295, 1951.
 [Ortiz and Kearns2003] Luis E Ortiz and Michael Kearns. Nash propagation for loopy graphical games. In Advances in Neural Information Processing Systems, pages 817–824, 2003.
 [Osborne and Rubinstein1994] Martin J Osborne and Ariel Rubinstein. A course in game theory. MIT press, 1994.
 [Palmieri and Lallouet2017] Anthony Palmieri and Arnaud Lallouet. Constraint games revisited. In International Joint Conference on Artificial Intelligence, IJCAI 2017, pages 729–735, 2017.
 [Rabin1993] Matthew Rabin. Incorporating fairness into game theory and economics. The American economic review, pages 1281–1302, 1993.
 [Roy et al.2010] Sankardas Roy, Charles Ellis, Sajjan Shiva, Dipankar Dasgupta, Vivek Shandilya, and Qishi Wu. A survey of game theory as applied to network security. In System Sciences (HICSS), 2010 43rd Hawaii International Conference on, pages 1–10. IEEE, 2010.
 [Soni et al.2007] Vishal Soni, Satinder Singh, and Michael P Wellman. Constraint satisfaction algorithms for graphical games. In Proc. of the 6th international joint conference on Autonomous agents and multiagent systems, page 67. ACM, 2007.
 [Su et al.2016] Juexiao Su, Tianheng Tu, and Lei He. A quantum annealing approach for boolean satisfiability problem. In Proc. of the 53rd Annual Design Automation Conference, page 148. ACM, 2016.
 [Vickrey and Koller2002] David Vickrey and Daphne Koller. Multiagent algorithms for solving graphical games. In AAAI/IAAI, pages 345–351, 2002.
 [Von Neumann and Morgenstern1944] John Von Neumann and Oskar Morgenstern. Theory of games and economic behavior. 1944.
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