Game theory models for communication between agents: a review

Abstract

In the real world, agents or entities are in a continuous state of interactions. These interactions lead to various types of complexity dynamics. One key difficulty in the study of complex agent interactions is the difficulty of modeling agent communication on the basis of rewards. Game theory offers a perspective of analysis and modeling these interactions. Previously, while a large amount of literature is available on game theory, most of it is from specific domains and does not cater for the concepts from an agent-based perspective. Here in this paper, we present a comprehensive multidisciplinary state-of-the-art review and taxonomy of game theory models of complex interactions between agents.

Farooqui and Niazi

Complex Adapt Syst Model (2016) 4:13 DOI 10.1186/s40294-016-0026-7

Game theory models forcommunication betweenagents: a review

Aisha D. Farooqui1 and Muaz A. Niazi2*

*Correspondence: [email protected]

2 Computer Science Department, COMSATS Institute of IT, Islamabad, PakistanFull list of author information is available at the end of the article

Background

In the real world, agents or entities are in a continuous state of interactions (Niazi etal. 2011). Examples of these include the continuously interacting agents in the stock market (Bonabeau 2002). These agents and systems can be adaptive in nature and can also evolve. Their current behavior can depend on the past so they often learn from history.

The interaction of agents leads to a wide variety ofcomplexity dynamics (McDaniel and Driebe 2001). Complexity arises due to non-linear agent interactions. The behavior of such non-linear systems can be chaotic and unpredictable. Complex adaptive systems (CAS) in the natural world (Niazi etal. 2011) and complex physical systems (CPS) (Winsberg 2001) in man-made systems are examples of such agent interactions.

One key difficulty faced by Complexity researchers isin the modeling of communication andcomplex agent interaction(Niazi and Hussain 2012). Modern communication systems are oftencomposed of hierarchical complex systems. These systems can be modeled as multiagent systems usingagent-based modeling (ABM). Modeling CAS and CPS using ABM not only allows for prediction of outcomes but also helps in terms of gaining an understanding of the complex inter-connnections and interactions(Epstein 2008). However, a key issue in such models is to understand the dynamics ofagent interaction. Game Theory oers techniques and tools for modeling communication problems among agents.

2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Game theory oers a perspective of analysis and modeling of these interactions (Carmichael 2005). It is a discipline that studies decision making of interactive entities (Dixit and Skeath 1999). We can say that strategic thinking is perhapsthe most recognized essence of game theory.

Previously, while a large amount of literature is available on game theory, most of it is focused onspecic domains like Biology, Economics, and Computer Science (Shoham and Leyton-Brown 2008). Game theoryhas also been used in business to model interactions of stakeholders etc.

To the best of our knowledge, there is an absence of a state-of-the-art reviews of game theoretical literature from the agent-based modeling perspective. This paper presents a comprehensive review of game theory models and their applications. Additionally, a taxonomy of classes of games is also presented.

The paper is organized as follows: rst, we give an overview of game theory and present a taxonomy of games. This is followed by literature review in the next section. Then, in the discussion we classify games and discuss open problems before concluding the paper.

Game theory overview

While the essence of game theory has perhaps practically applied itself since life presented itself on this planet, formal literature on the topic can be traced back tothe work of Von Neumann and Morgenstern (1944). They worked on zero-sum games. Then in the 1950s, Nashs work resulted in signicant advancement of this eld (Nash 1950). Subsequently, Game theory has since been used in many dierent elds like biology (Hofbauer and Sigmund 1998), politics and other domains (Morrow 1994).

Game theory presents a technical analysis of strategic interactions (Shoham and Leyton-Brown 2008). These strategic interactions are concerned with the interaction of decision makers in the game (Geckil and Anderson 2009). The behavior of a decision maker in game theory models is called strategic and the action performed while making any move is called a strategy. Strategy considers how agents act, what they prefer, how they make their decisions, and their behaviors etc. These interactions can be complex as the action of even a single agent can inuence other agents and vice versa. Game theory can thus be considered as a powerful tool to model and understand complex interactions.

One way of classifying game theory models is to divide them into cooperative and non-cooperative games (Shoham and Leyton-Brown 2008). In cooperative games, we focus on a set of agents. Whereas,in non-cooperative games the focus is on the development ofmodelsof interactions, preferences, and so on, with a focus onindividual agents.1 It can model dierent types of games including zero-sum (Shoham and Leyton-Brown 2008), stochastic (Mertens and Neyman 1981), repeated Aumann and Maschler (game of fairness as if player cuts unequ), Bayesian (Bge and Eisele 1979) and congestion (Rosenthal 1973).

1 Literature usually considers cooperative and non-cooperative as conicting and non-conicting game theory. But we are following denition of Shoham etal. that cooperative game theory focus on modeling set of players and non-cooperative models individual player (Shoham and Leyton-Brown 2008).

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Multidisciplinary nature ofgame theory

Game theory can be seen everywhere in living systems, in general, and human society, in particular. Inpersonal life as well as inprofessional life, every day we are faced with decisions which often can be simplied using game theory.There are dierent areas where game theory has beenapplied such as Economics, Politics etc.

(Shoham and Leyton-Brown 2008). Algorithmic game theory is an example of application in computer science (Roughgarden 2010). Biologists haveused it to learn species behaviors (Hofbauer and Sigmund 1998). In mathematics, there is a complete branch that studies decision-making process (Mazalov 2014). It also has its inuences in business (Geckil and Anderson 2009). It can model interactions of stakeholders, dynamics in interest rates etc.

Dixit and Skeath (1999) note that we can use game theory mainly in three ways that are an explanation, prediction, and prescription.

Explanation

Game theory can be used to explain insights of a situation like why that happened, what were the causes, Eects of that happening etc. We can do a complete case study by using game theory.

Prediction

Game theory studies decision makers (autonomous agents) that have actions to take, preferences that what they want, dierent options which they can choose etc. By analyzing these actions, preferences, options etc we can predict dierent moves of agents on dierent types of situation.

Prescription

If we can analyze agent actions, strategies etc to predict its moves, then we can denitely give advice about dierent moves to agents. It means we can provide a sophisticated model for future decision-makings.

Now let us consider basic concepts of game theory.

Basic concepts

Dixit and Nalebu (1993) have dened Game theory as:

Denition 1 The branch of social science that studies strategic decision-making.

Another denition is by Hutton (1996):

Denition 2 An intellectual framework for examining what various parties to a decision should do given their possession of inadequate information and dierent objectives.

Shoham and Leyton-Brown (2008) have dened game theory as:

Denition 3 Game theory is the mathematical study of interaction among independent, self-interested agents.

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In the oxford dictionary, self-interested means self-seeking or self-serving. Anyone who is self-interested is concerned strongly with own interests. This seems selshness of someone who do not consider others interests.

However, in game theory, these are actually intelligent agents and their behavior is based on articial intelligence models (Wooldridge 2009). These are autonomous entities, with their own description of world states and they behave accordingly (Shoham and Leyton-Brown 2008). Unfortunately, there is no universal denition of the agent but autonomy is one of the basic properties of the agent.

In Computer Science, Algorithmic game theory is used (Roughgarden 2010). It combines game theory together with computer science. It focuses on creating algorithms for strategic interactions, calculating Nash equilibrium etc.

Game

Carmichael (2005) has dened games as:

Denition 4 A scenario or situation where for two or more individuals, their choice of action or behavior has an impact on the other (or others).

The game consists of several things such as

players

strategies (actions taken while interactions)

payos (utilities gained)

payo function (calculates utility against each strategy)

and of course, game rules.

Geckil and Anderson (2009) has dened game as:

Denition 5 A game-theoretic model is an environment where each decision makers actions interact with those of others

Game representation

There are mainly two ways to represent the game. Normal-form is simply a matrix that describes strategies and payos of the games (Morrow 1994). Another representation is extensive-form, which is a tree-like structure (Morrow 1994). Extensive-form contains more information than normal-form like a sequence of player moves. However, there are games that require richer representation such as innite repeated games. To represent such games we have Beyond Normal-Extensive form (Shoham and Leyton-Brown 2008).

Decision theorem

Game theory has two decision theorems known as maximin and minimax (Mazalov 2014). The minimax theorem minimizes the loss of a player. The maximin theorem used to maximize the benet gain by the player.

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Games taxonomy

We saw dierent types of games in the literature review. These games were presented using three types of game representations. Normal-form, extensive-form and beyond normal and extensive-form games (Shoham and Leyton-Brown 2008). We proposed a taxonomy of games based on these three game representation types. See Fig.1.

The taxonomy mainly classies games into three types, as there are three types of representations. Then it further classies games that are included in both normal-form games and extensive-form games. Games included in both because a normal-form representation can be derived from extensive-form games. Beyond normal and extensive form includes those games that need richer representation. These games can be innite and undetermined. Therefore, that it is difficult to represent them in rst two representations.

These games have been discussed in literature according to game representation types but is not presented as the taxonomy in this paper demonstrates. There are previously given taxonomies, but these are specic to the two-player game. Kilgour and Fraser have presented a taxonomy discussing ordinal games (Kilgour and Fraser 1988). Rapoport and Guyer (1978) have presented another taxonomy considering 22 games. The taxonomy given in this paper is not specic and is based on the type of game representation.

Normalform games

It is conceptually straightforward strategic representation (Morrow 1994). It describes all observable and possible strategies and the utility against each strategy. It can represent all nite games and taken as a universal representation of games. It uses a matrix to represent strategic interactions of players in a matrix form. It consists of

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Set of players

Strategy space, a set of all strategies of a player

Payo function, it calculates the utility against each strategy.

Table1 adapted from Morrow (1994) shows a normal form representation of Matching Pennies game. If both P1 and P2 get heads, P1 will take both coins else P2 will win and take both coins. The numbers 1 and 1 shows the utility gained or loosed by players.

Extensiveform games

It is an alternative way of representing games in a tree-like structure. It denes dierent stages of the game. Moves, choices, and actions dened according to each stage. We can derive a normal-form representation from extensive representation. Morrow (1994) described Matching pennies game in extensive form representation. See Fig.2.

Beyond normal/extensive games

There are games needs richer representation like repeated games (Shoham and Leyton-Brown 2008). These can be nite or innite. Therefore, that it is difficult to represent them in normal/extensive forms. The games included here are.

Repeated games: These are also called stage games. Players play these games multiple times (Aumann and Maschler 1995).

Stochastic games: These are also called Markov games. There are stages in the game. Every stage represents the state of a game from a nite set of game states. The player has a set of actions that consists of many nite actions (Mertens and Neyman 1981).

Bayesian games: These are games of incomplete information. Players select their strategies according to Bayes Rule (Bge and Eisele 1979).

Congestion games: These games are the class of non-conicting games (Rosenthal 1973). In these games, all the players have same strategy set. The result of every player relies upon the strategy it picks and all other players picking the same strategy.

Complex adaptive systems

Complex systems have special types of systems known as Complex adaptive systems (Mitchell 2009). These systems have the dynamic environment and non-linear interaction of components. The amazing thing for researchers is that these systems are composed of so simple components and exhibits emergent behavior when combined together. Such systems can be understood only by considering all components collectively.

Table 1 Matching pennies: game in normal-form (This table is adapted from Morrow (1994))

P1 P2

H T

H (1, 1) (1, 1)

T (1, 1) (1, 1)

In this game if both players show same side of coins that is both shows head or tail then P1 wins and P2 looses both coins. If both players shows dierent sides then P2 wins and P1 looses both coins

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Nonlinear agent interaction

Complex adaptive systems are subset of dynamic non-linear system (McDaniel and Driebe 2001). In non-linear agent interactions, the inputs are inversely proportional to output (Lansing 2003). In these amazing systems, small changes can results in a big change and vice versa. Mathematically, the behavior of the non-linear system can be described as non-linear polynomial equations.

There can be more than one attracters in non-linear systems (Socolar 2006). These attractors are of dierent types with complicated limit cycles. The trajectories are restricted to areas that have unstable limit cycles.

Agentbased computing

Agent-based computing is a wide domain (Niazi and Hussain 2011). The agent here can simply a software providing any service. Or it can be fully autonomous agent whose behavior based on articial intelligence. Agent-based computing should not be confused with other terms in articial intelligence. Such terms are agent-oriented programming, multi-agent oriented programming, and agent-based modeling. These all are actually collected together in agent-based computing.

Now in the next section, we will present a review on available game theoretic literature.

Review

In the previous section, we gave an overview of game theory and presented a taxonomy of games. In this section, we will explore available game theoretic literature.

Zerosum game theoretic models

Zero-sum games are the mathematical representation of conicting situations (Washburn 2003). In these games, the total of gains and losses is equal to zero. Application of

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these game theoretic models can be seen in dierent elds like network security (Perea and Puerto 2013) and resource allocation (Zhou et al. 2011). There are also dierent types of games. Such as zero-sum games with incomplete information and large Zero-sum games.

Al-Tamimi etal. (2007) have discussed Q-learning designs for the zero-sum game. By using a model-free approach they obtained a solution for the game. Autopilot design for the F-16 plane is performed that shows productiveness of method.

Daskalakis etal. (2015) have proposed no-regret algorithm. This zero-sum game theoretic algorithm achieves regret when applying against an adversary. After using the algorithm, quadratic improvement can be identied on convergence rate to game value. The lower bound for all distributed dynamics is optimal. This happens when payo matrix information is unknown to both players. But if they know they can compute minimax strategies privately.

Bopardikar etal. (2013) have studied larger zero-sum games. In these games, players have a large number of options. It proposes two algorithms. The Sampled Security Policy algorithm is to compute optimal policies. Then Sampled Security Value algorithm computes the level of condence on the given policy.

Moulin and Vial (1978) have proposed a class of games called strategically zero-sum games. These games have special payo structure. The mixed equilibrium of these games cannot be improved. The properties of games via a large body of correlation scheme is also described.

Sorin (2011) have worked on repeated zero-sum games. They described current advancement in these games especially together with dierential games. They rst dene models of repeated games and dierential games. Then they discuss issues related to these models.

Seo and Lee (2007) have considered conicting zero-sum game that involves decision-making process. This is an experimental study on trained monkeys. Monkeys take binary choices in the computer-simulated conicting game. The study described the decision-making process adaptive in both human and animals.

Zoroa et al. (2012) have modeled a perimeter patrol problem. They used Zero-sum discrete search games as a framework for their study. They studied problem occurred in cylindrical surface. The problem in the linear set having cyclic order is also studied. Optimal strategies are found via computer code.

Xu and Mizukami (1994) have studied systems of state space. They obtained saddle-point by a constructive method. It describes that there can be several saddle-point solutions for the system. When several saddle-points exist, this universal system diers from the state space system. They found possible conditions for the existence of saddle-point.

Ponssard and Sorin (1980) have discussed zero-sum games with incomplete information. They discussed two ways to determine information of states. It can be obtained via independent chance moves or the unique one. Unique moves cause dependence in state information. Thus, it is complicated to analyze. Several results acquired in the independent case have their equivalent in dependent one.

Chen and Larbani (2006) have proposed undetermined utility matrix game. They worked for the solution of decision-making problem (MADM). This decision making deals with prioritization of alternatives considering several attributes. Here weights of an

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MADM problem obtained with a fuzzy decision matrix. Finally, equilibrium solution is also obtained.

Li and Cruz (2009) have studied deception. They used a zero-sum game model with an asymmetrical structure. This paper considers the relationship between information and decision-making to understand deception. In these games, the rst player gets extra information. Whereas the second player has the power to inject deception. The paper also classies deception into active deception and passive deception.

Ponssard (1975) have worked on the zero-sum game in the normal form. They described that these games are equal to a linear program (LP). In these games, the players behavioral strategies are represented in variables. In normal form game variables are used to represent the players mixed strategies.

Wang and Chen (2013) have obtained feedback saddle-point for the zero-sum dierential game. The game is between counter-terror measure and economic growth. It uses Hamilton-Jacobi-Isaacs equation to obtain saddle-point. The saddle-point obtained, strengthens the government counter-terror and weakens the terrorist organizations.

Van Zandt and Zhang (2011) have studied equilibrium value for Bayesian zero-sum games. The conditions are characterized for equilibrium value and strategies. These games have a parameter to obtain payo function and strategies for every player. The information of every player is modeled as a sub- ~ -eld to obtain optimal strategies.

Marlow and Peart (2014) have studied soil acidication. They described a zero-sum game between a sugar maple and American beech. The negative impact of soil acidication on sugar maple supports beech in the game. The model lay down the ndings of this study and other evidence of soil acidication. The results suggest re-examining the cost-eectiveness of chemical remediation.

2player zerosum games

Mertens and Zamir (1971) have also discussed the two-person zero-sum game with incomplete information. These games are studied in a repetitive form. As a result, the game value is obtained with n repetitions. This is previously discussed by Harsanyi. However, still this paper is completely independent on its own.

Chang and Marcus (2003) have studied two-person zero-sum game. They considered optimal equilibrium game value and then analyzed error bounds. After that, they discussed methods that calculate the value of subgame.

Mndez-Naya (1996) have discussed 2-players continuous games. These games have set of pure strategies. These games also have right-sided semi-open real intervals and continuous payo functions. The paper described conditions for game value in the mixed game. It is proved that there is no assurance that mixed extended Zero-sum game has a value but there can be a value.

Qing-Lai etal. (2009) have proposed an algorithm for 2-D systems. It solves two-players zero-sum games. It obtains saddle-point by using adaptive critic technique. The optimal control policies have been computed using neural networks. The algorithm can be implemented without system model.

Zhang etal. (2011) have proposed an iterative algorithm. It obtains optimal solutions for the non-affine nonlinear zero-sum game. This is a two-player game with quadratic performance index. One player minimizes the performance index while other maximizes

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it. This study held to facilitate this minimax problem. The optimal strategy has obtained an order of state trajectories and Riccati dierential equations. Finally, the simulation shows successful results of this iterative method.

Gensbittel (2014) has worked on zero-sum incomplete information games. The author extended the CAV (U) Theorem of AumaanMaschler (Aumann et al. 1995). In this paper, the presented results are for innite repeated games. Finally, the paper provides optimal strategies for players in 2-players game having length n.

Bettiol et al. (2006) have considered Zero-sum state constrained dierential games. The study proves bolza problem for two-player dierential games. It shows that lower semi-continuous value function exists in dierential games. The optimal strategy is created and the value function is characterized by viscosity solutions.

Beyond 2player zerosum games

Initially, the zero-sum game is a 2-player game (Von Neumann and Morgenstern 1953). In which one player has to win and other has to loose the game. The following papers show that researchers have worked on beyond 2-player game.

Moulin (1976) has worked on beyond 2-player Zero-sum games. First, this study describes a large family of abstract extension. Then these extensions are classied based on information exchanged. Finally, characterization of all possible values gained from this abstract extension is described.

Okamura etal. (1984) have studied three-player zero-sum games. They investigated the learning of the behavior of variable-structure stochastic automata in a game. These automata have learning capabilities and can update their actions. The players have a lack of information of payo matrix. After every play, the environment, responds to automaton actions. After this, players update their strategies.

Decision theorems

Sauder and Geraniotis (1994) have worked on maximin and minimax theorems. They formulated signal detection process as two-players zero-sum game. The two-players are the detector designer and the signal designer. The signal detection problem arises when analyzing the signal is genuine or deceptive. Finally, results are validated via simulation.

Hellman (2013) have focused on rational belief system. The study got the basis from the work of Aumann and Dreze. They described that players have common knowledge of rationality. Whereas in this article, it is argued that there is no need of common rationality. Finally, it is shown that the expected payo in the game is only the minimax value.

Ponssard (1976) have discussed minimax strategies. These are prohibited to give particular solutions in optimal zero-sum game play. This study nds a strategy to be used after the mistake carries out in play. There are two approaches proposed to get optimal strategies. The rst approach arrived from perturbed games. The second approach established on the basis of the lexicographic application. If the opponent ignores mistakes, the strategy will remain optimal as it does not turn to give a loss.

Gawlitza etal. (2012) have proposed two strategy improvement algorithms for static program analysis. One is max-strategy and the other is min-strategy for static program analysis. These algorithms perform within a common general framework to solve v-cam cave equations.

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Rock paper scissors

Rock paper scissors (RPS) is a cyclic game with three strategies. See Fig.3. The game is 2-players zero-sum. The game rules are rock wins over scissors, scissors over paper and paper over rock. Following papers considers RPS game theoretic model.

Sinervo and Lively (1996) have used cyclic RPS game in a biological study. By using this zero-sum model they studied three dierent strategies of male side-blotch lizards. It studies territory use and patterns of sexual selection on male side-blotch lizards.

Bahel and Haller (2013) have computed Nash equilibria of cyclic RPS game. They characterized Nash equilibria into two sets. With an even number of actions, an innity of Nash equilibria exists. On the second set with an odd number of actions unique Nash equilibria is found. This paper studies the strength of Nash equilibria.

Frey etal. (2013) have studied complex dynamics in social and economic systems. This is realized by analyzing agents independently playing a multiplayer mod game. The game is like the rock paper scissors. The behavior of players in human groups is non-uctuating and eective. In this game the periodic behavior is stable.

Batt (1999) has also studied the model of Rock Paper Scissors gameandhas presentedinsights of the game having an efficient outcome with few conicts. The game players are biased for being a winner. This game is not efficient with major conicts. For that other approaches like coin-ip is the best choice.

Neumann and Schuster (2007) have used a zero-sum rock scissor paper game as a framework. By which they modeled the process of bacteriocin producing bacteria. The game is examined for three strains. These are of E. coli, bacteriocin producer, resistant and sensitive. They derived stability criteria for these strains. The paper actually proposes LotkaVolterra system model of the RPS game.

Duersch et al. (2012) have obtained Nash equilibrium for the 2-player symmetric game. There is no pure equilibrium exists in RPS game. They found that pure equilibrium strategy exists only in non-generalized rock paper scissors game. It also showed that pure equilibrium exists for the 2-player nite symmetric game.

Cake cutting

Cake cutting is a simple child game. See Fig.4. In this game, the rst player has to cut the cake and then the second player has to choose the piece. The rst player has to cut pieces

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equally. Otherwise, the second player has the choice to choose either the bigger piece or the smaller one. This is to accomplish honesty in the game.

Procaccia (2013) have discussed cake cutting game. They described that it is a powerful tool to divide heterogeneous goods and resources. Cake cutting algorithm looks for formal fairness in the division of heterogeneous divisible goods. But the design of these algorithms is a complex task for computer scientists.

Edmonds and Pruhs (2006) have proposed a randomized algorithm that considers cake cutting algorithm. It equally allocates resources between n numbers of players. This algorithm needs honesty of players.

Matching penny

Matching penny is also a zero-sum 2-player game. Both players secretly turn their coins and then compare with each other. If both are heads or tails then the rst player will win else player 2 will win both coins. See Fig.5.

McCabe etal. (2000) have studied three-person matching pennies game. It examines knowledge of player about other players payos and actions. The Naive Bayesian learning and sophisticated Bayesian learning are studied in this context. These approaches examine that estimated mixed strategies can be played or not. Results showed that players do not use sophisticated Bayesian learning to obtain Nash equilibrium.

Stein et al. (2010) have studied mixed extension of matching pennies, a zero-sum game. This study constructs examples to support polynomial games. Here Nash equilibria are representable as nitely moments. Whereas polynomial games cannot be represented as nitely moments.

Colonel Blotto

Colonel Blotto is a universal game providing a way for resource allocation. See Fig.6. The two colonels simultaneously distribute resources over battleelds. The player devoting the most resources wins that battleeld. The payo is equal to the total number of battleelds won.

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Roberson (2006) described the remarkable equilibrium payos in the Colonel Blotto game. It considers both symmetric and asymmetric cases of the zero-sum game. The proportion of won battleelds is the payo of player.

Hart (2008) have studied Discrete Colonel Blotto game. This is a Zero-sum game with the symmetric case for which optimal strategy is obtained. Both of these games deal with the conicting environment.

Kuhn Poker

Kuhn Poker is a simplied form of Poker developed by Harold W. Kuhn (Tucker 1959). In this 2-player game, the deck includes only three cards. One card is distributed to each player. The rst player has to bet or pass then the second player may bet or pass. On a bet, the next player must bet also. When both players pass or bet then the player with the highest card will win the pot.

Southey etal. (2009) have studied Kuhn Poker game. There main concern is opponent modeling in the game. They studied two algorithms, expert and parameter estimation. Their experiment showed that learning methods do not give good results in the small game.

Princess Monster

Rufus Isaac formulated a game Princess Monster in his book Dierential Games (Isaaks 1952). This is a Zero-sum game between two players, Princess and Monster. The game played on 2-D search set. See Fig.7. When the distance between both players is less than r then Princess got captured and Monster wins.

Wilson (1972) has developed this game on a circle. Princess and Monster move on a circle either clockwise or anti-clockwise. If both players move in the same direction, the

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game state does not change. But if they move in opposite directions then there will be a point on the circle on which both reach at the same time. At that point, Princess got captured and Monster wins.

Solution concepts

We have discussed before that game describes strategic interactions. In game theory, the solution concept is like a rule by which game theorists seeks how the game will be played. The Nash equilibrium, Pareto optimality, and Shapley values are dierent known solution concepts. These concepts are used to formally predict that how the game will be played.

Nash equilibrium ofgames

Nash (1951) dened Nash equilibrium. In the Nash equilibrium, all players know each others equilibrium strategy. And no utility a player can have by changing its own strategy only. For example, there is a game battle of sexes (Shah etal. 2012). The game is between husband and wife. Husband prefers to go for football match and wife wants to go for a concert. Also, they want to go together. The payo table is shown in Table2. The solution for the game can be either both go for a football match or go to a concert.

Singh and Hemachandra (2014) have studied Nash equilibrium for stochastic games with independent state processes. This study got basis from the work of Altman etal. 2008. They worked on N-player Constrained Stochastic games.

Grauberger and Kimms (2014) have computed Nash equilibria for network revenue management games. This study investigates network management competition. A heuristic is presented for computing Optimal Capacity allocations. It also computes Nash

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equilibria in non-zero-sum games. It computes approximate to exact Nash equilibrium. They used the linear continuous model to reduced computational time.

Gharesifard and Cortes (2013) have considered a network based scenario and obtained a Nash solution. Networks aim is to maximize or minimize a common objective function. The two players are two network agents. They have their objectives to achieve networks aim. Both agents with opposite aims make a zero-sum game between them. Each networks saddle-point dynamics implemented by both networks through local interactions. The saddle-point dynamics for concave-convex class converges to Nash equilibrium. This saddle-point dynamics do not work to converge directed networks.

Porter et al. (2008) have proposed two search methods that calculate Nash equilibrium. One method is for the two-player game and the second method is for the n-player game. Both methods uses backtracking approaches to search the space of small and balanced support. These methods are tested on dierent games. Results showed positive performance of these methods. Another approach the LemkeHouson algorithm for two-player games also discussed here.

Rosenthal (1974) have obtained correlated equilibria for 2-player games. These are more general strategies than Nash equilibrium known as correlated equilibrium. There can be a player who prefers correlated equilibria on Nash equilibrium. If this so, then correlated equilibria is a convenient solution. If the game is the best response then the correlated equilibria are not the right solution. It is good for the competitive games.

Hu and Wellman (2003) have computed Nash equilibrium for the general-sum stochastic game. They proposed a method for a multiagent Q-learning. The method Nash-Q

Table 2 Payo table ofbattle ofsexes (Adapted from Shah etal. (2012))

Husband Wife

Football Music

Footbal (3,1) (0,0) Music (0,0) (1,3)

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generalizes Q-learning of single-agent to the multiagent environment. It updates its Q-function by assuming Nash equilibrium actions as a choice of agents. It is shown that Nash Q provides efficiency to get equilibrium on single-agent Q-learning. This is an offline learning process. The online version of this learning process is also implemented.

Maeda (2003) have considered games that have fuzzy payos. They rst characterize equilibrium strategies as Nash equilibrium strategies. Then they examine characteristics of game values of fuzzy matrix games. Finally, they demonstrated this approach via numerical example.

Athey (2001) have studied games known as games of incomplete information. They proposed a restriction called single crossing condition (SCC) for these games. The Pure Strategy Nash equilibrium with a nite set of actions exists if SCC is satised. In these games, players have private information of their own. The results of this study show nondecreasing Pure Strategy Nash equilibrium. The proposed approach is constructive. So that the equilibria can be calculated for nite action games easily.

Pareto optimality

Pareto optimality introduced by Vilfredo Pareto (Yeung 2006). In Pareto optimal game, there exists a strategy that increases players gain without damaging others. For example, when Economy is competitive perfectly then it is Pareto optimal. This is because no changes in the Economy can make better the gain of one person and can make worse the gain of another person at the same time.

Feldman (1973) has discussed Pareto Optimality in bilateral barter. The proved the constraints under which trade moves go on to pairwise optimal allocation. Then this paper discussed some general conditions by which these allocations are Pareto optimal.

Kacem et al. (2002) have solved the exible job-shop scheduling problem.by using hybrid Pareto approach. Their proposed approach combines Fuzzy logic and evolutionary algorithms. This combination minimizes machine workloads and completion time.

Guesnerie (1975) have discussed insights of non-convex economics. The paper characterizes Pareto-optimal states. Then analyze how to achieve them in distributed economy. The focus of this paper mainly concerns with conditions needed for optimality, marginal cost pricing rules, and decentralized non-convex economy.

Shapley values

There is a Shapley value another solution concept used in cooperative game theory (Shapely 1953). It allocates a distribution to all players in a game. The distribution is unique and the game value depends on some desirable abstract characteristics. In simple words, Shapley value assigns credit among a group of cooperating players. For example, there are three red, blue and green players. The red player cooperates more than blue and green players. The goal is to form a pair and then assign credits to them. Each pair must have a red player as it cooperates more than others. So there can be two possible pairs. The two pairs are:

1. Red player, blue player2. Red player, green player.

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The red player cooperates more, so it will get more prot than player blue in the rst pair. Similarly, it will get more prot than a green player in the second pair.

Littlechild and Owen (1973) discussed the problem of computing Shapley value for large games. They considered the work of Broker and Thompson of about aircraft landing charges on the airport. This paper presents an expression that can be calculated when the cost function is a characteristics function. The costs of the biggest player in any subset of players is equal to the cost of that subset.

Gul (1989) has worked on the bargaining problem in a transferable utility economy. A framework is established by which the two approaches, cooperative and noncooperative, are compared. The stationary subgame perfect Nash equilibrium is used and with small time intervals, the gain is the Shapley value for the agent.

Prez-Castrillo and Wettstein (2001) have proposed a mechanism to analyze how cooperation produces surplus. It is a two-phased play. The rst phase is of bidding that gives the winner of the game. In the second phase the winner is rejected then the game is again played without that winner. This paper describes that the payo of the game coexists with Shapley value.

Decision theory

Parsons and Wooldridge (2002) have discussed both game and decision theories. As game theory studies agents interaction, it is closed relative to decision theory. Decision theory seeks to get the most favorable choice. That can maximize utilities of decision makers. Whereas the game theory also studies self-interested agents. It takes agents as greedy players want to maximize their own gain. This paper reviewed existing literature. Then it revealed issues related to autonomous agents and multi-agent system.

Hart etal. (1994) have worked on the two-person zero-sum game. They obtained game value and derived utility simultaneously by using decision theory. They found the gap between the axioms and presumption about expected utility maximization. Axioms characterize expected utility maximization, considering risk, in the individual decision. The presumption is that expected utility maximizers evaluate the game by their value. This study does not ll this gap completely. Because rationality involves playing maximin strategies is not proved.

Game theory incomputer science

Roughgarden (2010) have described Algorithmic Game Theory (AGT), a game theory applications in computer science. This paper explores current research formats in AGT. The research theme is dierent here than classical game theory. AGT receives the computational difficulty as a coupling requirement which makes it unique.

Wooldridge (2012) have explored the feasibility of game theory applications in computer science. They discussed issues related to the application of game theoretic models. They revealed the incorrect use of game theory model. They also mentioned that more research is needed in this area.

Ahmad and Luo (2006) have proposed an algorithm for video coding. It considers optimization of rate control. In this two-level algorithm, the rst level is about the target bits allocation. In the second level, each MB computes to share bits fairly. So that its quantization scale can be optimized.

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Games insocial systems

In this section, we will discuss game theory applications in social groups and others. In social groups, people interact and communicate each other. To model behaviors in such communication, game theory has been used.

Chen and Liu (2012) have modeled human behavior in social networks by using game theory. This is the study of the impact of social networks in our daily life. This generalized approach can be used for several social networks. The efficiency and fairness between users are main considerations of the model design.

Hand (1986) has discussed social conicts and social dominance. The social dominance based on Leverage is considered here. There are personals having greater resources and personals having fewer resources as well. The paper describes that game theory can be used to make less dominant individuals equal or greater to others.

Markov games

Altman (1994) have used Markov games to control the ow of arriving packets. These are the collection of normal-form games that agents play repeatedly. These games together with a value iteration algorithm are used for single controller. The controller design policies to control the ow. Markov games is another name of stochastic games. This study reveals the existence of the stationary optimal policy.

Ghosh and Goswami (2008) have studied semi-Markov game. They rst transformed the model into the completely observed semi-Markov game. Then they worked and obtained saddle-point. They showed the existence of saddle-point but with some conditions.

Laraki etal. (2013) have discussed stochastic games, subgame perfect and Borel sets. It describes conditions for the existence of game value. With these conditions the player 2 gets an optimal strategy for subgame perfect. The conditions described that payo is a bounded function f. The function f is measurable and is lower semi-continuous.

Deshmukh and Winston (1978) have developed zero-sum model for products price setting in two rms. The model is based on some assumptions. That is the current price of product and market positions inuenced future market positions. This provides a way to get balance benets gained from price variations.

Sirbu (2014) has studied zero-sum games. The paper discussed stochastic dierential game restricted to elementary strategies. The result shows the existence of value in a game with these strategies.

Pham and Zhang (2014) have studied 2-player zero-sum weak formulation game. The game discussed is Stochastic and Dierential game. The game value is obtained by visocsity solution. The paper showed the value of the game as a random process.

Hernandez-Hernandez etal. (2015) have studied Stochastic Dierential Equation. The game is between controller called minimizer and stopper called maximizer. The controller selects a nite-variation process. And the stopper selects time at which the game will stop. The study described that the obtained optimal strategies are not unique.

Oliu-Barton (2014) has worked on Finite Stochastic game. This is a zero-sum game. The paper proves the presence of value in the game. The aim of the study is to provide asymptotic behavior of strategies.

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Hamadne and Wang (2009) have studied Backward Stochastic Dierential Equations. These equations have terms. Their resulted solution is also a stochastic or random process. The paper presents a remarkable solution and showed the value in the game.

Shmaya (2006) have studied an interesting game with one informed player. It is a two-player zero-sum game with stochastic signals. The value of the game is taken as a function of player ones information structure. The properties of this function, examined, shows that every player has a positive value of information in zero-sum game.

Nonzerosum game models

In non-zero-sum games, there exists a universally agreed solution. It means there is no single optimal solution as zero-sum games have. These games model cooperation instead of conicts. There can be a win-win solution of game where everyone is a winner. The players can play a game while cooperating each other to achieve a common goal.

Sullivan and Purushotham (2011) have discussed a high-level summit on non-communicable disease (NCD). The summit held in New York on September 2011 in which they discussed cancer policies. The summit recognized cancer a rst high-level disease. This paper critically examined these policies. It gives an alternative solution based on a nonzero-sum game model for international cancer policy.

Bensoussan et al. (2014) have worked on the non-zero-sum stochastic dierential game. They modeled performance of two insurance companies. Each company is greedy to maximize its own utility. The surplus process modeled by a continuous-time Markov chain and an independent market-index process. The game solved by a dynamic programming principle. It is also mentioned that the presented game can be extended to several directions.

Carlson and Wilson (2004) have considered failure in the management of U.S. national forest. At rst, this seems a pure conict between US National Forest Service and Environmentalists. But in this paper, a non-zero-sum game theoretical model is developed. It examines the eects of these changes on outcomes. It is analyzed that some changes do not aect outcomes and some have potential impact.

Shenoy and Yu (1981) have studied partial conict games. This study examines the reciprocative strategy to induce cooperation. Reciprocative behavior is dened as Non-Zero-sum games. It describes conditions for cooperative behavior to give an optimal response to reciprocative behavior. The feasibility of playing reciprocative strategy is also determined. Finally, conditions are given for reciprocative strategy that results to Nash equilibrium.

Mussa (2002) have studied two monetary units, euro, and dollar. This article argues that there is a non-zero-sum game between both units. It denes euro benecial for both the euro area itself and rest of the world. Euro eects worlds economy indirectly. It is described that euro and the dollar are co-equal monetary standards. And is benecial to the United States, euro area itself and rest of the world.

Semsar-Kazerooni and Khorasani (2009) have studied multi-agent system that considers cooperative game theory. The common goal of the multi-agent team is to have consensus. Consensus can be accomplished over a common value for the agents output. This paper is a series of work. In this paper, a previously introduced strategy is used called semi-decentralized optimal control strategy.

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Khosravifar etal. (2013) have used an agent-based game theoretic model to analyze web services. There is a distributed environment in which agent cooperates each other. The performance of agents is analyzed by using non-zero-sum model. The decision-making process is also analyzed.

Radzik (1991) have obtained pure-strategy and Nash equilibrium for 2-player nonzero-sum games. The payo functions are upper semicontinuous. Agents are not allowed to interact each other in the model considers here. The optimality criterion dominant is the NE vector. This vector computes optimal actions of all players considering their payo function. The paper emphasizes solutions in pure strategies.

Radzik (1993) have computed Nash equilibria for discontinuous two-person non-zero-sum games. This study examines two classes of these games on the unit square. Here the payo function of the rst player is convex or concave in the rst variable. This supposition combined with bounded payo function entail the presence of Nash equilibria.

Games innetworks

The networks provide an excellent way of communication as well as support for distributed environments. The Game theory models have their obvious applications in network-based systems. The following papers use game theory to get optimal strategies for network problems.

Transport networks

Bell et al. (2014) have proposed a game theoretic approach for modeling degradable transport networks. By this approach, hyperpaths are generated between population centers and depot locations. They used a case study in the province of China to facilitate the proposal. Optimal hyperpaths are dened by using mixed strategy Nash equilibrium. Which give ultimate depot locations. These depot locations are found by using two forms of drop heuristic. These heuristics gives optimal solution except in one case. That is when the most appropriate location for only one rescue center is obtained.

Alpcan and Buchegger (2011) have studied vehicular networks. They examine security of network for the improvement of transportation. It is to provide optimal strategies to defend malicious threats. Three types of security games are studied here. When players knows the payo matrices the game is a zero-sum. When they know approximate payos the game is a fuzzy game. When players do not know each others payos, strategies can be improved via ctitious play.

Network security andreliability

Perea and Puerto (2013) have used game theory approach in network security. The game is between the network operator and attacker. The operator establishes network to achieve some goals. While the attacker wants to place damages in the network. The optimal strategy for the operator is building a network. The optimal strategy for attacker is nding edges to be attacked. This paper revealed dynamic aspects of the game.

Bell (2003) has proposed a novel method to identify failure nodes. It is a two-player game between a router and virtual network tester. Router has to nd a least-cost path, whereas network tester wants to increase trip-cost. The link in use are optimal for router and failure links are optimal for network tester. Network tester fails link to increase

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trip-cost. So the given maximin method is to identify those links that threaten to network.

Kashyap etal. (2004) have modeled multiple-input/output fading channel communication problem as a Zero-sum game. The players, maximizer and minimizer, have mutual information. On both maximizer and minimizer there is total power constraint. They obtained saddle-point of the game. It is shown that minimizer has no need of channel input knowledge.

Wei etal. (2012) have applied game theoretic approach for a non-correlated jamming problem. In this problem jammer has a lack of information about actually transmitted signals. There is a Zero-sum game between transceiver pair and jammer in the parallel fading channel. This paper explored CSI and solved problems related to it. The study nds equilibrium based on pure strategy. The game model adopts frequency hopping to defend against jam threats.

Chen etal. (2013) have used the zero-sum game model to analyze the performance of system. The approach examines communication across cooperative and malicious relays. It also analyzes the impact of this communication. The malicious relays can jam the network and they intentionally interrupt the system. The Nash equilibrium is determined to get optimal signaling strategies for cooperative relays.

Venkitasubramaniam and Tong (2012) have studied network communication. They used zero-sum game theoretic approach to provide anonymity. Optimizing anonymity problem is a game between network designer and adversary. The model showed the presence of saddle-point. The approach obtained optimal strategies by using parallel Relay networks. It explores throughput tradeos in large networks.

Wang and Georgios (2008) have considered Jammer and Relay problem. They modeled the problem between them as zero-sum mutual information game. By assuming source and destination being unaware optimal strategies are derived for both players. In non-fading scenario Linear Relay (LR) and Linear Jammer (LJ) are optimal strategies. In fading scenario, J cannot distinguish between Jamming and source signal. So the best strategy is to jam with Gaussian noise only. Here R forward with full power when jam link is worst. They derived optimal parameters on the basis of exact Nash equilibrium.

Zhao etal. (2008) have studied Wireless Mesh Networks. They used game theoretic approach for increasing performance of MAC protocols. This is an iterative game having two steps. In the rst step current state of the game is determined on each node. In a second step, the equilibrium strategy of the node is adjusted to the determined state of the game. The process is repeated till the desired performance is achieved. Finally, results are validated via simulation.

Larsson etal. (2009) have studied signal processing and communications in a game theoretic way. They demonstrated basic concepts of conicting and cooperative game theory through three examples of interference channel model. These are SISO IFC, MISO IFC, and MIMO IFC. For conicting case the study is limited to Nash equilibrium and price of anarchy (PoA). The Price of anarchy gives cost measures that system paid to operate without cooperation.

Nguyen et al. (2013) have used game theory to integrate distributed agent-based functions. They proposed an agent-based conceptual strategy. Which resolves the conicting interests between product agents and network agents. The method is based on

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cooperative game theory that integrates and solves conicting interests. Finally, the approach is veried by simulation with two case studies. First is like micro grid example and the second is the more complex case.

Quer etal. (2013) have used game theoretic approach to study inter-network cooperation. The scenario is about two ad hoc wireless networks. Both cooperates together to gain some benets. Statistical correlation between local parameters and performance is computed by Bayesian networks method. Both networks share their nodes to achieve cooperation. Game theory is used in nodes selection process. The system level simulator is used to conrm results. Results showed that increase in performance can be achieved by accurate selection of nodes.

Spyridopoulos (2013) have modeled problem of cyber-attacks. For that, they used Zero-sum one-shot game theoretic model. Single-shot games are opposed to repeated games. These models can be used when cooperation cannot be possible among players. The study explored adjustments and ideal techniques for both assailant and keeper. The study revealed a solitary ideal method for the keeper. The ns2 network simulator is used for the simulation of the model.

Khouzani etal. (2012) have studied software-based operations against malware attackers. Malware has to maximize the damage. And the network has to take robust defensive strategies against attacks. This makes the game a Zero-sum game. Simple robust defensive strategies are shown via dynamic game formulation. Finally, performance is evaluated through simulation.

Discretetime/continuoustime

Ye et al. (2013) have proposed a discrete-time Markov chain Parrondos model. They analyzed model theoretically and veried via simulation. One can realize rationality and adaptability from a macro level. They showed that agitating eect of rewiring is eective than the zero-sum game.

Al-Tamimi et al. (2007) have proposed an algorithm for the solution of a zero-sum game. The algorithm provides a solution for Riccati equation. They discussed two schemes of programming. One is heuristic dynamic and second is dual. These schemes used for the solution of the value function and game costate.

Liu etal. (2013) have proposed an algorithm based on nding approximate optimal controller. It is based on the class of discrete-time constrained systems. This iterative adaptive dynamic programming algorithm provides a solution for near-optimal control problem. The control scheme has three neural networks. These networks are taken as parametric structures to assist the proposed algorithm. This is described by two examples that showed the practicality and concurrence of the algorithm.

Wu and Luo (2013) have modeled H~ state feedback control problem as the two-person Zero-sum game. An algorithm is proposed for solving algebra rectaii equation. They developed two versions, offline and online. An offline version is a model-based approach. The online version is a model-free approach but partially. These approaches are validated through simulation.

Abu-Khalaf etal. (2008) have used policy iteration approach together with neural networks. They provide practical solution method for suboptimal control of constrained input systems. They modeled the problem as a continuous-time zero-sum game. The

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study showed new results and creates a least-squares-based algorithm for a practical solution. The proposed algorithm is applied to the RTAC nonlinear benchmark problem.

Resource allocation

Zhou etal. (2011) have modeled energy allocation problem in two phased training-based transmission. The model is based on the zero-sum game between two phases. The two phases are training phase and transmission phase. This study is about optimal energy allocation between these two phases. The closed-form solutions are derived from jammers view. The study proves the presence of NE for xed training length. Finally, it discusses channel state information.

Tan etal. (2011) have discussed radio networks. They used game theory approach for fair sub-carriers allocation and power allocation. The sub-carrier allocation and power allocation are based on colonel blotto game. The secondary users allocate budget wisely to transmit power to win sub-carriers. Power allocation and budget allocation are strategies used for fair sharing among secondary users. This paper proposed algorithms and conditions for the presence of unique NE. Finally, the results are validated through simulation.

Belmega etal. (2009) have discussed power allocation in fast fading multiple access channels. In these channels transmitters and receiver have many antennas. The study gives unique Nash equilibrium. It also gives best power allocation policies. The paper discussed two dierent games. In the rst game, the users can adapt their temporal power allocation to their decoding rank at the receiver. The other is to optimize their spatial power allocation between their transmit antennas. Finally, results are shown via simulation.

In the next section, we will classify games in tabular structures. Then will discuss some open problems.

Discussion

We discussed game theory and its applications in dierent domains by exploring dierent papers. We described how game theory models strategic and complex interactions of self-interested agents. We also proposed a general taxonomy of games, based on the types of game representation. The three types of game representation are Normal-form, Extensive-form, and Beyond Normal/Extensive form. Then we classify games according to these representation types.

We have seen dierent games while reviewing literature. Such as Markov games, Zero-sum game, Stochastic game, Bayesian games etc. These are actually dierent classes of games having dierent properties. We summarized dierent games, by their dierent types. See Table3. The legend used in the table is summarized in Table4.

We also summarized games discussed in dierent papers according to representation forms. The representation forms are Normal, Extensive and Beyond normal/extensive form. See Table5.

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Open problems

We have noted that while researchers applied game theory in dierent domains, there is still need to further exploit game theory in the modeling of complex systems research. In computer science, there is also a need to apply game theory in the domain of resource allocation algorithms such as in clouds, Internet of Things, Cyber physical systems, and others. Cake Cutting and Colonel Blotto arequite possibly good game-theoretic resource allocation modelsand can thus be used in such domains. However, they have not previously been used much in these areas. Furthermore,fair allocation is still a complex task in distributed systems. With the advent of mobile, pervasive computing, and cloud-based systems, practical distributed computing requires the resolution of such dilemmas on a regular basis. In other words, there is a growing need to use game theory for practical applications in the technological domains rather than restrict it to purely theoretical applications and those too, limited to very specic and niche areas of research.

Another open area for further research is in the development of taxonomies for specic game theoretic areas.We haveproposed a general taxonomy of games. We havealso mentioned few previously dened taxonomies. However, there is a need for the developmentof more taxonomies of games. These include the development of taxonomies and review of papers and gamessuch as in the domain ofBayesian games, Congestion games among others.

Conclusions andfuture work

This paper presents a review of game theory models from the agent-based modeling perspective. We havediscussed dierent classes ofgames such as Zero-sum, Perfect information, Bayesian, Congestion etc. We have also explored theimportance and nature of game theoryby means of a novel taxonomy.The presented taxonomy of game classes has beenbased on types of game representation. In the review, game theory applications in dierent elds has also been discussed. We believe that this review will help multidiscplinary researchers in expanding their knowledge about the state-of-the-art in game theory. In particular, it will help researchers to look at game-theoreticliterature analyzed from the perspective of agents and complexity.

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NZINoNoNoNo

Extended RPSBahel and Haller (2013)NZINoNoNoNo

Mod gameFrey et al. (2013)BNZINoYesNoNo

Continuous RPSNeumann and Schuster

(2007)

NZINoNoNoNo

BZPNoYesNoNo

BNZPYesYesNoNo

Railway networkPerea and Puerto (2013)NZPNoNoNoNo

VANET security modelAlpcan and Buchegger

(2011)

BZPNoYesNoNo

BZINoYesNoNo

Table 3 This table lists games dened indierent publications

GamesReferencesFormsZerosumPerfectStochasticRepeatedBayesianCongestion

Venkitasubramaniam

and Tong (2012)

Cake cutting

Balls and binsEdmonds and Pruhs

(2006)

Matching pennies

3-player MPMcCabe et al. (2000)BZP/INoYesYesNo

Blotto games

Colonel BlottoRoberson (2006)BZINoNoNoYes

Discrete Colonel BlottoHart (2008)BZINoYesNoNo

Princess Monster

PM on circleWilson (1972)BZINoYesNoNo

Poker

Kuhn PokerSouthey et al. (2009)EZINoNoNoNo

Networks

Flow controlAltman (1994)BZPYesYesNoNo

Network revenueGrauberger and Kimms

(2014)

Anonymous network-

ing

RPS

Three-morph matingSinervo and Lively

(1996)

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BZP/INoYesNoNo

Dynamic gameKhouzani et al. (2012)BZPYesYesNoNo

Parrondos model

Link A + game BYe et al. (2013)EZPYesYesNoNo

Transmission

E-D vs jammerKashyap et al. (2004)BZINoYesNoNo

Transmission securityChen et al. (2013)BZINoYesNoNo

Payo games

Average payoGhosh and Goswami

(2008)

BZIYesYesNoNo

Semicontinuous payoLaraki et al. (2013)BZIYesYesNoYes

Symmetric

Symmetric gameDuersch et al. (2012)NZINoNoNoNo

Mixed zero-sum

Mixed-strategySeo and Lee (2007)BZPNoYesNoNo

Mixed zero-sumHamadne and Wang

(2009)

BZIYesYesNoNo

BZIYesYesNoNo

Jammer-relayWang and Georgios

(2008)

Table 3 continued

GamesReferencesFormsZerosumPerfectStochasticRepeatedBayesianCongestion

Searching

AGTCS2-player searchZoroa et al. (2012)BZPNoYesNoNo

Investments

Insurance gamesBensoussan et al. (2014)BNZPYesYesNoNo

Duopoly

Duopoly gameDeshmukh and Winston

(1978)

Others

Web servicesKhosravifar et al. (2013)BNZINoNoNoYes

Farooqui and Niazi Complex Adapt Syst Model (2016) 4:13

Table 4 Legends used inTable3

Legends Name

N Normal-formE Extensive-formB Beyond normal/extensive Z Zero-sumNZ Non-zero-sumP PerfectI Imperfect

Table 5 Games indierent forms ofrepresentation

S. no Ref Games Normal Extensive Beyond N/E

1 Three-morph mating Sinervo and Lively (1996) Yes No No2 Extended RPS Bahel and Haller (2013) Yes No No3 Mod game Frey et al. (2013) No No Yes4 Continuous RPS Neumann and Schuster (2007) Yes No No5 Balls and bins Edmonds and Pruhs (2006) No No Yes 6 3-player MP McCabe (2000) No No Yes7 Colonel Blotto Roberson (2006) No No Yes8 Discrete colonel Blotto Hart (2008) No No Yes9 PM on circle Wilson (1972) No No Yes10 Kuhn Poker Southey et al. (2009) No Yes No11 Flow control Altman (1994) No No Yes12 Network revenue Grauberger and Kimms (2014) No No Yes13 Railway network Perea and Puerto (2013) Yes No No14 VANET security model Alpcan and Buchegger (2011) No No Yes15 Anonymous networking Venkitasubramaniam and Tong(2012)

Authors contributions

AF and MN both contributed equally in the paper. Both authors read and approved the nal manuscript.

Author details

1 Software Engineering Department, Bahria University, Islamabad, Pakistan. 2 Computer Science Department, COMSATS Institute of IT, Islamabad, Pakistan.

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No No Yes

16 Jammer-relay Wang and Georgios (2008) No No Yes17 Network-malware dynamic game Khouzani et al. (2012) No No Yes18 Link A + game B Ye et al. (2013) No Yes No

19 E-D vs jammer Kashyap et al. (2004) No No Yes20 Transmission security Chen et al. (2013) No No Yes21 Average payo Ghosh and Goswami (2008) No No Yes22 Semicontinuous payo Laraki et al. (2013) No No Yes23 Symmetric game Duersch et al. (2012) Yes No No24 Mixed-strategy Seo and Lee (2007) No No Yes25 Mixed zero-sum Hamadne and Wang (2009) No No Yes26 AGTCS2-player search Zoroa et al. (2012) No No Yes27 Insurance games Bensoussan et al. (2014) No No Yes28 Duopoly game Deshmukh and Wayne (1978) No No Yes29 Web services Khosravifar et al. (2013) No No Yes

Farooqui and Niazi Complex Adapt Syst Model (2016) 4:13

Competing interests

The authors declare that they have no competing interests.

Received: 22 December 2015 Accepted: 6 July 2016

References

Abu-Khalaf M, Lewis FL, Huang J (2008) Neurodynamic programming and zero-sum games for constrained control systems. IEEE Trans Neural Netw 19(7):12431252

Ahmad I, Luo J (2006) On using game theory to optimize the rate control in video coding. IEEE Trans Circuits Syst Video

Technol 16(2):209219

Al-Tamimi A, Abu-Khalaf M, Lewis FL (2007) Adaptive critic designs for discrete-time zero-sum games with application to control. IEEE Trans Syst Man Cybern Part B Cybern 37(1):240247Al-Tamimi A, Lewis FL, Abu-Khalaf M (2007) Model-free q-learning designs for linear discrete-time zero-sum games with application to h-innity control. Automatica 43(3):473481Alpcan T, Buchegger S (2011) Security games for vehicular networks. IEEE Trans Mobile Comput 10(2):280290Altman E (1994) Flow control using the theory of zero sum Markov games. IEEE Trans Autom Control 39(4):814818 Altman E, Avrachenkov K, Bonneau N, Debbah M, El-Azouzi R, Menasche DS (2008) Constrained cost-coupled stochastic games with independent state processes. Oper Res Lett 36:160164Athey S (2001) Single crossing properties and the existence of pure strategy equilibria in games of incomplete information. Econometrica 69(4):861889Aumann RF, Maschler M, Stearns RE (1995) Repeated games with incomplete information. MIT press, CambridgeBahel E, Haller H (2013) Cycles with undistinguished actions and extended rock-paper-scissors games. Econ Lett

120(3):588591

Batt C (1999) Rock, paper, scissors. Food Microbiol 16(1):1Bell MGF (2003) The use of game theory to measure the vulnerability of stochastic networks. IEEE Trans Reliab

52(1):6368

Bell MGH, Fonzone A, Polyzoni C (2014) Depot location in degradable transport networks. Transp Res Part B Methodol

66:148161

Belmega EV, Lasaulce S, Debbah M (2009) Power allocation games for mimo multiple access channels with coordination.

IEEE Trans Wirel Commun 8(6):31823192

Bensoussan A, Siu CC, Yam SCP, Yang H (2014) A class of non-zero-sum stochastic dierential investment and reinsurance games. Automatica 50(8):20252037Bettiol P, Cardaliaguet P, Quincampoix M (2006) Zero-sum state constrained dierential games: existence of value for

Bolza problem. Int J Game Theory 34(4):495527

Bge W, Eisele T (1979) On solutions of Bayesian games. Int J Game Theory 8(4):193215Bonabeau E (2002) Agent-based modeling: methods and techniques for simulating human systems. Proc Natl Acad Sci

USA 99(suppl 3):72807287

Bopardikar SD, Borri A, Hespanha JP, Prandini M, Di Benedetto MD (2013) Randomized sampling for large zero-sum games. Automatica 49(5):11841194Carlson LJ, Wilson PI (2004) Beyond zero-sum: game theory and national forest management. Soc Sci J 41(4):637650 Carmichael F (2005) A guide to game theory. Pearson Education, New YorkChang HS, Marcus SI (2003) Two-person zero-sum markov games: receding horizon approach. IEEE Trans Autom Control

48(11):19511961

Chen YW, Larbani M (2006) Two-person zero-sum game approach for fuzzy multiple attribute decision making problems.

Fuzzy Sets Syst 157(1):3451

Chen Y, Liu KJ (2012) Understanding microeconomic behaviors in social networking: an engineering view. IEEE Signal

Process Mag 29(2):5364

Chen MH, Lin SC, Hong YW, Zhou X (2013) On cooperative and malicious behaviors in multirelay fading channels. IEEE

Trans Inf Forensics Secur 8(7):11261139

Daskalakis C, Deckelbaum A, Kim A (2015) Near-optimal no-regret algorithms for zero-sum games. Games Econ Behav

92:327348

Deshmukh SD, Winston W (1978) A zero-sum stochastic game model of duopoly. Int J Game Theory 7(1):1930 Dixit AK, Nalebu BJ (1993) Thinking strategically: the competitive edge in business, politics, and everyday life. WW

Norton & Company, New York City

Dixit AK, Skeath S (1999) Games of strategy. Norton, New YorkDuersch P, Oechssler J, Schipper BC (2012) Pure strategy equilibria in symmetric two-player zero-sum games. Int J Game

Theory 41(3):553564

Edmonds J, Pruhs K (2006) Balanced allocations of cake. In: Null, IEEE, New York, p 623634Epstein JM (2008) Why model? J Artif Soc Soc Simul 11(4):12Feldman AM (1973) Bilateral trading processes, pairwise optimality, and pareto optimality. Rev Econ Stud 40(4):463473 Frey S, Goldstone RL, Szolnoki A (2013) Cyclic game dynamics driven by iterated reasoning. PloS one 8(2):e56416 Gawlitza TM, Seidl H, Adj A, Gaubert S, Goubault (2012) Abstract interpretation meets convex optimization. Journal

Symb Comput 47(12):14161446

Geckil IK, Anderson PL (2009) Applied game theory and strategic behavior. CRC Press, Boca RatonGensbittel F (2014) Extensions of the cav (u) theorem for repeated games with incomplete information on one side.

Math Oper Res 40(1):80104

Page 28 of 31

Farooqui and Niazi Complex Adapt Syst Model (2016) 4:13

Gharesifard B, Cortes J (2013) Distributed convergence to Nash equilibria in two-network zero-sum games. Automatica

49(6):16831692

Ghosh MK, Goswami A (2008) Partially observed semi-Markov zero-sum games with average payo. J Math Anal Appl

345(1):2639

Grauberger W, Kimms A (2014) Computing approximate nash equilibria in general network revenue management games. Eur J Oper Res 237(3):10081020Guesnerie R (1975) Pareto optimality in non-convex economies. Econom J Econom Soc 129Gul F (1989) Bargaining foundations of shapley value. Econom J Econom Soc 8195Hamadne S, Wang H (2009) BSDEs with two RCLL reecting obstacles driven by Brownian motion and poisson measure and a related mixed zero-sum game. Stoch Process Appl 119(9):28812912Hand JL (1986) Resolution of social conicts: dominance, egalitarianism, spheres of dominance, and game theory. Q Rev

Biol 201220

Hart S (2008) Discrete colonel blotto and general lotto games. Int J Game Theory 36(34):441460Hart S, Modica S, Schmeidler D (1994) A neo2 Bayesian foundation of the maxmin value for two-person zero-sum games.

Int J Game Theory 23(4):347358

Hellman Z (2013) Weakly rational expectations. J Math Econ 49(6):496500Hernandez-Hernandez D, Simon RS, Zervos M et al (2015) A zero-sum game between a singular stochastic controller and a discretionary stopper. Ann Appl Probab 25(1):4680Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeHu J, Wellman MP (2003) Nash q-learning for general-sum stochastic games. J Mach Learn Res 4:10391069Hutton W (1996) The state we are in London: VintageIsaaks R (1952) A mathematical theory with applications to warfare and pursuit, control, and optimization. Wiley, New

York

Kacem I, Hammadi S, Borne P (2002) Pareto-optimality approach for exible job-shop scheduling problems: hybridization of evolutionary algorithms and fuzzy logic. Math Computers Simul 60(3):245276Kashyap A, Basar T, Srikant R (2004) Correlated jamming on MIMO Gaussian fading channels. IEEE Trans Inf Theory

50(9):21192123

Khosravifar B, Bentahar J, Mizouni R, Otrok H, Mahsa Alishahi, Philippe Thiran (2013) Agent-based game-theoretic model for collaborative web services: decision making analysis. Expert Syst Appl 40(8):32073219Khouzani MHR, Sarkar S, Altman E (2012) Saddle-point strategies in malware attack. IEEE J Sel Areas Commun 30(1):3143 Kilgour DM, Fraser NM (1988) A taxonomy of all ordinal 2 2 games. Theory Decis 24(2):99117Lansing JS (2003) Complex adaptive systems. Annu Rev Anthropol 183204Laraki R, Maitra AP, Sudderth WD (2013) Two-person zero-sum stochastic games with semicontinuous payo. Dyn Games

Appl 3(2):162171

Larsson EG, Jorswieck EA, Lindblom J, Mochaourab R et al (2009) Game theory and the at-fading gaussian interference channel. IEEE Signal Process Mag 26(5):1827Li D, Cruz JB (2009) Information, decision-making and deception in games. Decis Support Syst 47(4):518527Littlechild SC, Owen G (1973) A simple expression for the shapley value in a special case. Manag Sci 20(3):370372Liu D, Li H, Wang D (2013) Neural-network-based zero-sum game for discrete-time nonlinear systems via iterative adaptive dynamic programming algorithm. Neurocomputing 110:92100Maeda T (2003) On characterization of equilibrium strategy of two-person zero-sum games with fuzzy payos. Fuzzy Sets

Syst 139(2):283296

Marlow J, Peart DR (2014) Experimental reversal of soil acidication in a deciduous forest: implications for seedling performance and changes in dominance of shade-tolerant species. For Ecol Manag 313:6368Mazalov V (2014) Mathematical game theory and applications. Wiley, New YorkMcCabe KA, Mukherji A, Runkle DE (2000) An experimental study of information and mixed-strategy play in the three-person matching-pennies game. Econ Theory 15(2):421462McDaniel RR, Driebe DJ (2001) Complexity science and health care management. Adv Health Care Manag 2(S11):37 Mndez-Naya L (1996) Zero-sum continuous games with no compact support. Int J Game Theory 25(1):93111Mertens JF, Neyman A (1981) Stochastic games. Int J Game Theory 10(2):5366Mertens JF, Zamir S (1971) The value of two-person zero-sum repeated games with lack of information on both sides. Int

J Game Theory 1(1):3964

Mitchell M (2009) Complexity: a guided tour. Oxford University Press, New YorkMorrow JD (1994) Game theory for political scientists. Princeton University Press, PrincetonMoulin H (1976) Extensions of two person zero sum games. J Math Anal Appl 55(2):490508Moulin H, Vial J-P (1978) Strategically zero-sum games: the class of games whose completely mixed equilibria cannot be improved upon. Int J Game Theory 7(34):201221Mussa M (2002) The euro versus the dollar: not a zero sum game. J Policy Model 24(4):361372Nash JF (1950) The bargaining problem. Econometrica 18(2):155162Nash J (1951) Non-cooperative games. Ann Math 286295Neumann G, Schuster S (2007) Continuous model for the rock-scissors-paper game between bacteriocin producing bacteria. J Math Biol 54(6):815846Nguyen PH, Kling WL, Ribeiro PF (2013) A game theory strategy to integrate distributed agent-based functions in smart grids. IEEE Trans Smart Grid 4(1):568576Niazi M, Hussain A (2011) Agent-based computing from multi-agent systems to agent-based models: a visual survey.

Scientometrics 89(2):479499

Niazi M, Hussain A et al (2011) A novel agent-based simulation framework for sensing in complex adaptive environments. IEEE Sens J 11(2):404412Niazi MA, Hussain A (2012) Cognitive agent-based computing-I: a unied framework for modeling complex adaptive systems using agent-based & complex network-based methods. Springer, Dordecht

Page 29 of 31

Farooqui and Niazi Complex Adapt Syst Model (2016) 4:13

Okamura K, Kanaoka T, Okada T, Tomita S (1984) Learning behavior of variable-structure stochastic automata in a three-person zero-sum game. IEEE Trans Syst Man Cybern 6:924932

Oliu-Barton M (2014) The asymptotic value in nite stochastic games. Math Oper Res 39(3):712721

Parsons S, Wooldridge M (2002) Game theory and decision theory in multi-agent systems. Auton Agents Multiagent Syst

5(3):243254

Perea F, Puerto J (2013) Revisiting a game theoretic framework for the robust railway network design against intentional attacks. Eur J Oper Res 226(2):286292Prez-Castrillo D, Wettstein D (2001) Bidding for the surplus: a non-cooperative approach to the shapley value. J Econ

Theory 100(2):274294

Pham T, Zhang J (2014) Two person zero-sum game in weak formulation and path dependent BellmanIsaacs equation.

SIAM J Control Optim 52(4):20902121

Ponssard J-P (1975) A note on the lp formulation of zero-sum sequential games with incomplete information. Int J Game

Theory 4(1):15

Ponssard J-P (1976) On the subject of non optimal play in zero sum extensive games: the trap phenomenon. Int J Game

Theory 5(23):107115

Ponssard JP, Sorin S (1980) Some results on zero-sum games with incomplete information: the dependent case. Int J

Game Theory 9(4):233245

Porter R, Nudelman E, Shoham Y (2008) Simple search methods for nding a Nash equilibrium. Games Econ Behav

63(2):642662

Procaccia AD (2013) Cake cutting: not just childs play. Commun ACM 56(7):7887Qing-Lai WEI, Zhang HG, Li-Li CUI (2009) Data-based optimal control for discrete-time zero-sum games of 2-d systems using adaptive critic designs. Acta Autom Sin 35(6):682692Quer G, Librino F, Canzian L, Badia L, Zorzi M (2013) Inter-network cooperation exploiting game theory and Bayesian networks. IEEE Trans Commun 61(10):43104321Radzik T (1991) Pure-strategy ~-Nash equilibrium in two-person non-zero-sum games. Games Econ Behav 3(3):356367 Radzik T (1993) Nash equilibria of discontinuous non-zero-sum two-person games. Int J Game Theory 21(4):429437 Rapoport A, Guyer M (1978) A taxonomy of 2x2 games. Gen Syst 23:125136Roberson B (2006) The colonel blotto game. Econ Theory 29(1):124Rosenthal RW (1973) A class of games possessing pure-strategy Nash equilibria. Int J Game Theory 2(1):6567 Rosenthal RW (1974) Correlated equilibria in some classes of two-person games. Int J Game Theory 3(3):119128 Roughgarden T (2010) Algorithmic game theory. Commun ACM 53(7):7886Sauder DW, Geraniotis E (1994) Signal detection games with power constraints. IEEE Trans Inf Theory 40(3):795807 Semsar-Kazerooni E, Khorasani K (2009) Multi-agent team cooperation: a game theory approach. Automatica

45(10):22052213

Seo H, Lee D (2007) Temporal ltering of reward signals in the dorsal anterior cingulate cortex during a mixed-strategy game. J Neurosci 27(31):83668377Shah IA, Jan S, Khan I, Qamar S (2012) An overview of game theory and its applications in communication networks. Int J

Multidiscip Sci Eng 3:511

Shapley LS (1953) A value for n-person games. Contrib Theory Games 2:307317Shenoy PP, Yu PL (1981) Inducing cooperation by reciprocative strategy in non-zero-sum games. J Math Anal Appl

80(1):6777

Shmaya E (2006) The value of information structures in zero-sum games with lack of information on one side. Int J Game

Theory 34(2):155165

Shoham Y, Leyton-Brown K (2008) Multiagent systems: algorithmic, game-theoretic, and logical foundations. Cambridge

University Press, New York

Sinervo B, Lively CM (1996) The rock-paper-scissors game and the evolution of alternative male strategies. Nature

380(6571):240243

Singh VV, Hemachandra N (2014) A characterization of stationary Nash equilibria of constrained stochastic games with independent state processes. Oper Res Lett 42(1):4852Sirbu M (2014) On martingale problems with continuous-time mixing and values of zero-sum games without the Isaacs condition. SIAM J Control Optim 52(5):28772890Socolar JES (2006) Nonlinear dynamical systems. In: Complex systems science in biomedicine. Springer, New York, pp

115140

Sorin S (2011) Zero-sum repeated games: recent advances and new links with dierential games. Dyn Games Appl

1(1):172207

Southey F, Hoehn B, Holte RC (2009) Eective short-term opponent exploitation in simplied poker. Mach Learn

74(2):159189

Spyridopoulos T (2013) A game theoretic defence framework against DoS/DDoS cyber attacks. Comput Secur 38:3950 Stein ND, Ozdaglar A, Parrilo PA (2010) Structure of extreme correlated equilibria: a zero-sum example and its implications. arXiv preprint http://arxiv.org/abs/1002.0035

Web End =arXiv:1002.0035 Sullivan R, Purushotham AD (2011) Avoiding the zero sum game in global cancer policy: beyond 2011 un high level summit. Eur J Cancer 47(16):23752380Tan CK, Chuah TC, Tan SW (2011) Fair subcarrier and power allocation for multiuser orthogonal frequency-division multiple access cognitive radio networks using a colonel Blotto game. IET Commun 5(11):16071618Tucker AW (1959) Contributions to the theory of games, vol 4. Princeton University Press, Princeton Venkitasubramaniam P, Tong L (2012) A game-theoretic approach to anonymous networking. IEEE/ACM Trans Netw

20(3):892905

Von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton Wang J, Chen F (2013) Feedback saddle point solution of counterterror measures and economic growth game. Oper Res

Lett 41(6):706709

Wang T, Georgios GB (2008) Mutual information jammer-relay games. IEEE Trans Inf Forensics Secur 3(2):290303

Page 30 of 31

Farooqui and Niazi Complex Adapt Syst Model (2016) 4:13

Washburn AR (2003) Two-person zero-sum games. Springer, Berlin

Wei S, Kannan R, Chakravarthy V, Rangaswamy M (2012) Csi usage over parallel fading channels under jamming attacks: a game theory study. IEEE Trans Commun 60(4):11671175Wilson DJ (1972) Isaacs princess and monster game on the circle. J Optim Theory Appl 9(4):265288Winsberg E (2001) Simulations, models, and theories: complex physical systems and their representations. Philos Sci

68(3):S442S454. http://www.jstor.org/stable/3080964

Web End =http://www.jstor.org/stable/3080964

Wooldridge M (2009) An introduction to multiagent systems. Wiley, West SussexWooldridge M (2012) Does game theory work? IEEE Intell Syst 27(6):7680Wu HN, Luo B (2013) Simultaneous policy update algorithms for learning the solution of linear continuous-time hinn state feedback control. Inf Sci 222:472485Xu H, Mizukami K (1994) Linear-quadratic zero-sum dierential games for generalized state space systems. IEEE Trans

Autom Control 39(1):143147

Ye Y, Lu NG, Cen YW (2013) The multi-agent Parrondos model based on the network evolution. Phys A Stat Mech Appl

392(21):54145421

Yeung DWK, Petrosjan LA (2006) Cooperative stochastic dierential games. Springer Science & Business Media, BerlinVan Zandt T, Zhang K (2011) A theorem of the maximin and applications to Bayesian zero-sum games. Int J Game Theory

40(2):289308

Zhang X, Zhang H, Wang X, Luo Y (2011) A new iteration approach to solve a class of nite-horizon continuous-time nonaffine nonlinear zero-sum game. Int J Innov Comput Inf Control 7(2):597608Zhao L, Zhang J, Zhang H (2008) Using incompletely cooperative game theory in wireless mesh networks. IEEE Netw

22(1):3944

Zhou X, Niyato D, Hjrungnes A (2011) Optimizing training-based transmission against smart jamming. IEEE Trans Veh

Technol 60(6):26442655

Zoroa N, Fernndez-Sez MJ, Zoroa P (2012) Patrolling a perimeter. Eur J Oper Res 222(3):571582

Page 31 of 31