Elements of Game Theory - Part I: Foundations, acts and mechanisms
EElements of Game Theory
Part I: Foundations, acts and mechanisms.
Harris V. Georgiou (MSc, PhD) ∗ Department of Informatics and Telecommunications,National & Kapodistrian University of Athens, Greece.
Abstract
In this paper, a gentle introduction to Game Theory is presented in the form of basicconcepts and examples. Minimax and Nash’s theorem are introduced as the formaldefinitions for optimal strategies and equilibria in zero-sum and nonzero-sum games.Several elements of cooperative gaming, coalitions, voting ensembles, voting power andcollective efficiency are described in brief. Analytical (matrix) and extended (tree-graph) forms of game representation is illustrated as the basic tools for identifyingoptimal strategies and “solutions” in games of any kind. Next, a typology of fourstandard nonzero-sum games is investigated, analyzing the Nash equilibria and theoptimal strategies in each case. Signaling, stance and third-party intermediates aredescribed as very important properties when analyzing strategic moves, while credibil-ity and reputation is described as crucial factors when signaling promises or threats.Utility is introduced as a generalization of typical cost/gain functions and it is usedto explain the incentives of irrational players under the scope of “rational irrational-ity”. Finally, a brief reference is presented for several other more advanced concepts ofgaming, including emergence of cooperation, evolutionary stable strategies, two-levelgames, metagames, hypergames and the Harsanyi transformation.
Keywords:
Game Theory, Minimax theorem, Nash equilibrium, coalitional gaming,indices of power, voting ensembles, signaling, bluff, credibility, promises, threats, util-ity function, two-level games, hypergames, evolutionary stable strategies, Harsanyitransformation, metagames. G AME THEORY is a vast scientific and research area, based almost entirely onMathematics and some experimental methods, with applications that vary fromsimple board games to Evolutionary Psychology and Sociology-Biology in groupbehavior of humans and animals. Conflict situations are presented everywherein the real world, every day, for thousands of years - not only in human societiesbut also in animals. The seller and the buyer have to come up with a mutuallyacceptable price for the grocery. The employer and the employee have to bargainin order to reach a mutually satisfying value for the salary. A buyer in an auction ∗ Email: [email protected] — URL: http://xgeorgio.info a r X i v : . [ c s . G T ] J un The building blocks has to continuously estimate the cost/gain value of making (or not) the nexthigher bid for some object. The primary adversaries in a wolf pack have todecide when it is beneficial to fight over the leadership and when to stop beforethey are severely wounded. A swarm of fish has to collectively “decide” what isthe optimal number and distance of the piket members or “scouts” that serveas the early warning for the group, perhaps even self-sacrificing if required. Allthese cases are typical examples, simpler or more complex, of conflict situationsthat depend on bargaining, coordination and evolutionary optimization. GameTheory provides a unified framework with robust mathematical foundations forthe proper formulation and analysis of such systems. In principle, the mathematical theory of games and gaming was first developedas a model for situations of conflict.
Game Theory is the area of research thatprovides mathematical formulations and a proper framework for studying ad-versarial situations. Although E. Borel looked at similar problems in the 1920s,John Von Neumann and Oskar Morgenstern provided two breakthrough papers(1928, 1937) as a kick-start of the field. Since the early 1940’s, with the endof World War II and stepping into the era of the Cold War that followed, thework of Von Neumann and Morgenstern has provided a solid foundation for themost simple types of games, as well as analytical forms for their solutions, withmany applications to Economics, Operations Research and Logistics. However,there are several limitations that fail to explain various aspects of real-worldconflicts [25], especially when the human factor is a major factor. The applica-tion of game-theoretic formulations in designing experiments in Psychology andSociology is usually referred to as gaming [46, 6].
The term game is the mathematical formulation of adversarial situations, wheretwo or more players are involved in competitive or cooperative acts. The zero-sum games are able to model situations of conflict between two or more players,where one’s gain is the other’s loss and vice versa. Most military problems canbe modeled as some form of two-player zero-sum game. When the structure ofthe game and the rationale of the players is known to all, then the game is oneof complete information , while if some of these information is somehow hiddenor unknown to some players, it is one of incomplete information . Furthermore,if all players are fully informed about their opponents’ decisions, the game isone of perfect information . In contrast, if some of the information about theother players’ moves, the game is one of partial or imperfect information . Suchgames of both complete and perfect information are all board games, like Chess,Go and Checkers, and they are all zero-sum by nature.Von Neumann and Morgenstern [48] proved that there is at least one optimalplan of decisions or strategy for each player in all zero-sum games, as well asa solution to the game that comes naturally as a result of all players followingtheir optimal strategies. At the game’s solution, each player can guarantee thatthe maximum gain an opponent can gain is kept under a specific minimal limit,defined only by this player’s own strategy. This assertion was formulated as a The building blocks theorem called Minimax and in the simple case of two opposing players withonly two strategies each the Minimax solution of the game can be calculatedanalytically as a solution of a 2x2 set of linear equations, which determine thestable solution or saddle-point .The consequences of the Minimax theorem have been thoroughly studiedfor many years after its proof. As an example, it mathematically proves theassertion that all board games, including the most complex ones like Chess,have at least one solution, i.e., an optimal (pure) strategy for both players thatcan be analytically calculated, at least in theory [44, 46, 37]. Of course, inthe case of Chess the game space is so huge that it is still unfeasible todayto calculate this theoretically optimal strategy, even with the help of parallelprocessing in supercomputers. In contrast, Checkers is a much smaller (3x3)and simpler game, making it possible to create the complete game space in anytypical desktop computer and calculate the exact optimal strategy - in fact, itis the same strategy that every child soon learns by trial-and-error, playing ina way that always leads to a win or a draw (never loose).In general, if the chosen strategy of one player is known to its opponent,then an optimal counter-strategy is always available. Hence, in simultaneous games where the opposing moves are conducted at the same time, each playerwould normally try not to employ a deterministic way of choosing its strategyand conceal this choice until the very last moment. However, the Minimaxtheorem provides a mathematically solid way of nullifying any stochastic aspectin determining the opponent’s choice and, in essence, make its exact choiceirrelevant: no matter what the opponent does, the Minimax solution ensuresthe minimum losses to each player, given a specific game setup. In other words,it provides an analytic way to determine the best defensive strategy, insteadof a preference to offensive strategies. In some zero-sum games this leads toone stable outcome or equilibrium , where each player would have no incentivenot to choose its Minimax strategy; however, if this choice leads to a negativehandicap for this player if it is known with complete certainty by the others,then this choice should not be manifested as certain. In practice this means thatthe Minimax solution would not be any single one of the player’s pure strategiesbut rather a weighted combination of them in a mixed strategy scheme, whereeach weight corresponds to the probability of choosing one of the available purestrategies via a random mechanism. This notion of using mixtures of purestrategies for randomly choosing between them leads to a false sense of securityin single-turn games, since the optimality of the expected outcome of the mixedstrategy scheme refers to the asymptotic (long-term) and not the “spot” (one-shot) payoff. Moreover, a game may involve an infinite number of strategiesfor the players, in a discrete or continuous set; in this case the game is labeledas continuous or infinite , while a finite game is one with a limited number of In Checkers, the board size is 3x3 and each position can be either empty or host the markof of one of the two players, “X” or “O”. Hence, if the two players are treated as interchangeable(i.e., who plays first) and no other symmetries are considered, the total number of all possibledistinct board setups is: · · . . . · · , . After applying the game rules andpruning the game tree for early stops (with incomplete boards), the true number of gamestates is about / of that set. Using simple tree-node representation for each board setup,e.g. a 3-value 9-positions vector dictionary ( = 3 (cid:39) . ≤ < = 2 bytes), such aprogram would only require about 484 KB or less than 0.5 MB. This is roughly the size of asmall-sized photo taken by the camera of a low-end smart-phone today, while in the ’80s thiswas almost the total size of RAM in a typical PC. The building blocks (discrete) strategies [14, 46].When the game is inherently repetitive or iterative , i.e., includes multipleturns and not just one, even the pure strategy suggested by Minimax shouldnot be chosen deterministically in every turn if according to the game setupthis information might provide a handicap to the opponent. This is a topic ofenthusiastic discussion about the optimality of the Minimax solution and itsinherent defensive nature, as it is not clear in general when information aboutan opponent’s next move is available and trustworthy enough to justify anydeviation from this Minimax strategy. Summary: • In zero-sum games, one player’s gains is another’s losses (and viceversa). • Information about the game structure and the opponents’ movesmay be complete or not, perfect or not. • All board games are inherently zero-sum, of complete and perfectinformation. • The
Minimax theorem assures that all board games have at leastone theoretically optimal way to play them, although its exactcalculation may be unfeasible in practice for some games (e.g.Chess, Go). • The
Minimax solution of a game is the combination of players’strategies that lead to an equilibrium or saddle-point . Although the Minimax theorem provided a solid base for solving many typesof games, it is only applicable in practice for the zero-sum type of games. Inreality, it is common that in a conflict not all players receive their opponents’looses as their own gain and vice versa. In other words, it is very commona specific combination of decisions between the players to result in a certainamount of “loss” to one and a corresponding “gain”, not of equal magnitude, toanother. In this case, the game is called nonzero-sum and it requires a newset of rules for estimating optimal strategies and solutions. As each player’sgains and losses are not directly related to the opponents’, the optimal solutionis only based on the assertion that it should be the one that ensures that theplayer has “no regrets” when choosing between possible decision options. Thisessentially means that, since each player is now interested in his/her own gainsand losses, the optimal solution should only focus on maximizing each player’sown expectations [33, 28, 13]. The Minimax property can still be applied inprinciple when the single most “secure” option must be identified, but now thesolution of the game gains a new meaning.During the early 1950’s, John Nash has focused primarily on the problemof finding a set of equilibrium points in nonzero-sum games, where the players
The building blocks eventually settle after a series of competitive rounds of the game [29, 30]. Thefailure of the Minimax approach to predict real-world outcomes in nonzero-sumgames comes from the fact that the players are assumed to act independentlyand simultaneously, while in reality they usually are not. Experience shows that possibly better payoffs with what a player might choose, after observing theopponent’s moves, is a very strong motivator when choosing its actual strategy[27]. In strict mathematical terms, these equilibrium points would not be thesame in essence with the Minimax solutions, as they would come as a result ofthe players’ competitive behavior over several “turns” of moves and not as analgebraic solution of the mathematical formulation in a single-turn game.In 1957 Nash has successfully proved that indeed such equilibrium points ex-ist in all nonzero-sum games, in a way that is analogous to the Minimax theoremassertion. This new type of stable outcome is referred to as Nash equilibrium after his name and can be considered a generalization of the corresponding Min-imax equilibrium in zero-sum games. In essence, they are the manifestation ofthe no regrets principle for all players, i.e., not regretting their final choice afterobserving their opponents’ behavior [44, 46]. However, although the Nash the-orem ensures that at least one such Nash equilibrium exists in all nonzero-sumgames, there is no clear indication on how the game’s solution can be analyti-cally calculated at this point. In other words, although a solution is known toexist, there is no closed form for nonzero-sum games until today. Seminal worksby C. Daskalakis & Ch. Papadimitriou in 2006-2007 and on have proved that,while Nash equilibria exist, they may be unattainable and/or practically impos-sible to calculate due to the inherent algorithmic complexity of this problem,e.g. see: [12, 34].It should be noted that players participating in a nonzero-sum game may ormay not have the same options available as alternative course of action, or thesame set of options may lead to different gains or payoffs between the players.When players are fully interchangeable and their ordering in the game makes notdifference to the game setup and its solutions, the game is called symmetrical .Otherwise, if exchanging players’ position does not yield a proportional exchangeof their payoffs, then the game is called asymmetrical . Naturally, symmetricalgames lead to Nash equilibrium points that appear in pairs, as an exchangebetween players creates its symmetrical counterpart.
Cooperation instead of competitiveness Summary: • In nonzero-sum games, the payoffs of the players are separated(although may be correlated). • If players are allowed to observe their opponents moves over sev-eral iterations, then the “no regrets” principle is a strong incentiveto revise their own strategies, even though their payoffs are sepa-rated. • The
Nash equilibrium theorem ensures that, under these condi-tions, there are indeed stable solutions in nonzero-sum games,similarly to the Minimax theorem for zero-sum games. • However, calculating the optimal strategies and the game solutionfor these Nash equilibria is a vastly more complex and generallyunfeasible task.
The seminal work of Nash and others in nonzero-sum games was a breakthroughin understanding the outcome in real-world adversarial situations. However, theNash equilibrium points are not always the globally optimal option for the play-ers. In fact, the Nash equilibrium is optimal only when players are strictlycompetitive, i.e., when there is no chance for a mutually agreed solution thatbenefits them more. These strictly competitive forms of games are called non-cooperative games. The alternative option, the one that allows communicationand prior arrangements between the players, is called a cooperative game andit is generally a much more complicated form of nonzero-sum gaming. Natu-rally, there is no option of having cooperative zero-sum games, since the gamestructure itself prohibits any other settlement between the players other thanthe Minimax solution.
The problem of cooperative or possibly cooperative gaming is the most commonform of conflict in real life situations. Since nonzero-sum games have at least oneequilibrium point when studied under the strictly competitive form, Nash hasextensively studied the cooperative option as an extension to it. However, thepossibility of finding and mutually adopting a solution that is better for bothplayers than the one suggested by the Nash equilibrium, essentially involves a setof behavioral rules regarding the players’ stance and “mental” state, rather thanstrict optimality procedures [27]. Nash named this process a bargain between theplayers, trying to mutually agree on one solution between multiple candidateswithin a bargaining set or negotiation set . In practice, each player should enter abargaining procedure if and only if there is a chance that a cooperative solutionexists and it provides at least the same gain as the best strictly competitive Cooperation instead of competitiveness solution, i.e., the best Nash equilibrium. In this case, if such a solution isagreed between the players, it is called bargaining solution of the game [28, 33].As mentioned earlier, each player acts upon the property of no regrets, i.e.,follow the decisions that maximize their own expectations. Nevertheless, thegame setup itself provides means of improving the final gain in an agreed solu-tion. In some cases, the bargaining process may involve the option of threats ,that is a player may express the intention to follow a strategy that is particularlycostly for the opponent. Of course, the opponent can do the same, focusing on asimilar threat. This procedure is still a cooperative bargaining process, with thethreshold of expectations raised for both players. The result of such a processmay be a mutually deterring solution, which in this case is called a threateningsolution or threat equilibrium . There is also evidence that, while cooperativestrategies do exist, in some cases “cooperation” may be the result of extortion between players with unbalanced power and choices [36].In his work, Nash has formulated a general and fairly logical set of six axioms,the Nash’s bargaining axioms , regarding the behavior of rational players, inorder to establish a non-empty bargaining set, i.e., to have at least one stablesolution (equilibrium) [28, 33, 29]. In non-strict form, these axioms can besummarized in the following propositions: • Any of the cooperative options under consideration must be feasible andyield at least the same payoff as the best strictly non-cooperative optionfor all players, i.e., cooperation must be mutually beneficial. • Strict (mathematical) constraints: Pareto optimality, independence of ir-relevant alternatives, invariance under linear transformations, symmetry[46, 33, 28].The first proposition essentially defines the term “rationality” for a player:he/she always acts with the goal of maximizing own gains and minimizing losses,regardless if this means strictly competitive or possibly cooperative behavior.The second proposition names a set of strict mathematical preconditions (notalways satisfied in practice), in order for such a bargaining set to exist. Havingsettled on these axioms, Nash was able to prove the corresponding bargainingtheorem : under these axioms, there exists such a bargaining process , it is uniqueand it leads to a bargaining solution, i.e., equilibrium. However, as in the generalcase of strictly competitive games, Nash’s bargaining theorem does not provideanalytical means of finding such solutions.The notion of bargaining sets and threat equilibrium is often extended inspecial forms of games that include iterative or recursive steps in gaming, eitherin the form of multi-step analysis ( meta-games ) or focusing on the transitionalaspects of the game ( differential games ). Modern research is focused on methodsthat introduce probabilistic models into games of multiple realizations and/ormultiple stages [33].
Cooperation instead of competitiveness Summary: • In nonzero-sum games, there may be non-competitive (coopera-tive) options that are mutually beneficial to all players. • Under some general rationality principles,
Nash’s bargaining theo-rem ensures that these cooperative outcomes may indeed becomethe game solution, provided that strict competitiveness yieldslower gains for all. • The procedure of structuring the “common ground” of cooperationbetween the players, normally conducted over several iterations,is the bargaining process.
Nash’s work on the Nash equilibrium and bargaining theorem provides the nec-essary means to study n -person non-cooperative and cooperative games undera unifying point of view. Specifically, a nonzero-sum game can be realized asa strictly competitive or a possibly cooperative form, according to the game’srules and restrictions. Therefore, the cooperative option can be viewed as ageneralization to the strictly competitive mode of gaming.When players are allowed to cooperate in order to agree on a mutuallybeneficial solution of game, they essentially choose one strategy over the othersand bargain this option with all the others in order to come to an agreement.For symmetrical games, this is like each player chooses to join a group of otherplayers with similar preference over their initial choice. Each of these groups iscalled a coalition and it constitutes the basic module in this new type of gaming:the members of each coalition act as cooperative players joined together and atthe same time each coalition competes over the others in order to impose its ownposition and become the winning coalition . This setup is very common whenmodeling voting schemes, where the group that captures the relative majorityof the votes becomes the winner.Coalition Theory is closely related to the classical Game Theory, especiallythe cooperating gaming [33, 28]. In essence, each player still tries to maximizeits own expectations, not individually any more but instead as part of a greateropposing term. Therefore, the individual gains and capabilities of each playeris now considered in close relation to the coalition this player belongs, as wellas how its individual decision to join or leave a coalition affects this coalition’swinning position. As in classic nonzero-sum games, the notion of equilibriumpoints and solutions is considered under the scope of domination or not in thegame at hand. Furthermore, the theoretical implications of having competingcoalitions of cooperative players is purely combinatorial in nature, thus makingits analysis very complex and cumbersome. There are also special cases ofcollective decision schemes where a single player is allowed to abstain completelyfrom the voting procedure, or prohibit a contrary outcome of the group via a veto option.In order to study the properties of a single player participating in a game Cooperation instead of competitiveness of coalitions, it is necessary to analyze the wining conditions of each coalition.Usually each player is assigned a fixed value of “importance” or “weight” whenparticipating in this type of games and each coalition’s power is measured as asum over the individual weights of all players participating in this coalition. Thecoalition that ends up with the highest cumulative value of power is the winningcoalition. Therefore, it is clear that, while each player’s power is related to itsindividual weight, this relation is not directly mapped on how the participationin any arbitrary coalition may affect this coalition’s winning or losing position.As this process stands true for all possible coalitions that can be formed, thiscompetitive type of “claiming” over the available pool of players/voters by eachcoalition suggests that there are indeed configurations that marginally favor theone or the other coalition, i.e., a set of “solutions”.The notion of solution in coalition games is somewhat different from the onesuggested for typical nonzero-sum games, as it identifies minimal settings forcoalitions that dominate all the others. In other words, they do not identifypoints of maximal gain for a player or even a coalition, but equilibrium “points”that determine which of the forming coalitions is the winning one. This typeof “solutions” in coalition games is defined in close relation to domination and stability of such points and they are often referred to as the Core . Von Neumannand Morgenstern have defined a somewhat more relaxed definition of such con-ditions and the corresponding solutions are called stable sets [33, 28]. It shouldbe noted that, in contrast to Nash’s theorems and the Minimax assertion ofsolutions, there is generally no guarantee that solutions in the context of theCore and stable sets need to exist in an arbitrary coalition game. Summary: • Players of similar preferences and mutual benefits may join ingroups or coalitions ; these coalitions may be competing with eachother, similarly to competitive games between single players. • The study of games between coalitions is inherently more complexthan with single players, as in this case every player contributesto the collective “power” and enjoys a share of the wins. • In general, coalitions are formed and structured under the scopeof voting ensembles , where the voting weight of each individualplayer contributes to the combined weight of the coalition.
The notion of the Core and stable sets in coalition gaming is of vital importancewhen trying to identify the winning conditions and the relative power of eachindividual player in affecting the outcome of the game. The observation thata player’s weight in a weighted system may not intuitively correspond to itsvoting “power” goes back at least to Shapley and Shubik (1954). For example, aspecific weight distribution to the players may make them relatively equivalentin terms of voting power, while only a slight variation of the weights may rendersome of them completely irrelevant on determining the winning coalition [45].
Cooperation instead of competitiveness Shapley and Shubik (1954) and later Banzhaf and Coleman (1965, 1971)suggested a set of well-defined equations for calculating the relative power ofeach player, as well as each forming coalitions as a whole [33, 28]. The
Shapley-Shubik index of power is based on the calculation of the actual contribution ofeach player entering a coalition, in terms of improving a coalition’s gain andwinning position. Similarly, the
Banzhaf-Coleman index of power calculateshow an individual player’s decision to join or leave a coalition (“swing vote”)results in a winning or loosing position for this coalition, accordingly. Bothindexes are basically means of translating each player’s individual importanceor weight within the coalition game into a quantitative measure of power interms of determining the winner. While both indices include combinatorialrealizations, the Banzhaf index is usually easier to calculate, as it is based onthe sum of “shifts” on the winning condition a player can incur [5]. Furthermore,its importance in coalition games is made clearer when the Banzhaf index isviewed as the direct result of calculating the derivatives of a weighted majoritygame (WMG).Seminal work by L. S. Penrose [35], as well as more recent studies with com-puter simulations [8], have shown that this discrepancy between voting weightsand actual voting power is clearly evident when there is large variance in theweighting profile and/or when the voting group has less than 12-15 members.Even in large voting pools, the task of designing optimal voting mechanismsand protocols with regard to some collective efficiency criterion is one of themost challenging topics in Decision Theory.
Summary: • Weighted majority games (WMG) are the typical theoretical struc-tures of the process of formulating the collective decision within acoalition. • In voting ensembles, each player’s voting weight is not directlyproportional to his/her true voting power within the group, i.e.,the level of steering the collective decision towards its own choices.
In most cases, majority functions that are employed in practice very simplisticwhen it comes to weighting distribution profile or they imply a completely uni-form weight distribution. However, a specific weighting profile usually producesbetter results, provided that is simple enough to be applied in practice andattain a consensus in accepting it as “fair” by the voters. Taylor and Zwicker[45] have defined a voting system as trade robust if an arbitrary series of tradesamong several winning coalitions can never simultaneously render them losing.Furthermore, they proved that a voting system is trade robust if and only if itis weighted. This means that, if appropriate weights are applied, at least onewinning coalition can benefit from this procedure.As an example, institutional policies usually apply a non-uniform votingscheme when it comes to collective board decisions. This is often referred to
Cooperation instead of competitiveness as the “inner cabinet rule”. In a hospital, senior staff members may attainincreased voting power or the chairman may hold the right of a tie-breakingvote. It has been proven both in theory and in practice that such schemes aremore efficient than simple majority rules or any restricted versions of them liketrimmed means. Nitzan and Paroush [32] have studied the problem of optimal weighted majority rules (WMR) extensively and they have proved that they areindeed the optimal decision rules for a group of decision makers in dichotomouschoice situations. This proof was later extended by Ben-Yashar and Paroush,from dichotomous to polychotomous choice situations [3]; hence, the optimalityof the WMR formulation has been proven theoretically for any n -label votingtask.The weight optimization procedure has been applied experimentally in trainedor other types of combination rules, but analytical solutions for the weights isnot commonly used. However, Shapley and Grofman [42] have established thatan analytical solution for the weighting profile exists and it is indeed relatedto the individual player skill levels or competencies [23]. Specifically, if deci-sion independence is assumed for the participating players, the optimal weightsin a WMR scheme can be calculated as the log-odds of their respective skillprobabilities, i.e.: w k = log ( O k ) = log (cid:18) p k − p k (cid:19) (2.1)where p k is the competency of player k and w k is its corresponding votingweight. Interestingly enough, this is exactly the solution found by analyticalBayesian-based approaches in the context of decision fusion of independent ex-perts in Machine Learning [24]. The optimality assertion regarding the WMR,together with an analytical solution for the optimal weighting profile, providesan extremely powerful tool for designing theoretically optimal collective deci-sion rules. Even when the independence assumption is only partially satisfiedin practice, studies have proved that WMR-based models employing log-oddsweighting profiles for combining pattern classifiers confirm these theoretical re-sults [19, 18]. Summary: • Weighted majority rules (WMR) have been proven theoreticallyas the optimal decision-making structures in weighted majoritygames. • The log-odds model has been proven both as the theoretically op-timal way to weight the individual player’s votes, provided thatthey decide independently. • The optimality of the log-odds weighting method has also beenproven experimentally, even when the independence assumptionis only partially satisfied.
Cooperation instead of competitiveness Condorcet (1785) [9] was the first to address the problem of how to designand evaluate an efficient voting system, in terms of fairness among the peoplethat participating in the voting process, as well as the optimal outcome forthe winner(s). This first attempt to create a probabilistic model of a votingbody is known today as the
Condorcet Jury Theorem [51]. In essence, thistheorem says that if each of the voting individuals is somewhat more likelythan not to make the “better” choice from a set of alternative options; and ifeach individual makes its own choice independently from all the others, then theprobability that the group majority is “correct” is greater than the individualprobabilities of the voters. Moreover, this probability of correct choice by thegroup increases as the number of independent voters increases. In practice, thismeans that if each voter decides independently and performs marginally higherthan 50%, then a group of such voters is guaranteed to perform better thaneach of the participating individuals. This assertion has been used in Socialsciences for decades as a proof that decentralized decision making, like in agroup of juries in a court, performs better than centralized expertise, i.e., a solejudge. The Condorcet Jury Theorem and its implications have been used as oneguideline for estimating the efficiency of any voting system and decision makingin general [51]. Under this context, the coalition games are studied by applyingquantitative measures on collective competence and optimal distribution of power in the ensemble, e.g. tools like the Banzhaf or Shapley indices of power. Thedegree of consistency of such a voting scheme on establishing the pair-wisewinner(s), as the Condorcet Jury Theorem indicates, is often referred to as the
Condorcet criterion .Shapley-Shubik and Banzhaf-Coleman are only two of several formulationsfor the indices of power in voting ensembles, each defining different payoff dis-tributions or realizations among the members of winning coalitions. In general,these formulations are collectively referred to as semivalue functions or semi-values and they are considered more or less equivalent in principle, althoughmay be different in exact values. Almost all of them are based on combinatorialfunctions (inclusion-exclusion operations in subsets) and, as a result, there isno easy way to formulate proper inverse functions that can be calculated inpolynomial time. Therefore, the design of exact voting profiles with weightsbased on semivalues, instead of competencies as described above (log-odds), isgenerally impractical even for ensembles of small sizes.For further insight on weighted majority games, weighted majority voting,collective decision efficiency and Condorcet efficiency, as well as applications toMachine Learning for designing pattern classifiers, see [17, 19, 18]. Game analysis & solution concepts Tab. 1:
Generic 2x2 zero-sum game in analytical form.
Gameexample
Player-2 y − y Player-1 x a b − x c d Summary: • Under the assumption of independent voters and that each de-cides “correctly” marginally higher than 50% of the time, thentheir collective decision as a group is theoretically proven to beasymptotically better any single member of the ensemble. • Furthermore, as the size of the ensemble increases, its collectivecompetency is guaranteed to increase too. • In the other hand, the problem of formulating an analytical so-lution for the optimal distribution of voting power within such agroup, i.e., the design of theoretically optimal voting mechanisms ,is still an open research topic.
One of the most important factors in understanding and analyzing games cor-rectly is the way they are represented. Games can be represented and analyzedin two generic formulations: (a) the analytical or normal form, where eachplayer is manifested as one dimension and its available choices (strategies) asoffsets on it, and (b) the extensive or tree-graph form, where each player’s “move”correspond to a node split in a tree representation. Each one of them has itsown advantages and disadvantages, but theoretically they are equivalent. In Table 1, an example of a zero-sum game in analytical form is presented.Player-1 is usually referred to as the “max” player and Player-2 is referred to asthe “min” player, while rows and columns correspond to each player’s availablestrategies, respectively. Since this is a zero-sum game and one player’s gainsis the other player’s losses, the “max” player tries to maximize the game value(outcome) while the “min” player tries to minimize it. In the context of theMinimax theorem, Player-2 chooses the maximum-of-minimums , while Player-2chooses the minimum-of-maximums . The x and y correspond to the weight orprobability of choosing the first strategy and, since this is a 2x2 game, the otherstrategies are attributed with the complementary probabilities, 1- x and 1- y .The exact Minimax solution for x and y depends solely on the values of theindividual payoffs for each of the four outcomes. Here, it is assumed that there isno domination in strategies, i.e., there is no row/column that is strictly “better” Game analysis & solution concepts Tab. 2:
Example 2x2 zero-sum game in analytical form.
Gameexample
Player-2(0) (1)Player-1 (0) 0 -3(1) 4 Tab. 3:
Example of a 2x2 nonzero-sum game in analytical form.
Gameexample
Player-2 y − y Player-1 x ( a , a ) ( b , b ) − x ( c , c ) ( d , d )than another row/column (column-wise/row-wise, respectively, all payoffs). Forexample, Player-1 would have a dominating strategy in the first row if and onlyif a ≥ c and b ≥ d . Based on this generic setup, this is a typical 2x2 system oflinear equations and, if no domination is present, its solution can be determinedanalytically as [44, 14, 26]: [ x, − x ] = (cid:20) d − ca − b − c + d , a − ba − b − c + d (cid:21) (3.1) [ y, − y ] = (cid:20) d − ba − b − c + d , a − ca − b − c + d (cid:21) (3.2) u = ad − bca − b − c + d (3.3)The Minimax solution [ x, y ] determines the saddle-point, i.e., the equilibriumthat is reached when both opponents play optimally in the Minimax sense, whenthe game has no pure (non-mixed) solution. In this case, the expected payoffor value of the game for both players is calculated by u (remember, this is azero-sum game). If the game has a pure solution, then it is determined as either0 or 1 for each probability x and y . Table 2 illustrates a zero-sum game and thecorresponding pure Minimax solution, by selecting the appropriate strategies foreach player. In this case, “max” Player-1 chooses the the maximum {1} betweenthe two minimum values {-3,1} from its own two possible worst-case outcomes,while “min” Player-2 chooses the the minimum {1} between the two maximumvalues {4,1} from its own two possible worst-case outcomes. Hence, the puresolution [1,1] is the Minimax outcome.In nonzero-sum games, the analytical form is still a matrix, but now the pay-offs for each player are separate, as illustrated in Table 3. Here, since the payoffsare separated, both players are treated as “max” and the Minimax solution foreach one is calculated by selecting the maximum-of-minimums as described be-fore for zero-sum games, focused solely on its own payoffs from each value pair.Although a (pure) Minimax solution can always be calculated for nonzero-sum games, the exact Nash equilibrium solution is a non-trivial task that cannotbe solved analytically in the general case. However, pure Nash equilibriumoutcomes can be identified by locating any payoff pairs ( z, w ) such that z is Game analysis & solution concepts Tab. 4:
Example of a 2x2 nonzero-sum game with one Nash equilibrium at[ A , B ]:(2,4). Gameexample
Player-2
A B
Player-1 A (3,3) (2*,4*) B (4*,1) (1,2*)the maximum of its column and w is the maximum of its row. In other words,every row for Player-1 is scanned and every entry in it is compared to the valuesin the same column, marking it if it is the maximum among them; the sameprocess is conducted for every column for Player-2, scanning each value row-wisefor its maximum; any payoff pair that has both values marked as maximumsis a Nash equilibrium in the game. Table 4 illustrates such an example, whereasterisk (*) marks the identified max-values and the single Nash equilibriumfor [ A , B ] at (2,4). Here, although the strategies are the same for both players,their (separated) payoffs are not, hence the game is referred to as asymmetric .According to the oddness theorem by Wilson (1971), the Nash equilibria almostalways appear in odd numbers [44, 33], at least for non-degenerate games, wheremixed strategies are calculated upon k linearly independent pure strategies. Summary: • Game representation in analytical form introduces a game matrix,with row and column positions associated to the strategies availableto the players and contents associated to the corresponding payoffs. • Analytical-form representation introduces very convenient ways toidentify Minimax solutions and Nash equilibria in games. • However, they are appropriate mostly for 2-player simultaneousgames, since any other configuration cannot be fully illustrated.
In the extensive form the game is represented as a tree-graph, where each nodeis a state labeled by a player’s number and each (directed) edge is a player’schoice or “move”. Strictly speaking, this is a form of state-transition diagramthat illustrates how the game evolves as the players choose their strategies.Figure 3.1 shows such a 2x2 nonzero-sum game of perfect information, whileFigure 3.2 shows a similar 2x2 game of imperfect information [46, 28, 49, 41,16, 14]. Nodes with numbers indicate players, edges with letters indicate chosenstrategies (here, symmetric) and separated payoffs (in parentheses) indicate thegame outcome after one full round. The dashed line between the two nodesfor Player-2 indicate that its current true state is not clearly defined due toimperfect information regarding Player-1’s move. In practice, these two statesform an information set for Player-2, which has no additional information todifferentiate between them. This is also valid in the case of simultaneous moves,
Game analysis & solution concepts Fig. 3.1:
Example of a 2x2 nonzero-sum game of perfect information.where Player-2 cannot observe Player-1’s move in advance of its own, and viceversa. In extensive form, an information set is indicated by a dotted line or bya loop, connecting all nodes in that set.The extensive form of game is usually the preferred way to represent thetree-graph of simple 2-player board games, where each node is clearly a stateand each edge is a player’s move. Even in single-player games, where a puzzlehas to be solved through a series of moves (e.g. Rubik’s cube) , the tree-graphis a very effective way to organize the game under an algorithmic perspective, inorder to program a “solver” in a computer. In practice, the problem is structuredas sequences of states and transitions in a tree-graph manner and the “game”is explored as it is evolving, move after move, expanding the tree-graph fromevery terminal node. The tree-graph can be expanded either by full a level(“breadth-first”), or from a branch all the way down to non-expandable terminalnodes (“depth-first”), or some hybrid scheme between these two alternatives.As described above, small games like Checkers can be structured and ex-panded fully, with their tree-graph having only internal (already expanded) andterminal nodes; however, in larger games like Chess or Go this is practicallyunfeasible even with super-computers. In such cases, the algorithm should as-sess the “optimality” of each expandable terminal node with regard to relevancetowards the predefined goal (“win” or “solution”), sort all these nodes accordingto their ranking and choose the “best” ones for expansion in the next iteration.This way, the search is sub-optimal but totally feasible with almost any mem-ory constraints - this is exactly how most computer players are programmedfor playing board games or solving complex puzzle games. In Artificial Intelli-gence, algorithms like A* and AB solve this type of problems as a path-findingoptimization procedure towards a specified goal [40, 31]. The combinatorial analysis of the classic 3x3x6 Rubik’s cube should take into accounttile permutations that can only be reached by the available shifts and turns of the slices ofthe device. Therefore, a totally “free” permutation scheme would produce: · · · =519 , , , , , , cube instances, while in practice the possible permutations areonly: · · (12! / · = 43 , , , , , , cube instances (about 12 times fewer)[50]. Game analysis & solution concepts Fig. 3.2:
Example of a 2x2 nonzero-sum game of imperfect information.Figure 3.3 illustrates the way a path-finding algorithm like A* would workin expanding a tree-graph as described above. The “root” node is the startingstate in a puzzle game (single-player) and each node represents a new stateafter a valid move. The numbers indicate the sequence in which the nodes areexpanded, according to some optimality-ranking function (not relevant here).For example, node “4” in the 3rd level is expanded before node “5” in the 2ndlevel, node “21” in the 5th level is expanded before node “22” in the 3rd level,etc. Here, node “30” in the 5th level is the last and most relevant terminal node(still expandable) towards the goal, hence the optimal path from the “root” stateis currently the: “5” → “7” → “11” → “30” and the next “best” single-step move isthe one towards “5”. The tree-graph can be expanded in an arbitrary numberof levels according to the current memory constraints for the program, but thesame path-finding procedure has to be reset and re-applied after the realizationof each step when two or more players are involved, since every response fromthe opponent effectively nullifies every other branch of the tree-graph.It should be mentioned that, although the extensive form of game represen-tation is often inefficient for large games like Chess, it can be used as a tool inthe proof of the existence of an optimal solution [15, 46]. Specifically, in everysuch game of complete and perfect information (all board games), each playerknows its exact position in the graph-tree prior to choosing the next move. Inother words, each player is not only aware of the complete structure of the gamebut also knows all the past moves of the game, including the ones of randomchoice. Hence, since there is no uncertainty in the moves, each player can removethe dominated strategies and subsequently identify the optimal choice, which isalways a pure strategy, i.e., the one that corresponds to the saddle-point of thegame. This proof actually ensures the existence of a (pure) optimal strategyin every typical board game, no matter how large or complex it is. Examplesinclude Tic-Tac-Toe, Chess, Backgammon, etc. The four interesting cases Fig. 3.3:
Example of the way a path-finding algorithm like A* would work inexpanding the tree-graph of a single-player “puzzle” game like Rubik’scube.
Summary: • Game representation in extended form introduces a tree-graph, withnodes associated to individual players and (directed) edges associatedto selected strategies (“moves”). • Extended-form representation introduces very convenient ways toidentify chains of moves and solution paths. • However, the calculation of Minimax solutions and Nash equilibria isnot straight-forward.
In the real world, games may be either zero-sum or nonzero-sum by nature.As described previously, the case of zero-sum games can be considered simplerand much easier to solve analytically, since it can be formulated as a typicalalgebraic set of linear equations that define the Minimax solution, regardlessif it contains pure or mixed strategies [44, 14]. However, nonzero-sum gamesare inherently much more complex and require non-trivial solution approaches,usually via some Linear Programming (constraint) optimization procedure, e.g.see: [20, 43]. In fact, it has been proven that the general task of finding theNash equilibria is algorithmically intractable [12, 10, 11, 34] - something that In their seminal works, Daskalakis, Goldberg and Papadimitriou have shown that the taskof finding a Nash equilibrium is PPAD-complete; informally, PPAD is the class of all search
The four interesting cases Tab. 5:
The general analytical (matrix) form of a 2x2 nonzero-sum symmetricgame.
Gametemplate
Player-2
C D
Player-1 C ( R,R ) (
S,T ) D ( T,S ) (
P,P )puts into a “philosophical” question the very nature and practical usefulness ofhaving proof of game solutions (i.e., stable outcomes) that we may not be ableto calculate.Some cases of nonzero-sum games are particularly interesting, especiallywhen they involve symmetric configurations. The players can switch places,the actual payoff values are usually of much less importance than their relativeordering as a simple preference list, the Minimax and Nash equilibria can beeasily identified, yet these simple games seem to capture the very essence ofbargaining and strategic play in a vast set of real-world conflict situations withno trivial outcomes.Table 5 shows a generic template for such very simple symmetric nonzero-sum games, employing only two strategies and four payoff values to completelydefine such games in analytical (matrix) form. Here, the game is symmetricbecause the players can switch roles without any effect in their correspondingpayoff pairs. Furthermore, they share two common strategies C and D , namedtypically after the choices of “cooperate” or “defect”, while constants P , R , S and T are the real-valued payoffs in each case [7].In practice, a player’s preference of strategies (and hence, the equilibria)depends only on the relative ordering of the corresponding payoffs and not theirexact values, which become of real importance only when the actual payoff valueof the game solution is to be calculated for each player. There is a finite numberof rank combinations, i.e., permutations, of these four constants, which produceall the possible unique game matrices of this type. Specifically, there are
4! = 24 different ways to order these four numbers, 12 of which can be discarded asqualitatively equivalent to other game configurations. Out of the 12 remaininggames, eight of them possess optimal pure strategies for both players, thereforethey can be considered trivial in terms of calculating their solution. The fourremaining configurations are the most interesting ones, as they do not possessany optimal pure strategy. These are the following: • Leader : T > S > R > P . • Battle of the Sexes : S > T > R > P . • Chicken : T > R > S > P . • Prisoner’s Dilemma : T > R > P > S .These four qualitatively unique games seem to capture the essence of mostof the majority real-world conflict situations historically. Although they have problems which always have a solution and whose proof is based on the parity argument fordirected graphs. Due to the proof of intractability, the existence of Nash equilibrium in allnonzero-sum games somewhat loses its credibility as a predictor of behavior.
The four interesting cases Tab. 6:
The typical setup of the
Leader game with two players. Nash equilibriaare marked with paired asterisks and the Minimax solution with boldnumbers.
Leadergame
Player-2
C D
Player-1 C ( , ) (3*,4*) D (4*,3*) (1,1)been studied extensively in the past, there are still many open research topicsregarding the feasibility, tractability and stability of the theoretical solutions. The
Leader or Coordination game [28, 33, 46, 44, 7, 16] is named after thetypical problem of two drivers attempting to enter a stream of increased trafficfrom opposite sides of an intersection. When the road is clear, each driver hasto decide whether to move in immediately or concede and wait for the otherdriver to move first. If both drivers move in (i.e., choose D ), they risk crashingonto each other, while if they both wait (i.e., choose C ), they will waste timeand possibly the opportunity to enter the traffic. The former case is the worst,hence the payoff of (1,1), while the later case is slightly more preferable with apayoff of (2,2). The best outcome is for one driver to become the “leader” andmove first, while the other becomes the “follower” and move second. There isstill some difference in their absolute gains, but now the deadlock is resolvedin the best possible way, no matter who is actually the leader and who is thefollower.Table 6 illustrates the analytical form of this game setup, where numbersindicate relative preferences rather than absolute gain values. There are twopure Nash equilibria, (3,4) and (4,3), which correspond to the proper assignmentof roles to the players, explicitly or implicitly, such that coordination is achieved.Since the game is symmetric the two players can switch roles, with only marginalincrease/decrease to their payoffs. In terms of Minimax strategies, each playeris free to choose the strategy that guarantees the maximum-of-the-minimumswithout any concern about the opponent’s payoffs, since this is a nonzero-sumgame. Hence, the Minimax solution is [ C , C ] at (2,2) marked in bold.In the real world, the assignment of roles as leader/follower is more effectivewhen applied explicitly, typically by some external mechanism or a predefinedset of rules. Street signs, traffic policemen and highway code for driving prop-erly are all such mechanisms for explicit resolution of deadlocks via priorityassignment in traffic. In the
Battle of the Sexes game [28, 33, 46, 7, 16], a married couple has to decidebetween entertainment options for the evening. The husband prefers one choice,while the wife prefers another. The problem is that they would both prefer toconcede to the same choice together even if it is not their own, rather than followtheir own choices alone. For example, of he wants to watch a sports match on
The four interesting cases Tab. 7:
The typical setup of the
Battle of the Sexes game with two players. Nashequilibria are marked with paired asterisks and the Minimax solutionwith bold numbers.
Battle ofthe Sexes
Player-2
C D
Player-1 C (1,1) (3*,4*) D (4*,3*) ( , )TV and she wants to go out for dinner, they both prefer either watching TV orgoing out for dinner as long as they are together.Table 7 illustrates the analytical form of the game, where strategy C is forconceding to the other’s preference and D is for defecting to his/her own choice.If they both concede the payoff (1,1) is the worst outcome, since they bothend up miserable and bored. If they both defect the payoff (2,2) is marginallybetter for both, but they end up being alone. The two other cases of someonefollowing the other yields the best payoffs for both, since the game is symmetricand they can switch places. The outcomes (3,4) and (4,3) are actually the twoNash equilibria, similarly to the Leader game; however, the Minimax solution(2,2) here corresponds to both players choosing D (not C as in Leader ) as theirbest Minimax strategy.
One of the most well-known strategic games is
Chicken [15, 26, 28, 33, 46, 7],dating back at least as far as the Homeric era. Two or more adversaries engagein a very dangerous or even lethal confrontation, each having a set of choices athis/her disposal and each of these choices producing more or less damage to allplayers if their choice is the same. Typically, this translates to the Hollywood’sfavorite version of two cars speeding towards each other, the drivers can chooseto turn and avoid collision or keep the course and risk death if the other driver donot turn either. The game seems simple enough, but there are several theoreticalimplications that make it one of the most challenging situations, appearing inmany real-world conflicts throughout History.Table 8 illustrates the typical
Chicken game setup with two players and twostrategic choices. Option C corresponds to turning away (“swerve”) and losingthe game, while option D corresponds to keeping the course and risk death.The worst possible outcome is at (1,1) when players persist in keeping courseand eventually crashing against each other. The mutually beneficial outcomeor “draw” is at (3,3) when both players decide to play safe and turn away; thisis actually the Minimax solution of the game, i.e., the most conservative and“rational” outcome if the game is a one-off round. On the other hand, there aretwo Nash equilibria for the two outcomes when only one player turns away andone persists.One particularly interesting feature of the Chicken game is that it is impos-sible to avoid playing it with some insistent adversary, since refusing to playis effectively equivalent to choosing C (swerve). Furthermore, the player whosucceeds in making his/her commitment to D adequately convincing is alwaysthe one that can win at the expense of the other player, assuming that the other The four interesting cases Tab. 8:
The typical setup of the
Chicken game with two players. Nash equilibriaare marked with paired asterisks and the Minimax solution with boldnumbers.
Chickengame
Player-2
C D
Player-1 C ( , ) (2*,4*) D (4*,2*) (1,1)player is rational and would inevitably decide to avoid disaster. In other words,the player that is somehow bounded to avoid losing at any cost and makes thiscommitment very clear to the opponent, is the one that will always win againstany rational player.This aspect of credible commitment is closely related to the notion of repu-tation , as well as the strange conclusion that in this game the most effectively“rational” strategy is the manifestation of “irrational” commitment to lethal risk.This becomes especially relevant in cases where the game is played a numberof times repeatedly and previous behaviors directly affect the players’ strategicchoices in the future: once the risky player starts winning he/she may maintainor even improve this advantage, as confidence and prior “risky” behavior makesit more and more difficult for future opponents to decide and deviate from theircautious Minimax choice of swerving. The Chicken game is perhaps the mostdescriptive and simple case where players’ previous behavior (i.e., reputation )is of such importance for predicting the actual outcome.
This forth basic type of non-trivial, nonzero-sum game is by far the most inter-esting one. The
Prisoner’s Dilemma game [15, 26, 28, 33, 46, 44, 7, 16] typicallyinvolves two prisoners who are accused of a crime. Each of them has the optionof remaining silent and withholding any information or confessing to the policeand accusing the other by disclosing details about the crime. The first choice C is effectively the cooperative option, while the second choice D corresponds topurely competitive behavior in order to reduce he/her own damages.Table 9 illustrates the typical Prisoner’s Dilemma game setup with two play-ers and two strategic choices. The payoffs here correspond simply to preferencesand not real gain/cost values, but the essence and the strategic properties ofthe game remain intact. In practice, what the game matrix says is that if thetwo prisoner’s remain silent, i.e., mutually cooperate, they will not be freed butthey will share an equal, relatively mild conviction. If they both talk and accuseeach other, i.e., mutually defect, they will share and equal but more severe con-viction. If only one of them talks to the police and the other remains silent, theone that talked is freed and the other serves a full-time conviction for both. Itis of course imperative that the two prisoners are immediately separated uponcapture and no communication between them is allowed; this does not nullifiesany prior arrangements they may have, but isolation after being captured meansthat neither of them can confirm they loyalty of the other. This is one of themain reasons why police always isolates suspects prior and during any similarinvestigation.
The four interesting cases Tab. 9:
The typical setup of the
Prisoner’s Dilemma game with two players.Nash equilibria are marked with paired asterisks and the Minimax so-lution with bold numbers.
Prisoner’sDilemma
Player-2
C D
Player-1 C (3,3) (1,4*) D (4*,1) ( *, *)The real beauty and singularity of the Prisoner’s Dilemma is that it impliesa paradox. A quick analysis of the payoffs in Table 9 yields two extremes at (1,4)and (4,1), corresponding to the two interchangeable cases one player cooperating( C ) and one not ( D ), but in contrast to the three previous games these are not Nash equilibria. There is only one Nash equilibrium at (2,2), which is in factthe Minimax solution too. This means that under the solution concepts of bothMinimax strategy and Nash equilibrium, theory suggests that the two prisoner’swill probably choose to betray one another, despite any previous arrangements.It is clearly evident that the outcome (3,3) is mutually beneficial and at thesame time unattainable due to lack of communication. However, in therms ofstrict personal gain, defecting ( D ) is the dominant strategy for both and neitherof them has any incentive to deviate from it. In other words, it appears thatdefecting is always the optimal choice regardless of what the other prisoner does- but if both adopt the same rationale, they will end up at (2,2) which is clearlyworse than the (3,3) that they could have gotten if they had chosen mutualcooperation.The essence of the paradox of Prisoner’s Dilemma lies in the inherent conflictbetween individual and collective rationality . While individual rationality iswell-understood, collective rationality deals with the scope of optimizing the mutual gain of the players. This is not a default behavior in strictly competitivesituations, as in zero-sum games, or nonzero-sum games that do not implycooperation. However, nonzero-sum games permit the idea of mutually optimalgains as a combination of simultaneously optimal separate payoffs. Under thisbroader scope, even (4,1) and (1,4) are worse than (3,3) since they yield a sumof 5 in gain value rather than 6, respectively.It should also be noted that the single Nash equilibrium in
Prisoner’s Dilemma is stable, while the corresponding pairs of Nash equilibria in the three previousgames are inherently unstable, since the players are not in agreement as towhich of the two equilibria is preferable. Furthermore, in the three previousgames the worst possible outcome comes when both players choose their non-Minimax strategy; in
Prisoner’s Dilemma this is not so. In fact,
Prisoner’sDilemma has produced lengthy academic debates and hundreds of studies in awide range of disciplines, from Game Theory and Mathematics to Sociology andEvolutionary Biology. The paradox of this game (as described above) has beenillustrated as a notorious example where theory often fails to predict the true“gaming” outcomes in the real world: cooperation can emerge spontaneously,even though theory says it should not [1, 2, 27, 7].
Signals, mechanisms & rationality Summary: • There are four basic nonzero-sum game types of particular interestnamely:
Leader (or
Coordination ), Battle of the Sexes , Chicken and
Prisoner’s Dilemma . • Three of these games (except
Prisoner’s Dilemma ) have two “mir-rored” pure Nash equilibria and players receive the worst possible pay-off when they choose to deviate from their optimal Minimax strategy. • Prisoner’s Dilemma is a very unique type of game, since neither Min-imax solution or Nash equilibrium (single one in this case) point tothe best mutually beneficial outcome; this is informally labeled as the paradox of this game.
Game formulation and representation in analytical or extensive form are imper-ative for proper analysis and identification of equilibria. However, they fail tocapture many elements of gaming as a multi-aspect process, especially in rela-tion to strategic moves ; these are actions performed by the players at differentplaces and times, even before the realization of the current game, with the goal ofenhancing strategic advantages and increasing the effectiveness of chosen strate-gies. Sometimes the “moves” are no more than message exchanges between theplayers, explicit or implicit, or simply tracking the history of previous choicesin iterated games. Formulating these factors into a proper mathematical modelcan be very difficult, but nevertheless they are matters of great importance inreal-world conflict situations.
The exchange of messages between the players is a very useful option when aplayer is trying to model or even predict the behavior of its opponent(s). Amessage or signal from one player to another may be voluntary or involuntary,direct or indirect, explicit or implicit [46, 44]. In any case, it carries some sortof strategic information, which is always valuable to the other player if it can beasserted as credible with a high degree of confidence. On the other hand, if thiscredibility can be manipulated and falsely asserted as such, the source playermay gain some strategic advantage by means of deceiving its opponent.Strategic signaling is the process of information exchange between two ormore players in a game, using any means or intermediate third-parties as car-riers. If the source player does this deliberately, the purpose is to project somestrategic preference or stance (“posturing”) in the game without making any ac-tual “move”, in order to intimidate or coordinate with the opponent(s). This isparticularly useful in situations where mutually beneficial equilibria are achiev-able but lack of preference ranking can lead to disastrous lack of coordination.The
Leader and
Battle of the Sexes games are such examples (see Tables 6 and
Signals, mechanisms & rationality Explicit signaling means that the source player sends out a clear messagewith undeniable association and content. An explicit signal may be volun-tary or involuntary; in the later case, the message is simply a “leak” with veryclear origin and content.
Implicit signaling happens when the origin or (mostcommonly) the content of the message is somehow inconclusive or “plausiblydeniable” as to the intentions of the source player. A signal exchange may occurdirectly between the players or via a third-party that performs the role of a car-rier. A number of combinations of these attributes are possible in practice, em-ploying direct/indirect messaging, voluntary/involuntary information exchange,with explicit/implicit messages. For example, a third-party carrier may sharean implicit signal or “leaked” (involuntary) information about a player’s stancewith another player, participating in the game only as a mediator, coordinatoror “referee”, rather than an actively involved player.A very special type of signaling is when the message exchange involves falseinformation, i.e., a bluff . This kind of signals is a very common practice ingames of imperfect and/or incomplete information, where the players do nothave a complete view of the game structure itself and/or the opponents’ choices,respectively. In this case, false signaling or bluffing is usually a strategic optionby itself, exploiting this uncertainty regarding the true status of the game toenhance advantages or mitigate disadvantages. A very common example ofsuch games is Poker, where a player with weaker deck of cards can project a false stance to its opponents, in order to avoid defeat or even secure a victoryagainst players with better decks of cards [46, 44]. Bluffing can be realizeddirectly between players or indirectly via a third-party carrier. In the later case,especially when the signaling is implicit and assumed involuntary, the credibilityof the assertion is strongly associated with the credibility of the carrier itself.In other words, even if the source player could not project a successful bluff onits own, a credible third-party carrier might be the necessary intermediate toachieve such a move. The role of third-party mediators in signaling is a specialtopic in the study of strategic moves and how they affect the final outcome ingames. Signals, mechanisms & rationality Summary: • A signal between players is a voluntary or involuntary, direct or indi-rect, explicit or implicit exchange of a message; it is usually a declara-tion of stance (“posture”) in the game, i.e., intent to include or excludea strategy from a set of open options. • Strategic moves , e.g. signaling, project some strategic preference with-out making any actual “move”, in order to intimidate or coordinatewith the other player(s). • A bluff is a projection of false information, i.e., exploiting the incom-plete/imperfect information structure of a game to gain some strate-gic advantage that could not be achievable if the game was of com-plete/perfect information. The effectiveness of projecting a strategic stance via signaling, regardless if itis true or bluff, depends heavily on the credibility of that signal, as well asthe credibility of the player itself [46, 44]. When it comes to a single signalor stance, the credibility is closely linked to the level of compatibility of thatsignal or stance with the rationality of the player. Although rationality per semay be only an assumption with regard to one’s opponent, in general terms itis fairly easy to examine the matrix or the tree-graph representation of a gameand establish whether a declared stance is beneficial or not to the associatedplayer. In other words, if that player is assumed to behave rationally, Minimaxstrategies and Nash equilibria can be used to filter out choices that are clearlyexcluded, at least with a high probability.The set of previous stances and/or moves, as well as their associated cred-ibility values, can be used as the history or reputation of that player, whichis in fact the a priori probability for any future stance and/or move of beingconsistent with its previous behavior [27]. Since games of complete and perfectinformation, e.g. Chess, are not compatible with false signaling and bluffs, thetrue theoretical aspect of credibility and reputation is relevant only in gamesof incomplete and/or imperfect information. Hence, Poker players are indeedcharacterized as being cautious or risk-takers according to their reputation onusing bluffs in lower or higher frequency, respectively.A player with a specific reputation can signal a specific stance to the others,projecting either a promise or a threat . A promise is a signal that usuallydeclares the intent to cooperate, i.e., choose the less aggressive approach. Thisis particularly useful when the players need to coordinate in order to avoid muchworse outcomes, as in the games
Leader and
Battle of the Sexes (see Tables 6and 7). On the other hand, a threat is a signal that usually declares the intentto compete, i.e., choose the more aggressive approach. This is still useful as themeans to enforce some kind of coordination, now in the form of extortion ratherthan willful cooperation. The
Chicken game is such any example (see Table 8),where one player must force the other to swerve, in order to naturally end up
Signals, mechanisms & rationality in one of the two Nash equilibria and avoid the worst outcome of crash.As it was mentioned earlier, Prisoner’s Dilemma is a very special type ofgame, since neither Minimax solution or Nash equilibrium points to the mutuallybeneficial option of cooperation; however, if signaling between the prisoners ispossible, i.e., if they are allowed to communicate with each other, cooperationbecomes much more plausible: all they have to do is to promise each other toremain silent and threat to accuse the other as a retaliation if they see the otherdoing such thing. One of the most interesting topics in modern Game Theory isthe study and analytical formulation of the conditions, the constraints and theexact processes of the evolution of cooperation in games like
Prisoner’s Dilemma ,where typical theory fails to predict optimal strategies, although such strategiesseem to exist, usually in accordance to some
Tit-for-Tat variation [1, 2, 27, 7].In any case, whether it is a promise or a threat, the signal or stance is labeledas credible or not. Hence, a credible promise is one that comes from a playerwith a reputation of being consistently reliable in fulfilling that promise, i.e.,actually choosing less aggressive strategies when signaling intent to cooperate.Similarly, a credible threat is one that comes from a player with a reputation ofbeing consistently reliable in fulfilling that threat, i.e., actually choosing moreaggressive strategies when signaling intent to compete [28, 33].
Summary: • Promise is a signal that usually declares the intent to cooperate, i.e.,choose the less aggressive approach; it is useful when players need tocoordinate in order to avoid much worse outcomes. • Threat is a signal that usually declares the intent to compete, i.e.,choose the more aggressive approach; it is useful a player wants toenforce some kind of coordination, in the form of extortion. • Credibility is closely linked to the level of compatibility of a signal orstance with the rationality of the player; in practice, it is a measure(probability) of whether the player will fulfill a promise or a threat, ifnecessary. • Reputation of a player is the a priori probability for any future stanceand/or move of being consistent with its previous behavior. • Credible promises and credible threats are associated with the reputa-tion and credibility of each player, as well as the actual payoffs in thecorresponding game matrix.
As it was mentioned earlier, if that player is assumed to behave rationally, i.e.,trying to minimize losses and maximize gains in terms of actual payoffs in eachoutcome, the credibility of a promise or a threat can be easily established witha high probability. Nevertheless, the fact that this is just a probability and nota perfect forecast comes from the fact that, in turn, the level of rationality of
Signals, mechanisms & rationality Tab. 10:
The typical setup of the
Hostage Situation game with two players.Player-1 is the assaulter and Player-2 is the rescuer-protector.
HostageSituation
Player-2
C D
Player-1 C (2,3) (1,4*) D (4*, *) ( *,1)that player cannot be evaluated perfectly and in exact terms.Rationality and incentives of a player emerge naturally from the exact for-mulation of its own utility function , which is nothing more than a generalizationof the loss/gain function that is described by the matrix or the tree-graph ofthe game [33, 16, 28]. If the formulation of the game’s payoff matrix is perfect,then it is clear when a strategy is optimal for a player and when it is not. How-ever, the truth is that these payoff values may not reflect the exact utility , i.e.,overall loss/gain value for that player, usually due to some “hidden” outcomesor side-effects. For example, a game matrix may describe the payoffs for eachoutcome and each player correctly, but with the assumption that these playersare rational in the same way: winning over their opponent; this may not betrue, e.g. when one player cares more about securing that their opponent doesnot win, rather than securing their own win. In other words, when the play-ers’ rationality is not symmetrically the same, then they do not share the sameutility function and the true payoffs in the game matrix may actually be quitedifferent.A very classic example of such games, assumed to be symmetric when theyare actually asymmetric by nature, is the Hostage Situation , described in ana-lytical form by Table 10. If the two opponents are treated as similarly rational,i.e., symmetric in terms of incentives and behavior, then the game is not muchdifferent than the classic
Chicken , where one must convince the other to swervefirst, in order to avoid the crash. This translates to either the authorities givein to the assaulter’s demands or the assaulter eventually surrenders to the au-thorities, both outcomes assumed to be equally rational, correspondingly, toeach player. However, if for some reason the assaulter is more determined thaninitially presumed, preferring to fight to the death rather than surrendering andending up in jail, then the game is inherently asymmetric and the payoff matrixis quite different, as illustrated in Table 10. What the matrix shows is thatnow Player-1, i.e., the assaulter, has a dominant strategy of always choosing themost aggressive stance, no matter what the authorities choose to do. There isno pure Minimax solution here, since there is no pure saddle-point (see payoffs“3” and “2” in bold); however, there is now a single Nash equilibrium at (4,2),i.e., aggressive assaulter and passive authorities - this is in fact the standard ap-proach internationally in all hostage situations: the authorities start with tryingto establish a communication link and negotiate with the assaulter, rather thanchoosing a rescue operation by direct action that could put the hostages indanger.As it is evident from the
Hostage Situation game of Table 10, the authoritiesare normally guided to a more passive and cooperative approach of negotiatingrather than using force, because the incentive is to protect the hostages at allcosts. This effectively translates to employing a utility function that includes
Signals, mechanisms & rationality Tab. 11:
The typical setup of the
Kamikaze game with two players. Player-1 isthe “kamikaze” and Player-2 is the defender.
Kamikaze
Player-2
C D
Player-1 C (2,3) (1,4*) D (4*, ) ( *, *)a high priority on the hostages’ lives, higher than the immediate capture orincapacitation of the assaulter. Hence, the rationality of Player-2 dictates a morepassive, cooperative stance. This changes drastically if, during this evolution,the lives of hostages are put in severe danger, e.g. when the assaulter poses avery credible threat or actually harms a hostage (assuming there are more). Inthis case, the authorities should change stance and employ the more aggressiveoption, because this is now the optimal response.Table 11 illustrates the Kamikaze game, which is actually a slightly modified
Hostage Situation game in terms of payoff matrix. The game is still asymmetricand the only variation is the swapping of payoff values {2} and {1} for Player-2 (marked in italics), which illustrates the new fact that at this point it ismore harmful for the hostages to remain idle rather than using direct force torescue them, even if this too poses some danger to them - again, this is exactlythe standard approach internationally in all hostage situations: the authoritiesfollow strict rules-of-engagement which state that, once it is established that thelives of hostages is in clear and severe danger, direct action is to be employedimmediately. The same setup emerges when the
Kamikaze game is studiedaccording to its name: when one player (assaulter) is more concerned aboutdamaging the opponent (defender) rather than protecting itself, then there isindeed a dominant strategy of always choosing the most aggressive stance, nomatter what the defender chooses to do. Likewise, the defender is now forced tochoose between its two worst outcomes and naturally chooses the less damagingone, i.e., direct counter-action rather than swerve. Here, the passive stanceis established as more damaging than all-out-conflict, exactly as in
HostageSituation with a very aggressive assaulter. In terms of game analysis, nowthere is indeed a pure Minimax solution at (3,2), which is also the single Nashequilibrium of the game. This explains why there is practically no other rational(strategically optimal) way to defend against a murderous hostage-taker or adesperate kamikaze than employing equally aggressive response.The concepts described along the strategic analysis and “rationalization”of the players in games like
Hostage Situation and
Kamikaze illustrate how aseemingly irrational course of actions can be easily explained and even classifiedas rational behavior, if the proper utility functions are employed. In other words,if the utility of each and every player is defined correctly, then all players in anygame can be labeled as “rational” ones. This proposition is often referred toas “rational irrationality” (valid/explainable behavior), rather than “irrationalrationality” (incomprehensible behavior) [27].
The frontier Summary: • Utility is the generalized cost/gain function of a player in a specificgame, depending on the outcomes but including any “hidden” regardsand side-effects. • Given a specific utility function , a player’s incentives emerge naturallyas the rational behavior of the underlying payoff-optimization process. • A player’s behavior may seem “irrational” if its utility function is in-complete; given a properly defined utility function, a player’s behaviorcan always be labeled as rational per se. • Hostage Situation and
Kamikaze are two examples of (asymmetric)stand-off games where the notion of “rational irrationality” is fullyexplained via proper definition of the corresponding utility functionsfor the assaulter.
This paper included only some of the most basic concepts of Game Theory,including solution methods and representations of typical games of special in-terest, like
Chicken and
Prisoner’s Dilemma . However, these are only a scratchon the surface of what lies beneath, the rigorous mathematical theory and thecomplex, some still unsolved, problems in this extremely interesting and usefulscientific area.All the games and setups presented thus far was somewhat “too perfect”,too simple compared to real-world situations of conflict. There are few caseswhere only two players are involved, their moves are full observable and theirincentives clear and consistent. In most conflicts, groups of players are spiralingin alternating rounds competing and cooperating, each knowing its own utilityfunction and very little about the others’, while signaling, third-party credibilityassertions and continuous bargaining are common things. Is there really a wayGame Theory can address all these aspects in the same clarity, mathematicalrobustness and universality as is does with simple cases of zero-sum and nonzero-sum games like the ones presented previously?The short answer is “No”. Game Theory is the mathematical way to approachsome of the most complex problems the human mind has ever encountered. Forexample, what are the prerequisites, the dynamics and the survivability of theevolution of cooperation as a strategy, in human or animal societies? What isthe asymptotic behavior of such “cooperative” groups? Can they survive in anenvironment of pure competition? These issues are addressed in other aspectsof the theory, namely the
Evolutionary Stable Strategies (ESS), not analyzed inthis study. In short, ESS are patterns of behavior in games of pure competitionand/or possible cooperation, such as the
Prisoner’s Dilemma , that not only mayemerge spontaneously but also survive as optimal strategies in iterative games.
Tit-for-Tat [1, 2] is such an example of ESS in iterated
Prisoner’s Dilemma :cooperation can emerge spontaneously given a set of conditions, primarily (a)
The frontier players “start nicely”, (b) continue with reciprocity, (c) don’t know when thegame finishes. Although it seems simple enough, spontaneous cooperation inconflict situations is one of the most intriguing and theoretically complex prob-lems in Game Theory today.In a slightly simpler scenario, a player may be involved in a game withanother player, while at the same time its strategic choices are relevant to asecond game, with some other player. For example, a politician may be in a“bargain” with voters, trying to gain their support by promising specific actionsif elected, while at the same time a second “bargain” may be taking place inparallel with the party’s main policies and governmental plan if it comes topower. If some of that politician’s promises are on conflict with the party’s mainlines, then as a player is involved in what is called a two-level game [39, 38].This form of gaming was first proposed by Putnam in the late ’70s and it modelstwo-level or multi-level conflict situations in general, where the strategic choicesof a player affect two or more simultaneous games. The solution concepts andequilibria are not much different than those of simple games, but now a strategyis optimal and produces a stable outcome only if it is such simultaneously in allthese games.Another very interesting aspect of gaming in general is the evolution ofstrategies and each player’s behavior as each observes the others’ moves. Insingle-step games, the Minimax solution (pure or mixed) is the one that dic-tates the optimal strategy for each player. The concept of iterative gaming ismuch more general, since it includes cases where the same players may face oneanother in the same single-step games many times in the future. In this case,Nash equilibria predict the most probable outcomes with much better accuracy.But the knowledge that there will be a “next round”, especially when playersalternate moves and one can observe the other before making its own (e.g. inChess), then the game analysis can expand to two or more steps ahead. In prac-tice, the player does not only take into account the strategic choices availableto the opponent(s) but also the “what if” combinations of moves-countermoves.Hence, the corresponding game matrix includes these combinations of compositestates on the opponent(s) side and the payoffs are estimated accordingly. Thistype of composite multi-step setup is often referred to as a metagame [46]. Theextended-form representation of metagames is more natural than the analytical(matrix) form, but the identification of equilibria and solutions is somewhat lessstraight-forward.Some games involve elements of chance regarding the game’s state or partialinformation regarding the observability of each player’s moves. In such gamesof imperfect information, modeling via a game matrix or a tree-graph can beproblematic, since many of the paths may be mutually exclusive and not justalternative choices. In the ’60s, very early on in the history of Game Theory,Harsanyi introduced the so-called Harsanyi transformation [21, 22, 28] for trans-forming a game of incomplete information to an equivalent game of completebut imperfect information. This may not seem much, but in reality there is avery distinct and important difference between them. If a random event dic-tates the exact structure and payoffs of the games, perhaps even the strategicbehavior of the players, then the analysis of such a game is inherently a very dif-ficult task. On the other hand, the Harsanyi transformation models this randomevent as a deterministic one, removing the element of chance and introducingthe notion of “hidden” information about it. In practice, this results in creating
The frontier multiple variations of the game, one for each possible configuration, and treatingthem separately. After they are individually analyzed, solutions and equilibriaare combined together within a probabilistic framework, introducing the moregeneralized concept of Bayesian Nash equilibria [28].In real-world conflict situations it is not uncommon that one or some of theplayers have a different knowledge or “view” of the game structure, its payoffsand the other players’ preferences. This means that each player acts upon itsown payoff matrix, possibly very different in structure and values than the oneused by the other players. Of course, all players are involved in the same, singlegame and the payoffs on each outcome is effectively a single one, despite eachplayer’s unique view of the game. This is extremely important if some of theplayers have a more complete view of the game, i.e., when they address thegame as one of (almost) complete information, while some opponents addressit as one of incomplete information. These special types of conflict are oftenreferred to as hypergames [47, 4]. Introduced by Bennett and Dando in late’70s and later revised in the ’00s by Vane and others, hypergames is a veryefficient way to describe games of asymmetric information between players byemploying different variations of the game matrix or tree-graph, according toeach player’s view. In practice, hypergames are treated the same way as simplegames, with each player deciding its strategic choices according to its own viewand, subsequently, combining the (partial) outcomes together.Game Theory is a vast scientific and research area, based almost entirely onMathematics and some experimental methods, with applications that vary fromsimple board games and auctions to Evolutionary Psychology and Sociology-Biology in group behavior of humans and animals. Although real-world situa-tions reveal that sometimes its predictive value is limited, the robust theoreticalframework and solution concepts provide an extremely valuable set of tools thatclarifies the inner workings and dynamics of conflict situations. The frontier Summary: • In accordance to Nash’s bargaining theorem, cooperation can emergespontaneously , even in competitive games, when a specific set of pre-requisites are satisfied. • Evolutionary stable strategies (ESS) are patterns of behavior in gamesof pure competition and/or possible cooperation that survive as opti-mal strategies in iterative games. • In two-level games , a player may be involved in a game with anotherplayer, while at the same time its strategic choices are relevant to asecond game, with some other player. • Metagames are multi-step game setups where the corresponding gamematrix includes combinations of “what if” composite states, regardingthe future strategic choices of the opponent(s). • The
Harsanyi transformation is used in games of incomplete infor-mation, e.g. when the game structure and payoffs depend on somerandom event, to transform it to an equivalent game of complete butimperfect information. • Hypergames is a very efficient way to describe games of asymmetricinformation between players by employing different variations of thegame matrix or tree-graph, according to each player’s view. • In general, Game Theory is a vast scientific and research area withrobust theoretical foundation that can be used as a predictive tool, aswell as (mostly) an extremely valuable approach to analyze conflictsituations.
Acknowledgement:
This work is dedicated to John F. Nash, pioneer andmathematical genius, who was killed earlier this month on May 23th 2015 in acar accident along with his wife Alicia. His inspirational work and breakthroughideas has changed Game Theory and Economics forever.
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