Abstract: Knowledge requires more than mere true belief, and we also tend to think it is more valuable. I explain the added value that knowledge contributes if its extra ingredient beyond true belief is tracking . I show that the tracking conditions are the unique conditions on knowledge that achieve for those who fulfill them a strict Nash Equilibrium and an Evolutionarily Stable Strategy in what I call the True Belief Game. The added value of these properties, intuitively, (...) includes preparedness and an expectation of survival advantage. On this view knowledge is valuable not because knowledge persists but because it makes the bearer more likely to maintain an appropriate belief state—possibly nonbelief—through time and changing circumstances. When Socrates concluded that knowledge of the road to Larissa was no more valuable than true belief for the purpose of getting to Larissa, he did not take into account that one might want to be prepared for a possible meeting with a misleading sophist along the way, or for the possibility of road work. (shrink)

We study a low-rationality learning dynamics called probe and adjust. Our emphasis is on its properties in games of information transfer such as the Lewis signaling game or the Bala-Goyal network game. These games fall into the class of weakly better reply games, in which, starting from any action profile, there is a weakly better reply path to a strict Nash equilibrium. We prove that probe and adjust will be close to strict Nash equilibria (...) in this class of games with arbitrarily high probability. In addition, we compare these asymptotic properties to short-run behavior. (shrink)

Using belief elicitation, the paper investigates the process of belief formation and evolution in a signaling game in which a common prior is not induced. Both prior and posterior beliefs of Receivers about Senders’ types are elicited, as well as beliefs of Senders about Receivers’ strategies. In the experiment, subjects often start with diffuse uniform beliefs and update them in view of observations. However, the speed of updating is influenced by the strength of initial beliefs. An interesting result (...) is that beliefs about the prior distribution of types are updated slower than posterior beliefs, which incorporate Senders’ strategies. In the medium run, for some specifications of game parameters, this leads to outcomes being significantly different from the outcomes of the game in which a common prior is induced. It is also shown that elicitation of beliefs does not considerably change the pattern of play in this game. (shrink)

David Lewis (1969) introduced sender-receiver games as a way of investigating how meaningful language might evolve from initially random signals. In this report I investigate the conditions under which Lewis signaling games evolve to perfect signaling systems under various learning dynamics. While the 2-state/2- term Lewis signaling game with basic urn learning always approaches a signaling system, I will show that with more than two states suboptimal pooling equilibria can evolve. Inhomogeneous state distributions increase (...) the likelihood of pooling equilibria, but learning strategies with negative reinforcement or certain sorts of mutation can decrease the likelihood of, and even eliminate, pooling equilibria. Both Moran and APR learning strategies (Bereby-Meyer and Erev 1998) are shown to promote successful convergence to signaling systems. A model is presented that illustrates how a language that codes state-act pairs in an order-based grammar might evolve in the context of a Lewis signaling game. The terms, grammar, and the corresponding partitions of the state space co-evolve to generate a language whose structure appears to reflect canonical natural kinds. The evolution of these apparent natural kinds, however, is entirely in service of the rewards that accompany successful distinctions between the sender and receiver. Any metaphysics grounded on the structure of a natural language that evolved in this way would track arbitrary, but pragmatically useful distinctions. (shrink)

Counterexamples to two results by Stalnaker (Theory and Decision, 1994) are given and a corrected version of one of the two results is proved. Stalnaker's proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition ‘strong rationalizability’), then the true (or actual) strategy profile is path equivalent to (...) a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C8 , the set of profiles that survive the iterative deletion of inferior profiles. (shrink)

The traditional solution concept for noncooperative game theory is the Nash equilibrium, which contains an implicit assumption that players’ probability distributions satisfy t probabilistic independence. However, in games with more than two players, relaxing this assumption results in a more general equilibrium concept based on joint beliefs. This article explores the implications of this joint-beliefs equilibrium concept for two kinds of conflictual coordination games: crisis bargaining and public goods provision. We find that, using updating consistent with Bayes’ rule, (...) players’ beliefs converge to equilibria in joint beliefs which do not satisfy probabilistic independence. In addition, joint beliefs greatly expand the set of mixed equilibria. On the face of it, allowing for joint beliefs might be expected to increase the prospects for coordination. However, we show that if players use joint beliefs, which may be more likely as the number of players increases, then the prospects for coordination in these games declines vis-à-vis independent beliefs. (shrink)

We consider modifications to the standard David Lewis signaling game and relax a number of unrealistic implicit assumptions that are often built into the framework. In particular, we motivate and explore various asymmetries that exist between the sender and receiver roles. We find that endowing receivers with a more realistic set of responses significantly decreases the likelihood of signaling, while allowing for unequal selection pressure often has the opposite effect. We argue that the results of this article (...) can also help make sense of a well-known evolutionary puzzle regarding the absence of an evolutionary arms race between sender and receiver in conflict-of-interest signaling games. (shrink)

The article studies properties of interchangeability of pure, mixed, strict, and strict mixed Nash equilibria. The main result is a sound and complete axiomatic system that describes properties of interchangeability in all four settings. It has been previously shown that the same axiomatic system also describes properties of independence in probability theory, nondeducibility in information flow, and non-interference in concurrency theory.

We develop a logical system that captures two different interpretations of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilibria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in (...) combination with a principle of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausible belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility. (shrink)

In classical game theory the idea that players randomize between their actions according to a particular optimal probability distribution has always been viewed as puzzling. In this paper, we establish a fundamental connection between n-person normal form games and quantum mechanics, which eliminates the conceptual problems of these random strategies. While the two theories have been regarded as distinct, our main theorem proves that if we do not give any other piece of information to a player in a (...) class='Hi'>game, than the payoff matrix—the axiom of “no-supplementary data” holds—then the state of mind of a rational player is algebraically isomorphic to a pure quantum state. The “no supplementary data” axiom is captured in a Lukasiewicz’s three-valued Kripke semantics wherein statements about whether a strategy or a belief of a player is rational are initially indeterminate i.e. neither true, nor false. As a corollary, we show that in a mixed Nash equilibrium, the knowledge structure of a player implies that probabilities must verify the standard “Born rule” postulate of QM. The puzzling “indifference condition” wherein each player must be rationally indifferent between all the pure actions of the support of his equilibrium strategy is resolved by his state of mind being described by a “quantum superposition” prior a player is asked to make a definite choice in a “measurement”. Finally, these results demonstrate that there is an intrinsic limitation to the predictions of game theory, on a par with the “irreducible randomness” of quantum physics. (shrink)

This paper examines the existence of strategic solutions to finite normal form games under the assumption that strategy choices can be described as choices among lotteries where players have security- and potential level preferences over lotteries (e.g., Cohen, Theory and Decision, 33, 101–104, 1992, Gilboa, Journal of Mathematical Psychology, 32, 405–420, 1988, Jaffray, Theory and Decision, 24, 169–200, 1988). Since security- and potential level preferences require discontinuous utility representations, standard existence results for Nash equilibria in mixed strategies ( (...) class='Hi'>Nash, Proceedings of the National Academy of Sciences, 36, 48–49, 1950a, Non-Cooperative Games, Ph.D. Dissertation, Princeton University Press, 1950b) or for equilibria in beliefs (Crawford, Journal of Economic Theory, 50, 127–154, 1990) do not apply. As a key insight this paper proves that non-existence of equilibria in beliefs, and therefore non-existence of Nash equilibria in mixed strategies, is possible in finite games with security- and potential level players. But, as this paper also shows, rationalizable strategies (Bernheim, Econometrica, 52, 1007–1028, 1984, Moulin, Mathematical Social Sciences, 7, 83–102, 1984, Pearce, Econometrica, 52, 1029–1050, 1984) exist for such games. Rationalizability rather than equilibrium in beliefs therefore appears to be a more favorable solution concept for games with security- and potential level players. (shrink)

ExtractBrian Skyrms's Signals has the virtues familiar from his Evolution of the Social Contract and The Stag Hunt. He begins with a very simple model of agents in interaction, and in a series of brief and beautifully clear chapters, this model and its successors are explored, elaborated, connected and illustrated through biological theory and the social sciences. Signals borrows its core model from David Lewis: it is Lewis's signalling game. In this game, two agents interact. One agent can (...) observe which of two equi-probable states the world is in, but that agent cannot act directly and profitably on that information. However, the informed agent can act in a way that will be perceptually salient to a second agent: say, by raising a red or a green flag. The second agent does have the capacity to respond appropriately to each state of the world. If that second agent chooses the right option, given the state of the world, both are rewarded. If the second agent fails to choose the right action, neither are. Obviously, the two agents are best off if they have a practice in which the informed agent regularly chooses a distinct, salient cue in response to each of the two world states, and in which the powerful agent uses that cue to select the rewarding act. Less obviously, agents with simple trial and error learning capacities can learn to signal and respond: neither explicit negotiation nor cognitive sophistication are required. Likewise, if individual agents do not have the capacity to learn, but if they breed true but with some variation, the evolutionary version of trial and error learning can take a population to one of the signalling system equilibria. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.A GLASS HALF-FULL: BRIAN SKYRMS'S SIGNALSVolume 28, Issue 1Kim Sterelny DOI: https://doi.org/10.1017/S0266267112000120Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. A GLASS HALF-FULL: BRIAN SKYRMS'S SIGNALSVolume 28, Issue 1Kim Sterelny DOI: https://doi.org/10.1017/S0266267112000120Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. A GLASS HALF-FULL: BRIAN SKYRMS'S SIGNALSVolume 28, Issue 1Kim Sterelny DOI: https://doi.org/10.1017/S0266267112000120Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Since the seminal paper of Nash (Proc Natl Acad Sci USA 36:48–49, 1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax) strategies. In a recent paper of Pruzhansky (Int J Game Theory 40:351–365, 2011) it was shown that under fairy general conditions maximin strategies in completely mixed games can guarantee the same expected payoff as completely mixed Nash equilibrium strategies. Based on this finding, the current paper argues that maximin strategies have (...) important properties. For instance, maximin strategies may refine Nash equilibria in subjective mixed strategies. Further, Bayesian rationality of the players may favor maximin strategies more often than Nash equilibrium strategies. The paper concludes with several suggestions for further experimental research that may shed more light on whether maximin behavior can explain reality better than Nash equilibrium. (shrink)

Common wisdom holds that communication is impossible when messages are costless and communicators have totally opposed interests. This article demonstrates that such wisdom is false. Non-convergent dynamics can sustain partial information transfer even in a zero-sum signalling game. In particular, I investigate a signalling game in which messages are free, the state-act payoffs resemble rock–paper–scissors, and senders and receivers adjust their strategies according to the replicator dynamic. This system exhibits Hamiltonian chaos and trajectories do not converge to (...) class='Hi'>equilibria. This persistent out-of-equilibrium behaviour results in messages that do not perfectly reveal the sender's private information, but do transfer information as quantified by the Kullback–Leibler divergence. This finding shows that adaptive dynamics can enable information transmission even though messages at equilibria are meaningless. This suggests a new explanation for the evolution or spontaneous emergence of meaning: non-convergent adaptive dynamics. (shrink)

The problem of cooperation for rational actors comprises two sub problems: the problem of the intentional object (under what description does each actor perceive the situation?) and the problem of common knowledge for finite minds (how much belief iteration is required?). I will argue that subdoxastic signalling can solve the problem of the intentional object as long as this is confined to a simple coordination problem. In a more complex environment like an assurance game signals may become unreliable. Mutual (...) beliefs can then bolster the earlier attained equilibrium. I will first address these two problems by means of an example, in order to draw some more general lessons about combining evolutionary theory and rationality later on. (shrink)

We introduce a new class of population games called stable games. These games are characterized by self-defeating externalities: when agents revise their strategies, the improvements in the payoffs of strategies to which revising players are switching are always exceeded by the improvements in the payoffs of strategies which revising players are abandoning. We show that stable games subsume many well-known classes of examples, including zero-sum games, games with an interior ESS, wars of attrition, and concave potential games. We prove that (...) the set of Nash equilibria of any stable game is convex, and offer an elementary proof of existence of equilibrium. Finally, we show that the set of Nash equilibria of a stable game is globally asymptotically stable under a variety of evolutionary dynamics. These convergence results are proved by constructing Lyapunov functions defined in terms of revision potentials—that is, potential functions for the protocols agents follow when they consider switching strategies. (shrink)

In this paper, I develop a syntactic framework for the analysis ofstrategic form games that is based on a straightforward combination ofstandard systems of doxastic, probabilistic and conditionalpropositional logic. In particular, for the probabilistic part I makeuse of the axiomatization provided in Fagin and Halpern (1994). The use ofconditionals allows to represent a strategic form game by a logicalformula in a very natural way. Also expected utility maximization can benaturally captured. I use this framework to prove a version of (...) a resulton Nash equilibrium conjectures first presented in Aumann and Brandenburger (1995). (shrink)

In this article three different types of loss aversion equilibria in bimatrix games are studied. Loss aversion equilibria are Nash equilibria of games where players are loss averse and where the reference points—points below which they consider payoffs to be losses—are endogenous to the equilibrium calculation. The first type is the fixed point loss aversion equilibrium, introduced in Shalev (2000; Int. J. Game Theory 29(2):269) under the name of ‘myopic loss aversion equilibrium.’ There, the players’ (...) reference points depend on the beliefs about their opponents’ strategies. The second type, the maximin loss aversion equilibrium, differs from the fixed point loss aversion equilibrium in that the reference points are only based on the carriers of the strategies, not on the exact probabilities. In the third type, the safety level loss aversion equilibrium, the reference points depend on the values of the own payoff matrices. Finally, a comparative statics analysis is carried out of all three equilibrium concepts in 2 × 2 bimatrix games. It is established when a player benefits from his opponent falsely believing that he is loss averse. (shrink)

The paper argues that the Nash Equilibrium Refinement Programme was less successful than its competitor, the Epistemic Programme. The prime criterion of success is the extent to which the programmes were able to reach the key objective guiding non-cooperative game theory for much of the twentieth century, namely, to develop a complete characterisation of the strategic rationality of economic agents in the form of the ultimate solution concept for any normal form and extensive game. The paper explains (...) this in terms of unjustified degrees of mathematisation in Nash Equilibrium Refinement Programme. While this programme’s mathematical models were often inspired by purely mathematical concerns rather than the economic phenomena they were intended to model, the Epistemic Programme’s models were developed with a keen eye to the role beliefs and desires play in strategic interaction between rational economic agents playing games; that is, their interactive epistemology. The Epistemic Programme succeeded in developing mathematical models formalising aspects of strategic interaction that remained implicit in the Nash Equilibrium Refinement Programme owing to an unjustified degree of mathematisation. As a result, the Epistemic Programme is the more successful theory.Keywords: Epistemic Programme; Game theory; Interactive epistemology; Mathematisation; Nash equilibrium. (shrink)

I introduce an extension of the Lewis-Skyrms signaling game, analysed from a dynamical perspective via simple reinforcement learning. In Lewis’ (Convention, Blackwell, Oxford, 1969) conception of a signaling game, salience is offered as an explanation for how individuals may come to agree upon a linguistic convention. Skyrms (Signals: evolution, learning & information, Oxford University Press, Oxford, 2010a) offers a dynamic explanation of how signaling conventions might arise presupposing no salience whatsoever. The extension of the atomic (...) signaling game examined here—which I will refer to as a salience game—introduces a variable parameter into the atomic signaling game which allows for degrees of salience, thus filling in the continuum between Skyrms’ and Lewis’ models. The model does not presuppose any salience at the outset, but illustrates a process by which accidentally evolved salience is amplified, to the benefit of the players. It is shown that increasing degrees of salience allow populations to avoid sub-optimal pooling equilibria and to coordinate upon conventions more quickly. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:"And Her Substance Would Be Mine":Envy, Hate, and Ontological Evacuation in Josephine Hart's SinA. Samuel Kimball (bio)Envy involuntarily testifies to a lack of being that puts the envious to shame.—René Girard, A Theatre of EnvySin, offspring of snt-ya, "that which is," in Germanic sun(d)jo, "it is true," "the sin is real," and ultimately from es-, "to be," source of am, is, sooth, soothe; of the Sanskrit roots sat- and (...) sant-, "existing, true, virtuous"; of the Latin, esse, "to be," from which is derived such words as essence, presence, absence, and represent; of the Greek roots einei, "to be" as well as ont-, "being," which provides the name for the philosophical study of being itself, ontology.—Adapted from The American Heritage Dictionaryof Indo-European RootsI: Ruth Garton's Envious NarrationThe etymology of sin suggests that the evil it names reveals something fundamental about being, something ruinous and yet unavoidable, something that endangers or even nullifies being and yet is inseparable from it, even essential to it. In Josephine Hart's second novel, Sin, this something is a species of Girardian desire in the form of a monstrous envy, which impels the first-person narrator, Ruth Garton, to a lifelong hatred of her cousin and stepsister, Elizabeth. The focus of Ruth's confessional narrative, envy, drives Ruth to take covert and increasingly virulent revenge against her quasi-sibling in a self-deluding attempt to appropriate not only the objects of Elizabeth's desire but ultimately Elizabeth's desire itself and the apparent ontological fullness that Ruth attributes to it. Over the course of her life, then, Ruth loses the promise [End Page 239] of her name in her ruthless pursuit of her enemy. As she becomes increasingly frustrated in her efforts to destroy her sister's life, Ruth finally turns her hatred of her nemesis against herself as well as the readers to whom she appeals for understanding. If envy heightens Ruth's consciousness to the point where she can perceive that her desire signals her ontological emptiness, nevertheless, it so captures her, so dominates her emotional and cognitive life, that she is unable to find a way to affirm her own—or anyone else's—being in the world. The result is a spectacular narrative record of the cost of recognizing but being unable to resist the mimeticism of desire.1II. The Girardian Dynamic of SinAs Girard has demonstrated, one of the most difficult existential challenges is to understand that desire is mimetically mediated. The result is that we come into our desire only through someone else's and always want something more and other than we think we want. Behind every move to possess a particular object is a largely unrecognized and in principle unsatisfiable desire for absolute satisfaction in the form of ontological completeness, self-sufficiency, intactness. On the one hand, Girard writes, "by the example of his own desire…the model conveys to the subject the supreme desirability of the object" (1977, 146). On the other hand, since to desire mimetically means to come into desire through the other rather than through oneself, by this same example the model conveys to the subject the contingency, the dependency, the secondariness, hence the emptiness of the subject's desire, even the subject's lack of desire, the lack that, paradoxically, constitutes his or her desire. Therefore, to be subjected to the other's want ultimately has the effect of signaling the seeming priority, fullness, and self-origination of this other person's desire. For this reason Girard asserts that "man... desires being, something he himself lacks and which some other person seems to possess" (146).If at first the other person models the desire that the subject lacks, it is in the logic of mimesis that the other will become a rival and, unless the rivalry is somehow interrupted or abrogated, inevitably an enemy to be destroyed. The subject will become locked into a potentially deadly game of trying to obtain the same object that the model desires, so as to experience the satisfactions—ultimately, the sense of being—that the object in question seems to provide... (shrink)

#1 NEW YORK TIMES BESTSELLER • A bold work from the author of The Black Swan that challenges many of our long-held beliefs about risk and reward, politics and religion, finance and personal responsibility In his most provocative and practical book yet, one of the foremost thinkers of our time redefines what it means to understand the world, succeed in a profession, contribute to a fair and just society, detect nonsense, and influence others. Citing examples ranging from Hammurabi to Seneca, (...) Antaeus the Giant to Donald Trump, Nassim Nicholas Taleb shows how the willingness to accept one’s own risks is an essential attribute of heroes, saints, and flourishing people in all walks of life. As always both accessible and iconoclastic, Taleb challenges long-held beliefs about the values of those who spearhead military interventions, make financial investments, and propagate religious faiths. Among his insights: • For social justice, focus on symmetry and risk sharing. You cannot make profits and transfer the risks to others, as bankers and large corporations do. You cannot get rich without owning your own risk and paying for your own losses. Forcing skin in the game corrects this asymmetry better than thousands of laws and regulations. • Ethical rules aren’t universal. You’re part of a group larger than you, but it’s still smaller than humanity in general. • Minorities, not majorities, run the world. The world is not run by consensus but by stubborn minorities imposing their tastes and ethics on others. • You can be an intellectual yet still be an idiot. “Educated philistines” have been wrong on everything from Stalinism to Iraq to low-carb diets. • Beware of complicated solutions (that someone was paid to find). A simple barbell can build muscle better than expensive new machines. • True religion is commitment, not just faith. How much you believe in something is manifested only by what you’re willing to risk for it. The phrase “skin in the game” is one we have often heard but rarely stopped to truly dissect. It is the backbone of risk management, but it’s also an astonishingly rich worldview that, as Taleb shows in this book, applies to all aspects of our lives. As Taleb says, “The symmetry of skin in the game is a simple rule that’s necessary for fairness and justice, and the ultimate BS-buster,” and “Never trust anyone who doesn’t have skin in the game. Without it, fools and crooks will benefit, and their mistakes will never come back to haunt them.”. (shrink)

Nash equilibria with identical supports are compared for bimatrix games that are different with respect to the risk aversion of player 2. For equilibria in 2× 2-bimatrix games and for equilibria with efficient supports in coordination games it is established for which cases increased risk aversion of player 2 benefits or hurts player 2.

Nash Equilibrium is a central concept ingame theory. It has been argued that playing NashEquilibrium strategies is rational advice for agentsinvolved in one-time strategic interactions capturedby non-cooperative game theory. This essaydiscusses arguments for that position: vonNeumann–Morgenstern's argument for their minimaxsolution, the argument from self-enforcingagreements, the argument from the absence ofprobabilities, the transparency-of-reasons argument,the argument from regret, and the argument fromcorrelated equilibrium. All of these argumentseither fail entirely or have a very limited scope.Whatever the use of Nash Equilibrium (...) is, therefore,it is not useful as a rational recommendation inone-time strategic interactions. This is good newsfor Bayesians: although this discussion does notargue directly for the Bayesian idea of rationalityas expected utility maximization, it argues againsta position that has been regarded as a contender insituations of strategic interaction. (shrink)

A strategic equilibrium is a profile of strategies that are each self-supporting given the profile. Strategic equilibria exist in games without Nash equilibria.

When von Neumann's and Morgenstern's Theory of Games and Economic Behavior appeared in 1944, one thought that a complete theory of strategic social behavior had appeared out of nowhere. However, game theory has, to this very day, remained a fast-growing assemblage of models which have gradually been united in a new social theory - a theory that is far from being completed even after recent advances in game theory, as evidenced by the work of the three Nobel Prize (...) winners, John F. Nash, John C. Harsanyi, and Reinhard Selten. Two of them, Harsanyi and Selten, have contributed important articles to the present volume. This book leaves no doubt that the game-theoretical models are on the right track to becoming a respectable new theory, just like the great theories of the twentieth century originated from formerly separate models which merged in the course of decades. For social scientists, the age of great discover ies is not over. The recent advances of today's game theory surpass by far the results of traditional game theory. For example, modem game theory has a new empirical and social foundation, namely, societal experiences; this has changed its methods, its "rationality. " Morgenstern (I worked together with him for four years) dreamed of an encompassing theory of social behavior. With the inclusion of the concept of evolution in mathematical form, this dream will become true. Perhaps the new foundation will even lead to a new name, "conflict theory" instead of "game theory. (shrink)

Brian Skyrms constructs a theory of "dynamic deliberation" and uses it to investigate rational decision-making in cases of strategic interaction. This illuminating book will be of great interest to all those in many disciplines who use decision theory and game theory to study human behavior and thought. Skyrms begins by discussing the Bayesian theory of individual rational decision and the classical theory of games, which at first glance seem antithetical in the criteria used for determining action. In his effort (...) to show how methods for dealing with information feedback can be productively combined, the author skillfully leads us through the mazes of equilibrium selection, the Nash equilibria for normal and extensive forms, structural stability, causal decision theory, dynamic probability, the revision of beliefs, and, finally, good habits for decision. The author provides many clarifying illustrations and a handy appendix called "Deliberational Dynamics on Your Personal Computer." His powerful model has important implications for understanding the rational origins of convention and the social contract, the logic of nuclear deterrence, the theory of good habits, and the varied strategies of political and economic behavior. (shrink)

Inspired by Shalev’s model of loss aversion, we investigate the effect of loss aversion on a bimatrix game where the payoffs in the bimatrix game are characterized by triangular fuzzy variables. First, we define three solution concepts of credibilistic loss aversion Nash equilibria, and their existence theorems are presented. Then, three sufficient and necessary conditions are given to find the credibilistic loss aversion Nash equilibria. Moreover, the relationship among the three credibilistic loss aversion (...) class='Hi'>Nash equilibria is discussed in detail. Finally, for2×2bimatix game with triangular fuzzy payoffs, we investigate the effect of loss aversion coefficients and confidence levels on the three credibilistic loss aversion Nash equilibria. It is found that an increase of loss aversion levels of a player leads to a decrease of his/her own payoff. We also find that the equilibrium utilities of players are decreasing as their own confidence levels when players employ the optimistic value criterion. (shrink)

This article deals with the reframing of the concept of universal mediated communication on a global scale. Subjects include the following: the universe has a conscious force field at all its scales, requiring continuous inter-scale communication of information; the field exhibits distinct electromagnetic frequencies associated with the building blocks of life; and advances in the technology of sound production with electronic synthesizers can be applied to study mechanisms of such universal communication. The question being addressed is how electronic synthesizers can (...) be used to create sonic xenolinguistic expressions when composing with sonified frequencies obtained from bio-organisms, human protein structures, phonemes and various types of interstellar frequencies that include, but are not limited to, gamma bursts, pulsars and neutron stars. The objective is to construct sonic xenolingusitic expressions that can act like feedback loops within the field of consciousness. These sonic expressions can include anomalous cognitive, emotive or spiritual experiences that may be emerging out of the universal field of consciousness in real time. Xenolinguistics is a hypothetical language used to communicate with other sentient beings in the universe. The rationale for composing sonic xenolingustic expressions is the belief that the universe is considered conscious and that there is a field of consciousness existing throughout the universe that in principle is also instrumental in how communication can evolve with extraterrestrials. Outlines of other attempts at extraterrestrial communication by other researchers will be offered based on current literature. The probability that other life forms have established some mode of information exchange that is detectable and that, in return, may be able to create a form of communication to respond will be covered. It is postulated that various energy fields existing before and after the creation of our universe interacted with one another becoming abiotic, and then over time, formed even more complex structures. It is these complex structures that were then able to communicate essential information for maintaining their stability and survivability by using specific signalling frequencies. It is hypothesized that first life in the universe emerged within an already existing field of information and that our consciousness is thereby able to establish communication with other forms of consciousness in the universe. It is on this basis that a novel scientific and philosophical paradigm is considered in which the universe is conceived of as an active information matrix that can entertain potential extraterrestrial communication. In this framework, the present article will offer a new approach to orchestrated electronic sounds as a means to such communication. By integrating quantum physical properties into that sonic framework, a new study emerged on how sonic xenolinguistic elements can be incorporated into electronic composition techniques. This new framework may produce new algorithms for musical expression that not only represent an innovative future of electronic music but may also be crucial in the formulation and induction of new states of consciousness that will enable humankind to directly participate in the further becoming of the universe as a result of the evolution of this universal field of consciousness. (shrink)

Lewis signaling games illustrate how language might evolve from random behavior. The probability of evolving an optimal signaling language is, in part, a function of what learning strategy the agents use. Here we investigate three learning strategies, each of which allows agents to forget old experience. In each case, we find that forgetting increases the probability of evolving an optimal language. It does this by making it less likely that past partial success will continue to reinforce suboptimal practice. (...) The learning strategies considered here show how forgetting past experience can promote learning in the context of games with suboptimal equilibria. (shrink)

Deliberation about what to do in any context requires reasoning about what will or would happen in various alternative situations, including situations that the agent knows will never in fact be realized. In contexts that involve two or more agents who have to take account of each others' deliberation, the counterfactual reasoning may become quite complex. When I deliberate, I have to consider not only what the causal effects would be of alternative choices that I might make, but also what (...) other agents might believe about the potential effects of my choices, and how their alternative possible actions might affect my beliefs. Counterfactual possibilities are implicit in the models that game theorists and decision theorists have developed – in the alternative branches in the trees that model extensive form games and the different cells of the matrices of strategic form representations – but much of the reasoning about those possibilities remains in the informal commentary on and motivation for the models developed. Puzzlement is sometimes expressed by game theorists about the relevance of what happens in a game ‘off the equilibrium path’: of what would happen if what is both true and known by the players to be true were instead false. (shrink)

This paper takes a novel approach to studying the existence and stability of Nash equilibria in N-firm Cournot–Bertrand oligopolies. First, we show that such games can be monotonically embedded into a game of strategic heterogeneity, so that each firm best responds to the choices of all other firms in a monotonic way. We then show that this monotonicity can be exploited to derive conditions which guarantee the existence of a unique, dominance solvable Nash equilibrium which is (...) stable under all adaptive dynamics. These conditions constitute a strict improvement over existing results in the literature. Finally, we examine the effect on these conditions resulting from additional firms entering the market. (shrink)

We observe that in certain two-player repeated games of incomplete information, where information may be incomplete on both sides, it is possible for an informed player to signal his status as an informed player to the other without revealing any information about the choice of chance. The key to obtaining such a class of games is to relax the assumption that the players’ moves are observable. We show that in such cases players can achieve a kind of signaling that (...) is “zero-knowledge”, in the sense that the other player becomes convinced that her opponent is informed without ever learning the choice of chance. Moreover, such “zero-knowledge signaling” has all of the statistical properties associated with zero-knowledge proofs in intereactive protocols. In particular, under the general assumption that moves are unobservable, such signaling leads to a class of equilibria in repeated games that are separatingin regard to the status of player 1–informed or uninformed–but only for player 2; any other player in a network, being unable to observe the moves of player 2, remains uncertain as to the status of player 1. (shrink)

The order of stages in a multistage game is often interpreted by looking at earlier stages as involving more long term decisions. For the purpose of making this interpretation precise, the notion of a delay supergame of a bounded multistage game is introduced. A multistage game is bounded if the length of play has an upper bound. A delay supergame is played over many periods. Decisions on all stages are made simultaneously, but with different delays until they (...) become effective. The earlier the stage the longer the delay. A subgame perfect equilibrium of a bounded multistage game generates a subgame perfect equilibrium in every one of its delay supergames. This is the first main conclusion of the paper. A subgame perfect equilibrium set is a set of subgame perfect equilibria all of which yield the same payoffs, not only in the game as a whole, but also in each of its subgames. The second xmain conclusion concerns multistage games with a unique subgame perfect equilibrium set and their delay supergames which are bounded in the sense that the number of periods is finite. If a bounded multistage game has a unique subgame perfect equilibrium set, then the same is true for every one of its bounded delay supergames. Finally the descriptive relevance of multistage game models and their subgame perfect equilibria is discussed in the light of the results obtained. (shrink)

In the standard formulation of game theory, agents use mixed strategies in the form of objective and probabilistically precise devices to conceal their actions. We introduce the larger set of probabilistically imprecise devices and study the consequences for the basic results on normal form games. While Nash equilibria remain equilibria in the extended game, there arise new Ellsberg equilibria with distinct outcomes, as we illustrate by negotiation games with three players. We characterize Ellsberg (...) class='Hi'>equilibria in two-person conflict and coordination games. These equilibria turn out to be related to experimental deviations from Nash equilibrium play. (shrink)

In studies of multi-agent interaction, especially in game theory, the notion of equilibrium often plays a prominent role. A typical scenario for the belief merging problem is one in which several agents pool their beliefs together to form a consistent "group" picture of the world. The aim of this paper is to define and study new notions of equilibria in belief merging. To do so, we assume the agents arrive at consistency via the use of a social belief (...) removal function, in which each agent, using his own individual removal function, removes some belief from his stock of beliefs. We examine several notions of equilibria in this setting, assuming a general framework for individual belief removal due to Booth et al. We look at their inter-relations as well as prove their existence or otherwise. We also show how our equilibria can be seen as a generalisation of the idea of taking maximal consistent subsets of agents. (shrink)

In its classical conception, game theory aspires to be a determinate decision theory for games, understood as elements of a structurally specified domain. Its aim is to determine for each game in the domain a complete solution to each player's decision problem, a solution valid for all real-world instantiations, regardless of context. "Permissiveness" would constrain the theory to designate as admissible for a player any conjecture consistent with the function's designation of admissible strategies for the other players. Given (...) permissiveness and other appropriate constraints, solution sets must contain only Nash equilibria and at least one pure-strategy equilibrium, and there is no solution to games in which no symmetry invariant set of pure-strategy equilibria forms a Cartesian product. These results imply that the classical program is unrealizable. Moreover, the program is implicitly committed to permissiveness, through its common-knowledge assumptions and its commitment to equilibrium. The resulting incoherence deeply undermines the classical conception in a way that consolidates a long series of contextualist criticisms. (shrink)

In this paper we present a concept of similarity in games, on which to ground alternative solution concepts, some of which differ from the classical notions in the field. In order to do this we impose a constraint on players’ beliefs that amounts to a variant of the well-known symmetry principle in classical bargaining theory. We show how this similarity relation helps to identify different Nash equilibria in games, and how these “similar Nash equilibria” can be (...) extended to non-symmetric games. While the notion is normative, it is nonetheless inspired by phenomena in which similarities between players lead to outcomes detected in behavioral studies. We study the strategic properties of the concept of similarity and discuss its relationships with Hofstadter’ notion of superrationality. (shrink)

Agents in a large population are randomly matched to play a certain game, payoffs in which represent fitness. Agents may have preferences that are different from fitness. They learn strategies according to their preferences, and evolution changes the preference distribution in the population according to fitness. When agents know the preferences of the opponent in a match, only efficient symmetric strategy profiles of the fitness game can be stable. When agents do not know the preferences of the opponent, (...) only Nash equilibria of the fitness game can be stable. For 2 × 2 symmetric games I characterize preferences that are stable. (shrink)

I consider games with ambiguous payoffs played by non-Expected Utility decision makers. Three equilibrium solutions are studied. Nash equilibrium in which equilibrium mixed strategies must be best responses, Crawford equilibrium in beliefs and pure equilibrium in beliefs in which equilibrium strategies are mixtures of best responses, with the latter restricting best responses to pure actions. I study the interactions between ambiguity preferences on one side and equilibrium properties on the other. I show how the equilibrium concepts differ, computing necessary (...) and sufficient conditions for existence and equivalence. I also show how these solution concepts fare against two fundamental principles of Nash equilibrium in standard games: the principle of indifference and the reduction principle. Given both are central to the computation of Nash equilibrium in games with Expected Utility players, their failure indicates how relaxing the Expected Utility hypothesis may disrupt standard game theoretic results such as the characterization of equilibria in two-player games. (shrink)

It is plausible that there are epistemic reasons bearing on a distinctively epistemic standard of correctness for belief. It is also plausible that there are a range of practical reasons bearing on what to believe. These theses are often thought to be in tension with each other. Most significantly for our purposes, it is obscure how epistemic reasons and practical reasons might interact in the explanation of what one ought to believe. We draw an analogy with a similar distinction between (...) types of reasons for actions in the context of activities. The analogy motivates a two-level account of the structure of normativity that explains the interaction of correctness-based and other reasons. This account relies upon a distinction between normative reasons and authoritatively normative reasons. Only the latter play the reasons role in explaining what state one ought to be in. All and only practical reasons are authoritative reasons. Hence, in one important sense, all reasons for belief are practical reasons. But this account also preserves the autonomy and importance of epistemic reasons. Given the importance of having true beliefs about the world, our epistemic standard typically plays a key role in many cases in explaining what we ought to believe. In addition to reconciling (versions of) evidentialism and pragmatism, this two-level account has implications for a range of important debates in normative theory, including the interaction of right and wrong reasons for actions and other attitudes, the significance of reasons in understanding normativity and authoritative normativity, the distinction between ‘formal’ and ‘substantive’ normativity, and whether there is a unified source of authoritative normativity. (shrink)

Game-theoretic solution concepts such as Nash and Bayesian equilibrium start from an assumption that the players’ sets of possible payoffs, measured in units of von Neumann–Morgenstern utility, are common knowledge, and they go on to define rational behavior in terms of equilibrium strategy profiles that are either pure or independently randomized and which, in applications, are often taken to be uniquely determined or at least tightly constrained. A mechanism through which to obtain a common knowledge of payoff functions (...) measured in units of utility is not part of the model. This paper describes an operational method of constructing a state of common knowledge of the key parameters of the players’ utility functions in terms of conditional small bets on the game’s outcome. When the rationality criterion of joint coherence is applied in this setting, the solution of a game is typically characterized by a convex set of correlated equilibria. In the most general case, where players are risk averse, the parameters of the equilibria are risk-neutral probabilities, interpretable as products of subjective probabilities and relative marginal utilities for money, as in financial markets. Risk aversion generally enlarges the set of equilibria and may present opportunities for Pareto-improving modifications of the rules of the game. (shrink)

We relax the assumption that priors are common knowledge, in the standard model of games of incomplete information. We make the realistic assumption that the players are boundedly rational: they base their actions on finite-order belief hierarchies. When the different layers of beliefs are independent of each other, we can retain Harsányi's type-space, and we can define straightforward generalizations of Bayesian Nash Equilibrium and Rationalizability in our context. Since neither of these concepts is quite satisfactory, we propose a hybrid (...) concept, Mirage Equilibrium, providing us with a practical tool to work with inconsistent belief hierarchies. When the different layers of beliefs are correlated, we must enlarge the type-space to include the parametric beliefs. This presents us with the difficulty of the inherent openness of finite belief subspaces. Appealing to bounded rationality once more, we posit that the players believe that their opponent holds a belief hierarchy one layer shorter than they do and we provide alternative generalizations of BNE and Rationalizability. Finally, we show that, when beliefs are degenerate point beliefs, the definition of Mirage Equilibrium coincides with that of the generalized BNE. (shrink)

A syntactic formalism for the modeling of belief revision in perfect information games is presented that allows to define the rationality of a player's choice of moves relative to the beliefs he holds as his respective decision nodes have been reached. In this setting, true common belief in the structure of the game and rationality held before the start of the game does not imply that backward induction will be played. To derive backward induction, a “forward belief” condition (...) is formulated in terms of revised rather than initial beliefs. Alternative notions of rationality as well as the use of knowledge instead of belief are also studied within this framework. Footnotes1 I would like to thank Wlodek Rabinowicz and three anonymous referees for very helpful comments. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:ESSE AS THE TARGET OF JUDGMENT IN RAHNER AND AQUINAS 0 NE OF THE commanding currents of thought in Catholic circles since the Second Vatican Council has been Transcendental Thomism. Though its proponents differ among themselves, it is safe to say that the common inspiration is that Thomistic metaphysical conclusions can be arrived at through a Kantian-style transcendental method. The emphasis is on the knower's conditions of knowing, not (...) upon the known. Karl Rahner was a significant figure in this movement. Rahner 's popufarity is undoubtedly connected with his ability cogently to work out attractive theological positions from his version of Transcendental Thomism. In any case, his theology does have this basis. It is a basis that receives extensive elaboration in Rahner's Spirit in the World. This article critically investigates that basis. My conclusion is that Spirit in the World contains a massive misconstrual of the esse grasped in judgment. This is no esoteric point of Thomistic exegesis. A correct appreciation of the esse targeted in judgment shows that no necessity for making a transcendental turn exists. As will be noted, such a turn, at least in Rahner's case, ineluctably leads to a curtailment of metaphysics. Kant is not defeated at his own game. I. The heart of Rahner's philosophy is his understanding of abstraction. Representative descriptions of this understanding are the following. Speaking of the function of the agent intellect, Rahner says, We must therefore ask how the agent intellect is to be understood so that it can know the form as limited, confined, and thus " ESSE " AS THE TARGET OF JUDGMENT 223 of itself embracing further possibilities in itself, as bordering upon a broader field of possibilities. Obviously this is possible only if, antecedent to and in addition to apprehending the individual form, the agent intellect comprehends of itself the whole field of these possibilities and thus, in the sensibly concretized form, experiences the concreteness as limitation of these possibilities, whereby it knows the form itself as able to be multiplied in this field. This transcending apprehension of further possibilities, through which the form possessed in a concretion in sensibility is apprehended as limited and so is abstracted, we call "pre-apprehension." 1 The agent intellect grasps the universal because it pre-apprehends the form so that the form's apparent limitation in the sensible thing reveals the form's in itself universality. Again, The intellect abstracts the forms given in concretion in sensibility insofar as it knows their limitation produced by the already realized synthesis of sensibility, and it does this by the fact that in apprehending them, it comprehends the breadth of their possibilities in a pre-apprehension.2 Finally, Therefore the form must be known as limited by the "this" whose form it is; only then can it be known that it is "broader" in itself and so able to be related to other "this's," and then for the first time can it appear as ontologically different from matter, from its supposit, and as universal in itself.3 I understand Rahner to be saying this. To gras:p a form as universal we must first apprehend it as limited in the individual. Seen as limited in the individual, we realize that in itself it is universal. But we see the individual as a limited instance of the form only because we have first pre-apprehended the form itself. Against this pre-apprehension, the individual appears as an instance of the form. For example, conscious of this thing as a dog, I realize that "dog" is limited here but in itself is broader. My awareness 1 Karl Rahner, Spirit in the World, translated by William Dych (New York: Herder and Herder, 1968), p. 142. 2 Ibid., p. 146. 3 Ibid., p. 140. 224 JOHN F. X. KNASAS of this as an instance of dog generates the awareness that of itself dog outstrips this instance. But this key awareness of the instance is itself generated by a pre-apprehension of dog in itself. Because of the pre-apprehension, I realize the limiting of dog here in Fido. The crucial factor in this understanding of abstraction is the pre-apprehension. Unless the... (shrink)

For the past 20 years or so the literature on noncooperative games has been centered on the search for an equilibrium concept that expresses the notion of rational behavior in interactive situations. A basic tenet in this literature is that if a “rational solution” exists, it must be a Nash equilibrium. The consensus view, however, is that not all Nash equilibria can be accepted as rational solutions. Consider, for example, the game of Figure 1.

Contents. Introduction. 1. Preliminaries. 2. Normal Form Games. 3. Extensive Games. 4. Applications of Game Theory. 5. The Methodology of Game Theory. Conclusion. Appendix. Bibliography. Index. Does game theory—the mathematical theory of strategic interaction—provide genuine explanations of human behaviour? Can game theory be used in economic consultancy or other normative contexts? Explaining Games: The Epistemic Programme in Game Theory—the first monograph on the philosophy of game theory—is an attempt to combine insights from epistemic logic (...) and the philosophy of science to investigate the applicability of game theory in such fields as economics, philosophy and strategic consultancy. I prove new mathematical theorems about the beliefs, desires and rationality principles of individual human beings, and explore in detail the logical form of game theory as it is used in explanatory and normative contexts. I argue that game theory reduces to rational choice theory if used as an explanatory device, and that game theory is nonsensical if used as a normative device. A provocative account of the history of game theory reveals that this is not bad news for all of game theory, though. Two central research programmes in game theory tried to find the ultimate characterisation of strategic interaction between rational agents. Yet, while the Nash Equilibrium Refinement Programme has done badly thanks to such research habits as overmathematisation, model-tinkering and introversion, the Epistemic Programme, I argue, has been rather successful in achieving this aim. "The 'epistemic' approach to game theory has emerged over the past twenty-five years. What is this approach? How does it differ from the conventional equilibrium-based approach to game theory? What have been its strengths and weaknesses to date? To find out, read this comprehensive and excellently written account". Adam Brandenburger, J. P. Valles Professor of Business Economics and Strategy, Stern School of Business, New York University "Reading Boudewijn de Bruin's book should be rewarding both for game theorists interested in the conceptual foundations of their discipline and for philosophers who want to learn more about formal analysis of strategic interaction. It provides an in-depth logical study of the currently dominant epistemic approaches to non-cooperative games, with an eye both to the attractions and to the serious challenges facing the Epistemic Programme". Wlodek Rabinowicz, Professor of Practical Philosophy, Department of Philosophy, Lund University . (shrink)

The paper attempts to rationalize cooperation in the one-shot prisoners' dilemma (PD). It starts by introducing (and preliminarily investigating) a new kind of equilibrium (differing from Aumann's correlated equilibria) according to which the players' actions may be correlated (sect. 2). In PD the Pareto-optimal among these equilibria is joint cooperation. Since these equilibria seem to contradict causal preconceptions, the paper continues with a standard analysis of the causal structure of decision situations (sect. 3). The analysis then raises (...) to a reflexive point of view according to which the agent integrates his own present and future decision situations into the causal picture of his situation (sect. 4). This reflexive structure is first applied to the toxin puzzle and then to Newcomb's problem, showing a way to rationalize drinking the toxin and taking only one box with-out assuming causal mystery (sect. 5). The latter result is finally extended to a rationalization of cooperation in PD (sect. 6). (shrink)

Here we use generalized signaling games to model how one’s beliefs might coevolve with the language one uses to characterize those beliefs. We will start by considering how one might individuate pragmatic notions of truth and how such notions might coevolve with a descriptive language. We will then consider how agents might evolve a language that allows them to characterize their beliefs and degrees of belief. Finally, we will consider evolutionary conditions under which a principle of indifference that assigns (...) equal prior probability to each possibility that one can describe in one’s evolved language might in fact be successful. (shrink)

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are developed.We begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly (...) introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued Logic.Then, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically detected.Lastly, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional Logic and Łukasiewicz Infinitely-valued Logic depending on whether only pure equilibria or also mixed equilibria are allowed. Such reductions rely upon the properties of semantics modulo satisfiability. We provide complexity and algorithmic discussion for the coherence problem and, also, for the problem of computing maximal and minimal probabilistic constraints on actions that preserves coherence.prepared by Sandro Márcio da Silva Preto.E-mail: [email protected]:https://doi.org/10.11606/T.45.2021.tde-17062021-163257. (shrink)