Common Knowledge Of Rationality Research Articles

Dropping rational expectations

We consider a two-period pure-exchange economy, where uncertainty prevails and agents, possibly asymmetrically informed, exchange commodities and securities of all kinds. Consumers’ characteristics, anticipations, beliefs and actions are all private and typically not known nor assessed by the other agents. This setting drops rational expectations along the ‘common knowledge of rationality and market clearing’ (CKRMC) assumptions, and, in particular, it drops the Radner (1979) price model and price inference assumptions. Unaware agents are shown to face an incompressible uncertainty over future states and prices, represented by a so-called ”minimum uncertainty set”. A sequential equilibrium obtains when agents expect the ‘true’ price as a possible outcome on every spot market, and elect optimal market-clearing strategies. This so-called ”correct foresight equilibrium” (CFE) is shown to exist whenever agents’ anticipation sets include the minimum uncertainty set. When the CKRMC assumptions are restored, the CFE is shown to lead to an overarching concept of rational expectations equilibrium, which generalizes the classical concepts.

Knowledge, Awareness and Probabilistic Beliefs

Abstract Bayesian game theory investigates strategic interaction of players with full awareness but incomplete information about their environment. We extend the analysis to players with incomplete awareness, who might not be able to reason about all contingencies in the first place. We develop three logical systems for knowledge, probabilistic beliefs and awareness, and characterize their axiom systems. Bayesian equilibrium is extended to games with incomplete awareness and we show that it is consistent with common prior and speculative trade, when common knowledge of rationality is violated.

An epistemic characterization of MACA

Dekel et al. (J Econ Theory 89 (1999) 165-185) offered a solution concept of “rationalizable self-confirming equilibrium (RSCE)” as the steady state where rational individuals observe the played actions and use the information about opponents’ payoffs in forming the beliefs about opponents’ behavior off the equilibrium path. In this paper we investigate epistemic conditions for RSCE from a decision-theoretic point of view. Within a standard semantic framework, we formulate and show, by using the notion of “conditional probability system (CPS),” that RSCE is the logical consequence of common knowledge of rationality and mutual knowledge of the actions along the path of play. In this paper, we also apply this epistemic approach to other related solution concepts such that self-confirming equilibrium (SCE) and sequential rationalizable self-confirming equilibrium (SRSCE).

Perfect Prediction in normal form: Superrational thinking extended to non-symmetric games

This paper introduces a new solution concept for non-cooperative games in normal form with no ties and pure strategies: the Perfectly Transparent Equilibrium. The players are rational in all possible worlds and know each other’s strategies in all possible worlds — which, together, we refer to as Perfect Prediction. The anticipation of a player’s decision by their opponents is counterfactually dependent on the decision, unlike in Nash Equilibria where the decisions are made independently. The equilibrium, when it exists, is unique and is Pareto optimal.This equilibrium is the normal-form counterpart of the Perfect Prediction Equilibrium; the prediction happens “in another room” rather than in the past. The equilibrium can also be seen as a natural extension of Hofstadter’s superrationality to non-symmetric games. Algorithmically, an iterated elimination of non-individually-rational strategy profiles is performed until at most one remains. An equilibrium is a strategy profile that is immune against knowledge of strategies in all possible worlds and rationality in all possible worlds, a stronger concept than common knowledge of rationality but also stronger than common counterfactual belief of rationality.We formalize and contrast the Non-Nashian Decision Theory paradigm, common to this and several other papers, with Causal Decision Theory and Evidential Decision Theory. We define the Perfectly Transparent Equilibrium algorithmically and prove (when it exists) that it is unique, that it is Pareto-optimal, and that it coincides with Hofstadter’s Superrationality on symmetric games. We relate the finding to concepts found in the literature such as Individual Rationality, Rationalizability, Minimax-Rationalizability, Second-Order Nash Equilibria, the Program Equilibrium, the Perfect Prediction Equilibrium, Shiffrin’s Joint-Selfish-Rational Equilibrium, the Stalnaker–Bonanno Equilibrium, the Perfect Cooperation Equilibrium, the Translucent Equilibrium, the Correlated Equilibrium, and Quantum Games. Finally, we specifically discuss inclusion relationships on the special case of symmetric games between Individual Rationality, Minimax-Rationalizability, Superrationality, and the Perfectly Transparent Equilibrium, and contrast them with asymmetric games.

Disagreement Under Almost Common Knowledge of Rationality

Two agents sincerely exchange their best guesses about the state of the world infinitely many times. When each agent places a small positive probability on the event that her opponent is of some finite level of reasoning and initial disagreement is large enough (that is, private signals are strong and different), permanent and large disagreement is possible even for infinitely sophisticated agents.

No trade and yes trade theorems for heterogeneous priors

We show that contrary to currently widely-held misperceptions, the classical no trade theorem obtains even under heterogeneous priors. That is, when priors are not common, speculative trade is still impossible under common knowledge of rationality. However, trade becomes mutually acceptable if at least one party to the trade puts at least some slight probability on the other party being irrational. We also derive bounds on disagreements in the case of heterogeneous priors and common p-beliefs.

Bounded rationality and correlated equilibria

We study an interactive framework that explicitly allows for non-rational behavior. We do not place any restrictions on how players can deviate from rational behavior. Instead we assume that there exists a lower bound p2 [0; 1] such that all players play and are believed to play rationally with a probability p or more. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we dene and characterize for arbitrary p2 [0; 1]. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. Thep-rational outcomes are easy to compute, also for games of incomplete information, and they can be applied to observed frequencies of play to compute a measure p that bounds from below the probability with which any given player is choosing actions consistent with payo maximization and common knowledge of payo maximization.

Lawrence A. Boland's Model building in economics: its purposes and limitations. Cambridge: Cambridge University Press, 2014, 298 pp.

Review of Lawrence A. Boland's Model building in economics: its purposes and limitations. Cambridge: Cambridge University Press, 2014, 298 pp.The purposes and limitations of all types of models used in economics, past and beyond. This is the rather ambitious topic of Lawrence Boland's new book. The book reviews standard microeconomics, macroeconomics, game theory (including evolutionary), experimental economics, and econometrics (both micro and macro). In its mission statement, the book is declared methodology with a small 'm', to be distinguished with a capitalized Methodology discussed by Philosophers. Methodology discussed by Philosophers deals with lofty issues of 'Realism', 'Testing', and 'Explanation', whereas methodologists are interested in, well, such things as the realisticness of assumptions, the testing of models, and explanation. In the end, the biggest drawback of the book ends up being a seemingly aesthetic aversion to completely imagined capital letters. It is clear that the book discusses the very same issues that have vexed philosophers of science and philosophers of economics for at least the last few decades, and arguably for centuries, yet it refuses to engage these resources. This is a regrettable, but perhaps inevitable, consequence of the limits of any one person trying to cover the whole of economics and the relevant philosophy of economics.Boland begins with a thesis about a paradigm and generational shift in economic modeling around the beginning of the 1980s. Economists having their postgraduate studies in the 1980s or later no longer conceive models as being models of a theory, built in order to provide an instrument with which to empirically test that theory. Instead, model building began to be equated with theorizing itself. This shift is argued for in a concise, yet convincing manner. And this shift does not sit all that well with the Popperian spirit of Boland's Methodology (yes, with a capital 'M'). Although the book does not attempt to systematically build a single argument or an encompassing philosophical position, the overall message emerging from the discussion seems to be a plea for more interest in the realisticness and empirical testing of modeling assumptions.The book first goes through the stylized history of the division between micro and macroeconomics. It dates the beginning of modern microeconomics to Marshall and proceeds from there to discuss the need for, and the plausible form of, microfoundations. Standard criticisms against the use of representative agent constructs and the sterility of general equilibrium theory are also raised. The biggest faults located by Boland in theoretical macro models based on equilibrium assumptions are their lack of dynamics and any idea of adjustment process for prices. There is nothing new in these accusations, but Boland raises the issues with a peculiar twist. It would not, in fact, be enough to endow the model with dynamics by having the endogenous variables be determined by their lagged values and additional exogenous variables. This would be merely a combination of exogenous shocks and a predetermined time path, not truly endogenous dynamics resulting from the actions of free agents. Approvingly quoting Hicks, Boland holds that economics must not only include time, but be in time.The argument here is that the problem is not just in the lack of an empirically credible adjustment process, but in the fact that the conceptions of learning and expectation formation implicit in any proposed sketch of an adjustment process are flawed, because they have always been based on inductive reasoning. And inductive reasoning was demonstrated to be impossible by Hume. This foundational problem of the groundlessness of inductive learning is also raised with respect to agents within game-theoretical models: common knowledge of rationality either assumes a foundationally irrational inductive learning process or simply assumes away the whole 'problem of induction'. …

Weakly rational expectations

Aumann and Drèze (2008) characterised the set of interim expected payoffs that players may have in rational belief systems, in which there is common knowledge of rationality and a common prior. We show here that common knowledge of rationality is not needed: when rationality is satisfied in the support of an action-consistent distribution (a concept introduced by Barelli (2009)), one obtains exactly the same set of rational expectations, despite the fact that in such ‘weakly rational belief systems’ there may not be mutual knowledge of rationality, let alone common knowledge of rationality. In the special case of two-player zero-sum games, the only expected payoff is the minmax value, even under these weak assumptions.

Robust Adversarial Risk Analysis: A Level-k Approach

Adversarial risk analysis is an active and important area of decision analytic research. Both single-actor decision analysis and multiple-actor game theory have been applied to this problem, with game theoretic methods being particularly popular. Although game theory models do explicitly capture strategic interactions between attackers and defenders, two of the key assumptions—decision making based on subjective expected utility maximization and common knowledge of rationality—are known to be descriptively inaccurate in some situations. This paper addresses these shortcomings by proposing, formulating, and illustrating the application of robust optimization methodologies to a level-k game theory model for adversarial risk analysis. Level-k game theory provides a practical method for modeling bounded rationality. Robust optimization provides an alternative way to model the actions of conservative players facing “deep” uncertainties about their environment—uncertainties that are possible to bound but that are difficult or impossible to represent using probability distributions. Our approach thus combines level-k and robust optimization insights to provide a computationally tractable model of boundedly rational players who are faced with significant and difficult to quantify uncertainties.

Common knowledge of rationality and market clearing in economies with asymmetric information

Abstract Consider an exchange economy with asymmetric information. What is the set of outcomes that are consistent with common knowledge of rationality and market clearing? To address this question we define an epistemic model for the economy that provides a complete description not only of the beliefs of each agent on the relationship between states of nature and prices but also of the whole system of interactive beliefs. The main result, Theorem 1, provides a characterization of outcomes that are consistent with common knowledge of rationality and market clearing (henceforth, CKRMC outcomes) in terms of a solution notion – Ex-Post Rationalizability – that is defined directly in terms of the parameters that define the economy. CKRMC manifests several intuitive properties that stand in contrast to the full revelation property of Rational Expectations Equilibrium. In particular, for a robust class of economies: (1) there is a continuum of prices that are consistent with CKRMC in every state of nature, and hence these prices do not reveal the true state, (2) the range of CKRMC outcomes is monotonically decreasing as agents become more informed about the economic fundamentals, and (3) trade is consistent with common knowledge of rationality and market clearing even when there is common knowledge that there are no mutual gains from trade.

Book Reviews

Book Reviews

Book Reviews

Arthur J. Robson of Simon Fraser University reviews “The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences” by Herbert Gintis. The EconLit Abstract of the reviewed work begins, “Explores how key concepts from the behavioral sciences can complement game theory in providing insights into human behavior. Discusses decision theory and human behavior; game theory--basic concepts; game theory and human behavior; rationalizability and common knowledge of rationality; extensive for….”

WHAT HAPPENED TO THE THREE-LEGGED CENTIPEDE GAME?

Abstract In the last two decades, several frameworks have been proposed to analyze the question of whether common knowledge of rationality is sufficient to justify the play of backward induction (BI) in games of perfect information. Three strands of literature have addressed this issue: the literature on equilibrium refinements, the literature on knowledge-based epistemology and the literature on interactive epistemology. This paper surveys seminal frameworks within the first two strands of research and assesses the extent to which they provide a satisfactory solution to the problem. These approaches are illustrated using a three-legged version of Rosenthal's centipede game, which is the classical example in the literature. The paper argues that some of these frameworks provide sensible answers to the riddle of BI or, at least, succeed in bringing the paradox to another level. The paper also points at consistency problems in the body of refinements of Nash equilibrium revealed by the surveyed literature.

Rational Expectations in Games

A player i's actions in a game are determined by her beliefs about other players; these depend on the game's real-life context, not only its formal description. Define a game situation as a game together with such beliefs; call the beliefs—and i's resulting expectation—rational if there is common knowledge of rationality and a common prior. In two-person zero-sum games, i's only rational expectation is the game's value. In an arbitrary game G, we characterize i's rational expectations in terms of the correlated equilibria of the doubled game 2G in which each of i's strategies in G appears twice. (JEL C72, D83, D84)

Mutual Knowledge of Rationality in the Electronic Mail Game

This paper reexamines the paradoxical aspect of the electronic mail game (Rubinstein, 1989). The electronic mail game is a coordination game with payo uncertainty. At a Bayesian Nash equilibrium of the game, players cannot achieve the desired coordination of actions even when a high order of mutual knowledge of payo functions obtains. We want to make explicit the role of knowledge about rationality of players, not only that of payo functions. For this purpose, we use an extended version of the belief system model developed by Aumann and Brandenburger (1995). We propose a certain way of embedding the electronic mail game in an belief system. And we show that for rational players to coordinate their actions, for any embedding belief systems, it is necessary that the upper bound order of mutual knowledge of payo functions exceeds the upper bound order of mutual knowledge of rationality. This result implies that under common knowledge of rationality, the coordination never occurs, which is similar to Rubinstein’s result. We point out, however, that there exists a class embedding belief systems for which the above condition is also sucient

Quel degré de rationalité favorise la coordination des agents ?

L’objet de cet article est de caractériser le comportement d’agents qui possèdent des biais psychologiques quand ils traitent l’information dans une économie représentée par un jeu de coordination en information incomplète. On s’intéresse en particulier à deux types de biais psychologiques: l’heuristique de disponibilité et l’optimisme des agents dans leurs capacités. On caractérise le comportement des agents et les conséquences sur l’équilibre de l’économie de la prise en compte de tels biais comme hypothèse de modélisation du comportement humain. Pour contraster les propriétés de tels équilibres avec les propriétés de l’équilibre à anticipations rationnelles, on retient l’hypothèse de cohérence des croyances. Le principal résultat que l’on montre est le suivant : Quand les agents possèdent de tels biais psychologiques dans la prise en compte de l’information quand ils forment leurs anticipations, il n’y a plus d’indetermination. Un équilibre unique est sélectionné qui est caractérisé par une règle de comportement simple dans laquelle les agents sont susceptibles de commettre des erreurs. L’équilibre obtenu peut être supérieur au sens de Pareto par rapport à l’équilibre bas. Quand les agents forment leurs anticipations de façon rationnelle, ils cherchent à anticiper le comportement des autres agents. En conséquence, ils ne prennent pas en compte leur information privée. La structure de l’équilibre est réflexive et l’indétermination apparaît à nouveau.

Counterfactuals in wonderland

The literary source of the main ideas in Aumann's article ``Backward Induction and Common Knowledge of Rationality is exposed and analyzed. The primordial archetypal images that underlie both this literary source and Aumann's work are delineated and are used to explain the great emotive impact that this work had on the community of game theorists.

Counterfactuals in Wonderland

The literary source of the main ideas in Aumann's article Backward Induction and Common Knowledge of Rationality is exposed and analyzed. The primordial archetypal images that underlie both this literary source and Aumann's work are delineated and are used to explain the great emotive impact that this work had on the community of game theorists.

Counterfactual Reasoning and Common Knowledge of Rationality in Normal Form Games

When evaluating the rationality of a player in a game one has to examine counterfactuals such as "what would happen if the player were to do what he does not do?" In this paper I develop a model of a normal form game where counterfactuals of this sort are evaluated as in the philosophical literature (cf. Lewis, 1973; Stalnaker, 1968). According to this method one evaluates a statement like ``what would the player believe if he were to do what he does not do'' at the world that is closest to the actual world where the hypothetical deviation occurs. I show that in this model common knowledge of rationality need not lead to rationalizability. I also present assumptions that allow rationalizability to follow from common knowledge of rationality. These assumptions suggest that rationalizability may not rely on weaker assumptions about belief consistency than Nash equilibrium.