Infinite games with finite knowledge gaps

Infinite games where several players seek to coordinate under imperfect information are deemed to be undecidable, unless the information is hierarchically ordered among the players. We identify a class of games for which joint winning strategies can be constructed effectively without restricting the direction of information flow. Instead, our condition requires that the players attain common knowledge about the actual state of the game over and over again along every play. We show that it is decidable whether a given game satisfies the condition, and prove tight complexity bounds for the strategy synthesis problem under $\omega$-regular winning conditions given by parity automata.Comment: 39 pages; 2nd revision; submitted to Information and Computatio

"The Folk Theorem with Private Monitoring and Uniform Sustainability"

This paper investigates infinitely repeated prisoner-dilemma games where the discount factor is less than but close to 1. We assume that monitoring is truly imperfect and truly private, there exist no public signals and no public randomization devices, and players cannot communicate and use only pure strategies. We show that implicit collusion can be sustained by Nash equilibria under a mild condition. We show that the Folk Theorem holds when playersf private signals are conditionally independent. These results are permissive, because we require no conditions concerning the accuracy of private signals such as the zero likelihood ratio condition. We also investigate the situation in which players play a Nash equilibrium of a machine game irrespective of their initial states, i.e., they play a uniform equilibrium. We show that there exists a unique payoff vector sustained by a uniform equilibrium, i.e., a unique uniformly sustainable payoff vector, which Pareto-dominates all other uniformly sustainable payoff vectors. We characterize this payoff vector by using the values of the minimum likelihood ratio. We show that this payoff vector is efficient if and only if the zero likelihood ratio condition is satisfied. These positive results hold even if each player has limited knowledge on her opponentfs private signal structure. Keywords: Repeated Prisoner-Dilemma Games, Private Monitoring, Conditional Independence, Folk Theorem, Uniform Sustainability, Zero Likelihood Ratio Condition, Limited Knowledge.

Bayesian games with a continuum of states

We show that every Bayesian game with purely atomic types has a measurable Bayesian equilibrium when the common knowl- edge relation is smooth. Conversely, for any common knowledge rela- tion that is not smooth, there exists a type space that yields this common knowledge relation and payoffs such that the resulting Bayesian game will not have any Bayesian equilibrium. We show that our smoothness condition also rules out two paradoxes involving Bayesian games with a continuum of types: the impossibility of having a common prior on components when a common prior over the entire state space exists, and the possibility of interim betting/trade even when no such trade can be supported ex ante

Endgame: The false destruction of the social imaginary

Vogl’s account of contemporary financial truth games suggests that a full understanding of our present condition requires the kind of knowledge produced by fiction and those who study it. But what kind of knowledge is that, and how does it escape the capitalist ontology of information

Payoff-Relevant States in Dynamic Games with Infinite Action Spaces

Maskin and Tirole have defined payoff-relevant states in discrete time dynamic games with observable actions in terms of a partition of the set of histories. Their proof that this partition is unique cannot be applied, when action spaces are infinite or when players are unable to condition on calendar time. This note provides a unified proof of existence and uniqueness for these cases. The method of proof is useful for problems other than the one treated here. To illustrate this, a well known characterization of common knowledge is generalized.

Measurable selection for purely atomic games

A general selection theorem is presented constructing a measurable mapping from a state space to a parameter space under the assumption that the state space can be decomposed as a collection of countable equivalence classes under a smooth equivalence relation. It is then shown how this selection theorem can be used as a general purpose tool for proving the existence of measurable equilibria in broad classes of several branches of games when an appropriate smoothness condition holds, including Bayesian games with atomic knowledge spaces, stochastic games with countable orbits, and graphical games of countable degree—examples of a subclass of games with uncountable state spaces that we term purely atomic games. Applications to repeated games with symmetric incomplete information and acceptable bets are also presented

Litotes, Irony and other Innocent Lies

In the following text we would like to present the philosophical discussion on untrusting lies, which introduces a space for innocent lie understood as figurative manipulation of the speech: a poetic trope that we would argue could not only be generously used to help us tolerating our sometime deceiving human condition—which is global and universally ours, that of the finitude of human capacity of knowledge and ethical action—but also to maximise our capacity for knowledge formation and adaptation to values. Concepts formation and communication relates to a collective interplay of different interiorized images, before it comes to the exterior in some well-chosen expressions, in self-mastered way; their origin remain in a mentally latent process of selection of content and ideas, as possible solutions of in a games of compatible propositions. These unconscious materials of our life relies on our capacity to identify and quickly switch between different spans, that enable us to focus on complex sets data, all depending very much on figurative manipulations, that should not be confounded with blameworthy and misleading representations

Subgame Perfect Correlated Equilibria in Repeated Games

Subgame Perfect Correlated Equilibria in Repeated Games by Pavlo Prokopovych and Lones Smith ABSTRACT This paper investigates discounted infinitely repeated games with observable actions extended with an extensive form correlation device. Such games capture situations of repeated interaction of many players who choose their individual actions conditional on both public and private information. At the beginning of each stage, the players observe correlated private messages sent by an extensive form correlation device. To secure a recursive structure, we assume that players condition their play on the prior history of action profiles and the latest private message they have received from the device. Given a public history, the probability distribution on the product of the players' message sets, according to which the device randomly selects private messages to the players, is common knowledge. This leads to the existence of proper subgames and the opportunity to utilize the techniques developed by Abreu, Pearce, Stacchetti (1990) for studying infinitely repeated games with imperfect monitoring. The extensive form correlation devices we consider send players messages confidentially and separately and are not necessarily direct devices. Proposition 1 asserts that, in infinitely repeated games, subgame perfect correlated equilibria have a simple intertemporal structure, where play at each stage constitutes a correlated equilibrium of the corresponding one-shot game. An important corollary is that the revelation principle holds for such games --- any subgame perfect correlated equilibrium payoff can be achieved as a subgame perfect direct correlated equilibrium payoff. We can therefore focus on the recursive structure of infinitely repeated games extended with an extensive form direct correlation device and characterize the set of subgame perfect direct correlated equilibrium payoffs. In the spirit of dynamic programming, we decompose an equilibrium into an admissible pair that consists of a probability distribution on the product of the players' action sets and a continuation value function. This generalization has allowed us to obtain a number of characterizations of the set of subgame perfect equilibrium payoffs. To illustrate a number of important properties of this set, we study two infinitely repeated prisoner's dilemma games. In the first game, the set of subgame perfect correlated equilibrium payoffs strictly includes not only the set of subgame perfect equilibrium payoffs but also the set of subgame perfect public randomization equilibrium payoffs. In the second game, the set of subgame perfect direct correlated equilibrium payoffs is not convex, strictly includes the set of subgame perfect equilibrium payoffs, and is strictly contained in the set of subgame perfect public randomization equilibrium payoffs. The latter is possible since, in the presence of a public randomization device, the history of public messages observed in previous stages is also common knowledge at the beginning of each stage, which is not the case when messages are private.repeated games with observable actions, correlated equilibrium, private information

Investigating the Effectiveness of an Educational Escape Game for Increasing Nutrition-Related Knowledge in Young Adolescents: A Pilot Study

Objective: As a pilot trial under the Games of Food consortium, this study assessed the effectiveness of an educational escape game alongside a self-study method as a nutrition knowledge intervention. Furthermore, this study explored the use of an escape game as an educational tool for young adolescents. Materials and Methods: Altogether three schools participated, one from Finland and two from the UK. Baseline questionnaires assessing knowledge were administered before intervention day. Participants from each class were randomly allocated by the researchers into either the escape game condition, where participants played a nutrition education escape game with a focus on plant-based protein sources, or the self-study condition, where participants received an educational leaflet with identical content. In addition to the knowledge post-assessment, the educational escape game condition answered an enjoyment and intrinsic motivation questionnaire to evaluate the game experience. Paired t-tests were used to determine significant changes within intervention conditions and ANCOVA was used to estimate the differences in knowledge. Results: The participants were 130 children (11-14 years), divided into educational escape game (n = 68) and self-study (n = 62) conditions. Both the educational escape game (20.7 vs. 23, p < 0.001) and self-study (21.1 vs. 23.1, p = 0.002) had improved overall knowledge scores. No significant differences in gained knowledge existed between groups. Of the educational escape game participants, 60% reported the game as mostly enjoyable and 46% reported added use and value for learning. Conclusion: The educational escape game condition was comparable to the self-study method for nutrition education in adolescents. However, since the educational escape game provides an enjoyable experience that may enhance intrinsic motivation to promote learning and possible behavior change, the use of escape games for nutrition education warrant further investigation.Peer reviewe