Free choice disjunction as a rational speech act

The so-called free choice inference (from You may take an apple or a pear to You may take an apple) is mysterious because it does not follow from ordinary modal logic. We show that this inference arises in the Rational Speech Act framework (Frank & Goodman 2012). Our basic idea is inspired by exhaustification-based models of free choice (Fox 2007) and by game-theoretic accounts based on iterated best response (Franke 2011). We assume that when the speaker utters You may take an apple or a pear, the hearer reasons about why the speaker did not choose alternative utterances such as You may take an apple. A crucial ingredient in our explanation is the idea of semantic uncertainty (Bergen, Levy & Goodman 2016). Specifically, we assume that the speaker is uncertain whether or not the hearer will interpret You may take an apple as forbidding them from taking a pear. This uncertainty can be thought of as resulting from Fox’s (2007) optional covert exhaustification. Uttering the disjunction is a way for the speaker to prevent the hearer from concluding that any fruit is forbidden to take. Knowing this, the hearer concludes that they may choose either fruit

Iterated dominance and iterated best response in experimental "p-beauty contests"

Picture a thin country 1000 miles long, running north and south, like Chile. Several natural attractions are located at the northern tip of the country. Suppose each of n resort developers plans to locate a resort somewhere on the country's coast (and all spots are equally attractive). After all the resort locations are chosen, an airport will be built to serve tourists, at the average of all the locations including the natural attractions. Suppose most tourists visit all the resorts equally often, except for lazy tourists who visit only the resort closest to the airport; so the developer who locates closest to the airport gets a fixed bonus of extra visitors. Where should the developer locate to be nearest to the airport? The surprising game-theoretic answer is that all the developers should locate exactly where the natural attractions are. This answer requires at least one natural attraction at the northern tip, but does not depend on the fraction of lazy tourists or the number of developers (as long as there is more than one)

Proof-theoretic Analysis of Rationality for Strategic Games with Arbitrary Strategy Sets

In the context of strategic games, we provide an axiomatic proof of the statement Common knowledge of rationality implies that the players will choose only strategies that survive the iterated elimination of strictly dominated strategies. Rationality here means playing only strategies one believes to be best responses. This involves looking at two formal languages. One is first-order, and is used to formalise optimality conditions, like avoiding strictly dominated strategies, or playing a best response. The other is a modal fixpoint language with expressions for optimality, rationality and belief. Fixpoints are used to form expressions for common belief and for iterated elimination of non-optimal strategies.Comment: 16 pages, Proc. 11th International Workshop on Computational Logic in Multi-Agent Systems (CLIMA XI). To appea

Hierarchical Reasoning versus Iterated Reasoning in p-Beauty Contest Guessing Games

This paper analyzes strategic choice in p-beauty contests. We first show that it is not generally a best reply to guess the expected target value (accounting for the own weight) even in games with n>2 players and that iterated best response sequences are not unique even after perfect/cautious refinement. This implies that standard formulations of ``level-k'' models are neither exactly nor uniquely rationalizable by belief systems based on iterated best response. Second, exact modeling of iterated reasoning weakens the fit considerably and reveals that equilibrium types dominate the populations. We also show that ``levels of reasoning'' cannot be measured regardless of the underlying model. Third, we consider a ``nested logit'' model where players choose their level. It dispenses with belief systems between players and is rationalized by a random utility model. Besides being internally consistent, nested logit equilibrium fits better than three variants of the level-k model in standard data sets.logit equilibrium, hierarchical response, level-k, beauty contest

Optimal cooperation-trap strategies for the iterated Rock-Paper-Scissors game

In an iterated non-cooperative game, if all the players act to maximize their individual accumulated payoff, the system as a whole usually converges to a Nash equilibrium that poorly benefits any player. Here we show that such an undesirable destiny is avoidable in an iterated Rock-Paper-Scissors (RPS) game involving two players X and Y. Player X has the option of proactively adopting a cooperation-trap strategy, which enforces complete cooperation from the rational player Y and leads to a highly beneficial as well as maximally fair situation to both players. That maximal degree of cooperation is achievable in such a competitive system with cyclic dominance of actions may stimulate creative thinking on how to resolve conflicts and enhance cooperation in human societies.Comment: 5 pages including 3 figure

Progress in Behavioral Game Theory

Is game theory meant to describe actual choices by people and institutions or not? It is remarkable how much game theory has been done while largely ignoring this question. The seminal book by von Neumann and Morgenstern, The Theory of Games and Economic Behavior, was clearly about how rational players would play against others they knew were rational. In more recent work, game theorists are not always explicit about what they aim to describe or advise. At one extreme, highly mathematical analyses have proposed rationality requirements that people and firms are probably not smart enough to satisfy in everyday decisions. At the other extreme, adaptive and evolutionary approaches use very simple models-mostly developed to describe nonhuman animals-in which players may not realize they are playing a game at all. When game theory does aim to describe behavior, it often proceeds with a disturbingly low ratio of careful observation to theorizing

Admissibility and Common Knowledge

The implications of assuming that it is commonly known that players consider only admissible best responses are investigated.Within a states-of-the-world model where a state, for each player, determines a strategy set rather than a strategy the concept of fully permissible sets is defined.General existence is established, and a finite algorithm (eliminating strategy sets instead of strategies) is provided.The concept refines rationalizability as well as the Dekel-Fudenberg procedure, and captures a notion of forward induction.When players consider all best responses, the same framework can be used to define the concept of rationalizable sets, which characterizes rationalizability.game theory

Self-tuning experience weighted attraction learning in games

Self-tuning experience weighted attraction (EWA) is a one-parameter theory of learning in games. It addresses a criticism that an earlier model (EWA) has too many parameters, by fixing some parameters at plausible values and replacing others with functions of experience so that they no longer need to be estimated. Consequently, it is econometrically simpler than the popular weighted fictitious play and reinforcement learning models. The functions of experience which replace free parameters “self-tune” over time, adjusting in a way that selects a sensible learning rule to capture subjects’ choice dynamics. For instance, the self-tuning EWA model can turn from a weighted fictitious play into an averaging reinforcement learning as subjects equilibrate and learn to ignore inferior foregone payoffs. The theory was tested on seven different games, and compared to the earlier parametric EWA model and a one-parameter stochastic equilibrium theory (QRE). Self-tuning EWA does as well as EWA in predicting behavior in new games, even though it has fewer parameters, and fits reliably better than the QRE equilibrium benchmark