Depth Of Reasoning Research Articles

Robust Predictions under Finite Depth of Reasoning

When players have a finite depth of reasoning, it is usually assumed that each player has a commonly known anchor behavior. This paper provides a general framework to examine whether predictions are robust to uncertainty about other players' anchors. We give two different sufficient conditions for the robustness. The first condition shows that any p-dominant equilibrium is robust if players put sufficiently small probability (decreasing in p) on high-depth types. This result highlights a distinction between two prominent finite depth of reasoning models: a risk dominated equilibrium is robust in the cognitive hierarchy model, but not in the level-k model. We also show that equilibria of dominance solvable models are robust.

Mechanism Design with Bounded Depth of Reasoning and Small Modeling Mistakes

We consider mechanism design in contexts in which agents exhibit bounded depth of reasoning (level k) instead of rational expectations. We use simple direct mechanisms, in which agents report only rst-order beliefs. While level 0 agents are assumed to be truth tellers, level k agents best-respond to their belief that other agents have at most k 1 levels of reasoning. We nd that incentive compatibility is necessary for implementation in this framework, while its strict version alone is sucient. Adding continuity to both directions, the same results are obtained for continuous implementation with respect to small modeling mistakes. We present examples to illustrate the permissiveness of our ndings in contrast to earlier related results under the assumption of rational expectations.

Level-[formula omitted] reasoning and time pressure in the 11–20 money request game

Arad and Rubinstein (2012a) have designed a novel game to study level-k reasoning experimentally. Just like them, we find that the depth of reasoning is very limited and clearly different from that in equilibrium play. We show that such behavior is even robust to repetitions; hence there is, at best, little learning. However, under time pressure, behavior is, perhaps coincidentally, closer to that in equilibrium play. We argue that time pressure evokes intuitive reasoning and reduces the focal attraction of choosing higher (and per se more profitable) numbers in the game.

The 11–20 Money Request Game: A Level-k Reasoning Study

We study experimentally a new two-player game: each player requests an amount between 11 and 20 shekels. He receives the requested amount and if he requests exactly one shekel less than the other player, he receives an additional 20 shekels. Level-k reasoning is appealing due to the natural starting point (requesting 20) and the straightforward best-response operation. Nevertheless, almost all subjects exhibit at most three levels of reasoning. Two variants of the game demonstrate that the depth of reasoning is not increased by enhancing the attractiveness of the level-0 strategy or by reducing the cost of undercutting the other player.

Bounded Reasoning and Higher-Order Uncertainty

The standard framework for analyzing games with incomplete information models players as if they have an innite depth of reasoning. This paper generalizes the type

Correlated equilibrium and higher order beliefs about play

We study a refinement of correlated equilibrium in which playersʼ actions are driven by their beliefs and higher order beliefs about the play of the game (beliefs over what other players will do, over what other players believe others will do, etc.). For any finite, complete-information game, we characterize the behavioral implications of this refinement with and without a common prior, and up to any a priori fixed depth of reasoning. In every finite game “most” correlated equilibrium distributions are consistent with this refinement; as a consequence, this refinement gives a classification of “most” correlated equilibrium distributions based on the maximum order of beliefs used by players in the equilibrium. On the other hand, in a generic two-player game any non-degenerate mixed-strategy Nash equilibrium is not consistent with this refinement.

Online discussions develop students’ clinical reasoning skills during fieldwork

Pilot study to examine if the addition of a structured framework improved the quality of clinical reasoning demonstrated within clinically based online discussions by occupational therapy students undertaking fieldwork. Cognitive elements of the clinical reasoning process present within students' discussions were compared before (2005 cohort) and after (2007 cohort) the introduction of a structured framework to guide discussions. Students' self reports (2007 cohort) of clinical reasoning skills were also analysed before and after two consecutive fieldwork placements. The structured framework increased the proportion of postings containing the advanced 'evaluate' element of clinical reasoning and decreased the proportion of postings containing only the basic 'understand' elements. Self-reported clinical reasoning skills improved following the first fieldwork placement and this improvement was sustained following the second placement. The structured framework appears useful in enhancing students' depth of clinical reasoning during fieldwork, although further investigation is required.

Error cascades in observational learning: An experiment on the Chinos game

The paper reports an experimental study based on a variant of the popular Chinos game, which is used as a simple but paradigmatic instance of observational learning. There are three players, arranged in sequence, each of which wins a fixed price if she manages to guess the total number of coins lying in everybodyʼs hands. Our evidence shows that, despite the remarkable frequency of equilibrium outcomes, deviations from optimal play are also significant. And when such deviations occur, we find that, for any given player position, the probability of a mistake is increasing in the probability of a mistake of her predecessors. This is what we call an error cascade, which we measure by evaluating the (heterogeneous) Quantal Response Equilibrium which better suits our data. We also check the robustness of our findings when we allow for belief heterogeneity by applying Kübler and Weizsäckerʼs (2004) cognitive frame of limited depth of reasoning.

Depth of Reasoning and Higher Order Beliefs

As demonstrated by the email game of Rubinstein (1989), the predictions of the standard equilibrium models of game theory are sensitive to assumptions about the fine details of the higher order beliefs. This paper shows that models of bounded depth of reasoning based on level-k thinking or cognitive hierarchy make predictions that are independent of the tail assumptions on the higher order beliefs. In addition to this finding, the tools developed in this paper offer a new direction for the analysis of models of bounded depth of reasoning and their applications to various economic settings.

Practising Social Work Law, 3rd edn, Suzy Braye and Michael Preston-Shoot, Basingstoke, Palgrave Macmillan, 2010, pp. xxxvii + 328, ISBN 978-0-230-54318-8 (pbk), 21.99

This is a long needed update of Suzy Braye and Preston-Shoot's 1997 book with considerable revision, indicating the plethora of legislation relevant to social workers, which has emerged in the intervening years, including not least the 1998 Human Rights Act. Since 1997, the number of laws relevant for social workers has more than doubled in number to seventy-five. This edition is a better structured book with a clearer emphasis on anti-discriminatory practice within a partnership approach to social work. The first three chapters present the critical wider social and political context in which social workers operate. There is a firm stance of viewing the legislation and legal processes critically, identifying weaknesses as well as strengths to ensure that practitioners do not assume that laws are always likely to further social justice. The depth of reasoning is admirably supported by a broad and detailed range of case law. The next two chapters use a variety of case scenarios to explore the relevant law and legal processes. This approach should assist social work practitioners at any level to understand the relevant issues and build appropriate knowledge. There is again a firm connection to the fundamental social work values behind the analysis. The processes involved should mirror the realities of a good practitioner, critically reflecting on the key issues and applying the legal knowledge in the best interests of her or his clients. These chapters are an improvement on the earlier edition, because the case studies develop a reflective application of law with a broader and larger range of individuals. Each case study draws on specific clusters of relevant and related legislation that is applied meticulously as different aspects of each case are explored. They tackle the complexities of people's situations such as a mother needing assistance in relation to domestic abuse where understanding of the law in relation to issues of violent offending, safeguarding of children as well as their significant need, housing and custody matters are all likely. My sole criticism of the studies concerned the visual presentation where differential background shading would have aided the reading.

The Professional Practice Knowledge of Nurse Preceptors

Significant research has been conducted and disseminated on preceptorship as an essential component in teaching and evaluating student learning in the final clinical practicum. The preceptorship triad-formally defined as an equilateral relationship among a nurse preceptor, faculty member, and undergraduate nursing student-overlooks the contextual challenges preceptors encounter. Preceptors develop relationships in the clinical setting to enhance teaching, and more importantly, to ensure student practice reflects the depth of clinical reasoning and skill acquisition necessary for safe and competent practice. Using descriptive surveys and qualitative focus groups, a research collaborative among academia, practice, and professional regulating bodies was established in southern Alberta, Canada, to research the types of knowledge nurses acquire and integrate to successfully assess, evaluate, and teach undergraduate nursing students in the clinical environment. This article describes the research findings and highlights the professional practice knowledge of nurse preceptors.

Neural correlates of depth of strategic reasoning in medial prefrontal cortex

We used functional MRI (fMRI) to investigate human mental processes in a competitive interactive setting--the "beauty contest" game. This game is well-suited for investigating whether and how a player's mental processing incorporates the thinking process of others in strategic reasoning. We apply a cognitive hierarchy model to classify subject's choices in the experimental game according to the degree of strategic reasoning so that we can identify the neural substrates of different levels of strategizing. According to this model, high-level reasoners expect the others to behave strategically, whereas low-level reasoners choose based on the expectation that others will choose randomly. The data show that high-level reasoning and a measure of strategic IQ (related to winning in the game) correlate with the neural activity in the medial prefrontal cortex, demonstrating its crucial role in successful mentalizing. This supports a cognitive hierarchy model of human brain and behavior.

Strength of dominance and depths of reasoning—An experimental study

Iterated elimination of strictly dominated strategies has a stronger epistemic foundation than iterated elimination of weakly dominated strategies. In this paper I investigate experimentally whether this translates into actual differences in behavior. In particular, I analyze whether it affects subjects’ depth of reasoning. This is done within two slightly different versions of a betting game under asymmetric information, in which very similar steps of reasoning can be applied to reach no-betting, or non-speculative, behavior, but of which only one can be solved by iterated elimination of strictly dominated strategies. The results indicate that strength of dominance matters indeed.

Correlated Equilibrium and Higher Order Beliefs about Play

We study a refinement of correlated equilibrium in which players' actions are driven by their beliefs and higher order beliefs about the play of the game (beliefs over what other players will do, over what other players believe others will do, etc.). For any finite, complete-information game, we characterize the behavioral implications of this refinement with and without a common prior, and up to any a priori fixed depth of reasoning. In every finite game most correlated equilibrium distributions are consistent with this refinement; as a consequence, this refinement gives a classification of most correlated equilibrium distributions based on the maximum order of beliefs used by players in the equilibrium. On the other hand, in a generic two-player game any non-degenerate mixed-strategy Nash equilibrium is not consistent with this refinement.

Revealing the Depth of Reasoning in p-Beauty Contest Games

The aim of this study is to evaluate the importance of the information about levels of reasoning on the individual's choices. I report the results from a series of experiments on p-beauty contest games with different subjects' pools and different information treatments. In experiments 2, 3 and 4, the winner(s) explained what reasoning he/she applied in choosing the target number. In each period, he/she wrote a short message (30 words maximum) and then stopped playing. The winner's message, and the winning number, were therefore displayed on all computer screens. In these sessions, therefore, subjects were informed on T, M (the target and value of the average in the previous period) and W (the level of reasoning of the winner in the previous period). In both the baseline sessions (S1 and S5), subjects were informed only on the values of T and M. The experimental evidence shows that non-winning players imitate the level of rationality of the winners, and there is a relevant proportion (Session 3 and 4) of the population who adopt strategies which are best responses to the imitators' behaviour rather than to the average level of rationality. Both the imitative strategies and the best responses to the imitative strategies produce a strong acceleration of the learning process.

Depth of reasoning in strategic form games

The depth of human reasoning in interactive decision-making situations can be assessed experimentally in dominance solvable non-cooperative games. We first review the studies employing two sorts of games—the Beauty Contest game and the Investment game. Studies employing these games show that: (i) people typically engage in relatively low levels of strategic reasoning; and (ii) there are individual differences in the depth of reasoning. There are other lines of research employing other types of games and definitions of the depth of reasoning that support the two conclusions above. We argue that these studies underestimate depth of reasoning because it is also necessary to apply a mental model about other players’ reasoning ability in order to play well. Implications of those findings to the theory-of-mind or Machiavellian Intelligence research are discussed.

Depth of Reasoning and Information Processing: A Predictive Model of SARS Behavior

This research is contextualized in a special crisis situation, the SARS epidemic, in Macao to test a model of behavior prediction. We tackle questions and specify conditions in four areas through the analysis of the interrelationships among patterns of media use, modes of information processing, evaluative responses, and behaviors. A total probability sample of 526 Macao residents aged 15 or above was interviewed during the peak SARS outbreak period from May 7 to 10, 2003 in Macao. Corresponding to our research hypothesis, the data shed light on four aspects: first, fear of SARS might imply a stronger affective element than what would be expected from an attitude based on rational reasoning; second, a clear society–personal distinction at the cognitive level is identified; third, cognitive activities of two kinds of information processors, central processing mode and peripheral processing mode, have been shown to predict fear and behavior in opposite directions; and last, the role of attitude serving as a mediating mechanism between information processing and behavior is supported, in line with the O-S-O-R conception.

Proportional Reasoning: Lessons from Research in Data and Chance

PRINCIPLES AND STANDARDS FOR SCHOOL Mathematics (NCTM 2000) places proportionality among the major concepts connecting different topics in the mathematics curriculum at the middle school level (p. 217). What concerns us about many of the problems presented to students, however, is that they are often posed purely as a ratio or proportion from the start. Often the statement of a problem is a giveaway that a proportion is involved. For example, the question “If 15 students out of 20 get a problem correct, how many students in a class of 28 would we expect to get the problem correct?” does not tap the depth of proportional reasoning that is required for meaningful problem solving.

Limited Depth of Reasoning and Failure of Cascade Formation in the Laboratory

We examine the robustness of information cascades in laboratory experiments. Apart from the situation in which each player can obtain a signal for free (as in the experiment by Anderson and Holt (1997), the case of costly signals is studied where players decide whether or not to obtain private information, at a small but positive cost. In the equilibrium of this game, only the first player buys a signal and makes a decision based on this information whereas all following players do not buy a signal and herd behind the first player. The experimental results show that too many signals are bought and the equilibrium prediction performs poorly. To explain these observations, the depth of the subjects' reasoning process is estimated, using a statistical error-rate model. Allowing for different error rates on different levels of reasoning, we find that the subjects' inferences become significantly more noisy on higher levels of the thought process, and that only short chains of reasoning are applied by the subjects.

Two paradigms for depth of strategic reasoning in games: Response to Colman

We contrast two approaches to probing the depth of theory-of-mind (ToM) reasoning by adults in matrix game settings: our own and that of Stahl and Wilson.