Level-k Reasoning Research Articles

Theories of reasoning and focal point play with a matched non-student sample

We present a coordination game experiment testing the robustness of the predictive power of level-k reasoning and team reasoning in a sample of Chinese tax administrators that is matched for likely socio-economic characteristics with our student sample. We show how the incidence of coordination game play is virtually identical between Chinese tax administrators and university students. However, compared to non-students, students are more attracted by the focal point when this offers equal payoffs while the other equilibria do not. Evidence from an additional experiment suggests that this could be because payoff equality promotes team reasoning more effectively among students.

Bounded rationality for relaxing best response and mutual consistency: the quantal hierarchy model of decision making

While game theory has been transformative for decision making, the assumptions made can be overly restrictive in certain instances. In this work, we investigate some of the underlying assumptions of rationality, such as mutual consistency and best response, and consider ways to relax these assumptions using concepts from level-k reasoning and quantal response equilibrium (QRE) respectively. Specifically, we propose an information-theoretic two-parameter model called the quantal hierarchy model, which can relax both mutual consistency and best response while still approximating level-k, QRE, or typical Nash equilibrium behavior in the limiting cases. The model is based on a recursive form of the variational free energy principle, representing higher-order reasoning as (pseudo) sequential decision-making in extensive-form game tree. This representation enables us to treat simultaneous games in a similar manner to sequential games, where reasoning resources deplete throughout the game-tree. Bounds in player processing abilities are captured as information costs, where future branches of reasoning are discounted, implying a hierarchy of players where lower-level players have fewer processing resources. We demonstrate the effectiveness of the quantal hierarchy model in several canonical economic games, both simultaneous and sequential, using out-of-sample modelling.

Level-k Reasoning and Biased Belief Updating in a Dutch Auction with Independent Private Values

Level-k Reasoning and Biased Belief Updating in a Dutch Auction with Independent Private Values

The role of personality beliefs and "small talk" in strategic behaviour.

Humans are predisposed to forming “first impressions” about the people we encounter including impressions about their personality traits. While the relationship between personality and strategic decision-making has been widely explored, we examine the role of personality impressions in predicting strategic behaviour and devising behavioural responses. In a laboratory setting, after only 4-minutes of “small talk”, subjects developed a sense of the personality of their partners, particularly extraversion, which consequently changed their behaviour in future interactions. Subjects cooperated more in public goods games when they believed their partner to be extraverted and found it more difficult to out-guess opponents they perceived as similar to themselves in a level-k reasoning task, having engaged in conversation with them. We trace how language can generate these effects using text analysis, showing that talking more makes individuals appear extraverted and pro-social which in turn engenders pro-social behaviour in others.

Level-k Models and Overspending in Contests

The experimental evidence on contests often reports overspending of contest participants compared to the theoretical Nash equilibrium outcome. We show that a standard level-k model may rationalize overspending in contests. This result complements the existing literature on overspending in contests, and it bridges an open gap between the contest and auction literature. In fact, the literature on auctions often runs parallel to that on contests.Overbidding in auctions has also been documented empirically, and it has been shown that, in private-value auctions, such overbidding can be rationalized by level-k reasoning. We bridge the existing gap between the auction and contest literature by showing that overbidding may also be true in a theoretical contest environment with level-k reasoning.

A 3D game theoretical framework for the evaluation of unmanned aircraft systems airspace integration concepts

Predicting the outcomes of integrating Unmanned Aerial System (UAS) into the National Airspace System (NAS) is a complex problem, which is required to be addressed by simulation studies before allowing the routine access of UAS into the NAS. This paper focuses on providing a 3-dimensional (3D) simulation framework using a game-theoretical methodology to evaluate integration concepts using scenarios where manned and unmanned air vehicles co-exist. In the proposed method, the human pilot interactive decision-making process is incorporated into airspace models which can fill the gap in the literature where the pilot behavior is generally assumed to be known a priori. The proposed human pilot behavior is modeled using a dynamic level-k reasoning concept and approximate reinforcement learning. The level-k reasoning concept is a notion in game theory and is based on the assumption that humans have various levels of decision making. In the conventional “static” approach, each agent makes assumptions about his or her opponents and chooses his or her actions accordingly. On the other hand, in the dynamic level-k reasoning, agents can update their beliefs about their opponents and revise their level-k rule. In this study, Neural Fitted Q Iteration, which is an approximate reinforcement learning method, is used to model time-extended decisions of pilots with 3D maneuvers. An analysis of UAS integration is conducted using an Example 3D scenario in the presence of manned aircraft and fully autonomous UAS equipped with sense and avoid algorithms.

Level-k reasoning in school choice

Boston Mechanism (BM) and Deferred Acceptance (DA) are two competing algorithms in school choice. BM is manipulable whereas DA encourages truth-telling. Recognizing heterogeneous sophistication of students, we use the level-k model to analyze students' strategies under BM in a complete information environment with strict school priorities. Because students tend to manipulate first choice under BM, we characterize the first choice reported by students at all possible sophistication levels. In a general framework we show that BM is no less efficient than DA, and in two specific belief settings we further show that it is as if students run DA in their minds when they make level-k reasoning under BM. We compare the relation between sophistication and welfare under BM in the two belief settings to emphasize the role of high sophistication and correct belief. We also compare our results with Nash equilibrium analysis.

Learning with Heterogeneous Misspecified Models: Characterization and Robustness

This paper develops a general framework to study how misinterpreting information impacts learning. Our main result is a simple criterion to characterize long-run beliefs based on the underlying form of misspecification. We present this characterization in the context of social learning, then highlight how it applies to other learning environments, including individual learning. A key contribution is that our characterization applies to settings with model heterogeneity and provides conditions for entrenched disagreement. Our characterization can be used to determine whether a representative agent approach is valid in the face of heterogeneity, study how differing levels of bias or unawareness of others’ biases impact learning, and explore whether the impact of a bias is sensitive to parametric specification or the source of information. This unified framework synthesizes insights gleaned from previously studied forms of misspecification and provides novel insights in specific applications, as we demonstrate in settings with partisan bias, overreaction, naive learning, and level-k reasoning.

Deterrence Using Peer Information

We propose a deterrence mechanism that utilizes insider information contained in criminal networks. Under this mechanism, a suspect, identified by the regulator, can nominate a connected person from the network for the crime, and only the more severe offender is penalized. We show theoretically that the equilibrium crime levels are lower with this mechanism than in the case where the first suspect is automatically punished. Crime levels depend on the network structure. Our experiment confirms the mechanism's deterrence effect, but also uncovers little sensitivity of the mechanism to network structure. Level-k reasoning helps explain these patterns.

Do people minimize regret in strategic situations? A level-k comparison

Regret minimization and level-k reasoning have been proposed as alternative models for rationalizing non-equilibrium behavior in games. We provide a theoretical and experimental analysis of the relationship between these two models, a relationship that has been neglected by economists. Both theories predict the same behavior in a surprisingly large number of experimentally tested games. We identify conditions under which this happens and use them to design a series of games to separate minimax regret from level-1. The experimental test of these games and data from Costa-Gomes and Crawford (2006) reveal that no one systematically minimizes regret, casting doubt on minimax regret as a relevant explanation for behavior in strategic situations.

Do People Minimize Regret in Strategic Situations? A Level-k Comparison

Regret minimization and level-k reasoning have been proposed as alternative models for rationalizing non-equilibrium behavior in games. We provide a theoretical and experimental analysis of the relationship between these two models, a relationship that has been neglected by economists. Both theories predict the same behavior in a surprisingly large number of experimentally tested games. We identify conditions under which this happens and use them to design a series of games to separate minimax regret from level-1. The experimental test of these games and data from Costa-Gomes and Crawford (2006) reveal that no one systematically minimizes regret, casting doubt on minimax regret as a relevant explanation for behavior in strategic situations.

Level-k Reasoning with Heterogeneous Information Signals

Level-k Reasoning with Heterogeneous Information Signals

Coordination games with asymmetric payoffs: An experimental study with intra-group communication

Two alternative modes of reasoning in coordination games are prominently discussed in the literature: level-k thinking and team reasoning. In order to differentiate between the two modes of reasoning, we experimentally investigate payoff-asymmetric coordination games using an intra-group communication design that incentivizes subjects to explain the reasoning behind their decisions. We find that the reasoning process is significantly different between games. In payoff-symmetric games, team reasoning plays an important role for coordination. In payoff-asymmetric games, level-k reasoning results in frequent miscoordination. Our study clearly illustrates how small differences between strategic situations have a strong influence on reasoning.

Payoff inequity reduces the effectiveness of correlated-equilibrium recommendations

We examine theoretically and experimentally how individuals’ willingness to follow third-party recommendations in 2 × 2 games is affected by payoff asymmetry. We consider six versions of Battle-of-the-Sexes. Recommendations imply monetary payoffs that are equal ex ante, but unequal ex post. So, although following recommendations constitutes a Nash equilibrium under standard preferences, sufficiently inequity-averse players can rationally disobey a recommendation that would lead to a very unfavourable payoff distribution, as long as the cost of doing so is not too large. Our theoretical model incorporates inequity aversion, along with level-k reasoning. Our main experimental result is consistent with the model: as either payoff asymmetry increases or the cost of disobeying an unfavourable recommendation decreases, subjects are more likely to disobey recommendations.

Cognitive Ability, Character Skills, and Learning to Play Equilibrium: A Level-kAnalysis

In this paper we investigate how cognitive ability and character skills influence behavior, success and the evolution of play towards Nash equilibrium in repeated strategic interactions. We study behavior in a p-beauty contest experiment and find striking differences according to cognitive ability: more cognitively able subjects choose numbers closer to equilibrium, converge more frequently to equilibrium play and earn more even as behavior approaches the equilibrium prediction. To understand better how subjects with different cognitive abilities learn differently, we estimate a structural model of learning based on level-k reasoning. We find a systematic positive relationship between cognitive ability and levels; furthermore, the average level of more cognitively able subjects responds positively to the cognitive ability of their opponents, while the average level of less cognitively able subjects does not respond. Finally, we compare the influence of cognitive ability to that of character skills, and find that both cognition and personality affect behavior and learning. More agreeable and emotionally stable subjects perform better and learn faster, although the effect of cognitive ability on behavior is stronger than that of character skills.

Level-K Reasoning in School Choice

Assuming students have heterogeneous sophistication and use the level-k reasoning, we compare the Boston mechanism (BM) and the student-optimal stable mechanism (SOSM). We show that for any distribution of students’ sophistication, BM is never less efficient than SOSM and sometimes is more efficient. We also show that more sophisticated students do not necessarily game BM better. Our results are different from those obtained from the Nash equilibrium analysis, and provide new insights on school choice.

Procrastination in Teams

Naively present-biased agents are known to be severe procrastinators. In team settings, procrastination can represent a form of free-riding that, in excess, may jeopardize a team’s ability to meet a deadline. Despite their reputations, we show how naivete and present bias can, in the right task environment, be desirable traits in a teammate that enable a team to optimize its performance while eliminating inefficient free-riding. These benefits emerge only from a more flexible specification (in comparison to existing models) as to how naive players reassess prior beliefs upon confronting present bias. By allowing the ‘depth’ and ‘direction’ of such reassessments to vary, our model links present-biased discounting theories to the recently-revived interest in modeling non-Bayesian reactions to null events, while offering a distinct approach reminiscent of level-k reasoning. Key themes from our analysis include the value of behavioral diversity, the opposite effects of ‘introspection’ and ‘extrospection’ on motivation, and that under- and over-thinking can both undermine efficiency.

On the persistence of strategic sophistication

We examine whether the “Level-k” model of strategic behavior generates reliable cross-game predictions at the individual level. We find no correlation in subjects' estimated levels of reasoning across two families of games. Furthermore, estimating a higher level for Ann than Bob in one family of games does not predict their ranking in the other. Direct tests of strategic reasoning generally do not predict estimated levels. Within families of games, we find that levels are fairly consistent within one family, but not the other. Our results suggest that the use of Level-k reasoning varies by game, making prediction difficult.

Cognitive Ability, Character Skills, and Learning to Play Equilibrium: A Level-k Analysis

In this paper we investigate how cognitive ability and character skills influence behavior, success and the evolution of play towards Nash equilibrium in repeated strategic interactions. We study behavior in a p-beauty contest experiment and find striking differences according to cognitive ability: more cognitively able subjects choose numbers closer to equilibrium, converge more frequently to equilibrium play and earn more even as behavior approaches the equilibrium prediction. To understand better how subjects with different cognitive abilities learn differently, we estimate a structural model of learning based on level-k reasoning. We find a systematic positive relationship between cognitive ability and levels; furthermore, the average level of more cognitively able subjects responds positively to the cognitive ability of their opponents, while the average level of less cognitively able subjects does not respond. Finally, we compare the influence of cognitive ability to that of character skills, and find that both cognition and personality affect behavior and learning. More agreeable and emotionally stable subjects perform better and learn faster, although the effect of cognitive ability on behavior is stronger than that of character skills.

Level-k reasoning in a generalized beauty contest

We study how the predictive power of level-k models changes as we perturb the classical beauty contest setting along two dimensions: the strength of the coordination motive and the information symmetry. We use a variation of the Morris and Shin (2002) model as the unified framework for our study, and find that the predictive power of level-k models varies considerably along these two dimensions. Level-k models are successful in predicting subject behavior in settings with symmetric information and a strong coordination motive. Their predictive power weakens significantly when either private information is introduced or the importance of the coordination motive is decreased.