Level-k Model Research Articles

Consistent Depth of Reasoning in Level-k Models

Level-k models often assume that individuals employ a fixed depth of reasoning across different games. We study this assumption by having subjects make choices in five classes of games chosen to identify inconsistent depth of reasoning. We demonstrate that depth of reasoning is pervasively inconsistent, changing both within and between classes of games. We show that this cannot easily be explained by factors such as subject confusion, changing beliefs about others’ depth of reasoning, stochastic choice, model misspecification, changing incentives, or low cognitive ability. We develop a simple model incorporating ambiguity aversion that predicts inconsistent depth of reasoning. (JEL C57, C90, D81, D83, D91)

How cognitive skills affect strategic behavior: Cognitive ability, fluid intelligence and judgment

We explore the influence of cognitive ability and judgment on strategic behavior in the beauty contest game (where the Nash equilibrium action is zero). Using the level-k model of bounded rationality, cognitive ability and judgment both predict higher level strategic thinking. However, individuals with better judgment choose zero less frequently, and we uncover a novel dynamic mechanism that sheds light on this pattern. Taken together, our results indicate that fluid (i.e., analytical) intelligence is a primary driver of strategic level-k thinking, while facets of judgment that are distinct from fluid intelligence drive the lower inclination of high judgment individuals to choose zero.

Cognitive reflection in experimental anchored guessing games

The cognitive reflection test or CRT (Frederick, 2005) has been found to be a reliable predictor of the degree of strategic sophistication of subjects in a variety of laboratory experiments. These studies have found that subjects who score higher in the CRT make choices that are closer to Nash equilibrium (i.e., Brañas-Garza et al., 2012). In an extended level-k model with free subjective beliefs, we theoretically decompose the closeness to equilibrium for the class of anchored guessing games introduced in Ballester et al. (2023) into two effects: subjects with a smaller distance to equilibrium must possess a higher reasoning level in the level-k hierarchy or their level-k iteration process must begin from a starting point (called “seed”) that is inherently more advantageously positioned, which translates into the concept of “seed distance” (or both). Our main experimental finding is that subjects with a higher CRT score play closer to equilibrium due to the fact that they iterate more often in their reasoning process (as in Brañas-Garza et al., 2012), yet we find no clear evidence that they have a smaller seed distance. We also find evidence of a learning or adaptation process, which can be characterized by a warm-up phase (in which subjects reduce their seed distance), followed by a learning phase (in which they increase their reasoning level, at a faster rate in subjects with higher CRT) and then a saturation phase in which no further improvements are made.

An experimental test of the global-game selection in coordination games with asymmetric players

In symmetric binary-choice coordination games, the global-game selection (GGS) has been proven to predict a high proportion of observed choices correctly. In these games, the GGS is identical to the best response to Laplacian beliefs about the fraction of players choosing either action. This paper presents an experiment on asymmetric games in which the GGS differs from the best response to Laplacian beliefs. In these games, the best response to Laplacian beliefs is a better predictor of individual actions than the GGS. In the considered games, the GGS provides poor guidance and also fails to give the right qualitative comparative statics predictions. Simple cognitive hierarchy models yield better predictions. The best response to a Laplacian belief about the distribution of other players’ actions yields the best prediction. Comparing maximum likelihood estimates for four probabilistic models shows that noisy Laplacian-belief or level-k models fit better than an estimated global-game equilibrium.

Predictable Effects of Visual Salience in Experimental Decisions and Games

AbstractBottom-up stimulus-driven visual salience is largely automatic, effortless, and independent of a person’s “top-down” perceptual goals; it depends only on features of a visual stimulus. Algorithms have been carefully trained to predict stimulus-driven salience values for each pixel in any image. The economic question we address is whether these salience values help explain economic decisions. Our first experimental analysis shows that when people pick between sets of fruits that have artificially induced value, predicted salience (which is uncorrelated with value by design) leads to mistakes. Our second analysis uses evidence from games in which choices are locations in images. When players are trying to cooperatively match locations, predicted salience is highly correlated with the success of matching (r = .57). In competitive hider-seeker location games, players choose salient locations more often than predicted by the unique Nash equilibrium. This tendency creates a disequilibrium “seeker’s advantage” (seekers win more often than predicted in equilibrium). The result can be explained by level-k models in which predicted stimulus-driven salience influences level-0 choices and thereby influences overall perceptions, beliefs, and choices of higher-level players. The third analysis shows that there is an effect of visual salience in matrix games, but it is small and statistically weak. Applications to behavioral IO, price and tax salience, nudges and design, and visually influenced beliefs are suggested.

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.

Learning in two-dimensional beauty contest games: Theory and experimental evidence

We extend the beauty contest game to two dimensions: each player chooses two numbers to be as close as possible to certain target values, which are linear functions of the averages of the two number choices. One of the targets depends on the averages of both numbers, making the choices interrelated. We report on an experiment where we vary the eigenvalues of the associated two-dimensional linear system and find that subjects can learn the Pareto-optimal Nash Equilibrium of the system if both eigenvalues are stable and cannot learn it if both eigenvalues are unstable. Interestingly, subjects can also learn it if the system has the saddlepath property – with one stable and one unstable eigenvalue — but only if the one unstable eigenvalue is negative. We show theoretically that our results cannot be explained by homogeneous level-k models where all agents apply the same level k depth of reasoning to their choices, including the naïve learning model. However, our results can be explained by a mixed cognitive-levels model, including the adaptive learning model. We also run a horserace between many models used in the literature with the winner being a simple mixed model with levels 0, 1, and equilibrium reasoning.

Beliefs, Learning, and Personality in the Indefinitely Repeated Prisoner's Dilemma

The indefinitely repeated prisoner’s dilemma (IRPD) captures the trade-off between the short-term payoff from exploiting economic partners and the long-term gain from building successful relationships. We aim to understand more about how people form and use beliefs about others in the IRPD. To do so, we elicit beliefs about the supergame strategies chosen by others. We find that initial beliefs match behavior quite well and that most subjects choose strategies that perform well given their beliefs. Motivated by belief clustering, we use beliefs to estimate a level-k model of boundedly rational thinking. We analyze how beliefs and behavior evolve with experience: beliefs become more accurate over time, and we use beliefs to provide new evidence about the mechanism that underlies learning from experience. Finally, we find that a survey measure of trust predicts cooperative behavior and optimism about others’ cooperation, which helps to explain how trust underpins successful economic exchanges.

Level-K Classification from EEG Signals Using Transfer Learning.

Tacit coordination games are games in which communication between the players is not allowed or not possible. In these games, the more salient solutions, that are often perceived as more prominent, are referred to as focal points. The level-k model states that players’ decisions in tacit coordination games are a consequence of applying different decision rules at different depths of reasoning (level-k). A player at will randomly pick a solution, whereas a player will apply their strategy based on their beliefs regarding the actions of the other players. The goal of this study was to examine, for the first time, the neural correlates of different reasoning levels in tacit coordination games. To that end, we have designed a combined behavioral-electrophysiological study with 3 different conditions, each resembling a different depth reasoning state: (1) resting state, (2) picking, and (3) coordination. By utilizing transfer learning and deep learning, we were able to achieve a precision of almost 100% (99.49%) for the resting-state condition, while for the picking and coordination conditions, the precision was 69.53% and 72.44%, respectively. The application of these findings and related future research options are discussed.

Information quality and regime change: Evidence from the lab

We experimentally test the effects of information quality in a global game of regime change. The game features a payoff structure such that more dispersed private information induces agents to attack more often and reduces regime stability in the Bayesian Nash Equilibrium. We show that subjects in the lab do not play as predicted by equilibrium theory. Instead, more dispersed information makes subjects more cautious, increasing regime stability. We show that this finding is consistent with a modified global game model in which agents engage in level-k thinking. In the level-k model, information quality affects agents’ actions through a novel channel, that enables a strategic attenuation effect. As information quality worsens, strategic complementarities between different level-k types weaken, generating a force that is capable of reversing the comparative statics from the equilibrium model.

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.

Are We Strategically Naïve or Guided by Trust and Trustworthiness in Cheap-Talk Communication?

Cheap-talk communication between parties with conflicting interests is common in many business and economic settings. Two distinct behavioral economics theories, the trust-embedded model and the level-k model, have emerged to explain how cheap talk works between human decision makers. The trust-embedded model considers that decision makers are motivated by nonpecuniary motives to be trusting and trustworthy. In contrast, the level-k model considers that decision makers are purely self-interested but limited in their ability to think strategically. Although both theories have been successful in explaining cheap-talk behaviors in separate contexts, they point to contrasting drivers for human behaviors. In this paper, we provide the first direct comparison of both theories within the same context. We show that, in a cheap-talk setting that well represents many practical situations, the two models make characteristically distinct and empirically distinguishable predictions. We leverage past experiment data from this setting to determine what aspects of cheap-talk behavior each model captures well and which model (or combination of models) has better explanatory power and predictive performance. We find that the trust-embedded model emerges as the dominant explanation. Our results, thus, highlight the importance of investing in systems and processes to foster trusting and trustworthy relationships in order to facilitate more effective cheap-talk interactions. This paper was accepted by Charles Corbett, operations management.

Efficient mechanisms for level-k bilateral trading

This paper revisits Myerson and Satterthwaite's (1983) classic analysis of mechanism design for bilateral trading, replacing equilibrium with a level-k model of strategic thinking and focusing on direct mechanisms. The revelation principle fails for level-k models, so restricting attention to direct mechanisms and imposing incentive-compatibility are not without loss of generality. If, however, only direct, level-k-incentive-compatible mechanisms are feasible and traders' levels are observable, Myerson and Satterthwaite's characterization of mechanisms that maximize traders' total surplus subject to incentive constraints generalizes qualitatively to level-k models. If only direct, level-k-incentive-compatible mechanisms are feasible but traders' levels are not observable, generically a particular posted-price mechanism maximizes traders' total expected surplus subject to incentive constraints. If direct, non-level-k-incentive-compatible mechanisms are feasible and traders best respond to them, total expected surplus-maximizing mechanisms may take completely different forms.

On the evolution of heuristics and bounded rational behavior in random games

The emergence of heuristics, sophisticated behavior—such as higher levels of Theory of Mind—and substantive rationality in strategic situations can be studied using evolutionary game theory to jointly account for the interactions between players and their environments. Research thus far has been relatively constrained by limiting analyses to a small number of simple games and by focusing on selection operating over actions rather than rules or policies that prescribe behavior in a wide range of games. As a result, the existing literature has not adequately reconciled the significant evidence amassed in the behavioral game theory literature that people use bounded-rational strategies, rather than the normative prescription (Nash equilibrium). I systematically explore the differential costs of implementing decision policies and how evolution is influenced by key environmental characteristics. Detailed maps linking the performance of policies to 3,060 environments are presented demonstrating that the steady state distribution of policies under replicator dynamics is strongly dependent on: a) complexity costs, derived from the opportunity cost associated with the decision time to execute the policies at a process-level and b) the environmental characteristics such as the size of the game, the degree of payoff uncertainty and the degree of mutual interests or conflict between players. Evolution rarely selects for substantive rationality (Nash equilibrium), favoring instead a plethora of bounded-rational policies with varying levels of theory of mind (Level-k models), and in some environmental niches other heuristics such as maxmin or maximization of joint payoffs to players. An intriguing finding is that there exist non-trivial environmental niches with moderate to high complexity costs, where random behavior can be selected for in conjunction with a policy that best-responds to randomness (Level-1). I conclude that while bounded-rational policies do not conform to the standards of substantive rationality, they exhibit both procedural and ecological rationality once the complexity of decision processes and interactions with environmental characteristics are accounted for.

A Bayesian Level-k Model in n-Person Games

In standard models of iterative thinking, players choose a fixed rule level from a fixed rule hierarchy. Nonequilibrium behavior emerges when players do not perform enough thinking steps. Existing approaches, however, are inherently static. This paper introduces a Bayesian level-k model, in which level-0 players adjust their actions in response to historical game play, whereas higher-level thinkers update their beliefs on opponents’ rule levels and best respond with different rule levels over time. As a consequence, players choose a dynamic rule level (i.e., sophisticated learning) from a varying rule hierarchy (i.e., adaptive learning). We apply our model to existing experimental data on three distinct games: the p-beauty contest, Cournot oligopoly, and private-value auction. We find that both types of learning are significant in p-beauty contest games, but only adaptive learning is significant in the Cournot oligopoly, and only sophisticated learning is significant in the private-value auction. We conclude that it is useful to have a unified framework that incorporates both types of learning to explain dynamic choice behavior across different settings. This paper was accepted by Manel Baucells, decision analysis.

Are We Strategically Naïve or Guided by Trust and Trustworthiness in Cheap-Talk Communication?

Cheap-talk communication between parties with conflicting interests is common in many business and economic settings. Two distinct behavioral economics theories, the trust-embedded model and the level-k model, have emerged to explain how cheap talk works between human decision makers. The trust-embedded model considers that decision makers are motivated by non-pecuniary motives to be trusting and trustworthy. In contrast, the level-k model considers that decision makers are purely self-interested, but limited in their ability to think strategically. While both theories have been successful in explaining cheap-talk behaviors in separate contexts, they point to contrasting drivers for human behaviors. In this paper, we provide the first direct comparison of both theories within the same context. We show that, in a cheap-talk setting that well represents many practical situations, the two models make characteristically distinct and empirically distinguishable predictions. We leverage past experiment data from this setting to determine what aspects of cheap-talk behavior each model captures well, and which model (or combination of models) has better explanatory power and predictive performance. We find that the trust-embedded model emerges as the dominant explanation. Our results thus highlight the importance of investing in systems and processes to foster trusting and trustworthy relationships in order to facilitate more effective cheap-talk interactions.

Beliefs, Learning, and Personality in the Indefinitely Repeated Prisoner's Dilemma

The indefinitely repeated prisoner's dilemma (IRPD) captures the trade-off between the short-term payoff from exploiting economic partners and the long-term gain from building successful relationships. We aim to understand more about how people form and use beliefs about others in the IRPD. To do so, we elicit beliefs about the supergame strategies chosen by others. We find that initial beliefs match behavior quite well and that most subjects choose strategies that perform well given their beliefs. Motivated by belief clustering, we use beliefs to estimate a level-k model of boundedly rational thinking. We analyze how beliefs and behavior evolve with experience: beliefs become more accurate over time, and we use beliefs to provide new evidence about the mechanism that underlies learning from experience. Finally, we find that a survey measure of trust predicts cooperative behavior and optimism about others' cooperation, which helps to explain how trust underpins successful economic exchanges.

Self-selection with non-equilibrium beliefs: Predicting behavior in a tournament experiment

In this study we use level-k thinking and a recently proposed model of non-equilibrium beliefs in dynamic games (PBNLK) to predict behavior in a tournament with self-selection. We find that the combination of level-k and PBNLK predicts both the population of types in the tournament, as well as the mean and variance of efforts better than Nash equilibrium, a static level-k model and other models of non-equilibrium beliefs. Our results show that non-equilibrium beliefs are an important determinant for the decision to compete in a tournament and the performance in that tournament. Moreover, a useful model of non-equilibrium beliefs should allow players to update their beliefs during the course of the competition.

Robust predictions under finite depth of reasoning

This paper studies a robustness of solutions in finite depth of reasoning models. The level-k and cognitive hierarchy models often assume that each player has a commonly known anchor behavior (so called a level-0 action) to obtain a sharp prediction. We provide a general framework to examine whether the obtained prediction is robust to small uncertainty about other players’ anchors. Our main result shows that any \(\mathbf p\)-dominant equilibrium is robust if players put sufficiently small probability (decreasing in \(\mathbf p\)) on other players’ having high reasoning levels. This result highlights a distinction between the two prominent models: a risk-dominated equilibrium is robust in the cognitive hierarchy model, but not in the level-k model.

Using Visual Salience in Empirical Game Theory

“Bottom-up” visual perception is fast, universal, and does not depend on personal goals or experience. Where exactly bottom-up attention is directed in a visual image-- usually based on contrast, color, feature orientation, and centrality-- can be accurately predicted by machine-learned algorithms. We test whether this type of salience can help explain economic decisions in four experimental analyses (three are new). When people pick between sets of valued fruits, salience can lead to mistakes (lower-value sets are sometimes chosen because they are salient). Two studies show strong and weak influence on choices, respectively, of salience in stock price charts and salient payoffs in game matrices. The central analysis is evidence from games in which choices are locations in images. When players are trying to cooperatively match locations, concentration of salience is associated with the success of matching (r=.57). In competitive hider-seeker location games, all players choose salient locations more often. This fact creates a “seeker’s advantage” (9% wins compared to the 7% predicted by unique equilibrium). The location game data are consistent with cognitive hierarchy and level-k models in which predicted salience influences nonstrategic level-0 choices. Model estimates from hider-seeker games can predict behavior in matching games despite their opposite strategic structures. Looking forward, bottom-up salience could be included as a predictable type of costless involuntary “pre-attention” in expanded models of Bayesian-rational inattention. Novel applications involving prices and taxes, disability, ethnic markers, and visually-influenced beliefs are suggested.