Level-k Model Research Articles

Dynamic Level-K and Cognitive Hierarchy Models of Repeated Normal-Form Games: A Note

This note investigates a set of dynamic versions of the level-k (LK) and cognitive hierarchy (CH) models in repeated normal-form games. Conventional LK and CH models assume a reasoning process that does not allow learning. This can be a restrictive assumption: When facing a repeated game, individuals tend to gather information from various sources. Even individuals with the lowest reasoning levels tend to improve their decisions over time. We propose dynamic level-k (DLK) and dynamic cognitive hierarchy (DCH) models to enable the interplay between within-period reasoning and between-period learning. In our models, players first update their beliefs about the choices made by their opponents across periods. Then, within each period, conditional on their reasoning levels, players stochastically best respond to the predicted play of their opponents. We select five publicly available datasets from four well-established papers to test the proposed models. Compared with other static and dynamic models, our models have better or similar performance both in terms of the in-sample fits and out-of-sample validations. Moreover, we provide a more detailed discussion linking each component of the models to the performance outcomes. The results underscore the importance of both learning and reasoning in understanding the experimental observations in repeated normal-form games.

The strategic environment effect in beauty contest games

Recent experimental studies have shown that observed outcomes deviate significantly more from the Nash equilibrium when actions are strategic complements than when they are strategic substitutes. This “strategic environment effect” offers promising insights into the aggregate consequences of interactions among heterogeneous boundedly rational agents, but its macroeconomic implications have been questioned because the underlying experiments involve a small number of agents. We studied beauty contest games with a unique interior Nash equilibrium to determine the critical group size for triggering the strategic environment effect, and we use both theory and experiments to shed light on its effectiveness. Based on cognitive hierarchy and level-K models, we show theoretically that the effect is operative for interactions among three or more agents. Our experimental results show a statistically significant strategic environment effect for groups of five or more agents, establishing its robustness against the increase in the population size.

False Consensus in Games: Embedding Level-k Models into Games of Incomplete Information

Level-k models have recently gained popularity as a framework for strategic behavior with bounded rationality. These models make two assumptions: agents of a higher level have richer beliefs, and can also perform more computations. I develop a model that incorporates Level-k models into games of incomplete information, by assuming that players’ beliefs exhibit the false consensus effect, which is the empirical observation that people think others share their beliefs, what allows players to believe others have the same level. A player’s level determines how complex her beliefs and I consider two solution concepts: Interim Correlated Rationalizability (ICR, with which level-0 behavior becomes endogenous), and k-th Best Response, which is the usual concept for Level-k games. My main result is that in this model the product topology and the uniform strategic topology coincide, what implies that players with similar beliefs behave similarly. Therefore, unlike for general type spaces, predictions will be robust to small specification errors. I show that in the Email game of Rubinstein (1989), when players receive few messages they never attack according to ICR; however, when they receive enough messages, they behave as if there was complete information. Other applications show that the model is consistent with experimental evidence of trading in “no-trade” contexts, and recovering multiplicity of equilibria in global games.

A window of cognition: Eyetracking the reasoning process in spatial beauty contest games

We study the reasoning process in an environment where final choices are well understood and the associated theory is procedural by introducing two-person beauty contest games played spatially on two-dimensional grid maps. Players choose locations and are rewarded by hitting targets dependent on opponents' choice locations. By tracking subjects' eye movements (lookups), we infer their reasoning process and classify subjects into various levels. More than a half of the subjects' classifications coincides with their classifications using final choices, supporting a literal interpretation of the level-k model for subject's reasoning process. Lookup analyses reveal that the center area is where most subjects initially look at. This sheds light on the level-0 belief. Moreover, learning lookups of a trial on average could increase payoffs of that trial and eliminates roughly 60% of the gap to empirical best response, indicating how valuable lookups can help predict choices.

The expected externality mechanism in a level-k environment

Mechanism design theory strongly relies on the concept of Nash equilibrium. However, studies of experimental games show that Nash equilibria are rarely played and that subjects may be thinking only a finite number of iterations. We study one of the most influential benchmarks of mechanism design theory, the expected externality mechanism (D’Aspremont and Gerard-Varet, J Public Econ 11:25–45, 1979) in a finite-depth environment described by the Level-k model. While the original mechanism may fail to implement the efficient rule in this environment, it can be adjusted to restore efficiency.

The Expected Externality Mechanism in a Level-k Environment

Mechanism design theory strongly relies on the concept of Nash equilibrium. However, studies of experimental games show that Nash equilibria are rarely played and that subjects may be thinking only a finite number of iterations. We study one of the most influential benchmarks of mechanism design theory, the expected externality mechanism (D’Aspremont and Gerard-Varet, J Public Econ 11:25–45, 1979) in a finite-depth environment described by the Level-k model. While the original mechanism may fail to implement the efficient rule in this environment, it can be adjusted to restore efficiency.

The nature of social learning: Experimental evidence

In the wide economic literature on social learning, many types of behavior – rational and non-rational – have been proposed. I suggest a level-k model that unifies many of them in one framework. This paper analyzes experimental data that is able to distinguish between levels of reasoning at the individual level. It relies on rich, existing data from Çelen and Kariv (2004) as well as new experimental data that includes written accounts of reasoning from incentivized intra-team communication. Three datasets provide consistent evidence that naïve inference in form of the best response to truthful play is the most common approach to social learning. The empirical type distributions feature heterogeneity similar to other level-k applications.

Second-price common value auctions with uncertainty, private and public information: Experimental evidence

We conduct a laboratory experiment of second-price sealed bid auctions of a common value good with two bidders. Bidders face three different types of information: common uncertainty (unknown information), private information (known by one bidder) and public information (known by both bidders), and auctions differ on the relative importance of these three types of information. We find that subjects barely differentiate between private and public information and deviate from the theoretical predictions with respect to all three types of information. There is under-reaction to both private and public information and systematic overbidding in all auctions above and beyond the standard winner’s curse. The Cursed Equilibrium and Level-k models successfully account for some features of the data but others remain largely unexplained.

An Experimental Study on Sequential Auctions with Privately Known Capacities

We experimentally study bidding behavior in sequential first-price procurement auctions where bidders’ capacity constraints are private information. Treatment differs in the ex-ante probability distribution of sellers’ capacities and in the (exogenous) probability that the second auction is actually implemented. Our results show that: (i) bidding behavior in the second auction conforms with sequential rationality; (ii) while first auction’s bids negatively depend on capacity, bidders seem unable to recognize this link when, at the end of the first auction, they state their beliefs on the opponent’s capacity. To rationalize this inconsistency between bids and beliefs, we conjecture that bidding in the first auction is also affected by a hidden, behavioral type – related to the strategic sophistication of bidders – that obfuscates the link between capacity and bids. Building on this intuition, we show that a simple level-k model may help explain the inconsistency.

Level-K Models Rationalize Overspending in Contests

The experimental evidence on contests often reports significant overspending of subjects as opposed to the Nash equilibrium. We complement the existing explanations by providing a theoretical rationale for overspending which is based on the level-k model.

A generalized cognitive hierarchy model of games

Subjects in simple games frequently exhibit non-equilibrium behaviors. Cognitive hierarchy (CH) and level k (LK) are two prevailing structural models that capture such behaviors well. This paper proposes a generalized CH (GCH) model that nests a variant of the LK model, called LM. GCH differs from CH in two ways. First, each lower level's actual frequency is exponentially weighted with α to form level-k's belief on relative proportions; α captures stereotype bias. CH assumes no stereotype bias (α=1) and LM assumes extreme bias (α=∞). Second, GCH replaces random choice with minimum aversion for level 0. Level 0s are more likely to choose strategies that never yield the minimum payoff for any of the opponent's strategies. GCH captures behaviors better than CH and LK in fifty-five n×m games from four datasets. Robustness tests using three new games further validate GCHs descriptive strength over CH and LK.

Designing Contests between Heterogeneous Contestants: An Experimental Study of Tie-Breaks and Bid-Caps in All-Pay Auctions

A well-known theoretical result in the contest literature is that greater heterogeneity decreases performance of contestants because of the “discouragement effect.” Leveling the playing field by favoring weaker contestants through bid-caps and favorable tie-breaking rules can reduce the discouragement effect and increase the designer’s revenue. We test these predictions in an experiment. Our data show that indeed, strengthening weaker contestants through tie-breaks and bid-caps significantly diminishes the discouragement effect. Bid-caps can also improve revenue. Most deviations from Nash equilibrium can be explained by the level-k model of reasoning.

What Model Best Describes Initial Choices in a Cournot Duopoly Experiment?

This paper tests empirically four models of bounded rationality using data from first responses in a Cournot duopoly experiment. Specifically, the models considered are Level-k, Quantal Response Equilibrium, Noisy Introspection and Logit Cognitive Hierarchy. It is found that the Level-k model (with proportions of Level-0, Level-1 and Level-8 given by 68.5%, 13.2% and 18.3% respectively) provides the best fit in terms of Log-Likelihood and BIC. Moreover, the robustness of our findings is corroborated analyzing subsets of the original data.

Elicited salience and salience-based level-[formula omitted

A level-k model based on a specific salience-pattern is the only model in the literature that accounts for behaviour in hide-and-seek games. This paper presents nine different experiments designed to measure salience. The elicited salience patterns tend to be similar, but none of them is similar to the pattern needed to allow the level-k model explain the hide-and-seek data. When based on any of the empirical salience measures, the salience-based level-k model does not fit the data well.

Designing Contests between Heterogeneous Contestants: An Experimental Study of Tie-Breaks and Bid-Caps in All-Pay Auctions

A well-known theoretical result in the contest literature is that greater heterogeneity decreases performance of contestants because of the Leveling the playing field by favoring weaker contestants through bid-caps and favorable tie-breaking rules can reduce the discouragement effect and increase the designer's revenue. We test these predictions in an experiment. Our data show that indeed, strengthening weaker contestants through tie-breaks and bid-caps significantly diminishes the discouragement effect. Bid-caps can also improve revenue. Most deviations from Nash equilibrium can be explained by the level-k model of reasoning.

Bounded Rationality in a Cournot Duopoly Game

This paper analyzes choices and welfare in a Cournot duopoly setting with linear demand using four models of bounded rationality. The models studied in this paper are Level-k, Cognitive Hierarchy, Asymmetric Quantal Response and Noisy Introspection. It is found that in the Level-k model choices, profits and welfare alternate around the Nash Equilibrium levels depending on whether the level is odd or even. In the Cognitive Hierarchy model the choices of the first two types (L-0 and L-1) coincide with the choices in the Level-k model, a L-2 produces a smaller quantity while the quantity of a L-3 is higher or lower depending on the value of a particular parameter in the model. Both in the Asymmetric Quantal Response and Noisy Introspection models, we find that choices are spread around the Nash Equilibrium level for all parameter values and thus welfare is below the Nash Equilibrium benchmark. We also use parameter estimates from other well-known experiments to obtain an approximation to empirically plausible welfare levels.

Stake size and the power of focal points in coordination games: Experimental evidence

We collect data from symmetric and asymmetric coordination games with a focal point and vary the stake size. The data show that in symmetric games coordination on the label-salient strategy increases with stake size. By contrast, in asymmetric games the coordination rates do not vary with stake size and are close to the levels predicted by both the mixed Nash equilibrium and the level-k model used by Crawford et al. (2008). These findings suggest that players' mode of reasoning, and the extent to which it can be explained by team reasoning or a level-k model, crucially depends on the symmetry or asymmetry of the coordination payoffs.

Cheap Talk with Two Audiences: An Experiment

In this paper we experimentally test strategic information transmission between one informed and two uninformed agents in a cheap-talk game. We find evidence of the “disciplining” effect of public communication as compared to private; however, it is much weaker than predicted by the theory. Adding a second receiver naturally increases the complexity of strategic thinking when communication is public. Using the level-k model, we exploit the within subject design to show how individuals decrease their level-k in public communication. Surprisingly, we find that individuals become more sophisticated when they communicate privately with two receivers rather than one.

Privacy concerns, voluntary disclosure of information, and unraveling: An experiment

We study the voluntary revelation of private information in a labor-market experiment where workers can reveal their productivity at a cost. While rational revelation improves a worker׳s payoff, it imposes a negative externality on others and may trigger further revelation. Such unraveling can be observed frequently in our data although less often than predicted. Equilibrium play is more likely when subjects are predicted to conceal their productivity than when they should reveal. This tendency of under-revelation, especially of low-productivity workers, is consistent with the level-k model. A loaded frame where the private information concerns the workers׳ health status leads to less revelation than a neutral frame.

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.