Level-k Thinking Research Articles

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.

Level-k thinking in the extensive form

AbstractLevel-k thinking has been widely applied as a solution concept for games in normal form in behavioral and experimental game theory. We consider level-k thinking in games in extensive form. Player’s may learn about levels of opponents’ thinking during the play of the game because some information sets may be inconsistent with certain levels. In particular, for any information set reached, a level-k player attaches the maximum level-$$\ell $$ ℓ thinking for $$\ell < k$$ ℓ < k to her opponents consistent with the information set. We compare our notion of strong level-k thinking with other solution concepts such as level-k thinking in the associated normal form, strong rationalizability, $$\Delta $$ Δ -rationalizability, iterated admissibility, backward rationalizability, backward level-k thinking, and backward induction. We use strong level-k thinking to reanalyze data from some prior experiments in the literature.

Level-[formula omitted] predatory trading

I incorporate the level-k thinking solution concept into a simplified (Brummermeier and Pedersen, 2005) predatory trading model to investigate the possibility of arbitraging arbitrageurs. While naive financial predators prey upon a single distressed investor, higher-level thinkers best respond to this and prey upon fellow predators. For some parameter values, sophisticated predators are able to reason their way to the Nash equilibrium strategy, and prices do not oscillate. As parameter values are perturbed, the system undergoes a bifurcation and predators select strategies from a mean-preserving spread of the Nash equilibrium strategy. In these settings, prices display excess volatility and a single shock can send predators into an oscillatory trading frenzy.

First Price Sealed-Bid Auctions with Bidders’ Heterogeneous Risk Behavior: An Adversarial Risk Analysis Approach

Bidders’ bidding behavior is analyzed in first price sealed-bid (FPSB) auctions using an adversarial risk analysis (ARA) framework. However, using nonstrategic play and level-k thinking solution concepts, modeling is performed by assuming only two bidders. Also, the aleatory and concept uncertainties have not been yet taken into account by using an ARA framework for these auctions. In this paper, we apply an ARA approach to model bidders’ bidding behavior in a more realistic way for FPSB auctions. We assume n bidders that may have different wealth and heterogeneous risk behaviors. We use nonstrategic play and level-k thinking solution concepts, and we take into account aleatory uncertainty in addition to epistemic uncertainty. Finally, concept uncertainty is taken into account to find ARA solutions for these auctions. We also provide numerical examples to illustrate our methodology.

Central Bank Balance Sheet Policies Without Rational Expectations

Abstract We study the effects of central bank balance sheet policies—namely, quantitative easing and foreign exchange interventions—in a model where people form expectations through an iterative level-k thinking process. We emphasize two main theoretical results. First, under a broad set of conditions, central bank interventions are effective under level-k thinking, while they are neutral in the rational expectations equilibrium. Second, when preferences exhibit constant relative risk aversion, asset purchases increase aggregate output if they target assets with pro-cyclical returns but reduce it if asset returns are counter-cyclical. Finally, we empirically show that forecast errors about future asset prices are predictable by balance sheet interventions, a property that differentiates our channel from popular alternatives, such as portfolio-balance and signaling channels.

Attack Identification for Cyber-Physical Security in Dynamic Games under Cognitive Hierarchy

This paper considers the problem of identifying the profiles and capabilities of attackers injecting adversarial inputs to a cyber-physical system. The system in question interacts with attackers of different levels of intelligence, each employing different feedback controllers against the system. Principles of behavioral game theory – specifically the concept of level-k thinking – is employed to construct a database of potential attack vectors. By observing the state trajectories under sequential interactions with different adversaries, the defender adaptively estimates both the number and profiles of the different attack signals using an online deterministic annealing approach. This information is used to dynamically estimate the level of intelligence of the attackers. Simulation results showcase the efficacy of the proposed method.

Level-K Thinking, Level-K-Value Strategy and Herd Behaviour Bias in Financial Decision-Making

The goal of the research is to analyse empirically if the financial decision-making is influenced by herd behaviour bias and if this bias is present during both bearish and bullish market trend, so if the investor act with the same Level-K thinking and herd behaviour bias, independently if the market is bearish or bullish. Keywords: Herd behaviour, decision making, Level-K thinking, Level-K-Value, loss aversion, prospect theory, disposition effect, bounded rationality, Keynesian beauty contest.

The Effect of The Company Size on the Intrinsic Value and Herd Behaviour

The goal of the research is to empirically investigate if the company size has an impact on the intrinsic value perceived by the investor and herd behaviour and if the media coverage and publicity influences the decision-making process of the investor. Consequently, the research will investigate if the investor prefers to invest in big size Company instead of to bet on small cap Company and investigate if media coverage and publicity imply an image value and consequently an intrinsic value associated to the Company. Keywords: company size bias, herd behaviour, intrinsic value bias, image value bias, regret bias, recency bias, strategic level-k thinking, beauty contest, bounded rationality.

Adversarial risk analysis for auctions using non-strategic play and level-k thinking: A general case of n bidders with regret

First-price sealed-bid auctions have been modeled using a utility function that takes into account bidders’ regret in case of winning or losing the auctioned item. However, that modeling does not consider bidders’ wealth which is an important determinant of bidders’ behavior in these auctions. In this paper, we apply an adversarial risk analysis approach for these auctions and find solutions using non-strategic play and level-k thinking solution concepts assuming n bidders. We define new regret parameters and a modified utility function to incorporate the effect of bidders’ wealth on their bidding behavior. In our modeling, we assume that the auctioned item is a normal item and has a reserve price. We give numerical examples to illustrate our methodology for each solution concept.

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.

Imperfect Macroeconomic Expectations: Evidence and Theory

We document a new fact about expectations: in response to the main shocks driving the business cycle, expectations under-react initially but over-shoot later on. We show how previous, seemingly conflicting, evidence can be understood as different facets of this fact. We finally explain what the cumulated evidence means for macroeconomic theory. There is little support for theories emphasizing under-extrapolation or two close cousins of it, cognitive discounting and level-K thinking. Instead, the evidence favors the combination of dispersed, noisy information and over-extrapolation. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Do You Know that I Know that You Know…? Higher-Order Beliefs in Survey Data

AbstractWe implement a new survey of firms, focusing on their higher-order macroeconomic expectations. The survey provides a novel set of stylized facts regarding the relationship between first-order and higher-order expectations of economic agents, including how they adjust their beliefs in response to a variety of information treatments. We show how these facts can be used to calibrate key parameters of noisy-information models with infinite regress as well as to test predictions made by this class of models. We also consider a range of extensions to the basic noisy-information model that can potentially better reconcile theory and empirics. Although some extensions like level-k thinking are unsuccessful, incorporating heterogeneous long-run priors can address the empirical shortcomings of the basic noisy-information model.

Feigning Ignorance for Long-term Gains

In dynamic strategic interactions, a player who might be able to spy the opponent’s actions might have incentives to feign ignorance and forgo immediate payoffs, so that he can earn higher future payoffs by manipulating the opponent’s suspicion. I model and experimentally implement this situation as a two-stage hide-and-seek game. About half of the informed experimental subjects fail to feign ignorance, and only some learn to through experience, a finding best explained by level-k thinking. The subjects who might be spied on underestimate the chance of spying upon observing ignorant behavior, yet mostly best respond to the opponents’ empirical behavior.

Non-equilibrium play in centipede games

Centipede games represent a classic example of a strategic situation, where the equilibrium prediction is at odds with human behavior. This study is explicitly designed to discriminate among the proposed explanations for initial responses in centipede games. Using many different centipede games, our approach determines endogenously whether one or more explanations are empirically relevant. We find that non-equilibrium behavior is too heterogeneous to be explained by a single model. However, most non-equilibrium choices can be fully explained by level-k thinking and quantal response equilibrium but each model for different subjects. Preference-based models play a negligible role in explaining non-equilibrium play.

Learning to Disclose: Disclosure Dynamics in the 1890s Streetcar Industry

We study the descriptiveness of the “unravelling” prediction in the 1890s streetcar industry. In this historical setting, capital-intensive streetcar companies gain the opportunity to disclose their earnings to dispersed investors via a new, quarterly newspaper supplement. We document that a quarter of the companies withhold their earnings from the first supplement, inconsistent with the “unravelling” prediction. However, almost all these companies start disclosing within the next couple of supplements, with the relatively-better companies among the remaining non-disclosers initiating disclosure and leaving the pool of non-disclosers each quarter. We interpret these stylized facts through the lens of a disclosure model featuring level-k thinking. Our model estimates that a substantial share of the companies employs a lower level of strategic thinking in the first supplement. This deviation from rational expectations appears to explain the initial failure of the “unravelling” prediction. Over time, companies appear to adopt higher levels of thinking, contributing to the rapid convergence to an (almost) full disclosure equilibrium. Collectively, our evidence is consistent with market forces yielding an (almost) full disclosure equilibrium in the medium to long run through repetition and learning.

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.

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.

Direct response and the strategy method in an experimental cheap talk game

In cheap talk games, equilibrium analysis predicts extreme limits on the information that can be transmitted when senders and receivers have different goals. Yet experimental evidence suggests that senders overcommunicate relative to this baseline, revealing more information than predicted in equilibrium. We propose that overcommunication may be due in part to limited cognitive engagement by subjects, captured by level-k thinking. To test this conjecture, we compare two elicitation methods, direct response and the strategy method, holding other elements of the game fixed. Existing experimental studies of cheap talk games use the standard direct response method, while the strategy method—in which subjects make selections for all contingent choices—is believed to encourage more thoughtful decisionmaking. We therefore expect senders to transmit less information with the strategy method than with direct response. In contrast, we find the reverse: the strategy method increased overcommunication. Further examination suggests that this occurred because senders played more naïvely with the strategy method than with direct response. Our findings suggest that the strategy method and direct response do not elicit the same choices in cheap talk games.

“Fatal Attraction” and Level-k thinking in games with Non-neutral frames

Traditional game theory assumes that if framing does not affect a game’s payoffs, it will not influence behavior. However, Rubinstein and Tversky (1993), Rubinstein, Tversky, and Heller (1996), and Rubinstein (1999) reported experiments eliciting initial responses to hide-and-seek and other types of game, in which subjects’ behavior responded systematically to non-neutral framing via decision labelings. Crawford and Iriberri (2007ab) proposed a level-k explanation of Rubinstein et al.’s results for hide-and-seek games. Heap, Rojo-Arjona, and Sugden’s (2014) criticized Crawford and Iriberri’s model on grounds of portability. This paper clarifies Heap et al.’s interpretation of their results and responds to their criticisms, suggesting a way forward.

Does Level-k Behavior Imply Level-k Thinking?

The level-k literature classifies subjects into different Lk types by their observed levels of reasoning in games. However, it remains unclear whether the observed level-k behavior is determined by belief or reasoning ability. This study proposes a strategy to identify the ability-bounded Lk subjects, who could not reason more than k steps of iterated best responses and thus have reached the upper bound of their reasoning capacity. The identification utilizes a combination of simultaneous and sequential ring games. In the sequential games, it requires an extra reasoning step to best respond as Lk, and thus the ability-bounded ones would fail the task. Results show that more than half of the L2 and L3 subjects are ability-bounded. Additionally, subjects' CRT scores, a measure of their cognitive ability, support the separation of these types. The findings suggest that both belief and reasoning ability could determine the observed levels, and thus one must be cautious when trying to infer belief or ability from the existing level-k data.