13.3 Quantal response equilibrium and level-k thinking
3 min read•august 7, 2024
and are game theory concepts that account for human limitations. They help explain why people don't always make perfect decisions in strategic situations, introducing randomness and varying levels of sophistication.
These models bridge the gap between theory and reality. By considering errors and different thinking depths, they provide more accurate predictions of how people actually behave in games and real-world scenarios.
Quantal Response Equilibrium Models
Quantal Response Equilibrium and Stochastic Choice
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- Quantal response equilibrium (QRE) extends the concept of to account for players making errors or exhibiting
- Players choose strategies probabilistically based on their expected payoffs
- Probability of choosing a strategy increases with its expected payoff relative to other strategies
- Stochastic choice models assume that players' decisions are subject to random errors or noise
- Players may not always choose the best response strategy due to cognitive limitations, inattention, or other factors
- Introduces a probabilistic element to the decision-making process (logit choice model)
Logit Equilibrium and Noisy Best Response
- Logit equilibrium is a specific type of QRE that uses the logit choice model to determine players' strategy probabilities
- Probability of choosing a strategy is proportional to the exponential of its expected payoff multiplied by a precision parameter
- Higher precision parameter implies more rational behavior, with players more likely to choose best response strategies
- Noisy best response models incorporate random errors into players' best response calculations
- Players aim to choose the best response strategy but may make mistakes or deviate from optimal play
- Degree of noise or error in the decision-making process can be adjusted (quantal response precision parameter)
Behavioral Equilibrium and Applications
- Behavioral equilibrium concepts, such as QRE and logit equilibrium, provide a framework for analyzing games with boundedly rational players
- Allows for the study of strategic interactions when players exhibit cognitive limitations or make stochastic choices
- Captures deviations from perfect rationality observed in experimental and real-world settings (auction bidding, voter turnout)
- QRE models have been applied to various domains, including industrial organization, political science, and experimental economics
- Explains observed behavior in games like the traveler's dilemma and the centipede game
- Provides insights into the role of noise and bounded rationality in strategic decision-making
Level-k Thinking and Cognitive Hierarchy
Level-k Thinking and Iterative Reasoning
- Level-k thinking is a model of strategic reasoning where players exhibit different levels of sophistication
- Level-0 players choose strategies randomly or based on a simple heuristic (uniform randomization)
- Level-1 players best respond to level-0 players, assuming they will play randomly
- Level-2 players best respond to level-1 players, and so on
- Iterative reasoning involves players considering the thought processes and strategies of their opponents
- Players engage in a series of "if I think, you think, I think" reasoning steps
- Depth of reasoning is determined by the player's level (level-1, level-2, etc.)
Cognitive Hierarchy Model and Strategic Sophistication
- model extends level-k thinking by assuming a distribution of player types with different levels of strategic sophistication
- Proportion of players at each level is determined by a Poisson distribution with mean τ (average level of sophistication)
- Higher values of τ indicate a more strategically sophisticated population
- Strategic sophistication refers to a player's ability to reason about the beliefs and strategies of other players
- More sophisticated players (higher levels) engage in deeper iterative reasoning
- Less sophisticated players (lower levels) rely on simpler heuristics or best respond to lower-level players
Applications and Experimental Evidence
- Level-k thinking and cognitive hierarchy models have been used to explain behavior in various strategic settings (beauty contest games, guessing games)
- Experimental evidence supports the existence of different levels of strategic sophistication among players
- Models capture the heterogeneity in players' reasoning abilities and the resulting outcomes
- These models provide insights into the role of bounded rationality and cognitive limitations in strategic interactions
- Help explain deviations from Nash equilibrium predictions in experimental games
- Offer a framework for understanding the distribution of strategic sophistication in a population (marketing campaigns, political campaigns)
Key Terms to Review (17)
Auction Theory: Auction theory studies how people bid in auctions and how these bidding processes affect the allocation of goods and services. This theory explores various auction formats, bidder strategies, and outcomes, providing insights into decision-making under competition and uncertainty. It connects to key concepts such as incomplete information and strategic interactions, making it essential for understanding economic behaviors in competitive markets.
Bargaining scenarios: Bargaining scenarios refer to situations in which two or more parties negotiate the terms of an agreement, often involving the distribution of resources or benefits. These scenarios are fundamental to understanding strategic interactions, as they require participants to consider not only their own preferences but also the potential responses and strategies of others involved. The analysis of bargaining scenarios is crucial for exploring concepts like quantal response equilibrium and level-k thinking, as these frameworks provide insights into how individuals might behave when making decisions in uncertain environments.
Bounded rationality: Bounded rationality refers to the concept that individuals, when making decisions, are limited by the information they have, cognitive limitations, and time constraints. This means that people do not always act with complete rationality, as their decision-making processes are influenced by various factors that restrict their ability to evaluate every possible option and outcome fully.
Cognitive Hierarchy: Cognitive hierarchy is a concept in game theory that describes how players' beliefs about others' reasoning processes influence their own decision-making. It suggests that individuals operate at different levels of reasoning, with some players thinking one step ahead while others may think several steps further. This multi-level thinking creates a structured hierarchy where each player's strategy depends on their expectations of how rationally others will behave.
Colin Camerer: Colin Camerer is a prominent behavioral economist and game theorist, known for his contributions to understanding how people make decisions in strategic situations. His work focuses on developing models that incorporate psychological insights into traditional economic and game theory frameworks, including concepts like quantal response equilibrium and level-k thinking, which help explain how individuals make choices when faced with uncertainty and competition.
David K. Levine: David K. Levine is an influential economist and researcher known for his contributions to game theory, particularly in relation to behavioral models and learning. His work often explores how individuals make decisions in strategic situations, integrating concepts like quantal response equilibrium and level-k thinking to better understand human behavior in economic contexts.
Expected Utility Theory: Expected utility theory is a framework for understanding how individuals make choices under risk, positing that people evaluate potential outcomes based on their probabilities and associated utilities. It helps explain decision-making processes in uncertain situations by calculating the expected utility for each option and choosing the one with the highest value. This theory is foundational in both economics and game theory, influencing concepts such as quantal response equilibrium and level-k thinking.
Field experiments: Field experiments are research methods used to test theories or hypotheses in real-world settings, as opposed to controlled laboratory conditions. They help researchers understand how individuals behave in natural environments and reveal insights into behavioral biases, decision-making processes, and social interactions. By collecting data in actual contexts, field experiments enhance the validity of findings related to human behavior and economic choices.
Iterated reasoning: Iterated reasoning is a concept in game theory where players think about what their opponents will do based on their own beliefs and expectations, and then adjust their strategies accordingly. This process involves multiple rounds of reasoning, as players anticipate the thoughts and actions of others in a strategic environment. The depth of this reasoning can influence decision-making, leading to various equilibria depending on how many levels of thinking each player employs.
Laboratory experiments: Laboratory experiments are controlled studies conducted in a structured environment where researchers manipulate variables to observe the effects on participants' behavior and decision-making. These experiments are crucial in understanding how individuals make choices, often revealing biases and deviations from traditional economic theories. By isolating variables, researchers can better examine the underlying psychological factors that influence decision-making, providing valuable insights into various behavioral phenomena.
Level-0 thinking: Level-0 thinking refers to the simplest form of reasoning in game theory, where players make decisions based solely on a random strategy or pure chance without considering the actions of others. This concept lays the groundwork for more complex decision-making strategies, as it represents the baseline level of reasoning that other levels build upon, particularly in the framework of level-k thinking and quantal response equilibrium.
Level-1 thinking: Level-1 thinking refers to a basic level of reasoning in strategic interactions, where individuals make decisions based solely on their own preferences and the actions they observe from others, without deeper analysis of potential higher-order strategies. This type of thinking underpins the initial layer of strategic behavior in contexts like quantal response equilibrium and level-k thinking, where players might not consider the full complexities of their opponents' reasoning.
Level-k thinking: Level-k thinking is a concept in game theory that describes how players make decisions based on their beliefs about the reasoning of other players. In this framework, players are categorized into different levels, with level-0 players choosing randomly, level-1 players best responding to level-0 players' actions, and so on. This hierarchical structure allows for more sophisticated predictions of behavior in strategic situations, recognizing that individuals may not always think multiple steps ahead.
Logit response model: The logit response model is a statistical method used to predict the probability of a binary outcome based on one or more predictor variables. It plays a crucial role in understanding decision-making under uncertainty, especially in contexts where individuals' choices can be influenced by various factors. This model helps to explain how people choose among alternatives and is particularly relevant when analyzing behaviors in games and strategic interactions.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where no player can benefit by unilaterally changing their strategy if the strategies of the other players remain unchanged. This means that each player's strategy is optimal given the strategies of all other players, resulting in a stable outcome where players have no incentive to deviate from their chosen strategies.
Quantal Response Equilibrium: Quantal response equilibrium is a solution concept in game theory that extends the traditional Nash equilibrium by allowing players to choose strategies probabilistically based on their expected payoffs. In this framework, players are assumed to make mistakes, leading to non-deterministic choices that reflect varying levels of rationality. This concept captures more realistic behavior in strategic interactions by acknowledging that not all players act perfectly rationally, and can be closely tied to level-k thinking, which also addresses how individuals strategize based on their beliefs about others' reasoning.
Rationalizability: Rationalizability is a concept in game theory that refers to the idea that players make decisions based on their beliefs about other players' strategies, assuming that everyone is rational and has common knowledge of rationality. This concept emphasizes that a player's choice can be justified by a belief that others are also making rational choices, leading to the formation of an equilibrium. It connects to several important ideas in strategic interactions, including the implications of strategy choices, the cognitive processes behind decision-making, and the iterative reasoning involved in eliminating non-viable options.