12.3 Models of bounded rationality and learning in games

Bounded rationality challenges the idea of perfect decision-making in games. It recognizes that players have limits on time, info, and brainpower. This leads to different outcomes than what classical game theory predicts, as people use shortcuts and learn as they go.

Models of learning in games show how players adjust their strategies over time. Some focus on reinforcement from past payoffs, while others look at beliefs about opponents. Hybrid models combine both approaches. These ideas help explain real-world behavior in strategic situations.

Bounded Rationality in Games

Concept and Implications

Top images from around the web for Concept and Implications

• 

  Herbert Simon: Racionalidad Limitada - Percepciones Económicas View original

  Is this image relevant?

• 

  Learning Approaches | Introduction to Psychology – Reinke View original

  Is this image relevant?

• 

  Making Decisions in Different Organizations | Organizational Behavior / Human Relations View original

  Is this image relevant?

• 

  Herbert Simon: Racionalidad Limitada - Percepciones Económicas View original

  Is this image relevant?

• 

  Learning Approaches | Introduction to Psychology – Reinke View original

  Is this image relevant?

1 of 3

Top images from around the web for Concept and Implications

• 

  Herbert Simon: Racionalidad Limitada - Percepciones Económicas View original

  Is this image relevant?

• 

  Learning Approaches | Introduction to Psychology – Reinke View original

  Is this image relevant?

• 

  Making Decisions in Different Organizations | Organizational Behavior / Human Relations View original

  Is this image relevant?

• 

  Herbert Simon: Racionalidad Limitada - Percepciones Económicas View original

  Is this image relevant?

• 

  Learning Approaches | Introduction to Psychology – Reinke View original

  Is this image relevant?

1 of 3

• Bounded rationality is the idea that decision-makers have limited cognitive abilities and face constraints such as time, information, and computational capacity when making decisions
  • Contrasts with the assumption of perfect rationality in classical game theory
• Bounded rationality can lead to deviations from the predictions of classical game theory
  • Players may not always choose the optimal strategy or reach the Nash equilibrium
  • Instead, they may use heuristics, satisficing, or other simplified decision rules
• The implications of bounded rationality for game-theoretic modeling include:
  • The need to consider the cognitive limitations of players and their impact on strategic behavior
  • The potential for multiple equilibria or non-equilibrium outcomes in games
  • The importance of learning and adaptation in shaping the dynamics of strategic interactions
  • The role of heuristics and simplified decision rules in guiding player behavior

Models and Goals

• Models of bounded rationality aim to capture these limitations and provide more realistic descriptions of human decision-making in strategic situations
  • Often incorporate cognitive constraints, learning, and adaptation
  • Examples include cognitive hierarchy models, quantal response equilibrium, level-k thinking models, and cursed equilibrium
• The goals of bounded rationality models include:
  • Explaining deviations from the predictions of classical game theory and Nash equilibrium
  • Providing more accurate predictions of human behavior in strategic situations
  • Incorporating the role of cognitive limitations, learning, and adaptation in shaping strategic decision-making
  • Offering insights into the design of institutions, markets, and incentives that account for bounded rationality

Models of Learning in Games

Reinforcement and Belief Learning

• Reinforcement learning is a model where players adjust their strategies based on the payoffs they receive
  • Players are more likely to repeat strategies that have yielded high payoffs in the past and less likely to use strategies that have led to low payoffs
  • Example: A firm that experiences increased profits after a price cut is more likely to continue using low-price strategies in the future
• Belief learning is a model where players form beliefs about the strategies of their opponents based on observed behavior and update these beliefs over time
  • Players choose their strategies based on their current beliefs about the likelihood of different opponent actions
  • Example: In a repeated prisoner's dilemma, a player who observes their opponent cooperating in previous rounds may form the belief that cooperation is more likely and adjust their strategy accordingly

Hybrid and Adaptive Learning Models

• Experience-weighted attraction (EWA) learning is a hybrid model that combines elements of reinforcement and belief learning
  • Players update both their propensities to play different strategies (reinforcement) and their beliefs about opponent strategies (belief) based on past experience
  • EWA can capture both the direct effect of payoffs on strategy choice and the indirect effect of beliefs about opponent behavior
• Adaptive learning models, such as weighted fictitious play, assume that players best-respond to a weighted average of their opponents' past actions
  • More recent actions receive greater weight in the player's decision-making process
  • Adaptive learning models can capture the idea that players place more emphasis on recent experiences when forming expectations about opponent behavior
• Other learning models in games include:
  • Imitation learning, where players copy the strategies of successful opponents
  • Aspiration-based learning, where players adjust their strategies based on whether their payoffs exceed or fall short of an aspiration level
  • Example: A firm that observes a competitor's successful marketing campaign may imitate this strategy in an attempt to improve its own performance

Bounded Rationality vs Nash Equilibrium

Explaining Deviations

• Models of bounded rationality can help explain why observed behavior in games often deviates from the predictions of Nash equilibrium, which assumes perfect rationality
• Cognitive hierarchy models assume that players have different levels of strategic sophistication, with some players being more sophisticated than others
  • These models can explain deviations from Nash equilibrium by accounting for the presence of less sophisticated players who may not best-respond to their opponents' strategies
  • Example: In a p-beauty contest game, where players choose numbers between 0 and 100 and the winner is the one closest to 2/3 of the average, the Nash equilibrium prediction is that all players choose 0. However, experiments show that players often choose higher numbers, which can be explained by the presence of less sophisticated players who do not fully iterate the best-response reasoning process
• Quantal response equilibrium (QRE) is a model that allows for stochastic choice, where players choose strategies with probabilities that are increasing in their expected payoffs
  • QRE can explain deviations from Nash equilibrium by allowing for "noisy" decision-making and the possibility of suboptimal choices
  • Example: In a game with multiple Nash equilibria, QRE can explain why players might not always coordinate on the most efficient equilibrium, as the probability of choosing each strategy depends on its relative payoff

Accounting for Heterogeneity in Strategic Thinking

• Level-k thinking models assume that players have different levels of strategic reasoning, with level-0 players choosing randomly, level-1 players best-responding to level-0, and so on
  • These models can explain deviations from Nash equilibrium by capturing the heterogeneity in players' strategic thinking
  • Example: In a game of rock-paper-scissors, the Nash equilibrium prediction is that players will choose each action with equal probability. However, level-k models can explain why some players might choose actions that exploit the anticipated choices of less sophisticated opponents
• Cursed equilibrium is a model where players fail to fully account for the correlation between their opponents' actions and their private information
  • This can lead to deviations from Nash equilibrium predictions, particularly in games with incomplete information
  • Example: In a common-value auction, where the value of the auctioned item is the same for all bidders but unknown at the time of bidding, cursed equilibrium can explain why players might overbid and fall prey to the winner's curse, as they fail to fully account for the information conveyed by winning the auction

Predictive Power of Bounded Rationality Models

Evaluating Predictive Power and Empirical Validity

• The predictive power of a model refers to its ability to accurately forecast behavior in new or out-of-sample situations. Empirical validity concerns the extent to which a model's predictions match observed data from experiments or real-world settings
• To evaluate the predictive power and empirical validity of models of bounded rationality and learning, researchers compare the models' predictions with data from controlled experiments or field studies
  • This involves designing experiments that can distinguish between the predictions of different models and collecting data on actual player behavior
  • Example: Researchers might design an experiment to test the predictions of reinforcement learning and belief learning models in a repeated game, and compare the models' performance in predicting the observed behavior of participants

Model Comparison and Cross-Validation

• Model comparison techniques, such as the Akaike information criterion (AIC) or the Bayesian information criterion (BIC), can be used to assess the relative fit of different models to the data while accounting for model complexity
  • Models with lower AIC or BIC values are preferred, as they strike a balance between goodness-of-fit and parsimony
  • Example: When comparing the performance of different learning models in a game, researchers might calculate the AIC or BIC values for each model and select the one with the lowest value as the best-fitting model
• Cross-validation methods, such as k-fold cross-validation or leave-one-out cross-validation, can be used to assess the out-of-sample predictive performance of models
  • These methods involve splitting the data into training and testing sets, fitting the models on the training set, and evaluating their predictions on the testing set
  • Example: In a study comparing the predictive power of different bounded rationality models, researchers might use k-fold cross-validation to estimate the models' performance on unseen data and select the model with the highest average performance across the folds

Robustness and Generalizability

• Robustness checks, such as testing the models' predictions under different experimental conditions or with different subject pools, can help establish the generalizability and external validity of the models
  • Example: To test the robustness of a bounded rationality model, researchers might replicate the experiment with different payoff structures, different subject populations (e.g., students vs. professionals), or in different cultural contexts
• The empirical validity of models of bounded rationality and learning is an ongoing area of research, with different models performing better in different contexts
  • Comparing the performance of multiple models across a range of strategic situations is important for understanding their strengths and limitations
  • Example: A model that performs well in predicting behavior in simple games may not necessarily generalize to more complex strategic environments, highlighting the need for testing models across a variety of contexts