Learning models in game theory explore how players adapt and improve their strategies over time. These models range from simple reinforcement learning to complex belief-based approaches, each offering unique insights into strategic behavior.

Reinforcement learning rewards successful actions, while belief-based models update expectations about opponents. Social learning and imitation add another layer, showing how strategies spread through populations. These models help explain real-world strategic behavior and decision-making processes.

Reinforcement and Q-Learning Models

Reinforcement Learning Fundamentals

• Reinforcement learning involves agents learning optimal strategies through trial and error interactions with their environment
• Agents receive rewards or punishments based on the outcomes of their actions and aim to maximize their cumulative reward over time
• Key components of reinforcement learning include states, actions, rewards, and a value function that estimates the long-term value of being in a particular state or taking a specific action
• Reinforcement learning algorithms update the value function based on the observed rewards and use it to guide the agent's decision-making process (Q-learning, SARSA)

Q-Learning Algorithm

• Q-learning is a model-free reinforcement learning algorithm that learns an optimal action-value function, denoted as Q(s,a), representing the expected cumulative reward of taking action a in state s
• The Q-function is updated iteratively based on the Bellman equation: $Q(s,a) \leftarrow Q(s,a) + \alpha[r + \gamma \max_{a'} Q(s',a') - Q(s,a)]$
  • $\alpha$ is the learning rate, $\gamma$ is the discount factor, $r$ is the immediate reward, and $s'$ is the next state
• Q-learning is an off-policy algorithm, meaning it learns the optimal policy independently of the policy being followed during the learning process
• The agent selects actions based on an exploration-exploitation trade-off, typically using strategies like $\epsilon$-greedy or softmax to balance exploring new actions and exploiting the current best action

Experience-Weighted Attraction (EWA) Learning

• EWA learning is a hybrid model that combines elements of reinforcement learning and belief-based learning
• Each strategy is assigned an attraction value, which is updated based on the payoffs received and the agent's experience
• The attraction update rule incorporates both the realized payoff and the forgone payoff of unchosen strategies, weighted by an experience factor
• The experience factor determines the relative importance of past and recent experiences in shaping the attraction values
• EWA learning can capture both the law of effect (reinforcement) and the power law of practice (experience-based learning) observed in human learning

Belief-Based Learning Models

Fictitious Play

• Fictitious play is a belief-based learning model where agents form beliefs about their opponents' strategies based on observed actions
• Each agent maintains a belief distribution over the possible strategies of their opponents and best-responds to these beliefs
• The belief distribution is updated using empirical frequencies of observed actions, assuming opponents are playing stationary strategies
• Fictitious play can converge to Nash equilibria in certain classes of games (zero-sum, potential, and supermodular games) but may fail to converge in general

Best Response Dynamics

• Best response dynamics is a learning model where agents repeatedly best-respond to their opponents' current strategies
• At each time step, one or more agents update their strategies to the best response given their opponents' current strategies
• Best response dynamics can be deterministic (agents always choose the best response) or stochastic (agents choose best responses with high probability)
• The convergence properties of best response dynamics depend on the game structure and the specific update rules used (simultaneous or asynchronous updates)

Bayesian Learning

• Bayesian learning is a belief-based model where agents update their beliefs about opponents' strategies using Bayes' rule
• Agents start with prior beliefs over the possible strategies of their opponents and update these beliefs based on observed actions
• The posterior belief distribution is obtained by multiplying the prior beliefs with the likelihood of observed actions, normalized by the total probability
• Bayesian learning allows agents to incorporate prior knowledge and handle uncertainty in a principled manner
• The convergence properties of Bayesian learning depend on the prior beliefs and the true strategies being played by the opponents

Imitation and Social Learning

Imitation Learning Mechanisms

• Imitation learning involves agents learning by observing and copying the behavior of other agents in the population
• Agents may imitate successful strategies based on their payoffs or popularity within the population
• Imitation can be random (copying a randomly selected agent) or guided by specific rules (copying the best-performing agent or the most common strategy)
• Imitation learning can lead to the spread of successful strategies and the emergence of social norms and conventions
• However, imitation can also result in suboptimal outcomes if agents copy inferior strategies or get stuck in local optima

Social Learning and Cultural Evolution

• Social learning refers to the process of acquiring knowledge, skills, or behaviors through observation, interaction, or communication with other agents
• Social learning can occur through various mechanisms, such as imitation, teaching, or language-mediated learning
• Cultural evolution studies how social learning shapes the distribution of behaviors, norms, and beliefs in a population over time
• Social learning can facilitate the accumulation of knowledge across generations and the adaptation of populations to changing environments
• However, social learning can also lead to the spread of maladaptive behaviors or the persistence of outdated practices (conformity bias, cultural inertia)
• The interplay between individual learning, social learning, and environmental factors determines the dynamics of cultural evolution in a population