🎱Game Theory Unit 13 – Statistical Methods in Game Theory Analysis

Key Concepts in Game Theory

• Game theory studies strategic decision-making between rational agents in situations where their actions affect each other's outcomes
• Players are the decision-makers in a game who choose strategies to maximize their payoffs
• Strategies are the actions or plans of action available to players in a game
• Payoffs are the outcomes or rewards players receive based on the combination of strategies chosen by all players
• Nash equilibrium is a stable state where no player can improve their payoff by unilaterally changing their strategy
  • Occurs when each player's strategy is a best response to the strategies of the other players
• Dominant strategy is a strategy that yields a higher payoff for a player regardless of the strategies chosen by other players
• Mixed strategies involve players randomizing over their available strategies according to certain probabilities

Probability Theory Foundations

• Probability theory provides the mathematical framework for quantifying uncertainty and analyzing random events in game theory
• Sample space is the set of all possible outcomes of a random experiment or game scenario
• Events are subsets of the sample space representing specific outcomes of interest
• Probability of an event is a measure of the likelihood of its occurrence, ranging from 0 (impossible) to 1 (certain)
• Conditional probability $P(A|B)$ measures the probability of event A occurring given that event B has already occurred
• Independent events are events where the occurrence of one does not affect the probability of the other
  • For independent events A and B, $P(A|B) = P(A)$ and $P(B|A) = P(B)$
• Random variables are functions that assign numerical values to outcomes in a sample space
  • Discrete random variables take on a countable number of distinct values (number of heads in 10 coin flips)
  • Continuous random variables can take on any value within a specified range (time until next goal scored in a soccer match)

Statistical Models in Game Theory

• Statistical models in game theory use probability distributions to represent uncertainty and variability in player strategies and payoffs
• Normal distribution is commonly used to model continuous random variables with symmetric bell-shaped probability density functions
  • Characterized by its mean $\mu$ and standard deviation $\sigma$
• Binomial distribution models the number of successes in a fixed number of independent trials with two possible outcomes (win or lose)
  • Characterized by the number of trials $n$ and the probability of success $p$ in each trial
• Poisson distribution models the number of rare events occurring in a fixed interval of time or space (goals scored in a soccer match)
  • Characterized by its rate parameter $\lambda$, representing the average number of events per interval
• Markov chains model systems that transition between different states over time, with future states depending only on the current state
  • Transition probabilities specify the likelihood of moving from one state to another in a single step

Data Collection and Sampling Methods

• Data collection involves gathering relevant information about player strategies, payoffs, and game outcomes to inform statistical analysis
• Sampling is the process of selecting a subset of individuals or observations from a larger population to draw inferences
• Simple random sampling ensures each member of the population has an equal probability of being selected in the sample
• Stratified sampling divides the population into distinct subgroups (strata) and then randomly samples from each stratum
  • Useful when subgroups have different characteristics that need to be represented in the sample
• Cluster sampling involves dividing the population into clusters, randomly selecting a subset of clusters, and including all members within those clusters in the sample
• Systematic sampling selects individuals from a population at regular intervals after a random starting point
• Sample size determination is crucial to ensure the sample is large enough to make reliable inferences about the population
  • Larger sample sizes generally lead to more precise estimates and higher statistical power

Hypothesis Testing in Game Scenarios

• Hypothesis testing is a statistical method for making decisions about population parameters based on sample data in game scenarios
• Null hypothesis ($H_0$) represents a default or status quo claim about a population parameter (no significant difference between two strategies)
• Alternative hypothesis ($H_a$) represents a claim that contradicts the null hypothesis (one strategy outperforms the other)
• Test statistic is a value calculated from the sample data used to make a decision about the null hypothesis
  • Common test statistics include z-score, t-score, and chi-square
• P-value is the probability of obtaining a test statistic as extreme as the observed value, assuming the null hypothesis is true
  • Smaller p-values provide stronger evidence against the null hypothesis
• Significance level ($\alpha$) is the threshold for rejecting the null hypothesis, typically set at 0.05 or 0.01
• Type I error occurs when the null hypothesis is rejected when it is actually true (false positive)
• Type II error occurs when the null hypothesis is not rejected when it is actually false (false negative)

Regression Analysis for Strategy Evaluation

• Regression analysis is a statistical method for modeling the relationship between a dependent variable and one or more independent variables in game strategy evaluation
• Simple linear regression models the relationship between two variables using a straight line equation: $y = \beta_0 + \beta_1x + \epsilon$
  • $y$ is the dependent variable, $x$ is the independent variable, $\beta_0$ is the y-intercept, $\beta_1$ is the slope, and $\epsilon$ is the error term
• Multiple linear regression extends simple linear regression to include multiple independent variables: $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_kx_k + \epsilon$
• Least squares estimation is a method for estimating the regression coefficients by minimizing the sum of squared residuals
• Coefficient of determination ($R^2$) measures the proportion of variance in the dependent variable explained by the independent variable(s)
  • Ranges from 0 (no explanatory power) to 1 (perfect fit)
• Residual analysis assesses the validity of regression assumptions, such as linearity, homoscedasticity, and normality of errors
• Logistic regression is used when the dependent variable is binary or categorical (win or lose)
  • Models the probability of an event occurring as a function of the independent variables using the logistic function

Bayesian Inference in Game Theory

• Bayesian inference is a statistical approach that combines prior beliefs with observed data to update probabilities and make inferences in game theory
• Prior probability distribution represents the initial beliefs about a parameter before observing any data
• Likelihood function quantifies the probability of observing the data given different values of the parameter
• Posterior probability distribution represents the updated beliefs about the parameter after incorporating the observed data
  • Obtained by combining the prior distribution and the likelihood function using Bayes' theorem: $P(\theta|D) = \frac{P(D|\theta)P(\theta)}{P(D)}$
• Conjugate priors are prior distributions that result in posterior distributions of the same family when combined with the likelihood function
  • Simplifies the computation of the posterior distribution
• Markov Chain Monte Carlo (MCMC) methods are used to sample from complex posterior distributions when analytical solutions are not available
  • Metropolis-Hastings algorithm and Gibbs sampling are common MCMC techniques
• Bayesian decision theory incorporates prior probabilities, likelihood functions, and utility functions to make optimal decisions under uncertainty

Advanced Statistical Techniques for Complex Games

• Advanced statistical techniques are required to analyze complex game scenarios with multiple players, strategies, and uncertain outcomes
• Multivariate analysis methods, such as principal component analysis (PCA) and factor analysis, reduce the dimensionality of high-dimensional strategy spaces
  • Identify latent variables or factors that explain the variability in player strategies
• Time series analysis models the temporal dependence and evolution of game outcomes over time
  • Autoregressive (AR) models express the current value of a variable as a linear combination of its past values
  • Moving average (MA) models express the current value of a variable as a linear combination of past forecast errors
  • Autoregressive integrated moving average (ARIMA) models combine AR and MA components to handle non-stationary time series
• Survival analysis methods, such as Kaplan-Meier estimator and Cox proportional hazards model, analyze the time until a specific event occurs in a game (player elimination)
• Spatial statistics techniques, such as kriging and spatial autocorrelation analysis, incorporate the spatial structure and dependence of game outcomes
  • Useful for analyzing games with a geographical component or spatial interactions between players
• Machine learning algorithms, such as decision trees, random forests, and support vector machines, can learn complex patterns and strategies from game data
  • Predict player behavior, classify game outcomes, and optimize decision-making in complex game environments