C Probability Essentials
Probability distributions, expectations, Bayes’ rule, and stochastic processes.
This appendix reviews the probability concepts that underpin mixed strategies, Bayesian games, and the simulation chapters. The emphasis is on intuition and R implementation rather than measure-theoretic rigour.
C.1 Sample spaces and events
A sample space \(\Omega\) is the set of all possible outcomes of a random experiment. An event \(E \subseteq \Omega\) is a subset of outcomes. A probability measure \(P\) assigns a number \(P(E) \in [0, 1]\) to every event, with \(P(\Omega) = 1\).
In game theory, the sample space often corresponds to the set of action profiles or the set of player types.
C.2 Discrete distributions
A discrete random variable \(X\) takes values in a countable set \(\{x_1, x_2, \ldots\}\) with probability mass function \(p(x_i) = P(X = x_i)\).
C.2.1 Common discrete distributions
# Bernoulli: coin flip (used in mixed strategies)
rbinom(10, size = 1, prob = 0.5)#> [1] 1 1 0 1 1 1 1 0 1 1
# Binomial: number of cooperators in n rounds
dbinom(3, size = 10, prob = 0.6) # P(X = 3)#> [1] 0.0425#> [1] "Scissors"
# Poisson: arrival counts (used in some mechanism design models)
rpois(10, lambda = 3)#> [1] 4 6 2 3 6 7 1 3 3 5C.2.2 Probability mass function and CDF
# PMF and CDF of Binomial(10, 0.4)
x <- 0:10
pmf <- dbinom(x, size = 10, prob = 0.4)
cdf <- pbinom(x, size = 10, prob = 0.4)
tibble(x = x, pmf = pmf, cdf = cdf) |>
head(6)#> # A tibble: 6 × 3
#> x pmf cdf
#> <int> <dbl> <dbl>
#> 1 0 0.00605 0.00605
#> 2 1 0.0403 0.0464
#> 3 2 0.121 0.167
#> 4 3 0.215 0.382
#> 5 4 0.251 0.633
#> 6 5 0.201 0.834C.3 Continuous distributions
A continuous random variable \(X\) has a probability density function \(f(x)\) such that \(P(a \leq X \leq b) = \int_a^b f(x)\, dx\).
C.3.1 Common continuous distributions
# Uniform on [0, 1]: used for types in Bayesian games
runif(5, min = 0, max = 1)#> [1] 0.1387 0.9889 0.9467 0.0824 0.5142
# Normal: payoff noise, error terms
rnorm(5, mean = 0, sd = 1)#> [1] -0.279 -0.133 0.636 -0.284 -2.656
# Exponential: waiting times, discount factors
rexp(5, rate = 1)#> [1] 4.996 0.224 1.211 0.719 1.308C.3.2 Density, CDF, and quantiles
R uses a consistent naming convention: d for density, p for CDF, q for quantile, and r for random draws.
# Normal distribution
dnorm(0, mean = 0, sd = 1) # density at 0#> [1] 0.399
pnorm(1.96, mean = 0, sd = 1) # P(X <= 1.96)#> [1] 0.975
qnorm(0.975, mean = 0, sd = 1) # 97.5th percentile#> [1] 1.96
rnorm(3, mean = 0, sd = 1) # three random draws#> [1] 0.358 0.302 -0.394C.4 Expected value and variance
The expected value of a discrete random variable is \(E[X] = \sum_i x_i \, p(x_i)\). For a continuous variable, \(E[X] = \int x \, f(x) \, dx\). In game theory, the expected payoff under a mixed strategy \(\mathbf{p}\) is:
\[ E[u_i] = \sum_{a \in A} p(a) \, u_i(a) \]
The variance \(\text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2\) measures the spread of outcomes.
# Expected payoff under a mixed strategy
payoffs <- c(3, 0, 5, 1)
probs <- c(0.3, 0.2, 0.1, 0.4)
# Expected value
ev <- sum(payoffs * probs)
ev#> [1] 1.8
# Variance
variance <- sum(probs * (payoffs - ev)^2)
variance#> [1] 2.36
# Using simulation to verify
simulated <- sample(payoffs, size = 100000, replace = TRUE, prob = probs)
mean(simulated)#> [1] 1.8
var(simulated)#> [1] 2.37C.5 Conditional probability
The conditional probability of \(A\) given \(B\) is:
\[ P(A \mid B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0 \]
Two events are independent if \(P(A \cap B) = P(A) P(B)\), equivalently \(P(A \mid B) = P(A)\).
# Simulation: P(both cooperate | Player 1 cooperates)
n <- 100000
p1 <- sample(c("C", "D"), n, replace = TRUE, prob = c(0.6, 0.4))
p2 <- sample(c("C", "D"), n, replace = TRUE, prob = c(0.5, 0.5))
# Joint and conditional (assuming independence here)
p_both_C <- mean(p1 == "C" & p2 == "C")
p_p1_C <- mean(p1 == "C")
p_both_C / p_p1_C # should be approx 0.5#> [1] 0.501C.6 Bayes’ theorem
Bayes’ theorem is the foundation of Bayesian games (7). It tells us how to update beliefs about a player’s type after observing their action:
\[ P(\theta \mid a) = \frac{P(a \mid \theta) \, P(\theta)}{P(a)} \]
where \(\theta\) is the player’s type, \(a\) is the observed action, \(P(\theta)\) is the prior, \(P(a \mid \theta)\) is the likelihood, and \(P(a) = \sum_{\theta'} P(a \mid \theta') P(\theta')\) is the marginal likelihood.
# Signaling game: worker chooses Education or No Education
# Types: High (prob 0.6), Low (prob 0.4)
# P(Education | High) = 0.9, P(Education | Low) = 0.2
prior_high <- 0.6
prior_low <- 0.4
p_edu_given_high <- 0.9
p_edu_given_low <- 0.2
# P(Education)
p_edu <- p_edu_given_high * prior_high + p_edu_given_low * prior_low
p_edu#> [1] 0.62
# P(High | Education) by Bayes' theorem
p_high_given_edu <- (p_edu_given_high * prior_high) / p_edu
p_high_given_edu#> [1] 0.871
# P(High | No Education)
p_no_edu <- 1 - p_edu
p_high_given_no_edu <- ((1 - p_edu_given_high) * prior_high) / p_no_edu
p_high_given_no_edu#> [1] 0.158C.6.1 Sequential updating
When multiple signals are observed, Bayes’ theorem can be applied sequentially: the posterior from one update becomes the prior for the next. This is used in repeated Bayesian games where players learn about opponents over time.
# Sequential Bayesian updating: observe actions round by round
# Prior: P(Cooperator type) = 0.5
# P(C action | Cooperator type) = 0.9
# P(C action | Defector type) = 0.1
update_belief <- function(prior, likelihood_coop, likelihood_defect, action) {
if (action == "C") {
l_coop <- likelihood_coop
l_defect <- likelihood_defect
} else {
l_coop <- 1 - likelihood_coop
l_defect <- 1 - likelihood_defect
}
numerator <- l_coop * prior
denominator <- l_coop * prior + l_defect * (1 - prior)
numerator / denominator
}
prior <- 0.5
actions <- c("C", "C", "D", "C", "C")
beliefs <- numeric(length(actions))
for (i in seq_along(actions)) {
prior <- update_belief(prior, 0.9, 0.1, actions[i])
beliefs[i] <- prior
}
tibble(round = seq_along(actions), action = actions, belief = beliefs)#> # A tibble: 5 × 3
#> round action belief
#> <int> <chr> <dbl>
#> 1 1 C 0.9
#> 2 2 C 0.988
#> 3 3 D 0.9
#> 4 4 C 0.988
#> 5 5 C 0.999C.7 Law of large numbers and Monte Carlo
The law of large numbers guarantees that the sample mean converges to the true mean as the sample size grows. This justifies the Monte Carlo methods used extensively in 16.
# Demonstrate convergence of sample mean
n_values <- c(10, 100, 1000, 10000, 100000)
true_mean <- 3.5 # E[Uniform die roll]
estimates <- vapply(n_values, function(n) {
mean(sample(1:6, n, replace = TRUE))
}, numeric(1))
tibble(n = n_values, estimate = estimates, true_mean = true_mean)#> # A tibble: 5 × 3
#> n estimate true_mean
#> <dbl> <dbl> <dbl>
#> 1 10 3.2 3.5
#> 2 100 3.58 3.5
#> 3 1000 3.48 3.5
#> 4 10000 3.49 3.5
#> 5 100000 3.50 3.5C.7.1 Monte Carlo estimation
To estimate a quantity \(\theta = E[g(X)]\), draw \(N\) independent samples \(X_1, \ldots, X_N\) and compute:
\[ \hat{\theta}_N = \frac{1}{N} \sum_{i=1}^{N} g(X_i) \]
The standard error of this estimate is \(\text{SE} = \sigma / \sqrt{N}\), where \(\sigma\) is the standard deviation of \(g(X)\).
# Estimate P(at least one cooperator in 5 players) via Monte Carlo
# Each player cooperates independently with probability 0.3
estimate_prob <- function(n_sims, n_players = 5, p_coop = 0.3) {
counts <- replicate(n_sims, {
actions <- rbinom(n_players, size = 1, prob = p_coop)
as.integer(sum(actions) >= 1)
})
c(estimate = mean(counts), se = sd(counts) / sqrt(n_sims))
}
# Exact answer: 1 - (1 - 0.3)^5
exact <- 1 - 0.7^5
set.seed(42)
mc <- estimate_prob(100000)
tibble(
method = c("Monte Carlo", "Exact"),
estimate = c(mc["estimate"], exact)
)#> # A tibble: 2 × 2
#> method estimate
#> <chr> <dbl>
#> 1 Monte Carlo 0.834
#> 2 Exact 0.832C.8 Random sampling for games
Several chapters use random sampling to simulate strategic interactions. The key functions are:
# sample(): draw from a discrete set
sample(c("Hawk", "Dove"), size = 10, replace = TRUE, prob = c(0.4, 0.6))#> [1] "Dove" "Hawk" "Dove" "Dove" "Hawk" "Hawk" "Dove" "Hawk" "Dove" "Hawk"#> [1] 4 6 5 6 7#> [1] 0.288 0.788 0.409#> [1] 0.288 0.788 0.409