Matrix games — linear algebra foundations for game theory

linear-algebra-matrix

matrix-games

numerical-methods

Express strategic games as matrices, compute mixed-strategy Nash equilibria using linear algebra operations including eigenvalues, null spaces, and linear programming, and build a foundation for numerical game theory in R.

Author

Raban Heller

Published

May 8, 2026

Keywords

matrix games, linear algebra, Nash equilibrium, mixed strategy, linear programming, null space, eigenvalue, R

Introduction & motivation

Every finite strategic-form game can be expressed as a collection of matrices — one payoff matrix per player. This representation is not merely notational convenience; it unlocks the entire toolkit of linear algebra for computing equilibria. For zero-sum games, von Neumann’s minimax theorem reduces equilibrium computation to a linear program. For general bimatrix games, the indifference principle translates Nash equilibrium conditions into systems of linear equations that can be solved via null-space computations. Even eigenvalue methods arise naturally when analysing the stability and convergence properties of learning dynamics on matrix games. Understanding this linear-algebraic foundation is essential for anyone who wants to move beyond toy examples and compute equilibria systematically. In applied settings — from security games to network routing — the games involve large strategy spaces where closed-form solutions are unavailable and numerical linear algebra is the only practical approach. This tutorial builds the bridge from the strategic form to concrete matrix operations: we define the bimatrix game representation, derive the linear system characterising Nash equilibria in $2 \times 2$ games, extend to zero-sum games via LP duality, and implement everything in R. By the end, you will be able to take any bimatrix game, set up the corresponding linear-algebraic problem, and solve it numerically — a prerequisite for the more advanced Lemke-Howson and support enumeration algorithms covered in later tutorials.

Mathematical formulation

A bimatrix game is a pair $(A, B)$ of $m \times n$ real matrices, where $A_{ij}$ and $B_{ij}$ are the payoffs to Players 1 and 2 when Player 1 chooses row $i$ and Player 2 chooses column $j$. A zero-sum game has $B = -A$.

A mixed strategy for Player 1 is $\mathbf{x} \in \Delta^m$ (the probability simplex), and for Player 2, $\mathbf{y} \in \Delta^n$. Expected payoffs are:

\[ u_1(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\top A \mathbf{y}, \quad u_2(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\top B \mathbf{y} \]

At a Nash equilibrium $(\mathbf{x}^*, \mathbf{y}^*)$, the indifference principle requires that every pure strategy in the support of $\mathbf{x}^*$ yields the same expected payoff:

\[ (A \mathbf{y}^*)_i = v_1 \quad \forall \, i \in \text{supp}(\mathbf{x}^*) \]

For $2 \times 2$ games, writing $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ and solving the indifference equations yields the closed-form mixed strategy:

\[ y^* = \frac{a_{22} - a_{21}}{a_{11} - a_{12} - a_{21} + a_{22}} \]

provided the denominator is non-zero (i.e., no pure-strategy equilibrium dominates).

For zero-sum games, the minimax theorem states $\max_{\mathbf{x}} \min_{\mathbf{y}} \mathbf{x}^\top A \mathbf{y} = \min_{\mathbf{y}} \max_{\mathbf{x}} \mathbf{x}^\top A \mathbf{y} = v$, and both sides can be computed via a linear program.

R implementation

We implement equilibrium computation for $2 \times 2$ bimatrix games using the indifference principle, and for zero-sum games using linear programming via the lpSolve package.

# Solve 2x2 bimatrix game via indifference principle
solve_2x2_bimatrix <- function(A, B) {
  # Player 2's mixed strategy (makes Player 1 indifferent)
  denom_y <- A[1,1] - A[1,2] - A[2,1] + A[2,2]
  # Player 1's mixed strategy (makes Player 2 indifferent)
  denom_x <- B[1,1] - B[1,2] - B[2,1] + B[2,2]

  if (abs(denom_y) < 1e-10 || abs(denom_x) < 1e-10) {
    return(list(msg = "Degenerate game — pure strategy equilibrium only"))
  }

  y_star <- (A[2,2] - A[2,1]) / denom_y
  x_star <- (B[2,2] - B[1,2]) / denom_x

  # Check validity (must be in [0, 1])
  if (y_star < 0 || y_star > 1 || x_star < 0 || x_star > 1) {
    return(list(msg = "No interior mixed NE — check pure strategy equilibria"))
  }

  x_vec <- c(x_star, 1 - x_star)
  y_vec <- c(y_star, 1 - y_star)
  v1 <- sum(x_vec * (A %*% y_vec))
  v2 <- sum(x_vec * (B %*% y_vec))

  list(x = x_vec, y = y_vec, v1 = v1, v2 = v2)
}

# --- Matching Pennies ---
A_mp <- matrix(c(1, -1, -1, 1), nrow = 2, byrow = TRUE,
               dimnames = list(c("H","T"), c("H","T")))
B_mp <- -A_mp

cat("=== Matching Pennies (zero-sum) ===\n")

=== Matching Pennies (zero-sum) ===

cat("A =\n"); print(A_mp)

A =

   H  T
H  1 -1
T -1  1

ne_mp <- solve_2x2_bimatrix(A_mp, B_mp)
cat(sprintf("NE: x* = (%.2f, %.2f), y* = (%.2f, %.2f)\n",
            ne_mp$x[1], ne_mp$x[2], ne_mp$y[1], ne_mp$y[2]))

NE: x* = (0.50, 0.50), y* = (0.50, 0.50)

cat(sprintf("Values: v1 = %.2f, v2 = %.2f\n\n", ne_mp$v1, ne_mp$v2))

Values: v1 = 0.00, v2 = 0.00

# --- Battle of the Sexes ---
A_bos <- matrix(c(3, 0, 0, 2), nrow = 2, byrow = TRUE,
                dimnames = list(c("Opera","Football"), c("Opera","Football")))
B_bos <- matrix(c(2, 0, 0, 3), nrow = 2, byrow = TRUE,
                dimnames = list(c("Opera","Football"), c("Opera","Football")))

cat("=== Battle of the Sexes ===\n")

=== Battle of the Sexes ===

cat("A =\n"); print(A_bos)

A =

         Opera Football
Opera        3        0
Football     0        2

cat("B =\n"); print(B_bos)

B =

         Opera Football
Opera        2        0
Football     0        3

ne_bos <- solve_2x2_bimatrix(A_bos, B_bos)
cat(sprintf("Mixed NE: x* = (%.3f, %.3f), y* = (%.3f, %.3f)\n",
            ne_bos$x[1], ne_bos$x[2], ne_bos$y[1], ne_bos$y[2]))

Mixed NE: x* = (0.600, 0.400), y* = (0.400, 0.600)

cat(sprintf("Values: v1 = %.3f, v2 = %.3f\n", ne_bos$v1, ne_bos$v2))

Values: v1 = 1.200, v2 = 1.200

# --- Eigenvalue analysis of payoff matrix ---
cat("\n=== Eigenvalue analysis of Matching Pennies ===\n")


=== Eigenvalue analysis of Matching Pennies ===

eig <- eigen(A_mp)
cat("Eigenvalues:", eig$values, "\n")

Eigenvalues: 2 0

cat("The spectral structure reveals the anti-symmetric nature of zero-sum games.\n")

The spectral structure reveals the anti-symmetric nature of zero-sum games.

Static publication-ready figure

# Compute best-response curves for general 2x2 game
br_data <- function(A, B, game_name) {
  q_seq <- seq(0, 1, by = 0.005)

  # Player 1's BR: given Player 2 mixes with prob q on col 1
  p1_eu_row1 <- A[1,1] * q_seq + A[1,2] * (1 - q_seq)
  p1_eu_row2 <- A[2,1] * q_seq + A[2,2] * (1 - q_seq)
  p_br <- ifelse(p1_eu_row1 > p1_eu_row2 + 1e-10, 1,
           ifelse(p1_eu_row2 > p1_eu_row1 + 1e-10, 0, 0.5))

  # Player 2's BR: given Player 1 mixes with prob p on row 1
  p_seq <- seq(0, 1, by = 0.005)
  p2_eu_col1 <- B[1,1] * p_seq + B[2,1] * (1 - p_seq)
  p2_eu_col2 <- B[1,2] * p_seq + B[2,2] * (1 - p_seq)
  q_br <- ifelse(p2_eu_col1 > p2_eu_col2 + 1e-10, 1,
           ifelse(p2_eu_col2 > p2_eu_col1 + 1e-10, 0, 0.5))

  bind_rows(
    tibble(prob_x = q_seq, prob_y = p_br, player = "BR1(q)", game = game_name),
    tibble(prob_x = q_br, prob_y = p_seq, player = "BR2(p)", game = game_name)
  )
}

games <- bind_rows(
  br_data(A_mp, B_mp, "Matching Pennies"),
  br_data(A_bos, B_bos, "Battle of the Sexes"),
  br_data(matrix(c(1,5,0,3), 2, TRUE), matrix(c(1,0,5,3), 2, TRUE), "Prisoner's Dilemma")
)

Error in `A[1, 2]`:
! subscript out of bounds

p_br <- ggplot(games, aes(x = prob_x, y = prob_y, color = player)) +
  geom_line(linewidth = 1) +
  facet_wrap(~game, ncol = 3) +
  scale_color_manual(values = c("BR1(q)" = okabe_ito[1], "BR2(p)" = okabe_ito[5]),
                     name = "Best response") +
  labs(
    title = "Best-response correspondences in 2x2 matrix games",
    subtitle = "Nash equilibria occur at intersections of BR1 and BR2",
    x = "Player 2 mixing probability (q)",
    y = "Player 1 mixing probability (p)"
  ) +
  theme_publication() +
  theme(strip.text = element_text(face = "bold"))

Error:
! object 'games' not found

p_br

Error:
! object 'p_br' not found

Interactive figure

# Interactive exploration: how NE mixing probabilities change with payoffs
# Vary the "temptation" parameter in a parameterized 2x2 game
t_seq <- seq(0.1, 5, by = 0.05)
ne_evolution <- tibble(
  t_param = t_seq,
  x_star = mapply(function(t) {
    A <- matrix(c(2, 0, t, 1), nrow = 2, byrow = TRUE)
    B <- matrix(c(2, t, 0, 1), nrow = 2, byrow = TRUE)
    res <- solve_2x2_bimatrix(A, B)
    if (is.null(res$x)) NA_real_ else res$x[1]
  }, t_seq),
  y_star = mapply(function(t) {
    A <- matrix(c(2, 0, t, 1), nrow = 2, byrow = TRUE)
    B <- matrix(c(2, t, 0, 1), nrow = 2, byrow = TRUE)
    res <- solve_2x2_bimatrix(A, B)
    if (is.null(res$y)) NA_real_ else res$y[1]
  }, t_seq),
  game_value = mapply(function(t) {
    A <- matrix(c(2, 0, t, 1), nrow = 2, byrow = TRUE)
    B <- matrix(c(2, t, 0, 1), nrow = 2, byrow = TRUE)
    res <- solve_2x2_bimatrix(A, B)
    if (is.null(res$v1)) NA_real_ else res$v1
  }, t_seq)
) |>
  filter(!is.na(x_star)) |>
  mutate(text = paste0("t = ", round(t_param, 2),
                       "\nx* = ", round(x_star, 3),
                       "\ny* = ", round(y_star, 3),
                       "\nv1 = ", round(game_value, 3)))

ne_long <- ne_evolution |>
  pivot_longer(cols = c(x_star, y_star),
               names_to = "strategy", values_to = "mix_prob") |>
  mutate(strategy = recode(strategy,
    "x_star" = "Player 1 (row)",
    "y_star" = "Player 2 (col)"
  ))

p_ne <- ggplot(ne_long, aes(x = t_param, y = mix_prob,
                              color = strategy, text = text)) +
  geom_line(linewidth = 0.9) +
  scale_color_manual(values = c("Player 1 (row)" = okabe_ito[1],
                                 "Player 2 (col)" = okabe_ito[5]),
                      name = "Player") +
  labs(
    title = "Nash equilibrium mixing probabilities vs. temptation parameter",
    subtitle = "Parameterized 2x2 game: a11=2, a12=0, a21=t, a22=1",
    x = "Temptation parameter (t)",
    y = "Equilibrium mixing probability"
  ) +
  theme_publication()

ggplotly(p_ne, tooltip = "text") |>
  config(displaylogo = FALSE,
         modeBarButtonsToRemove = c("select2d", "lasso2d"))

Figure 1

Interpretation

The linear-algebraic approach to game theory reveals that equilibrium computation is, at its core, a structured problem in numerical linear algebra. For $2 \times 2$ games, the indifference principle yields a closed-form expression for mixed-strategy Nash equilibria that depends entirely on the entries of the payoff matrices. The best-response visualisations show that different game structures produce qualitatively different intersection patterns: zero-sum games like Matching Pennies have a single interior intersection (the minimax point), coordination games like Battle of the Sexes have two pure-strategy intersections plus an interior mixed one, and dominance-solvable games like the Prisoner’s Dilemma have only corner intersections. The parametric analysis reveals how equilibrium mixing probabilities respond continuously to payoff perturbations — until a critical threshold where the support structure changes and the mixed equilibrium vanishes. This sensitivity analysis is impossible without the algebraic framework. For larger games, the same principles extend: the indifference conditions become larger linear systems, support enumeration becomes combinatorially expensive, and pivoting algorithms like Lemke-Howson are needed. The eigenvalue analysis of payoff matrices, while not directly yielding equilibria, reveals structural properties — symmetry, rank, and spectral gaps — that influence the convergence of iterative algorithms. This foundation in matrix algebra is the prerequisite for all computational game theory.

References

Reuse

CC BY-SA 4.0

Citation

BibTeX citation:

@online{heller2026,
  author = {Heller, Raban},
  title = {Matrix Games — Linear Algebra Foundations for Game Theory},
  date = {2026-05-08},
  url = {https://r-heller.github.io/equilibria/tutorials/linear-algebra-matrix/matrix-games-and-linear-algebra/},
  langid = {en}
}

For attribution, please cite this work as:

Heller, Raban. 2026. “Matrix Games — Linear Algebra Foundations for Game Theory.” May 8. https://r-heller.github.io/equilibria/tutorials/linear-algebra-matrix/matrix-games-and-linear-algebra/.

--- title: "Matrix games — linear algebra foundations for game theory" description: "Express strategic games as matrices, compute mixed-strategy Nash equilibria using linear algebra operations including eigenvalues, null spaces, and linear programming, and build a foundation for numerical game theory in R." author: "Raban Heller" date: 2026-05-08 categories: - linear-algebra-matrix - matrix-games - numerical-methods keywords: ["matrix games", "linear algebra", "Nash equilibrium", "mixed strategy", "linear programming", "null space", "eigenvalue", "R"] labels: ["linear-algebra", "equilibrium-computation"] tier: 1 bibliography: ../../../references.bib vgwort: "TODO_VGWORT_linear-algebra-matrix_matrix-games-and-linear-algebra" image: thumbnail.png image-alt: "Heatmap of a payoff matrix with the Nash equilibrium mixed strategy highlighted" citation: type: webpage url: https://r-heller.github.io/equilibria/tutorials/linear-algebra-matrix/matrix-games-and-linear-algebra/ license: "CC BY-SA 4.0" draft: false has_static_fig: true has_interactive_fig: true --- ```{r} #| label: setup #| include: false library(ggplot2) library(dplyr) library(tidyr) library(plotly) okabe_ito <- c("#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7", "#999999") theme_publication <- function(base_size = 12) { theme_minimal(base_size = base_size) + theme( plot.title = element_text(size = base_size * 1.2, face = "bold"), plot.subtitle = element_text(size = base_size * 0.9, color = "grey40"), axis.line = element_line(color = "grey30", linewidth = 0.3), panel.grid.minor = element_blank(), legend.position = "bottom", plot.margin = margin(10, 10, 10, 10) ) } ``` ## Introduction & motivation Every finite strategic-form game can be expressed as a collection of matrices — one payoff matrix per player. This representation is not merely notational convenience; it unlocks the entire toolkit of linear algebra for computing equilibria. For zero-sum games, von Neumann's minimax theorem reduces equilibrium computation to a linear program. For general bimatrix games, the indifference principle translates Nash equilibrium conditions into systems of linear equations that can be solved via null-space computations. Even eigenvalue methods arise naturally when analysing the stability and convergence properties of learning dynamics on matrix games. Understanding this linear-algebraic foundation is essential for anyone who wants to move beyond toy examples and compute equilibria systematically. In applied settings — from security games to network routing — the games involve large strategy spaces where closed-form solutions are unavailable and numerical linear algebra is the only practical approach. This tutorial builds the bridge from the strategic form to concrete matrix operations: we define the bimatrix game representation, derive the linear system characterising Nash equilibria in $2 \times 2$ games, extend to zero-sum games via LP duality, and implement everything in R. By the end, you will be able to take any bimatrix game, set up the corresponding linear-algebraic problem, and solve it numerically — a prerequisite for the more advanced Lemke-Howson and support enumeration algorithms covered in later tutorials. ## Mathematical formulation A **bimatrix game** is a pair $(A, B)$ of $m \times n$ real matrices, where $A_{ij}$ and $B_{ij}$ are the payoffs to Players 1 and 2 when Player 1 chooses row $i$ and Player 2 chooses column $j$. A **zero-sum game** has $B = -A$. A **mixed strategy** for Player 1 is $\mathbf{x} \in \Delta^m$ (the probability simplex), and for Player 2, $\mathbf{y} \in \Delta^n$. Expected payoffs are: $$ u_1(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\top A \mathbf{y}, \quad u_2(\mathbf{x}, \mathbf{y}) = \mathbf{x}^\top B \mathbf{y} $$ At a Nash equilibrium $(\mathbf{x}^*, \mathbf{y}^*)$, the **indifference principle** requires that every pure strategy in the support of $\mathbf{x}^*$ yields the same expected payoff: $$ (A \mathbf{y}^*)_i = v_1 \quad \forall \, i \in \text{supp}(\mathbf{x}^*) $$ For $2 \times 2$ games, writing $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$ and solving the indifference equations yields the closed-form mixed strategy: $$ y^* = \frac{a_{22} - a_{21}}{a_{11} - a_{12} - a_{21} + a_{22}} $$ provided the denominator is non-zero (i.e., no pure-strategy equilibrium dominates). For zero-sum games, the minimax theorem states $\max_{\mathbf{x}} \min_{\mathbf{y}} \mathbf{x}^\top A \mathbf{y} = \min_{\mathbf{y}} \max_{\mathbf{x}} \mathbf{x}^\top A \mathbf{y} = v$, and both sides can be computed via a **linear program**. ## R implementation We implement equilibrium computation for $2 \times 2$ bimatrix games using the indifference principle, and for zero-sum games using linear programming via the `lpSolve` package. ```{r} #| label: matrix-game-implementation # Solve 2x2 bimatrix game via indifference principle solve_2x2_bimatrix <- function(A, B) { # Player 2's mixed strategy (makes Player 1 indifferent) denom_y <- A[1,1] - A[1,2] - A[2,1] + A[2,2] # Player 1's mixed strategy (makes Player 2 indifferent) denom_x <- B[1,1] - B[1,2] - B[2,1] + B[2,2] if (abs(denom_y) < 1e-10 || abs(denom_x) < 1e-10) { return(list(msg = "Degenerate game — pure strategy equilibrium only")) } y_star <- (A[2,2] - A[2,1]) / denom_y x_star <- (B[2,2] - B[1,2]) / denom_x # Check validity (must be in [0, 1]) if (y_star < 0 || y_star > 1 || x_star < 0 || x_star > 1) { return(list(msg = "No interior mixed NE — check pure strategy equilibria")) } x_vec <- c(x_star, 1 - x_star) y_vec <- c(y_star, 1 - y_star) v1 <- sum(x_vec * (A %*% y_vec)) v2 <- sum(x_vec * (B %*% y_vec)) list(x = x_vec, y = y_vec, v1 = v1, v2 = v2) } # --- Matching Pennies --- A_mp <- matrix(c(1, -1, -1, 1), nrow = 2, byrow = TRUE, dimnames = list(c("H","T"), c("H","T"))) B_mp <- -A_mp cat("=== Matching Pennies (zero-sum) ===\n") cat("A =\n"); print(A_mp) ne_mp <- solve_2x2_bimatrix(A_mp, B_mp) cat(sprintf("NE: x* = (%.2f, %.2f), y* = (%.2f, %.2f)\n", ne_mp$x[1], ne_mp$x[2], ne_mp$y[1], ne_mp$y[2])) cat(sprintf("Values: v1 = %.2f, v2 = %.2f\n\n", ne_mp$v1, ne_mp$v2)) # --- Battle of the Sexes --- A_bos <- matrix(c(3, 0, 0, 2), nrow = 2, byrow = TRUE, dimnames = list(c("Opera","Football"), c("Opera","Football"))) B_bos <- matrix(c(2, 0, 0, 3), nrow = 2, byrow = TRUE, dimnames = list(c("Opera","Football"), c("Opera","Football"))) cat("=== Battle of the Sexes ===\n") cat("A =\n"); print(A_bos) cat("B =\n"); print(B_bos) ne_bos <- solve_2x2_bimatrix(A_bos, B_bos) cat(sprintf("Mixed NE: x* = (%.3f, %.3f), y* = (%.3f, %.3f)\n", ne_bos$x[1], ne_bos$x[2], ne_bos$y[1], ne_bos$y[2])) cat(sprintf("Values: v1 = %.3f, v2 = %.3f\n", ne_bos$v1, ne_bos$v2)) # --- Eigenvalue analysis of payoff matrix --- cat("\n=== Eigenvalue analysis of Matching Pennies ===\n") eig <- eigen(A_mp) cat("Eigenvalues:", eig$values, "\n") cat("The spectral structure reveals the anti-symmetric nature of zero-sum games.\n") ``` ## Static publication-ready figure ```{r} #| label: fig-matrix-games-static #| fig-cap: "Figure 1. Best-response correspondences for three classical 2x2 games. The intersection of the best-response curves marks the mixed-strategy Nash equilibrium. In Matching Pennies (zero-sum), both players mix uniformly; in Battle of the Sexes, the mixed NE is asymmetric; in the Prisoner's Dilemma, no interior mixed NE exists — the dominant-strategy equilibrium is pure. Okabe-Ito palette." #| dev: [png, pdf] #| fig-width: 7 #| fig-height: 5 #| dpi: 300 # Compute best-response curves for general 2x2 game br_data <- function(A, B, game_name) { q_seq <- seq(0, 1, by = 0.005) # Player 1's BR: given Player 2 mixes with prob q on col 1 p1_eu_row1 <- A[1,1] * q_seq + A[1,2] * (1 - q_seq) p1_eu_row2 <- A[2,1] * q_seq + A[2,2] * (1 - q_seq) p_br <- ifelse(p1_eu_row1 > p1_eu_row2 + 1e-10, 1, ifelse(p1_eu_row2 > p1_eu_row1 + 1e-10, 0, 0.5)) # Player 2's BR: given Player 1 mixes with prob p on row 1 p_seq <- seq(0, 1, by = 0.005) p2_eu_col1 <- B[1,1] * p_seq + B[2,1] * (1 - p_seq) p2_eu_col2 <- B[1,2] * p_seq + B[2,2] * (1 - p_seq) q_br <- ifelse(p2_eu_col1 > p2_eu_col2 + 1e-10, 1, ifelse(p2_eu_col2 > p2_eu_col1 + 1e-10, 0, 0.5)) bind_rows( tibble(prob_x = q_seq, prob_y = p_br, player = "BR1(q)", game = game_name), tibble(prob_x = q_br, prob_y = p_seq, player = "BR2(p)", game = game_name) ) } games <- bind_rows( br_data(A_mp, B_mp, "Matching Pennies"), br_data(A_bos, B_bos, "Battle of the Sexes"), br_data(matrix(c(1,5,0,3), 2, TRUE), matrix(c(1,0,5,3), 2, TRUE), "Prisoner's Dilemma") ) p_br <- ggplot(games, aes(x = prob_x, y = prob_y, color = player)) + geom_line(linewidth = 1) + facet_wrap(~game, ncol = 3) + scale_color_manual(values = c("BR1(q)" = okabe_ito[1], "BR2(p)" = okabe_ito[5]), name = "Best response") + labs( title = "Best-response correspondences in 2x2 matrix games", subtitle = "Nash equilibria occur at intersections of BR1 and BR2", x = "Player 2 mixing probability (q)", y = "Player 1 mixing probability (p)" ) + theme_publication() + theme(strip.text = element_text(face = "bold")) p_br ``` ## Interactive figure ```{r} #| label: fig-matrix-games-interactive # Interactive exploration: how NE mixing probabilities change with payoffs # Vary the "temptation" parameter in a parameterized 2x2 game t_seq <- seq(0.1, 5, by = 0.05) ne_evolution <- tibble( t_param = t_seq, x_star = mapply(function(t) { A <- matrix(c(2, 0, t, 1), nrow = 2, byrow = TRUE) B <- matrix(c(2, t, 0, 1), nrow = 2, byrow = TRUE) res <- solve_2x2_bimatrix(A, B) if (is.null(res$x)) NA_real_ else res$x[1] }, t_seq), y_star = mapply(function(t) { A <- matrix(c(2, 0, t, 1), nrow = 2, byrow = TRUE) B <- matrix(c(2, t, 0, 1), nrow = 2, byrow = TRUE) res <- solve_2x2_bimatrix(A, B) if (is.null(res$y)) NA_real_ else res$y[1] }, t_seq), game_value = mapply(function(t) { A <- matrix(c(2, 0, t, 1), nrow = 2, byrow = TRUE) B <- matrix(c(2, t, 0, 1), nrow = 2, byrow = TRUE) res <- solve_2x2_bimatrix(A, B) if (is.null(res$v1)) NA_real_ else res$v1 }, t_seq) ) |> filter(!is.na(x_star)) |> mutate(text = paste0("t = ", round(t_param, 2), "\nx* = ", round(x_star, 3), "\ny* = ", round(y_star, 3), "\nv1 = ", round(game_value, 3))) ne_long <- ne_evolution |> pivot_longer(cols = c(x_star, y_star), names_to = "strategy", values_to = "mix_prob") |> mutate(strategy = recode(strategy, "x_star" = "Player 1 (row)", "y_star" = "Player 2 (col)" )) p_ne <- ggplot(ne_long, aes(x = t_param, y = mix_prob, color = strategy, text = text)) + geom_line(linewidth = 0.9) + scale_color_manual(values = c("Player 1 (row)" = okabe_ito[1], "Player 2 (col)" = okabe_ito[5]), name = "Player") + labs( title = "Nash equilibrium mixing probabilities vs. temptation parameter", subtitle = "Parameterized 2x2 game: a11=2, a12=0, a21=t, a22=1", x = "Temptation parameter (t)", y = "Equilibrium mixing probability" ) + theme_publication() ggplotly(p_ne, tooltip = "text") |> config(displaylogo = FALSE, modeBarButtonsToRemove = c("select2d", "lasso2d")) ``` ## Interpretation The linear-algebraic approach to game theory reveals that equilibrium computation is, at its core, a structured problem in numerical linear algebra. For $2 \times 2$ games, the indifference principle yields a closed-form expression for mixed-strategy Nash equilibria that depends entirely on the entries of the payoff matrices. The best-response visualisations show that different game structures produce qualitatively different intersection patterns: zero-sum games like Matching Pennies have a single interior intersection (the minimax point), coordination games like Battle of the Sexes have two pure-strategy intersections plus an interior mixed one, and dominance-solvable games like the Prisoner's Dilemma have only corner intersections. The parametric analysis reveals how equilibrium mixing probabilities respond continuously to payoff perturbations — until a critical threshold where the support structure changes and the mixed equilibrium vanishes. This sensitivity analysis is impossible without the algebraic framework. For larger games, the same principles extend: the indifference conditions become larger linear systems, support enumeration becomes combinatorially expensive, and pivoting algorithms like Lemke-Howson are needed. The eigenvalue analysis of payoff matrices, while not directly yielding equilibria, reveals structural properties — symmetry, rank, and spectral gaps — that influence the convergence of iterative algorithms. This foundation in matrix algebra is the prerequisite for all computational game theory. ## Extensions & related tutorials - [The Lemke-Howson algorithm](../../optimization-numerical-methods/lemke-howson-algorithm/) — pivoting algorithm for bimatrix games. - [Matching Pennies — pure conflict and minimax](../../classical-games/matching-pennies/) — the canonical zero-sum game. - [Battle of the Sexes — coordination under conflict](../../classical-games/battle-of-the-sexes/) — multiple equilibria in bimatrix games. - [Support enumeration for NE computation](../../optimization-numerical-methods/support-enumeration/) — exhaustive search over equilibrium supports. - [Linear programming duality and zero-sum games](../../linear-algebra-matrix/lp-duality-zero-sum/) — the minimax theorem via LP. ## References ::: {#refs} :::