game_theory.tex

\documentclass[a4paper, 11pt]{report}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{siunitx}
\usepackage{booktabs}
\usepackage{xcolor}
\usepackage[makestderr]{pythontex}

\definecolor{nash}{RGB}{46, 204, 113}
\definecolor{player1}{RGB}{52, 152, 219}
\definecolor{player2}{RGB}{231, 76, 60}

\title{Game Theory:\\
Nash Equilibrium, Mixed Strategies, and Payoff Analysis}
\author{Department of Economics\\Technical Report EC-2024-001}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
This report presents a computational analysis of game theory concepts. We implement Nash equilibrium finding for 2-player games, analyze mixed strategies in zero-sum and non-zero-sum games, visualize payoff matrices, and explore classic games including Prisoner's Dilemma, Battle of the Sexes, and Matching Pennies.
\end{abstract}

\tableofcontents

\chapter{Introduction}

Game theory studies strategic interactions where outcomes depend on choices of multiple decision-makers (players).

\section{Key Concepts}
\begin{itemize}
    \item \textbf{Nash Equilibrium}: No player can improve by unilaterally changing strategy
    \item \textbf{Dominant Strategy}: Optimal regardless of opponent's choice
    \item \textbf{Mixed Strategy}: Randomizing over pure strategies
\end{itemize}

\chapter{Two-Player Normal Form Games}

\section{Payoff Matrix Representation}
For players 1 and 2 with strategies $S_1 = \{s_1^1, ..., s_1^m\}$ and $S_2 = \{s_2^1, ..., s_2^n\}$:
\begin{equation}
\text{Payoffs: } (u_1(s_1^i, s_2^j), u_2(s_1^i, s_2^j))
\end{equation}

\begin{pycode}
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import linprog
plt.rc('text', usetex=True)
plt.rc('font', family='serif')

np.random.seed(42)

def find_pure_nash(payoff1, payoff2):
    m, n = payoff1.shape
    nash = []
    for i in range(m):
        for j in range(n):
            best_resp_1 = np.argmax(payoff1[:, j]) == i
            best_resp_2 = np.argmax(payoff2[i, :]) == j
            if best_resp_1 and best_resp_2:
                nash.append((i, j))
    return nash

def find_mixed_nash_2x2(payoff1, payoff2):
    # For 2x2 games
    # Player 2's mix makes Player 1 indifferent
    a, b = payoff1[0, 0], payoff1[0, 1]
    c, d = payoff1[1, 0], payoff1[1, 1]
    if (a - b - c + d) != 0:
        q = (d - b) / (a - b - c + d)
    else:
        q = 0.5

    # Player 1's mix makes Player 2 indifferent
    a, b = payoff2[0, 0], payoff2[1, 0]
    c, d = payoff2[0, 1], payoff2[1, 1]
    if (a - b - c + d) != 0:
        p = (d - c) / (a - b - c + d)
    else:
        p = 0.5

    p = np.clip(p, 0, 1)
    q = np.clip(q, 0, 1)
    return p, q

def expected_payoff(p, q, payoff):
    return p * q * payoff[0, 0] + p * (1-q) * payoff[0, 1] + \
           (1-p) * q * payoff[1, 0] + (1-p) * (1-q) * payoff[1, 1]

# Classic games
games = {
    "Prisoner's Dilemma": {
        'payoff1': np.array([[-1, -3], [0, -2]]),
        'payoff2': np.array([[-1, 0], [-3, -2]]),
        'strategies': ['Cooperate', 'Defect']
    },
    'Battle of Sexes': {
        'payoff1': np.array([[3, 0], [0, 2]]),
        'payoff2': np.array([[2, 0], [0, 3]]),
        'strategies': ['Opera', 'Football']
    },
    'Matching Pennies': {
        'payoff1': np.array([[1, -1], [-1, 1]]),
        'payoff2': np.array([[-1, 1], [1, -1]]),
        'strategies': ['Heads', 'Tails']
    }
}
\end{pycode}

\chapter{Classic Games Analysis}

\section{Prisoner's Dilemma}

\begin{pycode}
fig, axes = plt.subplots(2, 3, figsize=(14, 8))

for idx, (name, game) in enumerate(games.items()):
    ax = axes[0, idx]
    payoff1 = game['payoff1']
    payoff2 = game['payoff2']

    # Plot payoff matrix
    im = ax.imshow(payoff1, cmap='RdYlGn', aspect='auto')

    # Annotate cells
    for i in range(2):
        for j in range(2):
            text = f'({payoff1[i,j]}, {payoff2[i,j]})'
            ax.text(j, i, text, ha='center', va='center', fontsize=10)

    ax.set_xticks([0, 1])
    ax.set_yticks([0, 1])
    ax.set_xticklabels(game['strategies'])
    ax.set_yticklabels(game['strategies'])
    ax.set_xlabel('Player 2')
    ax.set_ylabel('Player 1')
    ax.set_title(name)

    # Find equilibria
    pure_nash = find_pure_nash(payoff1, payoff2)
    for eq in pure_nash:
        rect = plt.Rectangle((eq[1]-0.5, eq[0]-0.5), 1, 1,
                              fill=False, edgecolor='green', linewidth=3)
        ax.add_patch(rect)

# Best response analysis for Prisoner's Dilemma
pd_payoff1 = games["Prisoner's Dilemma"]['payoff1']
pd_payoff2 = games["Prisoner's Dilemma"]['payoff2']

ax = axes[1, 0]
p_range = np.linspace(0, 1, 100)
# Player 1's expected payoff vs strategy
for j, s2 in enumerate(['Cooperate', 'Defect']):
    eu = p_range * pd_payoff1[0, j] + (1 - p_range) * pd_payoff1[1, j]
    ax.plot(p_range, eu, label=f'P2 plays {s2}')
ax.set_xlabel('P1 probability of Cooperate')
ax.set_ylabel('P1 Expected Payoff')
ax.set_title("Prisoner's Dilemma: P1 Payoffs")
ax.legend()
ax.grid(True, alpha=0.3)

# Battle of Sexes mixed equilibrium
bos_payoff1 = games['Battle of Sexes']['payoff1']
bos_payoff2 = games['Battle of Sexes']['payoff2']
p_mix, q_mix = find_mixed_nash_2x2(bos_payoff1, bos_payoff2)

ax = axes[1, 1]
p_range = np.linspace(0, 1, 100)
q_range = np.linspace(0, 1, 100)
P, Q = np.meshgrid(p_range, q_range)
EU1 = expected_payoff(P, Q, bos_payoff1)
cs = ax.contourf(P, Q, EU1, levels=20, cmap='viridis')
ax.plot(p_mix, q_mix, 'ro', markersize=10, label='Mixed NE')
ax.scatter([1, 0], [1, 0], color='green', s=100, marker='s', label='Pure NE')
ax.set_xlabel('P1: prob(Opera)')
ax.set_ylabel('P2: prob(Opera)')
ax.set_title('Battle of Sexes: P1 Payoff')
ax.legend()
plt.colorbar(cs, ax=ax)

# Matching Pennies
mp_payoff1 = games['Matching Pennies']['payoff1']
mp_payoff2 = games['Matching Pennies']['payoff2']
p_mp, q_mp = find_mixed_nash_2x2(mp_payoff1, mp_payoff2)

ax = axes[1, 2]
EU1 = expected_payoff(P, Q, mp_payoff1)
cs = ax.contourf(P, Q, EU1, levels=20, cmap='RdBu')
ax.plot(p_mp, q_mp, 'ko', markersize=10, label=f'Mixed NE ({p_mp:.1f}, {q_mp:.1f})')
ax.set_xlabel('P1: prob(Heads)')
ax.set_ylabel('P2: prob(Heads)')
ax.set_title('Matching Pennies: P1 Payoff')
ax.legend()
plt.colorbar(cs, ax=ax)

plt.tight_layout()
plt.savefig('game_theory_analysis.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{game_theory_analysis.pdf}
\caption{Game theory analysis: payoff matrices (top), equilibrium analysis (bottom).}
\end{figure}

\chapter{Mixed Strategy Equilibria}

\section{Computing Mixed Nash Equilibrium}
For a 2x2 game, player 1 mixes to make player 2 indifferent:
\begin{equation}
p \cdot u_2(s_1^1, s_2^1) + (1-p) \cdot u_2(s_1^2, s_2^1) = p \cdot u_2(s_1^1, s_2^2) + (1-p) \cdot u_2(s_1^2, s_2^2)
\end{equation}

\begin{pycode}
# Compute all equilibria
results = []
for name, game in games.items():
    pure = find_pure_nash(game['payoff1'], game['payoff2'])
    p, q = find_mixed_nash_2x2(game['payoff1'], game['payoff2'])

    # Expected payoffs at mixed equilibrium
    eu1 = expected_payoff(p, q, game['payoff1'])
    eu2 = expected_payoff(p, q, game['payoff2'])

    results.append({
        'name': name,
        'pure': pure,
        'mixed_p': p,
        'mixed_q': q,
        'eu1': eu1,
        'eu2': eu2
    })

# Visualize dynamics
fig, axes = plt.subplots(1, 2, figsize=(12, 4))

# Replicator dynamics for Matching Pennies
ax = axes[0]
dt = 0.01
T = 100
p_hist = [0.6]
q_hist = [0.4]

for _ in range(int(T/dt)):
    p, q = p_hist[-1], q_hist[-1]

    # Fitness calculations
    f1_H = q * mp_payoff1[0, 0] + (1-q) * mp_payoff1[0, 1]
    f1_T = q * mp_payoff1[1, 0] + (1-q) * mp_payoff1[1, 1]
    f1_avg = p * f1_H + (1-p) * f1_T

    f2_H = p * mp_payoff2[0, 0] + (1-p) * mp_payoff2[1, 0]
    f2_T = p * mp_payoff2[0, 1] + (1-p) * mp_payoff2[1, 1]
    f2_avg = q * f2_H + (1-q) * f2_T

    dp = p * (f1_H - f1_avg) * dt
    dq = q * (f2_H - f2_avg) * dt

    p_new = np.clip(p + dp, 0.01, 0.99)
    q_new = np.clip(q + dq, 0.01, 0.99)

    p_hist.append(p_new)
    q_hist.append(q_new)

ax.plot(p_hist, q_hist, 'b-', alpha=0.7)
ax.plot(p_hist[0], q_hist[0], 'go', markersize=10, label='Start')
ax.plot(0.5, 0.5, 'ro', markersize=10, label='NE')
ax.set_xlabel('P1: prob(Heads)')
ax.set_ylabel('P2: prob(Heads)')
ax.set_title('Replicator Dynamics: Matching Pennies')
ax.legend()
ax.grid(True, alpha=0.3)

# Iterated elimination
ax = axes[1]
# Show Cournot duopoly convergence
q1_hist = [10]
q2_hist = [10]
a, b, c = 100, 1, 10  # Demand: P = a - b*Q, cost = c

for _ in range(20):
    # Best response functions
    q1_br = (a - c - b * q2_hist[-1]) / (2 * b)
    q2_br = (a - c - b * q1_hist[-1]) / (2 * b)
    q1_hist.append(max(0, q1_br))
    q2_hist.append(max(0, q2_br))

ax.plot(q1_hist, 'b-o', label='Firm 1')
ax.plot(q2_hist, 'r-s', label='Firm 2')
ax.axhline((a - c) / (3 * b), color='green', linestyle='--', label='NE')
ax.set_xlabel('Iteration')
ax.set_ylabel('Quantity')
ax.set_title('Cournot Duopoly Convergence')
ax.legend()
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('game_dynamics.pdf', dpi=150, bbox_inches='tight')
plt.close()
\end{pycode}

\begin{figure}[htbp]
\centering
\includegraphics[width=0.95\textwidth]{game_dynamics.pdf}
\caption{Game dynamics: replicator dynamics cycling (left), Cournot convergence (right).}
\end{figure}

\chapter{Numerical Results}

\begin{table}[htbp]
\centering
\caption{Nash equilibria for classic games}
\begin{tabular}{@{}lccc@{}}
\toprule
Game & Pure NE & Mixed $(p, q)$ & Expected Payoffs \\
\midrule
\py{results[0]['name']} & \py{str(results[0]['pure'])} & (\py{f"{results[0]['mixed_p']:.2f}"}, \py{f"{results[0]['mixed_q']:.2f}"}) & (\py{f"{results[0]['eu1']:.2f}"}, \py{f"{results[0]['eu2']:.2f}"}) \\
\py{results[1]['name']} & \py{str(results[1]['pure'])} & (\py{f"{results[1]['mixed_p']:.2f}"}, \py{f"{results[1]['mixed_q']:.2f}"}) & (\py{f"{results[1]['eu1']:.2f}"}, \py{f"{results[1]['eu2']:.2f}"}) \\
\py{results[2]['name']} & \py{str(results[2]['pure'])} & (\py{f"{results[2]['mixed_p']:.2f}"}, \py{f"{results[2]['mixed_q']:.2f}"}) & (\py{f"{results[2]['eu1']:.2f}"}, \py{f"{results[2]['eu2']:.2f}"}) \\
\bottomrule
\end{tabular}
\end{table}

\chapter{Conclusions}

\begin{enumerate}
    \item Prisoner's Dilemma: Dominant strategy leads to Pareto-inferior outcome
    \item Battle of Sexes: Multiple equilibria require coordination
    \item Matching Pennies: Only mixed strategy equilibrium exists
    \item Replicator dynamics may cycle rather than converge
    \item Nash equilibrium provides predictions for strategic behavior
\end{enumerate}

\end{document}