Session 1 and 2

Review of Linear Transformations

• A linear transformation $T$ from vector space $V$ over field $F$ to vector space $W$ must satisfy $T(cx) = cT(x)$ and $T(x+y) = T(x) + T(y)$.
• If $T: V \to W$ is a linear transformation, then $T(0) = 0$, $T(-v) = -T(v)$, and $T(x-y) = T(x) - T(y)$.
• The null space of $T$, $N(T)$, contains vectors $x \in V$ where $T(x) = 0$.
• The range of $T$, $R(T)$, contains vectors $w \in W$ where $w = T(x)$ for some $x \in V$.
• For a linear transformation $T: V \to W$, $N(T)$ is a subspace of $V$ and $R(T)$ is a subspace of $W$.

Linear Transformation Examples

• For an $m \times n$ matrix $A$ with entries in $F$, $L_A : F^n \to F^m$ is defined by $L_A(x) = Ax$.
• $T: P(R) \to R$ defined by $T(f) = f(2)$ is a linear transformation.
• $T: C[a,b] \to R$ defined by $T(f) = \int_a^b f(t) dt$ is a linear transformation, where $C[a,b]$ is the vector space of real-valued continuous functions on $[a,b]$.
• $T: C[a,b] \to C[a,b]$ defined by $T(f) = \int_a^x f(t) dt$ is a linear transformation.
• $T: V \to V$ defined by $T(x) = cx$, where $c$ is a constant, is a linear transformation.

Null Space and Range Examples

• For the matrix $A = \begin{bmatrix} 1 & 2 \ 1 & 2 \end{bmatrix}$ and $L_A : R^2 \to R^2$, $N(L_A) = { x_2 \begin{bmatrix} -2 \ 1 \end{bmatrix} : x_2 \in R } = span( \begin{bmatrix} -2 \ 1 \end{bmatrix} )$ and $R(L_A) = { c \begin{bmatrix} 1 \ 1 \end{bmatrix} : c \in R } = span( \begin{bmatrix} 1 \ 1 \end{bmatrix} )$.
• For $T: P(R) \to R$ defined by $T(f) = f(2)$, $N(T) = { f \in P(R) : f(2) = 0 }$ and $R(T) = R$.
• For $T: C[a,b] \to R$ defined by $T(f) = \int_a^b f(t) dt$, $N(T) = { f \in C[a,b] : \int_a^b f(t) dt = 0 }$ and $R(T) = R$.

Natural Logarithm Function

• The natural logarithm function, denoted as ln, is defined on $]0; +\infty[$.
• It is the primitive of the function $x \mapsto \frac{1}{x}$ that equals zero at 1.

Algebraic Properties of Natural Logarithms

• $\ln(1) = 0$
• $\ln(e) = 1$
• $\ln(ab) = \ln(a) + \ln(b)$
• $\ln(\frac{1}{a}) = -\ln(a)$
• $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
• $\ln(a^n) = n\ln(a)$
• $\ln(\sqrt{a}) = \frac{1}{2}\ln(a)$

Study of the Natural Logarithm Function

• The function ln is differentiable on $]0; +\infty[$, and its derivative is $x \mapsto \frac{1}{x}$.
• The function ln is strictly increasing on $]0; +\infty[$, since $\frac{1}{x} > 0$ for all $x > 0$.

Limits of the Natural Logarithm Function

• $\lim_{x \to +\infty} \ln(x) = +\infty$
• $\lim_{x \to 0} \ln(x) = -\infty$

Derivative of ln(u)

• $(\ln(u))' = \frac{u'}{u}$

Equations and Inequalities Involving Natural Logarithms

• For all strictly positive real numbers $a$ and $b$:
• $\ln(a) = \ln(b) \Leftrightarrow a = b$
• $\ln(a) < \ln(b) \Leftrightarrow a < b$, due to ln being strictly increasing on $]0; +\infty[$.

Statics: General Principles

• Mechanics studies how bodies react to forces.
• Statics deals with bodies at rest, contrasted with dynamics, which deals with bodies in motion.

Fundamental Concepts in Statics

• Basic quantities include length (defines size of space), time (defines event sequence), mass (compares body actions), and force (body action on another).
• Force magnitude, direction, and application point are crucial.
• Idealizations simplify analysis, including particles (mass, no size), rigid bodies (no deformation), and concentrated forces (point loads).

Newton's Laws of Motion

• First Law: Objects at rest or in uniform motion stay that way unless acted upon by an unbalanced force.
• Second Law: $F = ma$, where F is the resultant unbalanced force, m is the mass, and a is the acceleration in the same direction as F.
• Third Law: Action and reaction forces between two particles are equal, opposite, and collinear.
• Gravitational Attraction Law: $F = G\frac{m_1 m_2}{r^2}$, with $G = 66.73(10^{-12}) m^3/(kg \cdot s^2)$.
• Weight: $W = mg$, where $g \approx 9.81 m/s^2$ (gravity acceleration).

Units of Measurement

• SI units include meter (m), second (s), kilogram (kg), and Newton (N).
• Newton is defined as $N = kg \cdot m/s^2$.

International System of Units (SI)

• Rules for use include representing quantities with symbols, using prefixes to reduce numerical values, avoiding compound prefixes and denominators, using a space between values and units, and omitting periods after symbols.

Numerical Calculations

• Dimensional Homogeneity: Each equation term must have the same dimensions to be valid.
• Significant Figures: Indicate the accuracy of a numerical value.
• Rounding: Final answers appropriately rounded for accuracy representation.

General Procedure for Analysis

• Steps include carefully reading the problem, drawing diagrams, applying relevant principles, solving equations, and checking the answer.

Lines in Calculus: Slope

• The slope, $m$, of a non-vertical line through points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$.

Point-Slope Form

• The equation of a line through $(x_1, y_1)$ with slope $m$ is given by $y - y_1 = m(x - x_1)$.

Slope-Intercept Form

• The equation of a line with slope $m$ and y-intercept $b$ is $y = mx + b$.

Equations for Vertical and Horizontal Lines

• A vertical line through $(a, b)$ has the equation $x = a$.
• A horizontal line through $(a, b)$ has the equation $y = b$.

Parallel and Perpendicular Lines

• Non-vertical lines with slopes $m_1$ and $m_2$ are parallel if $m_1 = m_2$. -Non-vertical lines are perpendicular if $m_1 = -\frac{1}{m_2}$ or $m_1m_2 = -1$.

Poisson Process Definition and Notation

• $N(t)$ represents the number of events occurring in the time interval $[0, t]$.

Poisson Process Definition

• ${N(t), t \geq 0}$ is a Poisson process with rate $\lambda, \lambda > 0$, when:
• $N(0) = 0$
• increments are independent.
• The number of events in time t follows a Poisson distribution with a mean of $\lambda t$: $P{N(t+s) - N(s) = n} = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n=0,1, \dots$ for all $s, t \geq 0$.

Proposition 6.1 for Poisson Processes

• Given a Poisson process ${N(t), t \geq 0}$ with rate $\lambda$:
• $E[N(t)] = \lambda t$
• $\operatorname{Var}(N(t)) = \lambda t$

Proof of Proposition 6.1

• $N(t)$ follows a Poisson distribution with parameter $\lambda t$.
• The expected value of $N(t)$ is $\lambda t$.
• The variance of $N(t)$ is $\lambda t$.

Stationary Increments Definition

• A stochastic process ${N(t), t \geq 0}$ has stationary increments if the distribution of $N(t+s) - N(s)$ is independent of $s$.

What is Game Theory?

• Game Theory is the study of settings with multiple agents.
• The agents have different preferences (utility functions) and different possible actions.
• Each agent is affected by their own actions, but also by the actions of other agents.

Examples of Game Theory Applications

• Auctions
• Routing in Networks
• Sponsored Search
• Security Games
• Fair Division
• Cost Sharing
• Social Networks

Selfish Routing Setting

• A network connects a source $s$ to a destination $t$.
• Each edge $e$ has a cost/latency function $\mathcal{l}_e(x)$, indicating the cost per user when x users use edge $e$.
• Infinitely many users control infinitesimally small traffic amounts and aim to travel from $s$ to $t$.
• Users choose the path that minimizes their cost.

Braess's Paradox - Example 1

• Scenario with two paths:
• $s \rightarrow a \rightarrow t$, cost: $x + 1$
• $s \rightarrow b \rightarrow t$, cost: $1 + x$
• Traffic splits evenly as each user chooses the cheaper path.
• Cost to each user: $0.5 + 1 = 1.5$.

Braess's Paradox - Example 2

• A zero-latency link is added from $a$ to $b$.
• $s \rightarrow a \rightarrow b \rightarrow t$ becomes better than $s \rightarrow b \rightarrow t$, because $x \le 1$.
• All traffic shifts to $s \rightarrow a \rightarrow b \rightarrow t$.
• Cost to each user becomes $1 + 1 = 2$.

Questions Addressed in Algorithmic Game Theory

• Existence: Does a Nash Equilibrium always exist?
• Computation: How hard is it to compute one?
• Efficiency: How "good" is a Nash Equilibrium?

Topics Covered in the Course

• Basics of Game Theory
• Solution Concepts for Coalitional Games
• Mechanism Design with Money
• Price of Anarchy
• Fair Division

Reading Material

• Nisan, Roughgarden, Tardos, Vazirani, Algorithmic Game Theory, Cambridge University Press, 2007.
• Easley & Kleinberg, Networks, Crowds, and Markets, Cambridge University Press, 2010.
• Osborne, An Introduction to Game Theory, Oxford University Press, 2003.
• Shoham & Leyton-Brown, Multiagent Systems: Algorithmic, Game-Theoretic and Logical Foundations, Cambridge University Press, 2009.

Definition of a Poisson Process

• A stochastic process ${N(t): t \geq 0}$ with rate $\lambda > 0$ is Poisson if:
  • $N(0) = 0$
  • It has independent increments
  • For $s, t \geq 0$, $P[N(t+s) - N(s) = n] = e^{-\lambda t} \frac{(\lambda t)^n}{n!}$ for $n = 0, 1, 2,...$

Properties of Poisson Processes

• Memoryless Property: Poisson processes are memoryless.
• Stationary Increments: The number of events in a time interval depends only on the interval's length.
• Relation to Exponential Distribution: Times between events are IID exponential with mean $1/\lambda$.

Key Results for Poisson Processes

• Superposition: Independent Poisson processes with rates $\lambda_1$ and $\lambda_2$ combine to form a Poisson process with rate $\lambda_1 + \lambda_2$.
• Decomposition: Events in a Poisson process with rate $\lambda$ are classified as type 1 (probability $p$) or type 2 (probability $1-p$), resulting in two independent Poisson processes with rates $\lambda p$ and $\lambda (1-p)$, respectively.

Applications of Poisson Processes

• Modeling customer arrivals
• Modeling phone calls at a call center
• Modeling radioactive decay

Examples of Calculations with Poisson Processes

• For customers arriving at a store according to a Poisson process with rate 10 per hour:
• The probability of exactly 5 customers arriving between 10:00 AM and 10:30 AM is approximately 0.1755.
• The probability of more than 2 customers arriving between 10:00 AM and 10:15 AM is approximately 0.4562.

Random Variables - Definition

• A random variable is a function assigning a real number to each outcome of a random experiment.

Types of Random Variables

• Discrete Random Variable:
• Takes isolated values. Usually integers.
  • Example: number of heads in three coin flips, number of children in a family.
• Continuous Random Variable:
• Can take any value within a given interval.
  • Example: height of a person, temperature.

Definition of Probability Distribution

• The probability distribution of a random variable describes the probability of each value (discrete) or interval of values (continuous).

Representations of Probability Distributions

• Discrete: Table, bar diagram.
• Continuous: Density function, histogram.

Parameters of a Random Variable

• Expected Value (Mean):
  • Denoted as $E(X)$ or $\mu$.
    • For a discrete variable: $E(X) = \sum_{i=1}^{n} x_i P(X = x_i)$
  • For a continuous variable: $E(X) = \int_{-\infty}^{+\infty} x f(x) dx$
• Variance:
  • Denoted as $V(X)$ or $\sigma^2$.
  • Measures the spread of values around the mean.
  • $V(X) = E[(X - E(X))^2] = E(X^2) - [E(X)]^2$
• Standard Deviation:
• Denoted as $\sigma$.
  • The square root of the variance.
• $\sigma = \sqrt{V(X)}$

Common Probability Distributions

• Bernoulli Distribution:
  • Models an experiment with two outcomes: success (1) or failure (0).
• Parameter: $p$ (probability of success).
  • $P(X = 1) = p$, $P(X = 0) = 1 - p$
• $E(X) = p$, $V(X) = p(1-p)$
• Binomial Distribution:
  • Models the number of successes in $n$ independent Bernoulli trials.
• Parameters: $n$ (number of trials), $p$ (probability of success).
  • $P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$
  • $E(X) = np$, $V(X) = np(1-p)$
• Poisson Distribution:
• Models the number of rare events occurring in a given interval of time or space.
• Parameter: $\lambda$ (average rate of occurrence).
• $P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$
• $E(X) = \lambda$, $V(X) = \lambda$
• Normal (Gaussian) Distribution:
  • A common continuous distribution. Models natural phenomena.
  • Parameters: $\mu$ (mean), $\sigma$ (standard deviation).
  • Probability density: $f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}$
  • Denoted as $N(\mu, \sigma^2)$
  • $E(X) = \mu$, $V(X) = \sigma^2$

Properties of Normal Distribution

• Approx. 68% of values are between $\mu - \sigma$ and $\mu + \sigma$.
• Approx. 95% of values are between $\mu - 2\sigma$ and $\mu + 2\sigma$.
• Approx. 99.7% of values are between $\mu - 3\sigma$ and $\mu + 3\sigma$.

Exponential Distribution

• Models the lifetime of an event or the waiting time between events in a Poisson process.
• Parameter: $\lambda$ (rate).
• Probability density: $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• $P(X > t) = e^{-\lambda t}$ (survival probability)
• $E(X) = \frac{1}{\lambda}$, $V(X) = \frac{1}{\lambda^2}$

Limit Theorems

• Central Limit Theorem:
• The sum of a large number of independent and identically distributed random variables tends towards a normal distribution.
• Useful for approximating complex distributions.
• Law of Large Numbers:
• The empirical mean of a sample of independent random variables converges to the expected value.
• Justifies using the sample mean to estimate the population mean.

Properties of Gases

• Gases are highly compressible, susceptible to thermal expansion, and have low viscosity.
• Pressure ($P$) is defined as force ($F$) per unit area ($A$), expressed as $P = \dfrac{F}{A}$.
• Pressure units: pascal (Pa), atmosphere (atm), torr, and mm Hg.
• $1 \text{ Pa} = 1 \frac{\text{N}}{\text{m}^2}$, $1 \text{ atm} = 101,325 \text{ Pa}$, $1 \text{ atm} = 760 \text{ torr}$, $1 \text{ atm} = 760 \text{ mm Hg}$

The Ideal-Gas Equation

• The ideal-gas equation relates pressure ($P$), volume ($V$), number of moles ($n$), and temperature ($T$): $PV = nRT$
• $R$ is the ideal-gas constant: $R = 0.08206 \frac{\text{L atm}}{\text{mol K}}$ or $R = 8.314 \frac{\text{J}}{\text{mol K}}$
• Standard temperature and pressure (STP) is 0°C (273.15 K) and 1 atm; ideal gas molar volume at STP is 22.41 L.

Applications of the Ideal-Gas Equation

• Gas density ($\rho$) can be calculated using $\rho = \dfrac{m}{V} = \dfrac{PM}{RT}$, where $M$ is the molar mass.
• The molar mass of a gas can be determined from its density using $M = \dfrac{\rho RT}{P}$.

Gas Mixtures and Partial Pressures

• In gas mixtures, partial pressure of each gas contributes to the total pressure.
• Dalton’s law states that total pressure ($P_T$) is the sum of each gas's partial pressures: $P_T = P_1 + P_2 + P_3 +...$
• Partial pressure is calculated from mole fraction ($X$) and total pressure: $P_i = X_i P_T$, where $X_i = \dfrac{n_i}{n_T}$.

Kinetic-Molecular Theory of Gases

• Describes gas behavior via molecular motion.
• Postulates:
• Gases are in continuous, random motion.
• Particle volume is negligible.
• Intermolecular forces are negligible.
• Collisions are elastic.
• Average kinetic energy is proportional to absolute temperature.
• Root-mean-square (rms) speed ($u_{rms}$) of gas molecules is: $u_{rms} = \sqrt{\dfrac{3RT}{M}}$.

Effusion and Diffusion

• Effusion is gas molecule escape through a small hole.
• Graham’s law: effusion rate is inversely proportional to the square root of molar mass: $\dfrac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\dfrac{M_2}{M_1}}$.
• Diffusion is one substance spreading through another.

Real Gases: Deviations from Ideal Behavior

• Real gases deviate from ideal behavior at high pressures and low temperatures.
• Due to intermolecular forces and the real volume of gas molecules.
• Van der Waals equation accounts for these deviations: $(P + a\dfrac{n^2}{V^2})(V - nb) = nRT$, where $a$ and $b$ are van der Waals constants.