Game Theory: Strategic Interactions

Game Theory Overview

• Game theory focuses on mathematical interpretations of strategic interactions involving self-interested agents.
• A "game" is any interaction among multiple players where each player's outcome is mutually dependent.
• Strategic interactions mean that each player's outcome is influenced by the actions of all the other players.
• The rationality assumption means that each player aims to maximize their own utility.
• Main components of a game are a set of players, a set of actions or strategies for each player, and a utility function for each player based on the actions of all players.

Examples of Games

• Penalty Kick Game: Two players (Kicker and Goalie) each choosing Left or Right with opposing utilities (Goalie wins if they choose the same direction, Kicker wins otherwise).
• The goalies choice of direction and that of the kicker determine the winning utility for each
• Prisoner's Dilemma: Two prisoners choosing to Cooperate or Defect, affecting their prison sentences, where both cooperating yields light sentences, both defecting yields moderate sentences, and one cooperating while the other defects results in freedom for the defector and a heavy sentence for the cooperator.

Algorithmic Game Theory Defined

• Algorithmic Game Theory (AGT) merges computer science and game theory. Two main directions exist

Algorithms in Games

• Games often face computational hardness.
• It focuses on designing efficient algorithms to solve games or ensure "good" outcomes.
• Examples include computing Nash equilibria, finding optimal strategies, and mechanism design.

Games in Algorithms

• Computer science settings are modeled as games
• Focus is on analyzing strategic participant behavior.
• Systems are designed to withstand strategic manipulation.
• Network routing, online auctions, and social networks are examples.

Selfish Routing Example

• A graph with source $s$ and destination $t$ is used
• Each edge $e$ has a cost function $c_e(x)$ that represents the edge's cost based on the fraction of traffic on the edge $x$.
• The objective is to route traffic from $s$ to $t$ to minimize the total cost.
• Selfish Routing: players control traffic volume & choose paths to minimize individual cost, possibly increasing the total cost.
• Price of Anarchy (PoA): Measures the inefficiency of selfish behavior.
• $PoA = \frac{\text{Total cost of the Nash equilibrium}}{\text{Optimal total cost}}$
• Braess's Paradox: Adding network capacity can ironically increase congestion.

Course Logistics Summary

• Ariel Procaccia is the instructor.
• Office hours are Tuesday from 4:30-5:30 pm in GHC 8121, or by appointment.
• Xiaohui Bei is the TA.
• Friday office hours from 4:30-5:30 pm in GHC 8001.
• Course website: https://www.cs.cmu.edu/~arielpro/15896S24/
• Piazza: piazza.com/cmu/spring2024/15896
• Grading breakdown: 50% Problem Sets, 25% Midterm, 25% Final Project.
• No required textbook; resources will be posted online.
• Prerequisites: Algorithms, Probability, and Mathematical maturity.

WKB Approximation: Basic Concept

• Considers a particle in a time-independent potential $V(x)$.
• The 1-D Schrödinger equation is:
  • $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi(x) = E\psi(x)$
  • $\frac{d^2\psi}{dx^2} = \frac{2m}{\hbar^2} (V(x) - E) \psi(x)$
• Classically allowed region: $E > V(x)$
• Classically forbidden region: $E < V(x)$

Simple Cases Examples

• $V(x) = 0$ (free particle)
  • $\frac{d^2\psi}{dx^2} = -k^2 \psi(x)$, where $k = \sqrt{\frac{2mE}{\hbar^2}}$
  • $\psi(x) = A e^{\pm ikx}$
• $V(x) = \infty$ (infinite potential)
  • $\psi(x) = 0$

WKB Approximation

• For a general potential $V(x)$, the solution can be approximated by assuming that $V(x)$ is "slowly varying" and is locally nearly constant.
• The wavefunction is approximated as $\psi(x) \approx A e^{\pm ikx}$, where $k(x) = \sqrt{\frac{2m(E - V(x))}{\hbar^2}}$ and $k$ is a function of $x$, amplitude $A$ must also be a function of $x$.
• The approximate solution is given by $\psi(x) = A(x) e^{\pm i \int k(x') dx'}$.

WKB Approximation: Derivation Details

• $\psi(x) = e^{\frac{i}{\hbar}S(x)}$, where $S(x)$ is a complex function to expand the approximation.
• Substituting this into the Schrödinger equation:
  • $(S'(x))^2 = 2m(E - V(x)) + i\hbar S''(x)$
• The approximation means that $S(x)$ is slowly varying, hence $S''(x)$ is small.
• $(S'(x))^2 \approx 2m(E - V(x))$
• To improve approximation expand 𝑆(𝑥) in powers of ℏ
  • 𝑆(𝑥)=𝑆0(𝑥)+ℏ𝑆1(𝑥)+ℏ2𝑆2(𝑥)+...

Zeroth Order Details

• $(S_0'(x))^2 = 2m(E - V(x))$
• $S_0'(x) = \pm \sqrt{2m(E - V(x))} = \pm \hbar k(x)$
• $S_0(x) = \pm \hbar \int k(x') dx'$

First Order Details

• $2 S_0'(x) S_1'(x) = i S_0''(x)$
• $S_1(x) = \frac{i}{2} \int \frac{S_0''(x')}{S_0'(x')} dx' = \frac{i}{2} \ln(S_0'(x))$
• $S_1(x) = \frac{i}{2} \ln(\hbar k(x)) = \frac{i}{2} \ln(k(x)) + constant$
• $e^{\frac{i}{\hbar} S(x)} = e^{\frac{i}{\hbar} (S_0(x) + \hbar S_1(x) +...)} = e^{\frac{i}{\hbar} S_0(x)} e^{i S_1(x)}...$
• $\psi(x) \approx e^{\pm i \int k(x') dx'} e^{-\frac{1}{2} \ln(k(x))} = \frac{1}{\sqrt{k(x)}} e^{\pm i \int k(x') dx'}$

WKB Solutions Defined

• In the classically allowed region where 𝐸>𝑉(𝑥):
• $\psi(x) \approx \frac{1}{\sqrt{k(x)}} \left[ C e^{+i \int k(x') dx'} + D e^{-i \int k(x') dx'} \right]$
• In the classically forbidden region where 𝐸<𝑉(𝑥):
  • $\psi(x) \approx \frac{1}{\sqrt{\kappa(x)}} \left[ A e^{+ \int \kappa(x') dx'} + B e^{- \int \kappa(x') dx'} \right]$

Validity of the WKB Approximation

• Valid when potential 𝑉(𝑥) is slowly varying

Validity of the WKB Approximation Explained

• $\lambda(x) = \frac{2\pi}{k(x)}$ is the local wavelength.
• $\left| \frac{d\lambda}{dx} \right| \ll 1$

WKB Approximation: Condition

• This can also be written as: $\left| \frac{p'(x)}{p(x)^2} \right| \ll \frac{1}{\hbar}$, where $p(x) = \hbar k(x) = \sqrt{2m(E - V(x))}$ is the classical momentum.
• The WKB approximation fails when $E \approx V(x)$ mainly at turning points.

Turning Points Defined

• Turning points are the points where 𝐸=𝑉(𝑥).
• WKB approximation isn't valid near these points because wavelength becomes infinite.

Elements of Life - Key Components

• Life is primarily composed of six elements: Carbon (C), Hydrogen (H), Oxygen (O), Nitrogen (N), Sulfur (S), and Phosphorus (P).

Isotopes Defined

• Atoms of an element share the same proton count, but differ in the neutron count
• Isotopes are different forms of the same chemical element

Radioactive Isotopes Explained

• Radioactive isotopes decay spontaneously - emitting particles and energy

Radioactive Isotopes Use-cases

• Applications of radioactive isotopes include dating fossils, tracing atoms through metabolic processes, and diagnosing medical disorders like cancer.

Covalent Bonds: Definitions

• Atoms sharing valence electrons for stability results in covalent bonds
• Covalent bonds yield molecules

Electronegativity Defined

• Electronegativity: The attraction level of an atom toward electrons in a covalent bond.
• High electronegativity = atom attracts shared electrons more strongly.

Covalent Bonds: Types

• A nonpolar covalent bond has atoms sharing electrons equally.
• A polar covalent bond occurs when one atom is more electronegative, leading to unequal sharing.

Ionic Bonds Defined

• Some atoms transfer electrons to others to generate an attraction force
• An ionic bond: an attraction force between anion and cation
• An ion: a charged molecule.
• A cation is a positively charged ion.
• An anion is a negatively charged ion.

Weak Chemical Bonds

• Hydrogen bonds and van der Waals interactions are weak interactions that reinforce the shapes of large molecules like the ones that form living matter

Hydrogen Bonds Explained

• A hydrogen bond forms when a hydrogen atom covalently bonded to one electronegative atom is also attracted to another electronegative atom

Van der Waals Interactions Explained

• Electrons are not evenly distributed, and accumulate by chance in one part of a molecule, which results in Van der Waals interactions.
• These interactions occur between closely positioned molecules with charges from electron distribution.

Molecular Shape and Function

• Molecular shape determines the function in the biological form.
• The shape determined by the atomic valence orbital placement.
• The s and p orbitals create molecular shapes.

Molecular Recognition

• Biological entities recognize and interact based on the specificity
• Molecules sharing shapes exhibit similar biological traits.

Chemical Reactions: Basics

• Chemical reactions produce and break chemical bonds
• Reactants are the starting components of chemical reactions
• Products are the result elements of a chemical reaction
• Reactions are reversible

Chemical Equilibrium Defined

• Chemical equilibrium: occurs if reverse/forward reactions match

Water Uniqueness for Life

• Water in living organisms has four chief attributes, which are critical for life on earth:
  • Cohesive behavior
  • Ability to moderate temperature
  • Expansion upon freezing
  • Versatility as a solvent

Polarity of Water

• A water molecule is polar because oxygen is more electronegative than hydrogen
• The slightly negative oxygen of one water molecule is attracted to the slightly positive hydrogen of a nearby water molecule forming a weak hydrogen bond

Water Properties

• Cohesion and adhesion are related
• Cohesion: Hydrogen bonds hold water molecules together
• Adhesion: An attraction between different substances. An example, between water and plant cell walls
• Surface tension: Measure of how hard it is to break the surface of a liquid

Moderation of Temperature

• Water absorbs heat from warmer air and releases stored heat to cooler air
• Water can absorb or release a large amount of heat with only a slight change in its own temperature

Heat and Temperature Definitions

• Kenetic energy: measurement of motion
• Heat: measure of overall amount of kenetic energy resulting from molecular motion
• Temperature: measures the degree of heat given the average kinetic energy of the molecules
• Specific heat: measurement of heat that must be absorbed or lost for 1 g of that substance to change its temperature by 1°C

Heat of Vaporization Details

• Heat of vaporization: quantity of heat a liquid consumes for conversion to gas
• Evaporative cooling: when a liquid evaporates, surface cools

Why Ice Water Floats

• Ice floats since it is less dense compared to liquid water
• Maximum water density = 4°C
• Hydrogen bonds in ice make it less dense

Solvent of Life

• Water is a solvent because it is polar
  • Solution: mixture that is homogeneous
  • Solvent: dissolving agent
  • Solute: is dissolved
  • Water is solvent = aqueous solution

Hydrophilic and Hydrophobic

• A hydrophilic substance has closeness to water
• A hydrophobic substance doesn't have closeness to water
• Oil compounds are hydrophobic because they have relatively nonpolar bonds

Acid and Base

• A hydrogen atom separates and is carried to proton(H^+)
• The molecule now known as hydronium (H3O^+)
• The other molecule after losing proton is hydroxide (OH^−)
• An acid = composition that boosts H^+
• A base = composition that decreases H^+

pH Scale

• Acids and bases are measured using the pH scale
• In a solution, pH of H and OH is H+][OH^−] = 10^{-14} and at 25 °C has formula pH=-log[H+]
• pH under 7 = acidic
• pH above 7 = basic
• Biological fluids have pH values in the range of 6 to 8

Buffers Described

• Cells need a sustained pH level (close to pH 7)
• An acid-base pair composes many buffers, reversibly pairing with H^+

Acidification Defined

• Pollutants mix in the air, causing acid rain, snow, or fog with a pH under 5.2
• Ocean acidification decreases carbonate
• As carbonate ions reduce, calcification decreases.

Hyperspectral Imaging Defined

• Hyperspectral imaging, also known as imaging spectroscopy, is a technique that captures a wide range of the electromagnetic spectrum with increased detail.
• Traditional cameras capture three color channels (red, green, and blue), but hyperspectral cameras can capture hundreds of narrow, contiguous spectral bands.

How Hyperspectral Imaging Works

• Illumination: Target illuminated by light source
• Capture: Hyperspectral camera collects emitted light.
• Spectral Separation: spectrometer is utilized to divide light - based on band
• Detection: Spectral band light is measured for each pixel
• Image Cube Formation: X/Y axes are arranged onto 3D cubes

Applications of Hyperspectral Imaging

• Agriculture: Yield prediction, disease/crop tracking
• Environmental Science: Pollution monitoring, water quality assessment, forestry
• Food Safety: Quality control, contamination detection
• Medical Imaging: Skin/cancer examination
• Surveillance: Material/target detection

Hyperspectral Imaging Advantages

• Detailed Spectral Information: Complete light signature captured
• Material Identification: Spectral properties
• Non-Destructive: Light measurement without target contact

Hyperspectral Imaging Disadvantages

• High Cost: Cameras are very costly to produce
• Large Data Volume: Images use hefty compute and storage
• Complex Data Analysis: Expert algorithms and knowledge requirements

What's the difference: Hyperspectral vs Multispectral Imaging

• Feature: Hyperspectral Imaging
  • Number of Bands : Multiple/Contiguous
  • Spectral Resolution: High
  • Data Volume: Large
  • Cost: High -Complexity of Analysis: Complex -Application Specificity: Detailed identification
• Feature: Multispectral Imaging
  • Number of Bands : Little/Broad
  • Spectral Resolution: Low
  • Data Volume: Smaller
  • Cost: Lower -Complexity of Analysis: Simpler -Application Specificity: General remote, land cover

Conclusion and Trends in Hyperspectral Imaging

• Hyperspectral imaging is a new tech, offering a variety of advantages and challenges for its users.
• Cost and analysis complexity are improving as tech advances

Linear Algebra: Scalars and Vectors

• A scalar is a real number ($\mathbb{R}$).
• A vector is an element of $\mathbb{R}^n$, represented by an ordered list of $n$ scalars (its components).

Vector Representation

• $\mathbf{u} = \begin{bmatrix} u_1 \ u_2 \ \vdots \ u_n \end{bmatrix} \in \mathbb{R}^n$ - represents vector u in given dimension
• A $\mathbb{R}^n$ is a dimension space (vector value)

Vector Operations

Addition

• Given vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$, addition is defined by:
  • $\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \ \vdots \ u_n + v_n \end{bmatrix}$

Scalar Multiplication

• Given vector $\mathbf{u} \in \mathbb{R}^n$ and scalar $c \in \mathbb{R}$, is defined by:
  • $c\mathbf{u} = \begin{bmatrix} cu_1 \ cu_2 \ \vdots \ cu_n \end{bmatrix}$

Vector Properties

Commutativity

• Vectors follow this rule- $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$

Associativity

• Vectors abide by this rule- $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$

Neutral vector

• Vector contains neutral scalar properties-$\mathbf{u} + \mathbf{0} = \mathbf{u}$

Inverse

• Inverse- $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$

Distributivity

• Scalar properties-
  • $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$
  • $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$

Compatibility

• Multiplication is compatible vector multiplication is defined by- $c(d\mathbf{u}) = (cd)\mathbf{u}$

Multiplication vectors by scalar

• Scalar multiplication- $1\mathbf{u} = \mathbf{u}$

Linear Combination

• Scalar $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_k \in \mathbb{R}^n$ is a combination using scalars:
  • $c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \dots + c_k\mathbf{u}_k$

Linear Independence

• Vectors Linear independence if equation has sole solution- $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_k \in \mathbb{R}^n$ are independent
  • $c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \dots + c_k\mathbf{u}_k = \mathbf{0}$
  • Otherwise they are dependent

Base

• Independent base-Vectors may be composed using base

Dot Product

• Vectors multiplication is known a dot- $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + \dots + u_nv_n = \sum_{i=1}^{n} u_i v_i$

Standard

• Scalar is determined to be standard- $||\mathbf{u}|| = \sqrt{\mathbf{u} \cdot \mathbf{u}} = \sqrt{u_1^2 + u_2^2 + \dots + u_n^2}$

Standard Properties

Scalability

• $||c\mathbf{u}|| = |c| \cdot ||\mathbf{u}||$

Triangle inequality

• $||\mathbf{u} + \mathbf{v}|| \le ||\mathbf{u}|| + ||\mathbf{v}||$

Gap/Distance

• Between Vectors- $d(\mathbf{u}, \mathbf{v}) = ||\mathbf{u} - \mathbf{v}||$

Axis-Aligned

• Vector are know as perpendicular if scalar is defined- $\mathbf{u} \cdot \mathbf{v} = 0$

Cast

• Projection onto- $\text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{v}||^2}\mathbf{v}$