Understanding Game Theory

Game Theory

• Game theory offers a mathematical approach to dissecting strategic interactions among decision-makers.
• It aids inforecasting rational behavior in interdependent scenarios.

Key Game Theory Elements

• Players: Individuals or entities making decisions.
• Strategies: Potential actions for players.
• Payoffs: Outcome rewards players get based on strategy.
• Rationality: Assuming players prioritize maximizing their gains.
• Equilibrium: A stable state where no player benefits from changing strategy alone.

Game Types

• Cooperative vs. Non-cooperative: Focuses on whether binding agreements can be formed among players.
• Zero-sum vs. Non-zero-sum: Differenciates gains where one player's gain equates to another's loss or those where all players can benefit.
• Complete vs. Incomplete Information: Focuses on players having full knowledge of game rules and payoffs.
• Static vs. Dynamic: Looks at simultaneous moves versus sequential turns players take.

Algorithmic Game Theory (AGT)

• AGT is where game theory meets computer science, emphasizing computation in strategic settings.
• It seeks efficient methods for finding Nash equilibria.
• It evaluates how incentives mold algorithm and system design.
• It designs mechanisms that promote desired behavior.

Main AGT Areas

• Equilibrium Computation: Algorithms to find Nash equilibria in games are developed.
• Mechanism Design: Focuses on rules and incentives to achieve objectives.
• Price of Anarchy: Measures inefficiency from selfish behavior compared with socially optimal outcomes.
• Social Choice Theory: It creates voting procedures resistant to manipulation for equitable decisions.
• Network Games: Strategic interactions, for example in network creation, are analyzed through networks.

Prisoner's Dilemma

• An illustration of conflicting individual interest versus shared welfare.
• Two suspects decide whether to cooperate (stay silent) or defect (betray the other).
• The outcome depends on mutual decisions, highlighting the tension between individual choice and potential collective benefits.
• The equilibrium leans towards both defecting, worse than if they both cooperated.

Real-World AGT Uses

• Internet and the Web: Used in online auction design and reputation systems.
• E-commerce: It optimizes pricing and detects fraud.
• Social Networks: Information diffusion and marketing are analyzed.
• Resource Allocation: In shared spaces, this ensures fair, efficient allocation.
• Political Science: Voting system design and campaign examination are included.

Summary

• By uniting game theory and computer science, AGT equips systems for strategic challenges.
• It finds application across diverse sectors.

Basic Definitions in Linear Algebra

• Linear algebra covers vector spaces and linear mappings between those spaces.

Vector Space

• A vector space includes a set E combined with vector addition and scalar multiplication operations.
• These operations must adhere to a set of axioms for E to qualify as a vector space.
• Vector addition combines two vectors to yield another: (u, v) -> u + v
• Scalar multiplication multiplies a vector by a scalar: (λ, u) -> λu
• Scalars originate from a field like real numbers or complex numbers.

Vector Space Axioms

• (E, +) is an Abelian group, exhibiting specific properties:

  • Associativity: (u + v) + w = u + (v + w)
  • Neutral element existence: a zero vector such that u + 0 = u for all u in E.
  • Inverse existence: ensures each vector u has an inverse -u, meeting u + (-u) = 0.
  • Commutativity: u + v = v + u

• Compatibility of scalar multiplication: λ(μu) = (λμ)u

• Scalar multiplication over vector addition: λ(u + v) = λu + λv

• Scalar multiplication over scalar addition: (λ + μ)u = λu + μu

• Scalar identity: 1u = u

Vector Subspace

• A subset F within a vector space E constitutes a subspace if it satisfies certain conditions:
  • Has to contain at least one element.
  • Vectors u, v in F must ensure that u + v remains in F.
  • Contains scalar multiples: λu stays within F for any u in F and scalar λ.
• This is checked through 0 is in F and λu + v belongs to F for any λ, u, v.

Linear Combination

• A linear combination defines a sum of vectors, each scaled by a scalar. $λ_1u_1 + ... + λ_nu_n$

Span or Vector Space Generated

• The vector space generated encompasses all possible linear combinations from its vector set.
• It is denoted Vect(u_1,..., u_n); and a vector subspace of E

Linearly Independent, Generating, and Bases

• A subset is linearly independent provided no vector is a combination of the others.
• A generating set means that all vectors in E can be written through those vectors linear combinations.
• Vectors qualify as a base if they're both linearly independent and span the space.

Dimension Defined

• All bases within a vector space share the same count of vectors.
• This count is termed space dimension, shown via “dim(E)”.

Linear transformation Defined

• Linear transformations uphold vector addition and scalar multiplication.
• $f(u + v) = f(u) + f(v)$ and $f(λu) = λf(u)$

Linear transformation Kernel and Image

• Kernel or null space gathers vectors converted to 0 following $f(u) = 0$.
• Image gathers vectors mapped to F for any vector from E
• The core is a subspace from E, while the image is a subspace within F.

Rank Explained

• It tells how many dimensions the transformation actually covers.
• Defined $rg(f) = dim(Im(f))$.

Linear Transformation Key Theorem

• Describes how a domain space splits into transformed parts by a mapping.
• Defined: $dim(E) = dim(Ker(f)) + rg(f)$

Isomorphisms defined

• Preserving structures, it bijectively connects two spaces.

Defining Matrices

• Matrices structure numbers, arranging cells into forms with rows and columns.
• Denoted through size, and $m \times n$ shows with rows and columns.

Properties of Matrix Operations

• Summing up matrices requires equal dimensions to combine cell values directly.
• Scalar multiplication scales each matrix element.
• Multiplying matrices requires column and row equality, where element $ij$ integrates values based on defined summation.

Matrix Inverse Defined

• When multiplying results uniquely within matrix identity: $AB = BA = I$

Determinant Defined

• A unique scalar per square matrix indicates invertibility, zero indicates non-invertible. $\newline$

Values and vectors defined

• Vectors stay aligned when undergoing transformation with a factor: $Av = λv$.

Scalar Dot Product Properties

• Multiplication must display particular relational traits.
• $\langle u, v \rangle = \overline{\langle v, u \rangle}$
• Scalar multiplication is $ \langleλu+μv,w> =λ\langle u,w\rangle + μ\langle v,w\rangle$
• Positive definite: $\langle u, u \rangle \geq 0$ and $\langle u, u \rangle = 0 \Leftrightarrow u = 0$.

Euclidean Space Defined

• Spaces include dot product abilities onto their vector make up

Orthogonal Defined.

• Relationship where their dot yield null result: $\langle u, v \rangle = 0$

Orthonormal Defined

• Basis exhibits unit scaling that are mutually perpendicular:$\langle e_i, e_j \rangle = \delta_{ij}

Gram-Schmidt Process

• Transforms to make simpler dot production by creating orthoganility.

Physics Basics

• Physics mathematically presents the natural or objective world.
• Defined physical measures are key to physics, are divided as vector and scalars

Vector properties

• Vectors need direction including magnitude.

Scalar properties

• Describes magnitude through physical measure $$

Scalar to Vector Types

Working with Two Dimensions Cartesians

• Vector $A$ describes parts on $x$ and $y$, with unit measure along $x = i$, and $measure along y=j$ represented as $\newline$
• Descirbed vector with formula: defined vector component $A = Ai + Aj$. Magnitude $vector|A| = root(Ax^{2} + Ay^{2})$, and degree $vector = tan^{-1} * Measure(A_y/ A_x)$

Described travel total or path

• Lets assume a travel scenario where a driver took a path that led twenty meters East, and after measure 10m going North, we calculate the path measure for Total movement $a = 20i+10j$
• Absolute travel measure: $ Absolute(a) path= root(20^{2}+10^{2})= 22.4m$

The travelled angle after path

• Angle $= tan * -1 * Measure(10 / 20) = to 27$ degrees, $vector = tan^{-1} * Measure(A_y/ A_x)$ $\newline$

Working with Two Dimensions Polars

• Using both absolute and degree measurements $vector A= (Measure A Angle)$ $\newline$

Calculating cartesian conversions

• Convert to cartesian, vector $A$ = measure $A, base cos vector" $\newline$
• Convert absolute, use trig absolute vector$=measure A base sin vector" $\newline$

Use trig to find the path absolute

• The bird and path: lets find bird velocity that flies 10 meters at 30 degrees
• Path Velocity is base vector $A.Cos = 8.Cos$
• Degree path is base $a sin theta is approximately equal to base 5 ms$ $\newline$
• So the measure of vector A is absolute .91 1+8.66j, when putting path together

Bernoulli's Principal

• A principle stating fluid speed increases with simultaneous pressure reduction or the decrease in fluid’s potential energy
• Applies for isentropic flows only. Effects of irreversible processes (turbulence) and non adiabatic processes (heat radiation) are small. $\newline$
• A constant streamline is to state the sum of all forms of energy in a fluid are constant.
• The principle is from the conservation of energy.

Incompressible flow equation

• Refers to a state where most densities of fluid can be constant.
• It has been performed with liquids.
• Bernoulli's equation in original form is valid only for incompressible flow.

Properties of Incompressible Equation

• $V^{2}/2 + gz + p/p = constant$

Where:

• v = fluid flow at a point
• g = gravity
• z = point elevation
• p = chosen pressure
• p = fluid density across all of the point locations on the fluid

Assumptions include.

• Consistent speed flow is steady
• Flow is constant in density
• No viscosity
• Applicable properties follow consistent properties following a set path
• friction is lost