Algorithmic Game Theory

Algorithmic Game Theory

• Focuses on mathematical models of strategic interactions among rational agents within social sciences, logic, and computer science.
• Originally addressed zero-sum games, but now covers logical decision-making in humans, animals, and computers.

Non-Cooperative Game

• Players make decisions independently.
• Defined by: a set of players, strategies for each player, and a payoff function for each player based on strategies.
• The Nash equilibrium is the fundamental solution.
• Nash equilibrium is a set of strategies where no player benefits from changing their strategy if others stick to theirs.

Prisoner's Dilemma (Example)

• Two suspects are arrested and cannot communicate.
• Confess/Confess: Each gets 5 years. Confess/Silent: Confessor is released, the other gets 10 years Silent/Silent: Each gets 1 year.
• Nash equilibrium: Both confess, despite it not being the best outcome for either.

Cooperative Game

• Players can make binding agreements.
• Defined by: a set of players and a characteristic function assigning a value to each player subset/coalition.
• The core is the set of stable payoff allocations, where no coalition can gain by forming on its own

Matching Game (Example)

• Players are divided into two preference-ordered groups.
• The goal is a stable matching where unmatched pairs don't both prefer each other over their current matches.

Algorithmic Game Theory

• Merges algorithm design and game theory to design algorithms for strategic settings.
• Addresses how to incentivize socially desirable behavior, compute Nash equilibria efficiently, and create algorithms robust to strategic actions.

Examples of Algorithmic Game Theory

• Sponsored search auctions need to incentivize truthful bidding and efficient ad allocation.
• Network routing protocols must withstand selfish behavior of participants.
• Social networks need manipulation resistance from bad actors.

Resource Allocation

• Assigns available resources to various uses in the context of algorithmic game theory.
• Is often examined through mechanism design, which aims to align player incentives with desired outcomes.

Examples of resource allocation:

• Fair division aims to fairly distribute resources among people
• Public goods provision aims to provide public goods when people can free-ride.
• Auctions are used to efficiently allocate resources and generate revenue.

Mechanism Design

• Is the design of game rules to achieve a desirable outcome with participants who have private information and act strategically.
• Designed to incentivize revelation of true preferences for efficient decision-making.

Key Concepts in Mechanism Design

• Social Choice Function: Specifies the desired outcome for every possible set of preferences or information.
• Mechanism: Rules defining participant actions and linking them to outcomes.
• Incentive Compatibility: Ensuring participants benefit by revealing their true information.
• Revelation Principle: Any implementable outcome can also be achieved with truthful information revelation.
• Vickrey-Clarke-Groves (VCG) Mechanism: Achieves resource allocation by charging participants for the impact they have on others.

Mechanism Design Goals

• Efficiency in resource allocation to those who value them most.
• Individual Rationality exists when participation is beneficial.
• Budget Balance: assures payments cover resource costs
• Truthfulness/Incentive Compatibility occurs incentivized accurate information sharing

Applications of Mechanism Design

• Auctions aim to efficiently allocate goods and generate revenue for the seller.
• Voting systems accurately need to reflect voter preferences
• Resource allocation should happen efficiently with private information on valuations/costs.
• Matching Markets should pair individuals/resources for mutual benefit.

Congestion Games

• Model non-cooperative situations where a player's payoff depends on the number of players using the same resource.
• The more players using a resource, the lower the player payoff.

Definition of Congestion Games

• An example of a congestion game is a tuple $(N, R, (c_r)_{r \in R})$:
• $N = {1, 2,..., n}$ is the set of players.
• $R$ is the set of resources.
• $S_i \subseteq 2^R$ is player’s $i$ strategy set from $N$, with each strategy being a resource subset.
• $c_r : {1, 2,..., n} \rightarrow \mathbb{R}$ cost function for using resource $r$ based on player count.

Network Routing Example

• Two paths exist from source to destination.
• Each path is a resource with a cost based on congestion
• Each player wants to minimize their cost.
• Players: Two
• Resources: Path 1 and Path 2.
• Strategies: Player 1/2 can choose {Path 1} or {Path 2}.
• Cost Functions: 𝑐_1(𝑥) = 𝑥, 𝑐_2(𝑥) = 3.

Nash Equilibrium in Congestion Games

• At Nash equilibrium each player's strategy is the best response to other player's strategies, meaning no player can improve their outcome with a change in their strategy.
• For the example above with the two players and the 2 paths, the Nash equilibrium has both players on Path 1.
• The cost being 2 for each player, they wont switch to Path 2 because that would have a cost of 3.

Price of Anarchy (PoA)

• Measures the inefficiency of game, equal to ratio of equilibrium social cost to optimal social cost.
• Can be high in congestion games, particularly with non-linear cost functions.

Social Cost

• Is calculated using the sum of players costs

Price Of Anarchy example

• Social cost of Nash equilibrium in the previous example: 𝑐_1(2) + 𝑐_1(2) = 2 + 2 = 4.
• Optimal social cost with one on Path 1, one on Path 2: 𝑐_1(1) + 𝑐_2(1) = 1 + 3 = 4.
• PoA = 4/4 = 1, indicating the Nash equilibrium is efficient in this specific case.
• This is because the Nash Equilibrium is the same outcome as the best outcome in this example.

Conclusion on Congestion Games

• Congestion games model situations where a player's payoff depends on the resource usage by others.
• Price of Anarchy is important for designing robust systems that accounts for selfish players.

Matplotlib Simple Plot Code

• Creates a plot with linear, quadratic, and cubic functions.
• Labels the x-axis, y-axis, and title.
• Includes a legend for clarity

Matplotlib Image Demo Code

• Loads and displays an image, turns off axis labels.
• The example shows how to load an image from a local device.

Matplotlib Histogram Code

• Generates a histogram of a normally distributed dataset.
• Sets the number of bins and appearance
• Labels axes clearly.

Modelo Atómico de Bohr (Bohr Atomic Model)

Postulados (Postulates)

• Electrons orbit the nucleus in allowed orbits without radiating.
• Angular momentum (L) is a multiple of (h/2\pi = \hbar):(L = mvr = n\hbar, \quad n = 1, 2, 3,...).
• Energy is emitted or absorbed only when an electron jumps orbits, with frequency determined by (E = h\nu)

Limitaciones (Limitations)

• Only compatible with Hydrogen atoms, inaccurate with polyelectronic atoms and fails to explain chemical bonding.

Modelo Atómico Actual (Current Atomic Model)

Limitaciones del modelo de Bohr (Limitations of Bohr's Model):

• Only Explains Hydrogen spector not Polyelectronic spectors. Fails to explain chemical bonding

Principio de Incertidumbre de Heisenberg (Heisenberg Uncertainty Principle):

• The exact position and velocity are impossible to know simultaneously. The equation is (\Delta x \Delta p \geq \frac{h}{4\pi}).

Ecuación de Schrödinger (Schrödinger Equation):

• The equation of (\hat{H}\Psi = E\Psi) describes subatomic particles, where (\hat{H}) is the Hamiltonian operator, (\Psi) is the wave function, and (E) is energy

Significado físico de (\Psi) (Physical Significance of (\Psi)):

• • While it lacks physical meaning alone, (|\Psi|^2) shows the probability density of finding an electron. Orbitals are regions of high probability.
• • The orbital is where electrons are likeliest to be

Números cuánticos (Quantum Numbers):

• Quantum numbers define the features atomic orbitals.
• • The principal quantum number (n) indicates energy level where (n = 1, 2, 3,...).
• The azimuthal quantum number (l) defines orbital shape where (l = 0, 1, 2,..., n-1) with (l = 0 \rightarrow) orbital, (l = 1 \rightarrow) orbital p, and (l = 2 \rightarrow) orbital d.
• The magnetic quantum number (m_l) indicates orbital spatial orientation and (m_l = -l, -l+1,..., 0,..., l-1, l).
• The spin quantum number (m_s) describes electron Intristic angular momentum whre (m_s = +1/2) for espín arriba and the other being Espin abajo.

Ejemplos (Examples):

• (n = 1, l = 0, m_l = 0): orbital 1s
• (n = 2, l = 0, m_l = 0): orbital 2s
• (n = 2, l = 1, m_l = -1, 0, 1): orbitales 2p

Reglas (Rules):

• Pauli Exclusion Principle states, each orbital holds two electrons max. Hund's Rule says electrons fill same-energy orbitals to maximize parallel spins

Configuración electrónica (Electronic Configuration):

• It shows Electron distribution across atomic orbitals.
• Example: Oxygen (Z = 8) is configured as (1s^2 2s^2 2p^4)
• Chart describes the electron configuration.

Tabla periódica (Periodic Table):

Configuración electrónica y la tabla periódica (Electronic Configuration and the Periodic Table):

• Elements are arranged into blocks by electron completion
• The s-block consists groups 1 & 2, p-block from 13 to 18, d-block groups from 3 to 12, blocks with lanthanides and actinides.
• Periods represent which principal energy level is being filled; groups denote chemical similarities from having same valence configurations.

Propiedades periódicas (Periodic Properties):

Radio atómico (Atomic Radius):

• It decreases moving across periods, but increases going down groups.

Energía de ionización (Ionization Energy):

• Increases across periods and decreases down groups.

Afinidad electrónica (Electron Affinity):

• Rising across and falling down table in trends.

Electronegatividad (Electronegativity):

• Electronegativity rises period to period and then decreases over any period when charted.

Quantum Mechanics (Physics 441/541)

• Instructor: Sergey Savrasov, Office: 237 Physics, Tel: 243-4224, Email: [email protected], Office hours: Tue, Thu 11 am - 12 am
• Teaching Assistant: Sean Barrett, Email: [email protected], Office hours: Mon 1-3 pm, 9 Physics

Course Description:

Course serves as an intro to quantum mechanics

Textbooks:

• D.J. Griffiths, Introduction to Quantum Mechanics, 2nd edition
• Lecture notes posted on Canvas

Grading:

• Homework 50%
• Midterm 20%
• Final Exam 30%

Homework:

• Assigned weekly, due Fridays in class.
• Late submissions not accepted; posted via Canvas.

Exams:

• Midterm: Thursday, February 8, in class.
• Final: Wednesday, March 21, 8-10 am, 194 Chem.

Course website:

The course website is located on Canvas. Here you will find the lecture notes, homework assignments, and other important information about the course.

Academic Honesty:

Everyone must to submit their on solutions!

Disability Resources:

Ensure to contact disability resources department so appropriate arrangements can be made

Química Orgánica (Organic Chemistry)

Nomenclatura de Alcanos (Alkane Nomenclature):

Reglas de la IUPAC (IUPAC Rules):

• It is important, to identify and name that longest chain alkanes. Ensure to Number where subsituents are as low as you can make them
• List names alphebetically and write out the complete compounds.

Alcanos Comunes(Common Alkanes):

Refer to the table for various Molecular Formula, the Name, the Structure and Abbreviation

Grupos Comunes(common isomer group)

Refer to the table for various Isomer Group, Formula, Isomer, and Abbreviation.

Ejemplos (Examples):

Here are some examples:

• 3-Metilpentano
• 2,3-Dimetilbutano
• 2-Cloro-4-Metilhexano
• 4-Etil-2-Metilhexano
• 5-Isobutil-2-Metilnonano

Nomenclatura de Alquenos y Alquinos(Aliphatic and Alka Nomenclature)

Reglas de la IUPAC (IUPAC Rules):

• It is important, to identify and name that longest chain containing double/triple bond. Identify positions of the double/triple bonds with numbers List names alphebetically and write out the complete compounds. If you see both double and triple, the lowest numered the priority rule

Ejemplos (Examples)):

Here are some examples:

• 2-Buteno
• 1-Buteno
• 2-Penteno
• 2-Metil-2-Buteno
• 3-Metil-1-Penteno
• 1-Butino
• 2-Butino
• 3-Metil-1-Butino

Isómeros Cis y Trans (Cis and Trans Isomers):

• • Cis: Same-side substituents on double bond. Trans: Opposite-side substituents.

Ejemplos(examples:)

Here are some examples:

• Cis-2-Buteno
• Trans-2-Buteno

Nomenclatura de Alcoholes(Alcohol nomenclature)

Reglas de la IUPAC (IUPAC Rules):

• It is important, to identify and name that longest chain containing group (-OH). Make sure the lowest numbered are the group ( -OH). And write a complete list alphabetically and write out to the compound with "-ol suffix".

Ejemplos (Examples):

Here are some examples: Metanol Etanol

• 1-Propanol -2-Propanol -2-Metil-2-Propanol -2-Butanol -1,2-Etanodiol (glicol)

Nomenclatura de Éteres(Ether nomenclature)

reglas de la IUPAC( Iupac Rules):

Make sure to find that Longest chain and be the first priority. Then make sure Alkoxy groups R-O, are as substituents..

Ejemplos(Examples):

Here are some examples:

• • Metoximetano
• • Etoxietano
• Metoxietano 2-Metoxipropano

Nomenclatura Común(common Nomenclatura)

Here are some examples:

• Dietil Éter
• Éter Metil Etílico

Nomenclatura de Aldehídos y Cetonas(Alde Nomenclature and Ketones)

Reglas de la IUPAC (IUPAC Rules):

• It is important, to identify and name that longest chain. Chain must contained the Carbonyl GroupC=O Numbering is key to the c=o Then "-al suffix" should be added .

Ejemplos(examples):

Here are some examples:

• • Metanal (formaldehído)
• • Etanal (acetaldehído)
• • Propanal
• Propanona (acetona)
• • 2-Butanona
• Butanal

Nomenclatura de Ácidos Carboxílicos(carboxylic Acid Nomenclature):

Reglas de la IUPAC (IUPAC Rules):

• Carbon atom is always the main priority which should be as group #1 Suffix is to be called "oico prefix"

AcidosCarboxilicosCommunes(Commune carboxylic Acid):

Refer to the table to various types of acids and names that go with them

Exemplos(Examples):

Here are some examples:

• • Ácido Etanoico
• • Ácido 2-Cloropropanoico
• Ácido 3-Metilbutanoico

Nomenclatura de Esteres(Ester Nomenclature):

reglas de la IUPAC( Iupac Rules):

You have to name Alkyl(R) for the Alchole group, And Carboxylic acid and suffix it with "-ato"

Ejemplos (Examples):

Here are some examples:

• Etanoato de Metilo -Etanoato de Etilo -Metanoato de Etilo

Nomenclature de Aminas(Amide Nomenclature):

Reglas de la IUPAC (IUPAC Rules)

The Nitrogens atom chain as you want.

Ejemplos (Examples):

Here are some examples:

• Metilamina -Etilamina -Propilamina Dimetilamina -Trimetilamina -Etilmetilamina

Nomenclature de Amidas(Amine Nomenclature):

reglas de la IUPAC( Iupac Rules):

Chain containing -CONH2 Amide chain is in first priority Number of substituents "amida prefix is

Ejemplos(examples)):

Here are some examples: -- Metanamia

Nomenclatura de Haluros de AlquiloHalurosAiquilo Nomenclature:

reglas de la IUPAC( Iupac Rules):

You want to find longest chain, number chain lowest number chain fluororo, clrodo, yodo

Matrices

Definition:

A matrix is a structured array consisting of symbols. Elements compose said matrix whether it is by row or column We define size of Matrix with two rows an three Columns and name Matrix

SpecialMatrix: There are different kind of Matrices, wheater Square Matrix, column Matrix, vector matrix, zero matrix or Identity matrix Each serves its purpose and they all require different ways to solve them Transposing: Switch the rows and the Columns

Matriz Arthmitic

Arthromitic requires elements that are of both A+B Scalar multi is defined as multiplyting "CA" Transpose switch columns and rows Chemical vs Chemistry Chemical vs chemistry requires the comparison based between each other, for example Chemistry covers Compositio,Structure, and Reactions of matter as Chemists Chemical Engineer covers things on Designing and Controlling and optimoizing how to create chemial process Both must have different sets of tools used and its important to know Thermodynamics is the study and application of the Thermodynamics studies and appliees relations to Properties to matter Heat Transfers has to do study with thermal transfer from one location to another

Fluid Mechanics

A force being applied and the resistance of the flow, and the area of the surface There are different types of Newtonian and Non Newtonian fluids with different properties Important also Is Flow Rqte, mass flow rate, Bernouilli's Equation, Reynolds Number

Resumen Ejecutivo (Executive Summary)

• Introducción (Introduction) This highlights everything being about the company.
• Oportunidad de Mercado (Market Opportunity) The company's markets is well being good
• Proposition de Valor (value proposition) The company's value is its innovation, and that means technology and solutions.
• modelo de Negocios(Business model is about what it does and what revenue will comes)
• Proyecciones Financiera(is about the how the company is going to have revenue.)

Explores mathematical models of strategic interactions. Covers non-cooperative games with independent player decisions and Nash equilibrium solutions. Includes examples like the Prisoner's Dilemma demonstrating strategic decision-making.