Algorithmic Game Theory: Selfish Routing

Algorithmic Game Theory

• Combines computer science, game theory, and economics.
• Focuses on designing and analyzing algorithms for strategic interactions and understanding strategic agent behavior in networked systems.

Selfish Routing

• Model involves a network of $n$ nodes and $m$ edges.
• Each edge $e$ has a cost function $\mathcal{l}_e(x)$, representing the cost (or delay) incurred by each traffic unit, with $\mathcal{l}_e(x)$ being non-negative, non-decreasing, and continuous.
• $k$ classes of users inject $r_i$ units of traffic from $s_i$ to $t_i$.
• A feasible flow $f$ ensures the traffic of each class is routed, with $\sum_{P:s_i \rightarrow t_i} f_p = r_i$ for every $i$.
• The total traffic on edge $e$ is defined as $f_e = \sum_{P:e \in P} f_p$.
• The cost incurred by traffic on edge $e$ is $f_e \cdot \mathcal{l}_e(f_e)$, with the total cost being $C(f) = \sum_e f_e \cdot \mathcal{l}_e(f_e)$.
• In a Nash equilibrium flow, no user can unilaterally switch paths to lower their cost, satisfying $\sum_{e \in P} \mathcal{l}_e(f_e) \le \sum_{e \in P'} \mathcal{l}_e(f_e)$ for every $s_i-t_i$ path $P'$ given $f_p > 0$.

Braess's Paradox

• Adding an edge to a network can increase the total cost at Nash equilibrium.

Price of Anarchy (PoA)

• Measures the inefficiency of Nash equilibrium.
• Defined as the ratio between the total cost at Nash equilibrium and the optimal total cost: $PoA = \frac{C(f)}{C(f^*)}$.
• $f$ is a Nash equilibrium flow and $f^*$ is an optimal flow.
• Price of Stability (PoS) is the ratio of the best Nash equilibrium cost to the optimal cost.

Mechanism Design

• A subfield of game theory designing games/mechanisms to achieve desired outcomes despite strategic player behavior.
• Also known as reverse game theory, starting with the intended outcome.

Elements of a Mechanism

• Consists of a set of agents, a type space (private information), an outcome space, and mechanism rules.
• Mechanism rules involve allocation rules (outcome determination) and payment rules (agent payments).

Single-Item Auction

• Agents are bidders with private item valuations (type space).
• Bidders win the item and pay a price (outcome space).
• Allocation rule dictates that the highest bidder secures the item.
• Payment methods include first-price (winner pays their bid) and second-price auctions (winner pays the second-highest bid).

Desirable Properties of Mechanisms

• Individual rationality where participation is beneficial to each agent.
• Truthfulness (incentive compatibility), so agents maximize their utility by revealing their true type.
• The goal is to maximize social welfare.
• Mechanisms may strive for budget balance.
• Computational efficiency is ideal.

Revelation Principle

• Any mechanism can be transformed into a direct mechanism where agents truthfully report their types.
• A direct mechanism is truthful if it's in each agent's best interest to report their actual type, regardless of others.

VCG Mechanism

• Guarantees truthfulness and social welfare maximization.
• Uses the principle that each agent pays the negative externality they impose on other agents (named after Vickrey, Clarke, and Groves).

Limitations of VCG

• Not always budget-balanced and can face computational challenges in certain contexts.

Coalitional Game Theory

• Studies of the behavior of coalitions of players.
• Focuses on how groups of players can cooperate to achieve a common goal and how benefits of cooperation should be divided among the players.

Key Concepts

• A coalition is a subset of players agreeing to cooperate.
• A characteristic function assigns a value to each coalition, representing its achievable total payoff.
• A solution concept specifies how payoffs should be distributed among players.

Cooperative vs. Non-Cooperative Game Theory

• Non-cooperative assumes independent, selfish players.
• Cooperative assumes players can form coalitions and make binding agreements.

Examples of Coalitional Games

• Voting games where players are voters and coalitions are majority groups.
• Cost-sharing games where players share costs.
• Matching games involving stable pairings.

Solution Concepts

• The Core represents stable payoff distributions with no coalition incentives to deviate.
• The Shapley Value offers a fair, efficient, and symmetric payoff distribution.
• The Nucleolus minimizes the maximum dissatisfaction of any coalition.

Applications of Coalitional Game Theory

• Applied in economics, political science, and computer science.

Social Choice Theory

• Studies collective decision-making and how individual preferences create a collective choice.
• Concerned with the properties of social choice rules and design.

Key Concepts

• Involves voters, alternatives, preference profiles, and social choice rules.

Examples of Social Choice Rules

• Majority rule, plurality rule, Borda count and Condorcet rule.

Desirable Properties of Social Choice Rules

• Pareto efficiency
• Condorcet winner criterion; independence of irrelevant alternatives (IIA).
• Non-dictatorship.

Arrow's Impossibility Theorem

• It is impossible to design a social choice rule that satisfies all of the desirable properties simultaneously.
• It highlights the fundamental difficulties of collective decision-making.

Gibbard-Satterthwaite Theorem

• Any social choice rule that is onto and non-dictatorial is manipulable.
• Manipulable describes that voters can obtain a better outcome by misrepresenting their preferences.

Applications of Social Choice Theory

• Important in designing voting systems and analyzing electoral outcomes.
• Helpful in the allocation of resources, and regulating markets.
• In computer science, used in designing recommendation systems.

Matrices

• A matrix is a rectangular array of real numbers, denoted by uppercase letters.
• Numbers inside the matrix are elements denoted by lowercase letters with subscripts for row and column. $$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

Types of Matrices

• A square matrix has the same number of rows and columns ($m = n$).
• A diagonal matrix is a square matrix with all elements outside of the main diagonal being zero.
• An identity matrix is a diagonal matrix with all elements on the main diagonal equal to one, denoted by $I$.

Operations with Matrices

• Adding involves summing corresponding elements of two matrices, $A$ and $B$, of the same size.
• Scalar multiplication involves multiplying each element of matrix $A$ by a scalar $c$.
• Matrix multiplication is defined as the product of the of two matrices $A_{m \times n}$ and $B_{n \times p}$ resulting in a matrix $C_{m \times p}$.

Linear Equation Systems

• A set of linear equations with the same variables, representable in matrix form as $Ax = b$.
• $A$ is the coefficient matrix.
• $x$ is the variable vector.
• $b$ is the independent terms vector.

Solving Methods

• The solution of small systems employ substitution and elimination.
• The Gauss-Jordan method systematically transforms the augmented matrix $[A | b]$.
• Cramer's rule can finds the solution using determinants, provided the coefficient matrix is invertible.

Vector Spaces

• Vector spaces have set of objects or vectors.
• Two operations: vector addition and scalar multiplication are defined.

Subspaces

• A subset of a vector space also obeying vector space rules, under the same two operations.

Base and Dimension

• A basis of a vector space is a set of linearly independent vectors that span the entire space.
• The dimension is the number of vectors in a basis.

Linear Transformations

• Functions between vector spaces that preserve vector addition and scalar multiplication.

Matrix Representation

• Every linear transformation can be represented by a matrix like $[T]_\alpha^\beta$.

Kernel and Range

• The kernel of a transformation is the set of vectors that map to the zero vector.
• Range is the result of applying the transformation.

Eigenvalues and Eigenvectors

• Eigenvectors of a matrix $A$ ($v$) have the property $Av = \lambda v$.
• Where $\lambda$ are scalars called eigenvalues.

Calculating Eigenvalues

• Values are by solving the characteristic equation $\det(A - \lambda I) = 0$.

Diagonalization

• A matrix $A$ is diagonalizable if an invertible matrix $P$.
• The matrix $P^{-1}AP$ results in a diagonal matrix.
• The eigenvectors of $A$ the diagonal.

Finite Element Analysis Quick Start

• Import or create geometry
• Create a mesh
• Create a simulation
• Assign materials and conditions
• Run the simulation and post-process the results

Executive Summary

• Financial results of ALFA, S.A. for 2022.
• Methodology involved audited statements, and key ratios.

Financial Status

• Adequate liquidity
• Moderate debt
• Good solvency

Operational Performance

• Good profitability (NM:12%; ROE: 15%)
• Good asset efficiency

Cash Flow

• Good generating capacity, major investments.

Conclusions and Recommendations

• Good financial situation, maintain management quality.

The Ideal Op Amp Characteristics

• $R_{in} = \infty$
• $R_{out} = 0$
• $A = \infty$
• Golden rules:
  • Inverting and non-inverting voltage amounts match
  • No current flows.

Example: Non-Inverting Amplifier

• $V_{+} = V_{in}$
• $V_{+} = V_{-} = V_{in}$
• $V_{-} = V_{out} \frac{R_1}{R_1 + R_2}$
• $V_{in} = V_{out} \frac{R_1}{R_1 + R_2}$
• $A = \frac{V_{out}}{V_{in}} = \frac{R_1 + R_2}{R_1} = 1 + \frac{R_2}{R_1}$

Example: Inverting Amplifier

• $V_{+} = 0$
• $V_{+} = V_{-} = 0$
• $\frac{V_{in}}{R_1} + \frac{V_{out}}{R_2} = 0$
• $A = \frac{V_{out}}{V_{in}} = - \frac{R_2}{R_1}$

Vocabulary of Probabilities

• An experiment aléatoire is an experiment with results we know, but unpredictable, like rolling a die.
• L'univers of an experiment is the set of results, like $\Omega = {1, 2, 3, 4, 5, 6}$.
• Un événement is a subset of the univers, such as rolling even numbers ie. A = {2,4,6}.
• La probabilité d'un événement (P(A)) measures the likelihood between 0 and 1.

Vocabulary of Statistics

• Une population is the set of individuals studied, like students in a school.
• Un échantillon is a subset of the population.
• Un caractère is an object of study.
• Variable qualitative: When the object measured is immeasurable.
  • example is a colour.
• Variable quantitative: When the object measured is measurable.
  • Variable discrète: finite numbers. - example is a number of children or shoe size.
  • Variable continue: full range on an interval. - is size and weight.
• L'effectif counts the occurrences .
• La fréquence counts the occurrences, in percentage
• Moyenne: The average of values.
• Médiane: A point separating half of each value -Etendue: The largest value minus the smallest value.

Introduction to Static Electricity

• Positive and negative charges exist that attract and repel each other.
• Electric Charge is conserved and quantized.

Conductors and Insulators

• Conductors allow free electron movement (e.g., copper, aluminum, silver).
• Insulators restrict electron movement (e.g., glass, rubber, wood).

Charging Objects

• Friction: Electrons transferred from rubbing
• Conduction: Electrons transferred from contact
• Induction: Charge redistribution without contact

Coulomb's Law

• Quantifies the electric force,
• $F = k \frac{|q_1 q_2|}{r^2}$.

Electric Field

• Electric field measures per unit charge.

Electric Potential Energy

• Electric potential energy is electrical force per charge unit.

Capacitance

• System stores charge via the the equation is expressed as with $C = \frac{Q}{V}$.

Intro to ANOVA

• ANOVA compares 2 or more population means. It is parametric, (data is normally distributed, variance is equal).
• ANOVA analyzed data from exp or obv studies.

Key Concepts

• A Factor is a variable that is used to group data. The independent variable.
• A factor level is a value of a factor.
• A treatment are combinations of factor levels.
• Response variable is continuous and measured for each subject.
• Response var is the dependent variable.

Hypotheses

• $H_0$ is when pop means are equal. $\mu_1 = \mu_2 = \dots = \mu_k$
• $H_0$ is satisfied. $H_1$ At least one , $\mu_i \neq \mu_j$

Assumptions

• Check the 1 Normality of data , 2 Homogeneity of Variance and 3 the Independence before proceeding with test.

One-Way ANOVA

• Tests only 1 factor
• Compares means

Partition of Variance

• Data breaks down to many forms
• $SST$ , or Total Sum of Squares =
• $SSTR$ or Sum of Squares Treatment
• $SSE$, or Sum of Squares Error
• ANOVA Identity: $SST = SSTR + SSE$.

Degrees of Freedom

• $DFT = N - 1$ (Total DOF)
• $DFTR = k - 1$ (Treatment DOF)
• $DFE = N - k$ (Error DF)

Mean Squares

• $MSTR = \frac{SSTR}{DFTR}$ (Treatment MS)
• $MSE = \frac{SSE}{DFE}$ (Error MS)

F-Statistic

• Statistic tests, based on equations from the mean squares.
• A large F-statistic signals something is wrong

ANOVA Table

• Has various sums, and the associated values with calculations to ensure that anova hypotheses stand

Decision Rule

• The hypothesis is invalid if P value is higher than alpha (fail to reject hypothesis), and valid vice versa.

Multiple Comparisons

• Null hypotheses are deemed false based on: - Bonferroni - Tukey's HSD (Honestly Significant Difference) - Fisher's LSD (Least Significant Difference)

Two Way AVONA

• Two factor presence.
• 2 hypotheses + 1 interaction hypothesis possible

Fraction Comparison Definitions

• Fraction: A whole is made up of fraction. Two key points: Numerator and Denominator.

Fraction Comparison Rules

To compare fractions, we compare directly or indirectly

1. Fractions with the same denominator
   • If equal denominators exist, largest numerator wins the comparison
2. Fractions with the same Numerator - If equal numerators exist, smallest denominator wins the comparison
3. Fractions Without equal numerator/denominators
   • If unequal numerator/denominators exist, convert to make numerators/denominators match
   • compare the fractions post conversion
4. Comparing to 1
   • A fraction is equal to 1, less than 1 or great than 1 based on numer/dem.

Summary Table