Algorithmic Game Theory: Braess's Paradox

Algorithmic Game Theory

• Focuses on using algorithm design to improve game-theoretic outcomes
• Also uses game-theoretic solution concepts to algorithm design

Selfish Routing

• A network of roads is used with a latency function for each edge dependent on traffic fraction
• Many players control an infinitesimal amount of traffic
• Each player wants to travel from $s$ to $t$
• Players choose the shortest path based on current traffic
• Nash Equilibrium (NE) is a traffic pattern where no player benefits from changing paths

Braess's Paradox

• Adding a road can increase latency for all players in NE
• Initial network includes two paths from $s$ to $t$: $s \rightarrow v \rightarrow t$ and $s \rightarrow w \rightarrow t$
• Latency of each path depends on traffic $x$: $l_{sv}(x) = x$, $l_{vt}(x) = 1$, $l_{sw}(x) = 1$, $l_{wt}(x) = x$
• Total Latency is $1 + \frac{x}{2}$
• Augmented Network includes a new road $v \rightarrow w$ with latency $l_{vw}(x) = 0$
• Total latency becomes $2x$

Price of Anarchy

• Calculated as the cost of worst Nash Equilibrium / Optimal Cost
• It measures how inefficient a Nash Equilibrium is

Topics in AGT

• Mechanism Design
• Auctions
• Fair Division
• Social Choice
• Learning in Games
• Complexity of Equilibria
• Network Games
• Congestion Games

Algorithmes gloutons

• A method for solving optimization problems
• Involves making locally optimal choices at each step

Principle:

• First, identify the optimization problem and define an objective function
• Secondly, choose an initial solution
• While the solution not complete: select the most promising element according to some criterion and then add this element to the solution
• Finally, return the solution

Advantages

• Simplicity of design and implementation
• Efficiency in terms of calculation time.

Inconvénients

• Does not always guarantee the optimal solution.
• Requires a proof of validity to ensure the optimality of the solution.

Exemples

• The set covering problem requires finding a minimal number of subsets whose union covers the initial set
• The fractional knapsack problem involves maximizing the value of objects included in a sac without exceeding capacity
• The problem of making change

Optimalité

• Greedy algorithms do not always guarantee the optimal solution.
• To prove the optimality of a greedy algorithm, we can use the following techniques: induction and argument d'échange

Conclusion

• Greedy algorithms are a simple and effective approach to solving certain optimization problems.
• However, it is important to verify whether the greedy algorithm provides an optimal solution for the given problem.

Statistical Inference

Definition

• Statistical inference is the process of drawing conclusions about a population based on sample data.
• It involves estimation and hypothesis testing.

Estimation

Estimator

• A function of the sample data used to estimate a population parameter.
• Point estimators and confidence intervals are two types of estimators.

Point Estimator

• A single number is calculated to estimate the population parameter
• Example: Sample mean $\bar{X}$ is a point estimator of the population mean $\mu$

Properties of Point Estimators

• An estimator is unbiased if its expected value equals the parameter it is trying to estimate.
• An estimator is efficient if it has a small variance.
• An estimator is consistent if its value approaches the true parameter as the sample size increases.

Confidence Interval

• A range of values calculated from the sample data that is likely to contain the true population parameter with a certain level of confidence.
• Factors Affecting Confidence Interval Width: sample Size, confidence Level, and population Variability

Hypothesis Testing

• A formal procedure for testing a claim about a population parameter using sample data.
• Steps in Hypothesis Testing: state the hypotheses, set the significance level, calculate the test statistic, determine the p-value, and determine to make a decision
• Significance Level: Usually denoted by $\alpha$
• Test statistics: The test statistics Example: $t = \frac{\bar{X} - \mu_0}{s/\sqrt{n}}$

Types of Errors

• Type I Error: Rejecting the null hypothesis when it is true. Probability of Type I error is $\alpha$.
• Type II Error: Failing to reject the null hypothesis when it is false. Probability of Type II error is $\beta$, and power of the test is $1 - \beta$.

Factors Affecting Power

• Sample Size: Larger sample sizes result in greater power.
• Significance Level: Higher significance levels result in greater power.
• Effect Size: Larger effect sizes result in greater power.
• Variability: Lower variability results in greater power.

Algorithmic Trading and Order Execution

Instructor

• Marcos Lopez de Prado
• Office Hours: Tuesday, Thursday, 1:00 - 2:00 PM

Teaching Assistants

• Gordon Lee ([email protected])
• Yuhan Zhang ([email protected])
• Office Hours: Monday, Wednesday, 1:00 - 2:00 PM

Course Description

• Detailed overview of the algorithmic trading industry
• Covers electronic exchanges, market microstructure, order placement, smart order routing, optimal execution etc

Course Prerequisites

• Familiarity with a programming language (preferably Python)
• Basic Calculus, Probability Theory, and Linear Algebra

Grading Policy

• Homework: 40%
• Midterm Exam: 30%
• Final Project: 30%