Algorithmic Game Theory

Algorithmic Game Theory

• A combination of game theory and algorithm design.
• Focuses on interactions between rational, self-interested agents.
• Important because strategic behavior is common on the internet and efficient algorithms are required for scale.

Example Scenarios

• Sponsored search auctions, where advertisers bid for keywords and the auction mechanism determines ad placement and pricing.
• Network routing, where users choose routes to minimize latency, impacting overall network performance.
• Online dating sites, where users may misrepresent preferences, requiring careful matching algorithm design.

Selfish Routing

• Involves $n$ players choosing paths in a network from source $s_i$ to target $t_i$ to minimize travel time.
• Edge latency depends on traffic.

Braess's Paradox

• Adding capacity to a network can increase latency.
• Demonstrates that improved local choices do not always lead to a globally better outcome.

Price of Anarchy

• Measures how selfish routing impacts network performance.
• Defined as the ratio of the cost of the worst Nash equilibrium to the cost of the optimal solution.
• $\text{PoA} = \frac{\text{Cost of worst Nash equilibrium}}{\text{Cost of optimal solution}}$

Change of Variable in Multiple Integrals

• Extending change of variable from 1D to 2D.
• Formula: $\iint_{D} f(x, y) dxdy = \iint_{D^*} f(x(u, v), y(u, v)) |\frac{\partial (x, y)}{\partial (u, v)}| dudv$
• $\frac{\partial (x, y)}{\partial (u, v)} = det \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}$ is the Jacobian determinant.

Example 1

• Compute $\iint_{D} xy dxdy$ where D is a region in the first quadrant bounded by $xy = 1$, $xy = 9$, $y = x$, $y = 4x$.

Example 2

• Compute $\iint_{D} e^{-x^2 - y^2} dxdy$ where D is the region bounded by the semi-circle $y = \sqrt{4 - x^2}$ and the $x$-axis.

Example 3

• Find the area of region $D$ enclosed by the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which is $\pi ab$.

Algorithmic Trading Overview

• Algorithmic trading involves executing orders based on pre-programmed instructions at a higher speed and efficiency.
• Quantitative strategies use mathematical and statistical techniques to identify and exploit trading opportunities.

Advantages of Algorithmic Trading

• Speed and Efficiency.
• Cost Reduction
• Reduced Emotional Influence.
• Backtesting and Optimization.
• Increased Order Execution Precision

Disadvantages of Algorithmic Trading

• Technical Expertise Required
• Model Overfitting Risks.
• Market Complexity
• Dependence on Data Quality
• Regulatory and Compliance Hurdles

Key Components of Algorithmic Trading Systems

• Data feeds and collection
• Strategy development
• Backtesting platform
• Risk management
• Order execution
• Related infastructure

Developing Quantitative Strategies

• Data Collection and Preparation
• Hypothesis Generation
• Model Development
• Backtesting and Validation
• Risk Management
• Implementation and Monitoring

Backtesting

• Evaluating a quantitative trading strategy by simulating its performance on historical data is backtesting.
• This is limited by overfitting, data quality, transaction costs, market regime changes and execution limitations.

Order Book

• Shows buy and sell orders for an asset, organized by price level.
• Market makers provide liquidity, limit orders are set to a specific price or better, and market orders are immediate at the best available price.

Risk Management

• Primary risks in algorithmic trading exist, including model risk, technical risk, data risk, liquidity risk and regulatory risk
• Mitigation methods include position sizing, stop-loss orders, diversification, stress testing, and real-time monitoring.
• Sharpe Ratio: Sharpe Ratio = $\frac{R_p - R_f}{\sigma_p}$ where $R_p$ = Return, $R_f$ = Risk-free return and $\sigma_p$ = Standard deviation

Market Impact

• The effect a trader's transactions have on an asset’s price is market impact.
• Algorithmic trading can influence market impact.
• Slippage, which relates to market impact, is the difference between expected and actual trade price.

Linear Momentum

• Linear momentum ($\vec{p}$) equals mass ($m$) times velocity ($\vec{v}$): $\vec{p} \equiv m\vec{v}$
• Expressed in components: $p_x = mv_x$, $p_y = mv_y$, $p_z = mv_z$

Impulse-Momentum Theorem

• $\vec{F} = \frac{d\vec{p}}{dt}$ (Newton's Second Law)
• Impulse ($\vec{I}$) is the integral ∫▒𝐹⃗ 𝑑𝑡 from $t_i$ to $t_f$.
• Impulse-Momentum Theorem: (\vec{I} = \Delta \vec{p})

Isolated System

• Total momentum remains constant: (\vec{p}_{total} = \vec{p}_1 + \vec{p}_2 = \text{constant})
• Conservation of momentum: (\vec{p}{1i} + \vec{p}{2i} = \vec{p}{1f} + \vec{p}{2f})

Collisions

• Elastic collisions conserve kinetic energy, and inelastic collisions do not.
• Perfectly inelastic collisions result in objects sticking together.

Collisions in One Dimension

• Momentum is conserved: (m_1 \vec{v}{1i} + m_2 \vec{v}{2i} = m_1 \vec{v}{1f} + m_2 \vec{v}{2f})
• For perfectly inelastic collisions: (\vec{v}f = \frac{m_1 \vec{v}{1i} + m_2 \vec{v}_{2i}}{m_1 + m_2})

Glancing Collisions

• Momentum is conserved in x and y dimensions.

Center of Mass

• Defined as (\vec{r}_{CM} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i} = \frac{1}{M} \sum_i m_i \vec{r}_i)
• Its acceleration determined by (\vec{F}{net} = M \vec{a}{CM} = M \frac{d^2 \vec{r}_{CM}}{dt^2})

Conservation of Momentum

• $\vec{p}{total} = M \vec{v}{CM}$.

Rocket Propulsion

• Governed by $M \Delta v = |v_e \Delta m|$, with thrust defined as $|v_e \frac{\Delta m}{\Delta t}|$.
• Rocket equation yields $\Delta v = v_e \ln{\left( \frac{M_i}{M_f} \right)}$.

Fonction logarithme népérien (Natural Logarithm Function)

• Defined on $]0; +\infty[$ as the primitive of $x \longmapsto \frac{1}{x}$ that equals zero at 1.

Propriétés algébriques (Algebraic Properties)

• $\ln(1) = 0$
• $\ln(e) = 1$
• $\ln(ab) = \ln(a) + \ln(b)$
• $\ln(\frac{1}{a}) = -\ln(a)$
• $\ln(\frac{a}{b}) = \ln(a) - \ln(b)$
• $\ln(a^n) = n\ln(a)$
• $\ln(\sqrt{a}) = \frac{1}{2}\ln(a)$

Étude de la fonction (Function Analysis)

• $\lim_{x \to +\infty} \ln(x) = +\infty$
• $\lim_{x \to 0} \ln(x) = -\infty$

Dérivée (Derivative)

• $\ln'(x) = \frac{1}{x}$

Variations (Variations)

• Strictly increasing on $]0; +\infty[$.

Convexité (Convexity)

• The natural logarithm function is concave on $]0; +\infty[$.

Applications (Applications)

• Can use to resolve equations and inequalities
• Can use to derive functions
• Can use to calculate integrals

Algorithmic Game Theory combines game theory and algorithm design, focusing on strategic interactions. Examples include sponsored search auctions and network routing. Braess's Paradox illustrates that adding network capacity can increase latency.