Algorithmic Game Theory: Strategic Interactions

• Algorithmic Game Theory analyzes strategic interactions between rational decision-makers using mathematical frameworks.
• Rationality in game theory assumes players act to maximize their expected payoff in their self-interest.
• Strategic interaction means a player's decision outcome is based on the decisions of other players.

Applications of Game Theory

• Game theory has widespread applications in economics, political science, and computer science.
• Economics: auctions, bargaining, market equilibrium.
• Political science: voting, lobbying, international relations.
• Computer science: network routing, mechanism design, e-commerce.

Normal-Form Games

• In a normal-form game, games can be represented by: Players, strategies available to each player and payoff for each possible combination of strategy

Prisoner's Dilemma Example

• A normal-form game example is the Prisoner's Dilemma.

Nash Equilibrium Explained

• Nash Equilibrium is a set of strategies (one for each player) where no player benefits from unilaterally changing strategy.
• Nash Equilibrium: Each player's strategy is optimal, given the other players' strategies.
• No single player can gain a better outcome by altering their own strategy alone.

Finding Nash Equilibria Techniques

• Finding Nash Equilibria can be achieved through: Dominant strategy elimination, Best response analysis, using Mixed strategies to allow players to randomize over pure strategies.

Mechanism Design

• Mechanism design is creating games to achieve specific goals with private player information.
• Examples of Mechanism Design include: Auction formats Voting Rules, Matching Markets

Challenges in Mechanism Design

• Key challenges are: incentive compatibility, efficiency, and budget balance
• Incentive compatibility: Ensure players truthfully reveal private information
• Efficiency: Ensure the mechanism outcome is socially efficient
• Budget Balance: Ensure the mechanism needs no external subsidies

Algorithmic Considerations

• Algorithmic considerations exist in: Computation Complexity, Communication complexity, and Learning.
• Computation Complexity: Efficiently computing Nash equilibria/optimal mechanisms.
• Communication complexity: Communication needed for agreement or mechanism implementation.
• Learning: How players learn to play optimally in repeated games/complex environments.

Current Research Areas

• Research is ongoing in these fields: Mechanism design without money, Fair division, Social networks, and Artificial intelligence
• Mechanism design without money: Non-monetary mechanism designs are the focus.
• Fair division: Dividing resources fairly among agents with various preferences.
• Social networks: Analyzing strategic interactions in social networks.
• Artificial intelligence: Game theory for AI agents that strategically interact.

Analisi Matematica I (a.a. 2013-2014)

Esercizio 1. Summation Formula Proof

• Goal: Prove $\sum_{k=1}^{n} k^{2}=\frac{n(n+1)(2 n+1)}{6}$ for all $n \geq 1$ using induction.
• Base Case: For $n=1$, $\sum_{k=1}^{1} k^{2}=1^{2}=1=\frac{1 \cdot 2 \cdot 3}{6}$, so the assertion $P(1)$ is true.
• Inductive Step: Assume $P(n)$ holds and show $P(n+1)$ is true, proving $\sum_{k=1}^{n+1} k^{2}=\frac{(n+1)(n+2)(2 n+3)}{6}$.
• Using the inductive hypothesis: $\sum_{k=1}^{n+1} k^{2} = \frac{n(n+1)(2n+1)}{6} + (n+1)^{2}$.
• Simplifying, $\frac{n(n+1)(2 n+1)}{6}+(n+1)^{2} = \frac{(n+1)[n(2 n+1)+6(n+1)]}{6}$.
• Further simplification yields $\frac{(n+1)(2 n^{2}+7 n+6)}{6}=\frac{(n+1)(n+2)(2 n+3)}{6}$, completing the inductive step.

Esercizio 2. Sequence Iteration

Objective

• Determine if the sequence $x_{n+1}=\sqrt{2 x_{n}}$ to know if it is monotonically bounded for $n \geq 1$
• Given $x_1 = 1$

1. Verifying $x_{n} < 2$ for all $n \geq 1$ by induction
2. Verifying is monotonically increasing

• Base Case: For $n=1$, $x_1=1 < 2$ is true, since $1<2$
• Inductive Step: $P(n+1)$ if $P(n)$ is true
• From $x_{n+1} = \sqrt{2x_n}$ we can show that $x_{n+1} < 2$ is the same as $\sqrt{2x_{n}} < 2$
• Under the assumption of $x_n <2$ and solving the equation, $x_n < 2$
• Prove sequence is monotonically increasing $\qquad x_{n+1} > x_n, n\geq 1$ or $\qquad x_{n+1} = \sqrt{2x_n} > x_n, n\geq 1$ or $\qquad \sqrt{2x_n} > x_n$ or $\qquad x_n < 2$.

Lecture 16: Particle Accelerators

Introduction

• Particle accelerators facilitate exploration by accelerating charged particles to high energies.
• The impact of accelerated particles helps probing matter structure at small scales.
• The impact helps recreate the early universe.

Brief History

• 1932: First accelerator by Cockcroft & Walton.
• 1930s: Cyclotron invented by E.O. Lawrence.
• Post WWII: Development of synchrotrons.
• Today: Large facilities, e.g., LHC at CERN.

Applications

• Particle accelerators facilitate fundamental research and have medical and industrial uses.
• Fundamental Research: Investigate particle structure and test particle physics theories.
• Medical Application: Provide medical imaging and cancer therapy.
• Industrial Application: Help with material processing and non-destructive testing

Accelerator Principles

• Particle acceleration and beam focusing uses electric and magnetic fields.

Acceleration - Formula

• Charged particles are accelerated by electric fields: $\qquad \vec{F} = q\vec{E}$
  • $\vec{F}$ is the force on the particle.
  • $q$ is the charge of the particle.
  • $\vec{E}$ is the electric field.

Beam Focusing - Formula

• Magnetic fields focus particle beams; Lorentz force is: $\qquad \vec{F} = q(\vec{v} \times \vec{B})$
  • $\vec{v}$ is the velocity of the particle.
  • $\vec{B}$ is the magnetic field.

Linear Accelerators (Linacs)

• Linear Accelerators accelerate particles along a straight path using RF waves.
• Example: SLAC at Stanford

Circular Accelerators

• Circular accelerators use magnetic fields to bend particle trajectories into a circular path.
• Examples: Cyclotrons, Synchrotrons (Tevatron, LHC).

Key Components

• RF Cavities, Magnets, Vacuum System, and injector are Key Components in acceleration.

RF Cavities

• RF (radio frequency) Cavities provide accelerating electric field and every time particles pass they get energy.

Magnets

• Dipole Magnets: Guide beam in circular path.
• Quadrupole Magnets: Focus beam to prevent spread.
• Sextupole Magnets: Correct for chromatic aberrations.

Vacuum System

• The Vacuum system helps to minimize collisions with gas molecules to help in the prevention of beam loss.

Injector

• Injectors create the source of particles that will be accelerated.
• It combines the Ion source and a pre-accelerator

Synchrotrons

• Synchrotrons are accelerators that accelerate particles at a constant radius.
• They use: Increasing magnetic field in sync with particle energy and RF cavities during Acceleration.

Key Parameters

• The following are key parameters: Energy and luminosity.
• Energy: The energy of accelerated particles (TeV or GeV) Luminosity- Measure of collision rate $L = \frac{N^2f}{4\pi\sigma_x\sigma_y}$
  • $N$ is the number of particles per bunch.f is the collision frequency.
  • $\sigma_x, \sigma_y$ are the horizontal and vertical beam sizes at the interaction point.

Challenges

• Key challenges include: Space Charge Effects, Synchrotron Radiation, and Magnet Technology. Space Charge Effects: Repulsive interactions between charged particles in the beam cause instability Synchrotron Radiation: Emitted when charged particles accelerate leading to energy loss. Magnet Technology: superconducting magnets are required to have high energies

Future Directions

• Energy frontier and Intensity Frontier are future directions moving forward.

Energy Frontier - Pushing Energy to Explore New Physics

 - Future Circular Collider (FCC) at CERN.

Intensity Frontier

 - Increasing beam intensity to enhance sensitivity for rare detection
 - Project X at Fermilab.

Conclusion

• Particle accelerators are very important for technological advancement.
• On-going improvements to technology will help further understand the universe.

Algorithmic Trading and Order Execution

• Algorithmic trading is the use of pre-programmed instructions to perform the order in a trade
• High frequency trading (HFT) to slow execution strategies are used in trading process

The Primary Goal - Execute Orders

• Achieve execution of (large) orders without affecting price.
• Orders given to the market are generally smaller than the trader intends
• Hide large scale orders sent

Why Use Algorithmic Trading?

• To help: Reduce transaction costs, Improved Order Execution, Access to Multiple Markets etc.

1. Reduce Transaction Costs: Helps finding the best price.
2. Improved Order Execution: More accurate and faster.
3. Access to Multiple Markets: Helps Automate across exchanges.
4. Increased Trading Speed: Helps React quickly to market changes.
5. Back-testing: Test trading strategies on historical data.
6. Reduced Emotional Influence: Remove human bias.

Example Strategy - VWAP (Volume Weighted Average Price)

• To minimize market impact, the strategy cuts a large order into smaller tranches.
• Trade in proportion to the historical trading volumes. $VWAP = \frac{\sum{Price * Volume}}{\sum Volume}$

Implementation

Implementation for strategy:

1. Data Collection: Gather historical and real-time volume data.
2. Order Slicing: Divide the total order into smaller tranches.
3. Timed Execution: Release tranches based on volume patterns.
4. Monitoring: Continuously monitor and adjust.
5. Dynamic Adjustment: Adjust based on real-time market conditions.

Advantages

• Strategy reduces price movements and offer benchmark performance
• Reduced Market Impact: Minimizes price movement.
• Benchmark: Serves as a performance comparison.

Disadvantages

• Strategy relies on accurate volume predictions and may not suit thinly traded stocks.

Other Algorithm - Strategies

• Time Weighed Price (TWAP)
• percentage of volume (POV)
• Implementation Shortfall
• Pairs Trading
• Mean Reversion
• Delta-Neutral Hedging
• High-Frequency Trading (HFT)

Order Execution

• The process of completing a buy or sell order for a security.

Key Considerations

• Speed, Price, Size, Market Impact, Information Leakage

1. Speed: How quickly the trade is executed.
2. Price: The price at which the trade is executed.
3. Size: Quantity of shares or contracts.
4. Market Impact: The effect of the order on the security's price.
5. Information Leakage: Risk of order details becoming public.

Order types

• Market Order, Fill or Kill (FOK), Limit Order etc are order types

1. Market Order: Executed at best available price.
2. Limit Order: Only at a specified price or better.
3. Stop Order: Becomes a market order when the stop price is reached.
4. Stop-Limit Order: Becomes a limit order when the stop price is reached.
5. Hidden Order (Iceberg Order)**: Only a portion of the order is displayed.
6. Fill or Kill (FOK): Order must be executed immediately and completely, or it is cancelled.
7. Immediate or Cancel (IOC): Any portion of the order that cannot be immediately filled is cancelled.
8. All or None (AON): Order must be executed completely.

Execution Venues

• Execution Venues are: Stock Exchanges and Electronic Communication Networks (ECNs)

1. Stock Exchanges: Centralized locations for trading.
2. Electronic Communication Networks (ECNs): Automated systems that match buy and sell orders
3. Dark Pools: Exchanges that do not publicly display order information.
4. Over-The-Counter (OTC) Markets: Decentralized markets not listed on exchanges.

Algorithmic Game Theory

• Involves Multi-agent decision making, rooted in Computer Science

Definition

• A game Is multi-agent decision situations

Components:

• Agents (players) Actions (strategies) available to each agent Agents' preferences (utilities) over outcomes All outcomes based on players strategy

Example: Prisoner's Dilemma

• Each player's dominant strategy is to betray
• If both play dominant strategy $\implies$ both worse off

Example: Stag Hunt

• Two Nash equilibria: (Stag, Stag) and (Hare, Hare)

Nash Equilibrium

• Strategy profile: a set of strategies, one for each player
• A strategy profile is a Nash Equilibrium if no player has incentive to unilaterally deviate $$u_i(s_i, s_{-i}) \geq u_i(s'i, s{-i})$$
• $s = (s_1,..., s_n)$ is a Nash Equilibrium if for every player $i$ and every strategy $s'_i$
• $s_{-i}$: strategies of all players except $i$
• $u_i$: utility function of player $i$

Existence of Nash Equilibria

• Nash's Theorem (1951): States game has a least one Nash Equilibrium

Proof Idea

• Based on Brouwer's Fixed Point Theorem
• States that if $f$ is a continuous function from set that there exists $x$ st. $f(x) = x$
• Define a function $f$ that maps each strategy profile to another strategy profile.
  • If player can improve, $f$ changes the strategy
  • If no player can improve, $f$ returns same profile
• Theorem guarantees fixed point $x$ St. $f(x) =x$.
• This fixed point corresponds to a Nash Equilibrium

Algorithmic Game Theory

• It comprises with algorithm design and accounting for limitation: Implementation of mechanisms with nice properties (ie truthful)

Algorithmic game:

• Involves computation complexity, Approximation and Dynamics

Example: Sponsored Search Auction

• $n$ Bidders
• $m$ Slots
• Each click is based on value $V_i$ for each click
• Each slot has a Click-Through rate $a_j$
• If bidder i gets slot utility $v_i \alpha_j -p_i$ ($p_i$ is price)- Social welfare
• Goal: Maximize what auction do & truthful what players are allocated

Bernoulli's Principle

• The principle states that when fluid increases, its pressure or potential energy decreases.

Explanation

• Swiss mathematician Daniel Bernoulli published this Book Hyddodynamica was named

Explanation

• Daniel Bernoulli published this in his b

Algorithmic Game Theory uses math to analyze strategic interactions between rational decision-makers, who aim to maximize their payoff. It has applications in economics, political science, and computer science, with normal-form games like the Prisoner's Dilemma.