Algorithmic Game Theory

Algorithmic Game Theory

• Classical game theory focuses on interactions between perfectly rational agents who aim to maximize their preferences.
• Algorithmic Game Theory (AGT) blends computer science and game theory examining algorithm design efficiency and the impact of incentives.

Examples of AGT

• Shortest Path: Involves finding the most efficient route between a source and destination, considering factors like congestion affecting travel time.
• Load Balancing: Deals with scheduling jobs on machines to minimize costs, with each job aiming to reduce its individual expense.
• Auctions: Aims to design fair auction mechanisms where bidders are encouraged to bid honestly to allocate an item to the highest valuing bidder.
• Social Networks: Explores how opinions spread and communities form, acknowledging that individuals strategically influence others and build connections for information.

Central Themes in AGT

• Computational Efficiency: Concerned with the difficulty of computing game-theoretic outcomes and designing systems for easy computation of beneficial results.
• Mechanism Design: Focuses on creating game rules that lead to positive outcomes, aligning participant interests with good behavior.
• Price of Anarchy: Measures the inefficiency caused by selfish behavior, comparing Nash equilibrium to the optimal solution to assess loss.
• Coalitions: Examines when player groups are motivated to deviate and how to ensure stability in such scenarios.
• Learning: Investigates how players improve their strategies in repeated games and adapt to changing circumstances.

Selfish Routing

• Model: Routing involves directing traffic through a graph with edges having cost functions that increase with traffic volume.
• Flows: Flows are paths from source to destination, and the amount of traffic on an edge is crucial in calculating path cost.
• Cost of Path: Determined by the sum of costs on edges along the route, influencing the overall cost of the flow.

Wardrop Equilibrium Defined

• Wardrop equilibrium: Flow state where traffic uses only shortest paths; ensuring no unilateral cost improvements are possible for any agent.

Algorithmic Trading

• Algorithmic trading is a method of executing orders using automated, pre-programmed instructions that consider price, timing, and volume.
• It is also known as Automated Trading, Black-Box Trading, Algo-Trading, or Robotic Trading.

Algorithmic Trading: Functionality

• Opportunity is identified.
• Algorithm is developed.
• Algorithm is optimized via backtesting.
• Algorithm is initiated on a trading platform.
• Algorithm monitors market data and executes trades automatically.

Advantages of automation in trading

-Speed and efficiency due to faster trade execution. -Reduced emotional bias leading to better investment decisions. -Effective backtesting to evaluate and optimize strategies. -Increased diversifications across various assets and markets. -Minimized costs through automation.

• Improved order execution to minimize market impact.

Disadvantages of automation in trading

• Technical glitches, software bugs and connectivity issues
• Over-optimization leading to poor trading decisions
• Difficulty adapting to changing conditions
• Constant monitoring, maintenance, and updates
• Flash crashes due to malfunctioning algorithms
• Regulatory compliance burdens.

Common strategies

• Trend following to capitalize on market momentum.
• Mean reversion to profit on expected price corrections.
• Arbitrage to exploit price discrepancies across markets.
• Market making to provide liquidity
• Statistical arbitrage to exploit inefficiencies.
• Execution algorithms to optimize transaction costs.

Software Libraries

• Alpaca Trade API: Provides commission-free stock brokerage.

• IBridgePy: Offers a Python framework connecting Interactive Brokers.

• QuantConnect: Cloud-based trading platform.

• MetaTrader 5: An application for trading forex, stocks, and futures.

• The provided Python example demonstrates placing a buy order using the Alpaca Trade API.

Liouville Theory

• Centers upon a field theory applicable in two dimensions, particularly with relevance to string theory and conformal field theory.

Liouville Action

• Defined by a specific integral incorporating the Liouville field and metric properties.
• The theory can simplify under conformal gauge with equations of motion being derivable from its action.

Aspects of stress and energy

• Involves the stress-energy tensor and conformal anomaly, which features a central charge related with the coupling constant.

Vertex operators and Conformal dimension

• Liouville theory presents vertex operators, shaping correlation functions which include dimensions vital for understanding the theory on Riemann sphere's three-point function.

Discrete Random Variables

• A random variable is a variable whose value is a numerical outcome of a random phenomenon.
• A discrete random variable has a finite or countable set of possible values.

Discrete Random Variable Examples

• Number of children in a family.
• Number of customers arriving at a checkout between 1 PM and 2 PM.

Key Definitions

• Sample space: all possible outcomes.
• Probability distribution: set of pairs of values and probabilities, where probabilities range from 0 to 1 and sum to 1.
• Cumulative distribution: function representing probabilities of a variable less than or equal to a specified value, increasing in steps.

Parameters of a discrete variable

• Expected value: average value of the variable (sum of values multiplied by probabilities).
• Variance: measures the spread of values around the mean.
• Standard deviation: square root of the variance, indicating dispersion in the same units as the variable.

Common discrete distributions

• Bernoulli: results in success (1) or failure (0), with calculations for expected value and variance.
• Binomial: the number of successes in a series of independent trials, with expected value and variance formulas.
• Poisson: events occurring over an interval, depending on the rate of events and formulas for its expected values and variances.
• Hypergeometric: accounts for successes, without replacement, with calculations of its expected value and variance.