Algorithmic Game Theory

Algorithmic Game Theory

• A study of multi-agent decision problems, viewed as a branch of applied mathematics used mostly in economics and political science.
• A game involves multiple decision-makers (agents, players), each with a set of possible actions.
• A game has a well-defined outcome based on the actions of all agents.
• Each agent has a utility or preference for each potential outcome.

Example Games

Prisoner's Dilemma

• Two suspects are separately interrogated.
• Remaining silent results in a minor offense conviction (1 year).
• If one confesses and the other remains silent, the confessor is released (0 years), while the other receives a major offense conviction (4 years).
• If both confess, they both receive major offense convictions (3 years).

Stag Hunt

• Hunters can hunt a stag or rabbits.
• Stag hunting requires cooperation and yields more food, while solo rabbit hunting yields less.
• Hunters starve if they try to hunt a stag alone.

Braess's Paradox

• Adding a road to a network can increase the average travel time.
• Without the new road:
  • Cost from S to E: fixed cost 1
  • Cost from S to N: number of drivers x/100
  • Cost from W to E: number of drivers y/100
  • Cost from W to N: fixed cost 1
  • At Nash equilibrium, all drivers take paths S->N->E and W->E->N
  • Total cost is 1+ x/100 + y/100 + 1 which equals 3.
• With the new road:
  • With the new road At Nash equilibrium, all drivers take paths S->W->N->E and W->N->S->E
  • Total cost is now x/100 + y/100 + x/100 + y/100 which equals 4.

Nash Equilibrium

• A set of strategies where no player has an incentive to unilaterally change their strategy.
• Each player's strategy is the best response given the other players' strategies.

Algorithmic Game Theory

• Applies algorithmic and computational ideas to game theory and game-theoretic ideas to computer science.
• Example topics include computing Nash equilibria, mechanism design, price of anarchy, learning in games, and game-theoretic aspects of network routing.

Algorithmic Problem Solving (Week 1: Introduction)

Algorithmic Problem Solving Definition

• Algorithm: A process to be followed in calculations or problem-solving operations, especially by a computer.
• Problem: A formal specification of something we want a computer to solve.
• An algorithm should solve all possible instances of a problem.

Sorting

• Input: A sequence of numbers.
• Output: A reordering of the sequence in monotonic order.

Chemical Kinetics

Reaction Rate

• The reaction rate is the change in concentration of a reactant or product with respect to time.
• For the reaction aA + bB -> cC + dD

Rate Law

• The rate law is an equation that relates the reaction rate to the concentrations of reactants.
• Rate = k[A]^m[B]^n
  • k is the rate constant
  • m and n are the orders of the reaction with respect to A and B, respectively.
  • m + n is the overall order of the reaction

Determining Rate Laws

• Method of Initial Rates: Measure the initial rate of a reaction for different initial concentrations of reactants. Compare the rates to determine the order of the reaction with respect to each reactant.
• Integrated Rate Laws: Use calculus to derive equations that relate the concentration of a reactant to time. Compare the experimental data to the integrated rate laws to determine the order of the reaction.

Factors Affecting Reaction Rate

• Concentration increases the reaction rate.
• Temperature increases the reaction rate.
• A catalyst speeds up a reaction without being consumed.
• Surface Area increases the reaction rate of a solid reactant.

Reaction Mechanisms

• Definition: A step-by-step sequence of elementary reactions that describe the overall reaction.

Elementary Reactions

• Definition: An elementary reaction is a reaction that occurs in a single step.

Rate-Determining Step

• Definition: The rate-determining step is the slowest step in the reaction mechanism.

Intermediates

• Definition: An intermediate is a species that is formed in one step of the reaction mechanism and consumed in a subsequent step.

Activation Energy

• Definition: The activation energy is the minimum amount of energy that molecules must have in order to react.

Arrhenius Equation

• Relates the rate constant to the activation energy and temperature: $k = Ae^{-E_a/RT}$
  • where k is the rate constant A is the frequency factor $E_a$ is the activation energy R is the ideal gas constant T is the temperature in Kelvin

Catalysis

• A catalyst lowers the activation energy of a reaction, thereby increasing the reaction rate.
  • Homogeneous catalyst: A catalyst that is in the same phase as the reactants.
  • Heterogeneous catalyst: A catalyst that is in a different phase as the reactants.

Practice Problems

• The rate constant for the first-order decomposition of ethyl iodide is $2.47 \times 10^{-5} s^{-1}$ at $327^\circ C$. Calculate the half-life of this reaction.

• The following data were obtained for the reaction: $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$

  Experiment	[NO] (M)	[O_2] (M)	Initial Rate (M/s)
  1	0.020	0.010	$2.8 \times 10^{-6}$
  2	0.020	0.020	$5.6 \times 10^{-6}$
  3	0.040	0.010	$11.2 \times 10^{-6}$

• Determine the rate law for this reaction.

• The activation energy for a reaction is 100 kJ/mol. If the rate constant for the reaction is $1.0 \times 10^{-3} s^{-1}$ at 300 K, what is the rate constant at 400 K?

Linear Algebra and Matrix Analysis

1.1 Systems of linear equations

Definition 1.1.1

• A system of m equations linear has n unknowns $x_1, x_2$ to $x_n$ and is set of equations in the form:

$\qquad \begin{cases} \begin{aligned} a_{11}x_1 + a_{12}x_2 +... + a_{1n}x_n &= b_1 \ a_{21}x_1 + a_{22}x_2 +... + a_{2n}x_n &= b_2 \... \ a_{m1}x_1 + a_{m2}x_2 +... + a_{mn}x_n &= b_m \end{aligned} \end{cases}$

• Where the coefficients $a_{ij}$ and the constants $b_i$ are are real or complex numbers.

Definition 1.1.2

• A solution to a set of linear equations is an n-uplet of real or complex numbers $(s_1, s_2,..., s_n)$ that satisfy simultaneously all sets of equations.

Definition 1.1.3

• The total number of solutions to a set of linear equations is called the set of solutions to the system.

Definition 1.1.4

• If a system of linear equations is compatible, it has at least one solution; when it doesn't it is incompatible

1.2 Matrices

Definition 1.2.1

• A matrix is a rectangular array of numbers called its coefficients. A matrix has $m$ rows and $n$ columns is called a dimension $m \times n$.

$A = \begin{bmatrix} a_{11} & a_{12} &... & a_{1n} \ a_{21} & a_{22} &... & a_{2n} \... \ a_{m1} & a_{m2} &... & a_{mn} \end{bmatrix}$

Definition 1.2.2

• A column matrix has a single column and a row matrix has a single row

Definition 1.2.3

• A square matrix has the same number of rows and columns, the diagonal being from the top left to bottom right

Definition 1.2.4

• A diagonal matrix is a square matrix has zero as values except for the diagonal

Definition 1.2.5

• The identity matrix $I_n$ is a diagonal matrix of $n \times n$ which have 1 as their diagonal value

$I_n = \begin{bmatrix} 1 & 0 &... & 0 \ 0 & 1 &... & 0 \... \ 0 & 0 &... & 1 \end{bmatrix}$

Definition 1.2.6

• The upper triangular matrix is a square matrix with values that are zero above its diagonal, and a lower matrix has zero value above the diagonal

Definition 1.2.7

• A transpose of $A$, notated $A^T$ obtained by exchanging the rows and column of $A$ if $A$ is an $m \times n$ then $A^T$ has dimension $n \times m$