Algorithmic Game Theory: Prisoner's Dilemma

Algorithmic Game Theory

• Game theory studies strategic interaction's mathematical and logical aspects among rational agents.
• An agent is a generic term for a decision-maker
• Rational means acting to maximize one's utility

The Prisoner's Dilemma

• Two suspects are arrested, but police lack sufficient evidence for conviction without a confession.
• The police offer the same deal to each suspect separately, if one confesses and the other doesn't, the confessor is freed and the other gets the max sentence.
• If both confess, they receive a reduced sentence; if neither confesses, they receive a light sentence.

Payoff Matrix Example

• If suspect B confesses, suspect A should also confess ($−5>−10$ ).
• If suspect B does not confess, suspect A should confess ($0>−1$).
• A dominant strategy is when is optimal regardless of what the other players do.
• A Nash Equilibrium is a set of strategies where no player can improve their payoff by unilaterally changing strategy.
• The Nash Equilibrium is not always the best outcome for all players.

Algorithmic Game Theory Details

• Traditional game theory assumes rational agents with unlimited computational resources.
• Algorithmic game theory considers the computational aspects of game theory.
• Considers the complexity of finding a Nash Equilibrium.
• Considers how to design games that are easy to play.
• Considers how to design mechanisms that incentivize agents to act in a socially desirable way.

Selfish Routing

• This involves a network of roads.
• Each agent wants to travel from A to B as quickly as possible.
• Each road has a cost (e.g., travel time) dependent on traffic.

Pigou's Example of Routing

• There are two roads from A to B.
• One road has a fixed cost of 1.
• The other road has a cost of $𝑥$, where $𝑥$ is traffic fraction.
• One unit of traffic wants to travel from A to B.
• In Nash Equilibrium, all traffic takes the road with cost $𝑥$.
• The total cost becomes $1 \cdot x + x \cdot (1-x) = x + x - x^2 = 2x - x^2$.
• The cost to each agent is $𝑥$.
• Socially optimal, half take the road with cost 1, half take the road with cost $𝑥$.
• The total cost is $1 \cdot 0.5 + 0.5 \cdot 0.5 = 0.5 + 0.25 = 0.75$.
• The cost to each agent is $0.75$.
• Price of Anarchy is the ratio between Nash Equilibrium cost and socially optimal cost.
• In Pigou's Example, Price of Anarchy is $\frac{1}{0.75} = \frac{4}{3}$.
• The Price of Anarchy can be arbitrarily high.

Topics in Algorithmic Game Theory

• Mechanism Design is used to design games to achieve desired outcomes.
• Coalitional Game Theory is used to study the formation of coalitions among agents.
• Social Choice Theory is used for designing voting systems to choose the best candidate.

Chemical Engineering Thermodynamics - Introduction

• Thermodynamics deals with the relationships between heat, work, and substance properties.
• Applications include engines, refrigerators, power plants, phase equilibria, and chemical reaction equilibria.

Dimensions and Units

• Mass (m) is measured in kg, g, lb, oz, ton
• Length (L) is measured in m, cm, ft, in
• Time (t) is measured in s, min, hr, day
• Temperature (T) is measured in K, °C, °F, °R

Force Equation and Units

• $F=ma$
• SI unit: $N=kg \cdot m/s^2$
• English unit: $lbf=32.174 lb \cdot ft/s^2$

Pressure Equation and Units

• $P=F/A$
• SI unit: $Pa=N/m^2$
• $1 bar = 10^5 Pa$
• English unit: $psi=lbf/in^2$
• $1 atm = 14.696 psi=1.01325 bar$

Volume Units

• SI unit: $m^3$
• English unit: $ft^3$
• $1 gal = 3.785 L$

Temperature Equations

• $T(K) = T(^{\circ}C) + 273.15$
• $T(^{\circ}R) = T(^{\circ}F) + 459.67$
• $T(^{\circ}R) = 1.8 T(K)$
• $T(^{\circ}F) = 1.8 T(^{\circ}C) + 32$

Measures of Amount or Size

• Density: Mass per unit volume ($\rho = m/V$)
• Molar volume: Volume per mole ($\hat{V} = V/n$)
• Specific volume: Volume per unit mass ($v = V/m = 1/\rho$)
• Mole fraction: $x_i = n_i / n$
• Mass fraction: $w_i = m_i / m$

Thermodynamic Systems

• A system is the part of the universe that we are interested in.
• The surroundings are everything else.
• The boundary separates the system from the surroundings.

Types of Systems

• Closed system: No mass transfer across the boundary.
• Open system: Mass transfer across the boundary.
• Isolated system: No mass or energy transfer across the boundary.

Properties of Systems

• A property is a characteristic of a system that can be quantified.
• Extensive properties: Depend on the size of the system (e.g., volume, mass, energy).
• Intensive properties: Independent of the size of the system (e.g., temperature, pressure, density).
• A system is in thermodynamic equilibrium when its properties are uniform throughout the system and do not change with time.

State and Path Functions

• The state of a system is defined by its properties.
• A state function is a property that depends only on the current state of the system, not on how it reached that state (e.g., temperature, pressure, energy).
• A path function is a property that depends on the path taken to reach a certain state (e.g., heat, work).

Equilibrium

• Thermodynamic equilibrium implies:
  • Thermal equilibrium (uniform temperature)
  • Mechanical equilibrium (uniform pressure)
  • Chemical equilibrium (uniform chemical potential)

The Phase Rule

• $F = 2 - \pi + N$ where:
  • $F$ = degrees of freedom
  • $\pi$ = number of phases
  • $N$ = number of chemical species

Phase Rule Example

• Water at its triple point: $\pi = 3$, $N = 1$
• $F = 2 - 3 + 1 = 0$

Reversible and Irreversible Processes

• A reversible process is one that can be reversed by an infinitesimal change in conditions.
• An irreversible process is one that cannot be reversed.
• All real processes are irreversible.

Constant Property Processes

• Isothermal process: Constant temperature
• Isobaric process: Constant pressure
• Isochoric process: Constant volume
• Adiabatic process: No heat transfer