Algorithmic Game Theory: Selfish Routing

Algorithmic Game Theory

• It involves algorithm design in systems with strategic participants
• It combines mathematical modeling, system design, and algorithm design

Selfish Routing

• Involves each agent trying to optimize their own route when routing
• Traffic Network is represented as Graph $G(V,E)$
• $l_e(x)$ represents the latency on edge $e$ with traffic $x$

Wardrop Equilibrium

• It exists when no agent can improve their latency by changing their path
• In a scenario, $x$ portion of traffic uses the upper edge, and $1-x$ uses the lower edge
• With upper edge latency $l_1(x) = x$ and lower edge latency $l_2(x) = 1-x$
• Equilibrium is reached when $x = 1-x$, resulting in $x = 0.5$
• Total latency at equilibrium is $0.5$

Social Optimum

• Goal is to minimize the total latency
• Example: If total latency is $x \cdot l_1(x) + (1-x) \cdot l_2(1-x) = x^2 + (1-x)^2$
• Minimum latency is achieved when $x = 0.5$

Braess's Paradox

• Adding a new road can increase total latency
• A shortcut from A to B through C and D becomes available
• With everyone using it A -> C latency is 1 and D -> B latency is 1
• Everyone ends up using A -> C -> D -> B path, total latency increases to 2

Price of Anarchy (PoA)

• Quantifies how much network performance degrades due to selfish routing
• Defined as $PoA = \frac{\text{Cost of worst-case Nash equilibrium}}{\text{Cost of optimal solution}}$

Chemical Kinetics

• It studies the rates of chemical processes
• Includes influence of experimental conditions on reaction speed
• It also yields information about the reaction's mechanism and transition states
• Involves constructing mathematical models for reaction characteristics

Factors Affecting Reaction Rates

• Reactant concentration: Reaction rate is proportional to the concentration of the reactants
• Physical state of reactants and surface area: Reactions occur at contact area in heterogeneous reactions
• Temperature: Higher temperature gives molecules more thermal energy
• Presence of a catalyst: Catalysts lower activation energy
• Pressure: Increased pressure in gaseous reactions increases collisions
• Light: Photochemical reactions are driven by absorption of light

Reaction Rate

• For a generalized reaction $aA + bB \rightarrow pP + qQ$
• Rate Expression: $-\dfrac{1}{a}\dfrac{d[A]}{dt} = -\dfrac{1}{b}\dfrac{d[B]}{dt} = \dfrac{1}{p}\dfrac{d[P]}{dt} = \dfrac{1}{q}\dfrac{d[Q]}{dt}$
• Here, [A], [B], [P], [Q] are concentrations of A, B, P, and Q respectively
• a, b, p, and q are the stoichiometric coefficients for the reaction.

Rate Law

• Rate law or equation links the reaction rate with concentrations or pressures of reactants
• For the reaction $aA + bB \rightarrow pP + qQ$, the rate law is usually $v = k[A]^m[B]^n$
• [A] and [B] express the concentration of the reactants
• m and n (not necessarily equal to a and b) are the reaction orders
• k signifies the rate constant

Zero, First and Second Order reactions

• Zero Order: Reaction rate is independent of reactant's concentration ($v = k$)
• First Order: Reaction rate is directly proportional to concentration ($v = k[A]$)
• Second Order: Reaction rate is proportional to concentration squared ($v = k[A]^2$ or $v = k[A][B]$)

Research Abstracts

• A research abstract is a concise summary of a research study
• It summarizes the study's purpose, methods, results, and conclusions

Use Cases for Research Abstracts

• Academic conferences use research abstracts as a concise representation of different research to be presented
• Research publications for a quick summary
• Funding requests

Types of Abstracts

• Descriptive abstracts offer an overview of the research, but doesn't include detailed results or a conclusion
• Informative abstracts include an overview, the methods, key results, and the conclusion

How to Write a Research Abstract

• Research abstracts are shorter when the research is finished
• Specific guidelines must be respected (conference guidelines)
• Use concise and precise words
• Focus on points you think are relevant
• Correct with scrutiny, must reflect the correct information

Structure of an Abstract

• The Title must be a clear title
• The Introduction gives a brief overview of the problem
• An outline of the methods used
• The Results section shows the key findings
• The Conclusion interprets the information

Impact of Social Media on Youths

• The title of a summary of the impact of social media on the healt of Young individual
• A survey with 500 students found a connection between the usage of social media and depression
• An increase of the usage of social media can drive negative issues with the individual

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Algorithmic complexity

• Measures the resources needed for an algorithm like time and space
• Often written as Big O notation - $O(1)$, $O(log n)$, $O(n)$, depends on the algorithm

Growth Rates

• Constant - $O(1)$
• Logarithmic - $O(log n)$
• Linear - $O(n)$
• Exponential - $O(2^n)$
• Factorial - $O(n!)$

Algorithmic Game Theory continued

• Selfish Routing with network congestion and agent optimization
• Braess's Paradox, adding resources can hurt the overall performance
• Price of Anarchy, how inefficient is the system? PoA

Mechanism Design

• Goal is to design rules to achieve the desired outcome
• Examples: Auctions, Voting, Matching
• Challenges involves strategical thinking and private information

Mechanism Design Concepts

• Mechanism Design without Money; some rules do not allow transfers
• Revelation Principle; incentivized true information must work with a social choice rule

Complexity

• Goal involves concepts, such as Nash equilibrium
• Challenges exists such as NP-hard to compute and PPAD-hard to compute
• Topics include auctions, social choice, fair division, learning in games, network search, voting systems