Math 36
Mathematical Models in the Social Sciences

Homework

Differential Equations
(Due Friday, January 14)
Exponential and Logistic models:
• Topics: the four steps involved in modeling; solving differential equations using the seperation of variables technique; assumptions, differential equation, and function for population model (both exponential and logistic); solving population growth and radioactive decay problems
• Homework 01: available
Arms Race models:
• Topics: arms race assumptions and corresponding terms in differential equations; creating slope graphs to analyze long-term behavior of equations: interpreting the lines dx/dt = 0 and dy/dt = 0, the regions they bound, and their intersection; analyzing effects of changing parameters in equations
• Homework 02: available
Predator-Prey models:
• Topics: assumptions for predators and prey, and corresponding terms in differential equations; finding a single equation relating predator and prey populations; modifying assumptions and differential equations (prey population with logistic growth, predator population dependent on "critical ratio")
• Homework 03: available
Epidemic models:
• Topics: stages of disease; assumptions for population and progression of disease, corresponding differential equations, and possible modifications; time at which demand for medical services is highest; how preventing an epidemic is reflected mathematically
• Homework 04: available
Problem Session:
• Extra problems: not yet available

Game Theory
(Due Wednesday, February 2)
Two-player zero-sum games (Part I):
• Read: Ferguson, Part II, sections 1 and 2
• Topics: mixed and pure strategies; solving 2 by 2 matrix games; the minimax theorem
• Homework 05: available
Two-player zero-sum games (Part II):
• Read: Ferguson, Part II, sections 1 and 2
• Topics: solving 2 by n games; basic poker endgame analysis
• Homework 06: available
Matrix games:
• Read: Ferguson, Part II, section 3
• Topics: matrix multiplication; solving n by n matrix games
• Homework 07: available
Impartial combinatorial games:
• Read: Ferguson, Part I, sections 1, 2, and 6
• Topics: examples of combinatorial games: take-away, nim, green hackenbush; how to win at nim using *n and binary addition; Sprague-Grundy theorem
• Homework 08: available
Two-player general-sum noncooperative games:
• Read: Poundstone, Chapters 6 and 12
• Topics: payoff matrices for general-sum games; goal-dependent strategies; the prisoner's dilemma and variations; communicating through repeated play; natural selection of survival strategies
• Homework 09: available
Two-player general-sum cooperative games:
• Read: Ferguson, Part III, section 4
• Topics: solving TU and NTU games; Nash bargaining axioms; the Nash arbitration function
• Homework 10: available
Many-player cooperative games:
• Read: Ferguson, Part IV, sections 1,2,3
• Topics: coalitions; value/power functions; multiplayer payoff matrices; payoff vector, group and individual rationality, stable and unstable solutions; Shapley axioms and computing the Shapley function
• Homework 11: available
Problem Session:
• Extra problems: not yet available

Discrete Mathematics
(Due Monday, February 14)
Stochastic Processes:
• Goals:
• Homework 12: not yet available
Applications of Markov Chains:
• Goals:
• Homework 13: not yet available
Network Flow:
• Goals:
• Homework 14: not yet available
Graph Stability and Balance:
• Goals:
• Homework 15: not yet available
Tournaments:
• Goals:
• Homework 16: not yet available
Problem Session:
• Worksheet: not yet available

Voting Theory
(Due Friday, February 25)
Voting Theory:
• Goals:
• Homework 17: not yet available
Voting Theory:
• Goals:
• Homework 18: not yet available
Voting Theory: