1. Ec1818 Economics of Discontinuous Change Section 1 [Lectures 1-4] Wei Huang Harvard University (Preliminary and subject to revisions)

2. Outline My understanding for this course Key Points in Lecture 1 – 4 – Intuition of Discontinuous Change – Cellular Automata (CA model) – Prisoner‘s Dilemma (PD) and Strategies – Evolutionary Stable Games/Strategies

3. About This Course, My understanding What’s important – Phenomena, Intuition, Understanding and Applications – I will follow these in Sections Lectures are intuitive and informative, Sections may be somehow more abstract. Role of Mathematics – Helps to understand – Some basic math is required

4. Intuition of Discontinuous Change Discontinuous Change – Sharp change that affects lives in important ways. – What generates discontinuities? Non-linearities, Positive feedbacks and Interactions. Negative feedbacks can produce stability – D(t) = 100 – W(t) – S(t) = 20 + 0.6W(t-1) – If W0 = 20, Based on D(t) = S(t), W(t) = 80 – 0.6W(t- 1)… (Next Slide) – Instead, if S(t) = 20 + 1.6W(t-1), W(t) = 80 – 1.6W(t-1), what will happen?

5. D or S W(t) or W(t-1) S(t) D(t)

6. Cellular Automata (CA model) A discrete dynamical system based on a lattice – Behavior depends on near neighbors. The Schelling Model – Neighborhood, Preferences, Moving rules – Complete Math Framework? More complex! Externality may generates segregation beyond what people want! – Many possible outcomes.

7. An example from Previous PPT

8. PD and Strategies A B Strategy: A cookbook to follow in any case. For example, Simply “Cooperate” is not a strategy. But “ALWAYS cooperate” or “ALWAYS defect” is a strategy. One important Strategy: Tit-for-Tat (TFT),cooperate until opponent defects, then defect until opponent changes.

9. Basic calculations We should know how to calculate the payoffs given strategies and the game rules, provided by the matrix. For example, the payoff matrix provided in last slide. – T periods with discount factor w. – What is the payoff to A and B if (All C, All C)? How about (TFT, TFT)? How about (D, D)? – Is (TFT, TFT) a Nash Equilibrium here?

10. Evolutionary Stable Strategy (ESS) A dominant strategy robust to small invasion by another one strategy. A strategy s is an evolutionary stable strategy (ESS) if, for sufficiently small p > 0, (1-p)U(s, s) + p U(s, t) > (1-p)U(t, s) + p U(t, t) holds, for any other strategy t. “Harm by neighbor” game: – Two Pure Strategy NEs: (A, A), (B, B) – (A, A) does not correspond to ESS but (B, B) does.

11. Another Example: PD (1) Consider three-period PD repeated game with discount factor 1. The stage game is Consider three strategies: TFT, All C and All D. – U(TFT, TFT) = U(TFT, All C) = U(All C, TFT) = U(All C, All C)=3+3+3 = 9; – U(TFT, All D) = 0 + 1 + 1 = 2, U(All D, TFT) = 5 + 1 + 1 = 7 and U(All D, All D) = 1 + 1 + 1 = 3.

12. Another Example: PD (2) TFT Strategy is dominant against a small invasion of All D Strategy – (1-p)U(TFT, TFT) + p U(TFT, All D) > (1-p)U(All D, TFT) + p U(All D, All D) holds for all p < 2/3 TFT Strategy is not dominant against a small invasion of All C Strategy – (1-p)U(TFT, TFT) + p U(TFT, All C) = (1-p) U(All C, TFT) + p U(All C, All DC)