section2(博弈论讲义(Harvard University))汇总

– symmetry requires that j*=1+ci
• If each is responding to the BR of the other
• j*=1+c(1+cj*), j*=1/(1-c) = i* also by symmetry • πi,j= 2[1/(1-c) + 1/(1-c) + c(1/(1-c))2] - (1/(1-c))2 • πi,j= (3-2c)/(1-c)2
– Those that survive iterated dominance – Motivated by common knowledge assumptions
• Congruity
– Weakly congruous strategies
• Each strategy is a best response to others’ strategy
Nash Eqm (TR & DC)
strategy profiles
Pareto Optima
Set of Rationalizable Strategies: {T,D} x {C,R} Set of Weakly Congruent Strategies: {T,D} x {C,R} Set of Best Response Complete: {T,D} x {C, R} or {T,D} x {L,C,R} Set of Congruent Strategies: {T,D} x {C, R}
• πi = 2(i + j + cij) - i2 • πj = 2(i + j + cij) - j2
• Best response functions:
– d (πi)/di = 2+2cj - 2i – set equal to zero: 2i = 2 + 2cj
• i*=1+cj … This is our BR function for agent i
Partnership Game: Pareto-Efficiency
In Strategic Setting, Equilibrium was i*,j*=1/(1-c), πi,j= (3-2c)/(1-c)2
• Now we leave the strategic setting (S.S.) and maximize total profits (pareto efficiency)
πF = 4(i + j + cij) - i2- j2
• d (πF)/di = 4 +4cj - 2i • d (πF)/dj = 4 +4ci - 2j
Set equal to Fra Baidu bibliotekero: 2i = 4 + 4cj
• • • • i* = 2 + 2cj, j* = 2 + 2ci, i* = 2 + 2c(2 + 2ci*) i* = 2/(1-2c) = j*, πF = 8/(1-2c), Profit per partner, πi.j= 4/(1-2c) > (3-2c)/(1-c)2 in S.S. Note that output is also lower in the strategic setting
– Partnership game – Cournot game
• Mixed strategy equilibria
– Conceptualization – How to solve – Application: Iran and IAEA
Key Terms and Definitions
• Set of rationalizable strategies
• Pareto efficiency
Partnership Game: Setting
• • • • 2 Partners in business Complete information about the situation Simultaneous moves (no verifiable contract) Two players: i, j
– Each expends effort: i, j – Payoffs are: πi = 2(i + j + cij) - i2 πj = 2(i + j + cij) - j2
• Interpret these equations
Partnership Game: Analysis
• Payoffs are:
Contextual Strategic Issues • Complementarity
– Agents share the benefits of each other’s actions
• Substitutability
– Agents share the costs of each other’s actions
Externalities: The Dirty Little Secret
When One Person’s Actions Affect Others’ Welfare
• • • • • •
Classic Examples Pollution Lightening rods Security Grades and curving LaTeX and presentations Cooperation and competition
– Best response complete
• Set includes i’s best response to all others’ strategies
Quick Congruity Quiz
Player 2 Left Player 1 Top Middle Down 3,4 2,5 8,2 Center 2,3 -1,8 4,4 Right 9,15 4,0 5,0
GOV 2005: Game Theory
Section 2: Externalities
Alexis Diamond adiamond@fas.harvard.edu
Agenda
• Key terms and definitions • Complementarity and cross-partial derivatives)