博弈论基础 10.21

Two Kinds of Societies
• Hunting stags is most beneficial for society but requires “trust” between the hunters in that each believes that the other is joining forces with him. • Stag Hunt game
Two Kinds of Societies
• a society of individualists who do not cooperate to achieve a better outcome • a cooperative society
• Prediction • Outcome Dominant or Risk Dominant • norms of behavior
The Tragedy of the Commons
• first-order conditions：
• giving people the freedom to make choices may make them all worse off than if those choices were somehow regulated.
Nash Equilibrium in Pure Strategies
• [D] The pure-strategy profile s∗ = (s1∗, s2∗, . . . , sn∗) ∈ S is a Nash equilibrium if s∗ is a best response to s-i∗ ,for all i∈N, that is, • vi(si∗, s-i∗ )≥vi(si′, s-i∗ ) for all s′∈Si and all i∈N.
The Tragedy of the Commons
• Hardin (1968) • the conflict over scarce resources that results from the tension between individual selfish interests and the common good • “这是一个悲剧。每个人都被锁定在一个迫使 他在有限范围内无节制地增加牲畜的制度中。 在一个信奉公地自由使用的社会中，每个人追 求他自己的最佳利益，毁灭是所有人共同奔向 的目的地 • environmental catastrophes
Pure-Strategy Nash Equilibrium in a Matrix
Evaluating the Nash Equilibria Solution
• Battle of Sex: Equilibria selection • Price Competition: Nash equilibrium Fails to exist • Tragedy of the Commons: Nash equilibrium does not guarantee Pareto optimality
Nash Equilibrium: Some Classic Applications
Two Kinds of Societies
• Two hunters, players 1 and 2, can each choose to hunt a stag (S), which provides a rather large and tasty meal, or hunt a hare (H)—also tasty, but much less filling. • Hunting stags is challenging and requires mutual cooperation. If either hunts a stag alone, the chance of success is negligible, while hunting hares is an individualistic enterprise that is not done in pairs.
博弈论基础
周亚 北京师范大学系统科学学院 zhouya@bnu.edu.cn
Pinning Down Beliefs: Nash Equilibrium
From Beliefs to Actions About dominant strategy
Nash Equilibrium in Pure Strategies
• Exercise 5.1
Nash Equilibrium in Pure Strategies
• The requirements for a Nash equilibrium: • 1. Each player is playing a best response to his beliefs. (rationality) • 2.The beliefs of the players about their opponents are correct. (Prediction)
The Tragedy of the Commons
• n players, say firms, in the world, each choosing how much to produce. Their production activity in turn consumes some of the clean air that surrounds our planet. • total amount of clean air: K • consumption of clean air for production： ki ≥ 0 • benefit ：ln(ki)
Bertrand Duopoly
Bertrand Duopoly
• Let Si = {0, ε, 2ε, 3ε, . . .} • ε = 0.01 • one monopolistic firm : • v(p) = pq − 10q = (100 − p)(p − 10) • p = 55 , q = 45 , v = $2025
Nash Equilibrium in Pure Strategies
• [P] Consider a strategy profile s∗ = (s1∗, s2∗, . . . , sn∗) . If s∗ is either • 1. a strict dominant strategy equilibrium, • 2. the unique survivor of IESDS, or • 3. the unique rationalizable strategy profile, then s∗ is the unique Nash equilibrium.
The Tragedy of the Commons
• first-order condition of his payoff function：
• player i’s best response function ：
The Tragedy of the Commons
• Two players：
ቤተ መጻሕፍቲ ባይዱ
The Tragedy of the Commons
• a pasture shared by local herders • Each herder wants to maximize his yield, increasing his herd size whenever possible. • Each additional animal has a positive effect for its herder, but the cost of that extra animal, namely degradation of the overall quality of the pasture, is shared by all the other herders. • As a consequence the individual incentive for each herder is to grow his own herd, and in the end this scenario causes tremendous losses for everyone. • Free-riders; atmosphere, oceans, rivers, fish stocks, national parks
Bertrand Duopoly
• duopoly with two firms:
• If pj > 55, BRi (pj)= 55 • If pj < 10, i will set a price that is higher than that set by firm j • If pj = 10, i will set a price that is higher than that set by firm • If pj = 10.01, BRi (pj) = 10.01 • If 55 ≥ pj ≥ 10.02 , BRi (pj) =pj - 0.01
The Tragedy of the Commons
• Pareto criterion: can we find another consumption profile that will make everyone better off? • world’s payoff function ：
• 1）A Nash equilibrium is a system of beliefs and a profile of actions for which each player is playing a best response to his beliefs and, moreover, players have correct beliefs. • 2）As a profile of strategies for which each player is choosing a best response to the strategies of all other players.
Cournot Duopoly
• i=1,2; P = a − bq; ci(qi) = ci× qi
Bertrand Duopoly
• Joseph Bertrand (1883) • firms often set prices and let consumers choose from where they will purchase • each firm posts a price for otherwise identical goods. • all buyers will buy from the firm whose price is the lowest. • p = 100 − q ; ci (qi ) = 10qi
Bertrand Duopoly
• duopoly with two firms:
• Symmetric ; Two Nash Equilibria: • How to check if one profile is a NE or not?
Bertrand Duopoly
• One firm may have monopoly power, but when we let one more firm compete, and they compete on prices, then the market will behave competitively—if both choose a price of $10 then price will equal marginal costs! If we add a third and a fourth firm this will not change the outcome • This is not the case for Cournot competition, in which firms manage to obtain some market power as long as the number of firms is not too large.