ECON450 FINAL 2017

Posted by ECON爱好者 on March 31, 2017   Game Theory   econ450

重要的事情说三遍。

Practice exam, Practice exam, Practice exam

刷题，刷题，刷题

What do top students do differently? | Douglas Barton |

  1. not IQ
  2. not work hard
  3. do Practice exam

计划

  1. 首先，把近些年 last final 做好。
  2. 然后，复习近些年的midterm。
  3. 最后，根据做错的题目，阅读slides。
  4. 考试的时候，永远从最简单的题目，做起。不会的，难得，就留到后面。交卷前，一定检查。

midterm 1

Types of games

  • prisoner’s dilemma,
  • a battle of the sexes,
    • a pure coordination game
    • 2017 midterm1 q6
    • 合作，就获利，不合作，吃亏
  • a zero sum game
    • 2017 Midterm1 q1 A zero sum game (when the sum of all players payoff s for any strategy pro le is always equal to zero) has Nash equilibrium.

Static Game

  • weakly dominated
  • strictly dominated,
    • 2017 midterm1 q8
  • Solve the game by iterated elimination of strictly dominated strategies,
  • Equilibrium not payoff, is strategies.

Credible threat

  • Final – 2011 q1 In a sequential game, it can be useful for a player to have the ability to credibly commit to an action that would result in a disastrous outcome for all players

5.Nash Equilibrium in Pure Strategies for Continuous Strategy Spaces - Cournot/lecnotes5

5.1 - 2017 midterm1 q9

  • 这一题，双方有不一样的成本，所以，best response 不一样。
  • each payoff function.
  • Best response function
  • the Nash Equilibrium quantities
  • quantity should produce

6.mixed strategy/lecnotes6

  • 2017 midterm1 q4 In a mixed strategy equilibrium all pure strategies that are played with a strictly positive probability provide the same expected payoff against the other players’ strategies.

6.1 The Blue Chicken /2011 final q4

  • pure strategy
  • mixed strategy
    • 如果有一方A的payoff 结构变了。那不是他A的probability p要变。而是对方B要变q，从而让他A在两个action之间indifferent。
  • 警察抓醉驾司机的例子，也是一样。

9.Coase Theorem and Property Rights/lecnotes9

9.1 2017 midterm1 q8

  • the payoffs under an assignment of property rights
  • describe the range of Coasean contracts that could be written between the two farmers
  • ensure the socially desirable outcome.
  • 加总payoff最大的是socially desirable outcome.
  • 双方可以谈判，一方让渡一些好处给另一方，从低收益的NE，跳到高收益的cell。

Midterm 2

10. Voting / lecnotes10

2017 Midterm2 q6

  • vote honestly (they vote for the alternative that they rank the highest among the choices available),

  • votes strategically (aiming to get a decision that is the highest in terms of their personal ranking)
    • 有时候，自己的候选人会输，被选调了。
    • 退而求其次，要确保自己次优选择，把票给次优选择。
    • 不让自己的最差选择赢。
  • maximize payoff, not only vote my first candidate, but also second candidate, not my least favorite candidate
  • a Condorcet candidate?
    • 单挑，都能赢
    • 被群殴，就输的人。
    • 因为猪队友，分走了票。和他类似的队友，会分走一部分同类型选民的票。

11. Sequential Game/ Intro - Subgame Perfection

11.1 2017 midterm2 q1-5

  • What is the strategy?
    • plan of actions
  • the path of play
    • 仅仅是最终的路径
  • the subgame perfect Nash Equilibrium
    • 要包括最终路径，和其他off path的strategy，确保大家在每一个subgame里面，都是best response。
  • Payoff function / preference
  • Best response function
    • 我的payoff是你的action的function。

11.2 Final – 2011 q2

写出subgame perfect strategy

11. Dynamic Games of Complete Information / lecnotes11

2017 midterm2 q7

dictator and ultimatum games， 独裁者和最后通牒

Dynamic games of complete information

  • Extensive-form representation

  • Subgame-perfect Nash equilibrium
  • 在每一个subgame，都是NE/不愿意deviat, 都是best response
  • 不是写结果的payoff
  • equilibrium是一组strategies。 甲会做什么，乙会做什么。
  • 理论上的结果和实验结果不一样。
    • 可能是payoff的假设不对。
      • Player 3 derives the following utility from a distribution (k1, k2, k3):
      • 这个也可能是错的 utility function form is misspecified.
    • 可能not rational optimizors, or make errors
  • Game tree
  • backward induction. 2017 midterm2 q7 p4

12. Infinite Strategy Space - Stackelberg /lecnotes12 挑战者和垄断者

  • simultaneous Cournot Game
    • separately found the best response functions of the two players and simultaneously solved them for q1 and q2
  • Stackelberg
    • In dynamic games with finite strategy spaces, we worked by backward induction to find the best responses of the players who move later in the game, and took that into account in determining what the best move of the player playing before that is.

12.1 2017 midterm2 q8

  • 各自有reservation value (100, 1000)。
  • Backward induction
    • firm 2 maximize profit,
    • FOC, Best response function w2 is a function of w1
  • Firm 1 choose. maximize profit looking forward with assumption of w2
    • FOC, Best response function
  • Discounting payoff

12.2 2011 final 其 Product Differentiation through Marketing

双发都有Infinite Strategy Space 选择各自的产量。 firm1 多了个是否打广告的strategy。

  • strategy space
  • game tree
  • backward induction
    • 先firm2 maximize profit， FOC
    • 然后，firm 1 look forward, take q2 into objective function. maximize. FOC
    • 算出marketing的profit和之前没有marketing的profit。比较发现成本够低，就可以了。

13.Bargaining/lecnotes13

13.1 基本设定 p2. 看图说话

  • k， h是比例，也是bargaining power。

  • y是第二个人，x是第一个人的收益。

  • b和a，是outside opportunity或者reservation value。

  • 最终结果。证明（一般是最后的人，得到所有surplus）， 但是如果有discount， 那么耐心就起作用了。 。

  • A player’s bargaining power is inversely proportional to its degree of impatience.
    • δ is called the discount rate and β=1/(1+δ) is called the discount factor.
    • Note that 0<β<1
    • one becomes very impatient (as δ goes to infinity) the limit of β goes to zero, meaning that the player does not care at all about future consumption.

  • 越耐心的人，获得的比例越高。

  • The Game Tree

    • Backward Induction (SPNE)
  • The Naïve Sequential Bargaining Game
    • Striking result – all gains to last person to make an offer (add one round of offer to see what happens!)
    • All of the bargaining power due to the time constraint
      • 耐心很重要

  • Sequential Bargaining with Discounting

  •  In this set up, having a lousy outside opportunity/low discount rate means that one is less affected by the loss of time implied by a rejection and moving to the next round. It implies MORE bargaining power!

13.2 2011 final Q6) 15 points. Pre‐Bargaining Investments

  • over a net surplus (V‐RF‐RU)=100.

  • RF and RU are the firm and union’s respective opportunity cost of reaching an agreement

  • Let pF and pU be the player’s respective bargaining powers.

    • pF + pU =1 . 比例，份额。

可以写出各自的payoff functions，

13.2.1 firm option

firm 是否愿意在投资，取决于他的bargaining power。 投资前后相比，谁赚更多的钱。

13.2.2 Union option

看是否找下一家企业谈判。

他会希望，越谈，价$R_u$越高。

14.Dynamic Games of Complete and Imperfect Information/lecnotes14

这个比较难，一般不考。如果要考，可以看一个例子，在笔记上或者slide上，

先写一个normal form给information set。

  • Backward induction

  • Starting with those smallest subgames

  • Then move backward until the root is reached

  • Dynamic games of complete and imperfect information

  • A dynamic game in which every information set contains exactly one node is called a game of perfect information.
  • A dynamic game in which some information sets contain more than one node is called a game of imperfect information.

15.repeat game /lecnotes15

15.1 如果一个stage game有两个以上的NE. 可以利用其中一个较低收益的NE （1，1），作为threat去要求对方，在前一期，选择social optimal的strategy。然后，最后一期，再都选择较高收益的NE（4，4），作为承诺。因为NE（1，1）和NE(4,4),都是NE， 所以，credible threat。

如果，不合作，那么，就用较低收益的NE，作为惩罚。

15.2 Grim strategy/ trigger strategy

前面一直合作，那么大家停留的收益较高的NE（4，4） 比如说各自收益4，只要对方cheating， 那就再也不合作了，一直用惩罚的NE（1，1）

比较，合作和cheating的收益，会发现。 如果，一个人的 $β$ 足够高，比如说 $β > 1/4$。也就是说， 比较耐心，那么不急于赚快钱。

如果，$β < 1/4$， 合作和cheating相比，就不如cheating了。就像是网红，喜欢立马变现。

15.3 Tit-For-Tat 以眼还眼以牙还牙 策略

cheating一次，就惩罚一次，如果接受惩罚，那么，继续回到合作轨道。

计算合作和cheating一次或者两次的收益，可以发现不同的SPNE。

而且，这些都和$β >< 1/4$有关。

16.Static Games of Incomplete Information/lecnotes16

  • one player is uncertain about the rules of the game „

    • Who the other players are„

    • The strategies available to other players

    • Or most commonly, the other player’s payoff function

16.1 The Harsanyi Transformation

  • Harsanyi suggested that introducing “Nature” as a first player who (randomly) decides on the player’s type was a simple way of handling this problem.

  • 让自然Nature成为第三个 player。nature决定概率 P。

  • 与此同时，其中某些不确定的信息，通过player的type来描述。

  • player是哪一类的type，由nature来决定概率 p。

  • The strategies of the Incumbent must cover the possibility that she is a low cost type or a high cost type.

16.2 Bayesian Nash Equilibrium

p12的例子，就很清楚了。

  • Expected Payoffs – The Normal Form

  • The Normal Form Representation
    • The first number of each cell is the Expected Payoff of Entrant

  • 用normal form加上 expected payoff 基于不同type的probability， 来找最优解。

  • A (pure strategy) Bayesian N.E. is a strategy profile such that all players are adopting strategies from which they do not wish to unilaterally deviate regardless of the other players’ type and choice of strategies

16.3 2011 final q3 / Traveler’s Dilemma in Monetary Payoffs

  • 要用到概率，算出在每一种情况下的 expected payoff。

17.Auction/lecnotes17

An auction is simply a “mechanism” or “institution” to facilitate the exchange of goods between buyers and sellers

Common Forms of Auctions

  •  Ascending; Descending or Sealed (Simultaneous) Bids Bids
  •  Examples:
    •  English Art Auction; Ebay (ascending first price)
    •  Dutch Flower Auction (descending first price) Pt(Fi
    •  Procurement (First Price Sealed Bid)

Why Use Auctions??

  • Single buyer or seller
  • Limited number of goods to be sold / purchased
  • Cheap and Convenient: avoids individual negotiations with a potentially large number of buyers/sellers
  • AUCTIONS CAN RESOLVE UNCERTAINTY a seller has about the value of a good to potential buyers; a buyer has about the production costs faced by potential suppliers
  • Cost minimization or profit maximization

17.3 最重要的分类： Private Value Auctions vs Common Value Auctions:

Common Value Auctions: All buyers have the same value (generally uncertain) for the object on sale (mineral rights) 容易猜错价值，出价赢了，反而亏了。赢者的诅咒。

17.4 Private Value Second Price Auction

因为，出价是决定你赢面大小，但不是你的最终出价（排第二的人的价格）。

最重要的结论

B=V is a dominant strategy.

保持初心。 B是bit， 你的offer出价。 V是你的初心，你的内在的评价。

17.5 Private Value First Price Sealed Bid Auction: Optimal Biding Strategy

很强的假设。

Assume that values are drawn from

 Given that values are uniformly distributed, if only 2 firms, you win with

  • Let $V_i$ be the (unknown) value of the OTHER firm
  • $α$ 是出价和内心评价的比例。 $ B = α * V $. 假设这是一般规律。

画图 p9， uniformly distribution 。 就清楚了。

是分布图的高度。

17.5.2 if n firms,

This probability is given by

最终结论。p21

firm maximize profit, FOC

记住这个就可以了。

P27 As N increase, the optimal bid approaches V but IT IS NEVER OPTIMAL TO BID ONE’S VALUE in a FIRST price auction

17.6 COMMON VALUE AUCTIONS and THE WINNER’S CURSE

Protection Against the Winner’s Curse

也是很强的假设。一块玉石，大家猜这个东西的价值。 赢了，也许猜错了。亏钱。

Value estimates are uniformly distributed around the true value X.

假设平均uniform发布。

[(X+5)+(X-5)]/2 = X

如果你赢了。 那么你出价大概在这个位置， Y = X-5+10 [N/(N+1)] 那么你的estimate of true value 大概是 X = Y+ 5 - 10 (N/N+1)

If you are in a group of 24 and you drew a value of $20

  • Your bid should be B=X=20 + 5 - 10 (24/25) = $15.40

17.7 总结一下 auction 最佳策略

  • 17.4
  • 17.5
  • 17.6

19.Incomplete information/Compute the Bayesian Nash Equilibrium/lecnotes19

Incomplete information

19.3. 2011 final Q5) Migrant Workers

  • standard approach is to draw normal form
  • calculate one’s expected payoff

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