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Game Theory

Game Theory. By Hu Honggang 11121926 Zong Jiahui 12122511 Ge Yao 12122522 Ge Yajing 12122489. Mathematics in daily life. Outline. The Tian Ji Racing Prisoners’ Dilemma Penalty kick Auctions . The Tian Ji Racing. Part one. By Hu Honggang

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Game Theory

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  1. Game Theory By HuHonggang 11121926 ZongJiahui 12122511 Ge Yao 12122522 GeYajing 12122489 Mathematics in daily life

  2. Outline • The Tian Ji Racing • Prisoners’ Dilemma • Penalty kick • Auctions

  3. The Tian Ji Racing Part one By HuHonggang 11121926

  4. 2-1 inferior superior superior medium medium inferior The Strategy of Sun Zi The Tian Ji Racing Tian Ji Qi Sun Zi

  5. The Tian Ji Racing

  6. The Tian Ji Racing • If both sides didn't know other's strategy in advance, How to make wise arrangements for both sides?

  7. The Tian Ji Racing Winner 1 point Loser -1point Tie 0 point Qi's Payoff Matrix

  8. A simple example for Analysis The Payoff Matrix of S1 Players: S1,S2 strategies: S1-4 strategies S2-3 strategies How to choose the most favorable strategy

  9. The Tian Ji Racing The Mathematical Idea The Problem Feature: Two-Person Game The assumptions Both sides are rational without fluke mind when playing a game. The method to choose the most favorable strategy: Making The Worst Situation The Best

  10. The Tian Ji Racing Winner 1 point Loser -1 point Tie 0 point Rock-Scissors-Paper

  11. Prisoners’ Dilemma Part two By ZongJiahui 12122511

  12. Let us play a game Premise: • Without showing your neighbor what you are doing, put it in the box below either the letter Alpha or the letter Beta. • Think of this of a grade bid. • You will be randomly paired form with another form and neither you nor your pair will know whom you were paired.

  13. Here’s how the grades may be assigned for the class: Pair Me

  14. Prisoners’ Dilemma • Emphasize : There may be bad reasons but there's no wrong answers.

  15. The classic example of game theory——Prisoners’ Dilemma Rational choices by rational players can lead to bad outcomes.

  16. Prisoners’ Dilemma • There are two accused crooks, they're in separate cells and they're being interviewed separately. Besides, they're both told that if neither of them rats the other guy out, they'll go to jail for a year. If they both rat each other out, they'll end up in jail for two years, but if you rat the other guy out and he doesn't rat you out, then you will go home free and he'll go to jail for five years.

  17. Prisoners’ Dilemma

  18. Gaming model——Mathematical analysis on the problems of the strategy • Players: I={1,2} • Strategies: Si • payoff function: Hi(S) situation set: S={S1, S2} Matrix game: G=(S1,S2;A)

  19. Examples in life——who tidy up the dorm

  20. Other examples • Divorce struggles • Price competition • Global warming • Carbon emission • ……

  21. Prisoners’ Dilemma What remedies do we see? • Communication • Contract • Repeated interaction

  22. Penalty kick Part three By Ge Yao 12122522

  23. Penalty kick

  24. Penalty kick Zero-sum game Mixed strategy Nash equilibrium

  25. Goalie shooter 1:Assuming these numbers are correct. 2:Ignore the possibility that the goal keeper could stay put. 3:The idea of dominant strategies,neither one has a dominated strategy.

  26. To figure out what my expected payoff is depending on what I believe the goalie will do 1:The horizontal axis is my belief which means the probability that the goalie dives to the right. 2:The vertical axes mean payoff. 10 10 E(L,p(r)) 8 8 6 6 E(M,p(r)) 4 4 E(R,p(r)) 2 2 0 1 Belief P(r)

  27. Penalty kick conclusions 1:Middle is not a best response to any belief. 2:Do not choose a stratrgy that is never a best response to any belief.

  28. In the reality What is missing here?

  29. Penalty kick consideration 1:you are right-handed or left-handed 2:speed

  30. Penalty kick Real numbers 1:Ignore middle 2:Left is natural direction

  31. Auctions Part four By GeYajing 12122489

  32. Auctions you don't necessarily know what are the payoffs of the other people involved in the game or strategic situation.

  33. Auctions The first thing I wanted to distinguish are two extremes. • common values • private values These are extremes and most things lie in between.

  34. Auctions common value Sale has the same value for whoever buys it. But that doesn't mean they're all going to be prepared to bid the same amount because they may not know what that value is.

  35. Auctions private value The idea is that the value of the good at hand, not only is it different for everybody, but my valuation of this good has no bearing whatsoever on your value for the good, and your value for the good has no bearing whatsoever on my value for the good.

  36. Auctions Let's talk about this auction for a jars. So what we're going to do is we're going to have people bid for the value in the jar. What we find, by a lot, is that the winning bid was much, much greater than the true value. The name is the "winner's curse."

  37. Auctions why it is we fall into a winner‘s curse？ the winner isn't going to be the person who estimated it correctly. he winner's going to be way out here somewhere. The winner is going to be way up in the right hand tail. On average, the winning bid is going to be much, much bigger than the truth. The biggest error is typically going to be way out in this right tail and that's going to mean people are going to lose money.

  38. Auctions So what's the relevant estimate? The relevant estimate of the number of coins in the jar for you when you're bidding, how many coins do I think is in this jar given my shaking of it given the supposition that I might win the auction. I should bid the number of coins I would think were in the jar if I won (less a few). Provided you bid as if you know you won, when you win you're not going to be disappointed

  39. Double auction Background: double auction, buyers and sellers of their valuation of goods, Vb and Vs, after the two sides also put forward their offer, Pb and Ps, when Pb>Ps transaction, and transaction prices for the average number of buyers and sellers offer. Hypothesis: Vb and Vs obey uniform distribution on [0,1] Question: what is the strategy? What deal?

  40. Auctions For the buyer, the utility: For the seller, the utility: The parties select their offer, so that the utility maximization

  41. Auctions not transaction For the buyer ： max(Vb-(Pb+Ps)/2)Prob{Pb≥Ps }+0·Prob{Pb<Ps } =max(Vb-(Pb+E(Ps(Vs)| Pb≥Ps))/2)Prob{Pb≥Ps } transaction The seller's expected price SurposePs(Vs)=as+csVs Pb(Vb)=ab+cbVb Then max(Vb-(Pb+ (as +Pb)/2)/2)( Pb- as)/ cs So： Pb(Vb)=1/3as+2/3Vb ①

  42. Auctions Similarly we can get for the seller : max( (Ps+E(Pb(Vb)| Pb≥Ps))/2- Vs)Prob{Pb≥Ps } max( (Ps+ (Ps+ab+cb)/2)/2- Vs)( ab+cb-Ps)/ cb So： Pb(Vb)=1/3(ab+cb)+2/3Vs ②

  43. Auctions ①and②, the solution: Pb=2/3Vb+1/12 Ps=2/3Vs+1/4 This solution for both sides of the bidding strategy. If the transaction succeed: Pb≥Ps So Vb- Vs≥1/4 It can occur when thebuyermore than the seller1/4 valuation.

  44. Auctions transaction Vb Vb- Vs=1/4 1 The potential transaction, transaction can be realized through negotiations between the two sides 1/4 Vs 0 1

  45. Game Theory Conclusions

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