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Game Theory Mathematics in daily life By Hu Honggang 11121926 Zong Jiahui 12122511 Ge Yao 12122522 Ge Yajing 12122489

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Outline The Tian Ji Racing Prisoners’ Dilemma Penalty kick Auctions

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The Tian Ji Racing By Hu Honggang 11121926

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

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The Tian Ji Racing

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If both sides didn't know other's strategy in advance, How to make wise arrangements for both sides?

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The Tian Ji Racing (3,2,1)(3,1,2)(2,3,1)(2,1,3)(1,2,3)(1,3,2) (III,II,I)31111 (III,I,II)13111 (II,III,I)13111 (II,I,III)11311 (I,II,III)11131 (I,III,II)11113 Qi's Payoff Matrix Winner 1 point Loser -1point Tie 0 point Winner 1 point Loser -1point Tie 0 point

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A simple example for Analysis Players: S1,S2 strategies: S1-4 strategies S2-3 strategies The Payoff Matrix of S1

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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:

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The Tian Ji Racing Rock-Scissors-Paper RockScissorsPaper Rock01 Scissors01 Paper10 Winner 1 point Loser -1 point Tie 0 point Winner 1 point Loser -1 point Tie 0 point

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Prisoners’ Dilemma By Zong Jiahui 12122511

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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.

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Here’s how the grades may be assigned for the class: αβ α B-, B-A, C β C, AB+, B+ Pair Me

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

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

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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.

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Prisoners’ Dilemma Prisoner B stays silent (cooperates) Prisoner B betrays (defects) Prisoner A stays silent (cooperates) Each serves 1 year Prisoner A: 5 years Prisoner B: goes free Prisoner A betrays (defects) Prisoner A: goes free Prisoner B: 5 years Each serves 2 years

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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} situation set: S={S1, S2} Matrix game: G=(S1,S2;A)

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Examples in life——who tidy up the dorm

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Other examples Divorce struggles Price competition Global warming Carbon emission ……

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Prisoners’ Dilemma What remedies do we see? CommunicationContract Repeated interaction

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Penalty kick By Ge Yao 12122522

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Penalty kick

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Zero-sum game Mixed strategy Nash equilibrium

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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. l r L 4, -4 9,-9 M 6,-6 R 9,-9 4,- 4 Goalie

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

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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.

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What is missing here? In the reality

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

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Penalty kick Real numbers 1:Ignore middle 2:Left is natural direction l r L 63.6 94.4 R 99.3 43.7

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Auctions By Ge Yajing 12122489

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

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

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Auctions common value Sale has the same value for whoever buys it. 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. 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.

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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. 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.

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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."

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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. 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. 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. 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.

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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. 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). 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 Provided you bid as if you know you won, when you win you're not going to be disappointed

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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?

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

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Auctions Similarly we can get for the seller : max( (P s +E(P b (V b )| P b ≥P s ))/2- V s )Prob{P b ≥P s } max( (P s + (P s +a b +c b )/2)/2- V s )( a b +c b -P s )/ c b So ： P b (V b )=1/3(a b +c b )+2/3V s ②

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Auctions ① and ②, the solution: P b =2/3V b +1/12 P s =2/3V s +1/4 This solution for both sides of the bidding strategy. If the transaction succeed: P b ≥P s So V b - V s ≥1/4 It can occur when the buyer more than the seller 1/4 valuation.

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Auctions VbVb VsVs 1/4 1 10 transaction The potential transaction, transaction can be realized through neg otiations between the two sides V b - V s =1/4

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Game Theory

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