Game Theory

Mathematics in daily life

By Hu Honggang 11121926

Zong Jiahui 12122511

Ge Yao 12122522

Ge Yajing 12122489

Outline

The Tian Ji Racing

Prisoners’ Dilemma

Penalty kick

Auctions

The Tian Ji Racing

Part one

By Hu Honggang 11121926

The Tian Ji Racing

Sun Zi

2-1

The Strategy of Sun Zi

inferior

superior

medium

superior

medium

inferior

Tian Ji Qi

The Tian Ji Racing

The Tian Ji Racing

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

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) 3 1 1 1 1 -1

(III,I,II) 1 3 1 1 -1 1

(II,III,I) 1 -1 3 1 1 1

(II,I,III) -1 1 1 3 1 1

(I,II,III) 1 1 -1 1 3 1

(I,III,II) 1 1 1 -1 1 3

Qi's Payoff Matrix

Winner 1 pointLoser -1point

Tie 0 point

Winner 1 pointLoser -1point

Tie 0 point

A simple example for Analysis

Players: S1,S2strategies: S1-4 strategies S2-3 strategies

6 1 8

3 2 4

9 1 10

3 0 6

The Payoff Matrix of S1

The Tian Ji RacingThe Mathematical Idea

The Problem Feature: Two-Person Game The assumptionsBoth sides are rational without fluke mind when playing a game. The method to choose the most favorable strategy:

The Tian Ji Racing

Rock-Scissors-Paper

Rock Scissors Paper

Rock 0 1 -1

Scissors -1 0 1

Paper 1 -1 0

Winner 1 pointLoser -1 point

Tie 0 point

Winner 1 pointLoser -1 point

Tie 0 point

Prisoners’ Dilemma

Part two

By Zong Jiahui12122511

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.

Here’s how the grades may be assigned for the class:

α β

α B-, B- A, C

β C, A B+, B+

Pair

Me

Prisoners’ Dilemma

Emphasize :

There may be bad reasons

but there's no wrong answers.

The classic example of game

theory

——Prisoners’ Dilemma Rational

choices by rational players can lead to bad outcomes.

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.

Prisoners’ DilemmaPrisoner B stays silent (cooperates)

Prisoner B betrays (defects)

Prisoner A stays silent (cooperates)

Each serves 1 year

Prisoner A: 5 yearsPrisoner B: goes free

Prisoner A betrays (defects)

Prisoner A: goes freePrisoner B: 5 years

Each serves 2 years

Gaming model——Mathematical analysis

on the problems of the strategyPlayers: I={1,2}

Strategies: Si

payoff function: Hi(S)

situation set: S={S1, S2}

Matrix game: G=(S1,S2;A)

)15

02(

A

Examples in life——who tidy up the dorm

Other examples

Divorce struggles

Price competition

Global warming

Carbon emission

……

Prisoners’ Dilemma

What remedies do we see?

Communication

Contract

Repeated interaction

Penalty kickPart three

By Ge Yao12122522

Penalty kick

Penalty kick

Zero-sum game

Mixed strategy Nash equilibrium

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 6,-6

R 9,-9 4,- 4

Goalie

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

10

12

BeliefP(r)

E(R,p(r))

E(M,p(r))

E(L,p(r))

Penalty kickconclusions

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.

What is missing here?

In the reality

Penalty kick

1:you are right-handed or left-handed2:speed

consideration

Penalty kick

Real numbers

1:Ignore middle2:Left is natural direction

l r L

63.6 94.4

R 99.3 43.7

Auctions

Part four

By Ge Yajing 12122489

Auctions

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

Auctions

The first thing I wanted to distinguish are two extremes.

common values

private values

These are extremes and most things lie in between.

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.

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.

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."

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.

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

Double auctionBackground: 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?

Auctions

For the buyer, the utility:

For the seller, the utility:

The parties select their offer, so that the utility maximization

Auctions 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 }

Surpose Ps(Vs)=as+csVs

Pb(Vb)=ab+cbVb

transaction not transaction

The seller's expected price

Then

max(Vb-(Pb+ (as +Pb)/2)/2)( Pb- as)/ cs

So ：

Pb(Vb)=1/3as+2/3Vb①

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)+2/3Vs②

Auctions ① and ② , 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

SoVb- Vs≥1/4

It can occur when the buyer more than the seller 1/4 valuation.

Auctions Vb

Vs

1/4

1

10

transaction

The potential transaction, transaction can be realized through negotiations between the two sides

Vb- Vs=1/4

Game Theory

Conclusions