20 July 2014

What a shame that I didn't write anything for the last 3 months here. I don't have any issues in mathematics to talk about here, so this is a rare entries with my own updates.

Firstly a faithful congratulation toward the schoolmate in my secondary school, who won a silver medal in IMO 2014 at a relatively young age. Without doubt that he is capable and will achieve more in the future.

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RAMMASUN (09W) whose forced a typhoon signal no. 3 has just been dissipated in Vietnam. This is a rare occasion when a typhoon can develop greatly in northern South China Sea close to the coastal line. Traditionally geographical environment is not very favouring for super typhoon because the sea body is not large enough (comparing with East Sea and north west Pacific Sea) and not deep enough (when the typhoon absorbs the heat in the sea surface, warmer sea water comes up to provide more energy) to support a strong typhoon. However RAMMASUN reached 135kt before landing under the assistance of most if not all the other positive factor, and once more we can appreciate the beauty of nature here. Of course we also hope that those who are injured, homeless, or lost their beloved due to the storm, will get well soon.

09W has gone, but MATMO (10W) is coming, also with 95W and 96W. For a typhoon fans it is reasonable for us to expect more. It is worthy to mention that HKO again has an almost perfect prediction on the track of RAMMASUN in the South China Sea. Although MATMO is still in the Pacific side, it is interesting to see which observatory can make an accurate prediction on the behaviour of the storm as well as the subtropical ridge.

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Chelsea my favourite team has bought a few stars this transfer window that we will expect more next year. We have to say, even after the World Cup, that Luiz is a crucial player against the strong teams. Without him the team can't really achieve a stunning 5W1L against the other top 4 teams. Of course, 50M pounds is a good deal, then we have to find more suitable players to substitute him. Yesterday I watched the 3-2 reversal against Wimbledon where Terry score twice --- I can see many young players with potentials but yet need more real bloody experience. Want some more,Vitesse?

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I have written a set of notes for my friends on probability. After some reforming I will probably release it here. Same as the complex analysis last semester, the courses that I will take this semester --- functional analysis and manifold analysis is quite specific and I may not be able to address something easy and meaningful here, but I will try, if this is possible.

No game, no life --- game theory case analysis

No game no life is a recent anime that highlights the world that takes game contract as the highest principle, and game theory plays a superior role in the story in a very natural way. Now I would like to analysis one of the game in episode 2 as follows. For simplicity I would use Bob and Alice --- the common name in cryptography --- to describe the whole story.

Bob "Hey Alice let's play paper-scissor-rocks :) I will always play paper."

Alice "What if you don't play paper?"

Bob "If I play don't play paper you win if it's a draw [under original rules] and you draw if you lose [under original rules]. If I win you have to obey my order."

Alice "What if I get a draw?"

Bob "Then you will have to do my little favour :P"

Alice "What favour?"

Bob:

The payout table is as follows

It turned out that Alice had scissor and Bob had rock --- a draw --- with consequences equivalent to a lose as Bob defined. The analysis from Alice: based on random choices there are a 2/3 chance to win for rocks and scissors, but Bob claimed to play scissor so she'd better play scissor. Bob made a conclusion that the risk for Alice is invariant despite the rules:

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Consider the following payout matrix.

$P_1 = \begin{bmatrix}1&0&1\\1&1&0\\-1&1&0\end{bmatrix}$, $P_2 = \begin{bmatrix}1&-1&1\\1&1&-1\\-1&1&-1\end{bmatrix}$

Where $P_1$ is the matrix that Alice thought to be but $P_2$ is the real one. The matrix $P$ is defined as follows: $[P]_{ij}$ is the amount you win if you play the j-th option (vertical) and the opp plays the i-th option (horizontal). (Some textbooks may switch between you and your opp but it's just a matter of perspective on whether you, the analyzer is taking part in the game.)

By strong duality we know that the optimal strategy for both Bob and Alice should have the same expectation. Now suppose Alice has a probability vector $\vec{x}$ to play the three choices with payout matrix $P$. Then the expected outcome for Bob is given by $P\vec{x}$ for different choices from Bob. He as the payer would of course want to minimize the paid. Then we can set up the following linear program for Alice, to maximize the gain when Bob minimizes her gain among the three choices.

$\max (\min ([P\vec{x}]_i)) ~~~\text{subject to}~~~\sum [\vec{x}]_i = 1, \vec{x}\geq \vec{0}$.

And by adding dummy variable we have:

$\max z ~~~\text{subject to}~~~P\vec{x} - z\vec{b}\geq \vec{0},~\sum [\vec{x}]_i = 1,~\vec{x}\geq \vec{0}$

Where $\vec{b} = (1,1,1)^T$. This will be the optimal strategy for Alice.

Using $P_1$ we have $\vec{x}^* = (0,0.5,0.5)^T$ with expected gain $0.5$. Using $P_2$ we have the same optimal vector, but the expected gain is zero. It can be concluded that Alice had the perception that she has an advantage in the game, while actually she had not.

The matrix with four -1 and five 1, turns out to be a fair game (mathematically).

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Gambling at high stakes are of course a totally different game. There are not enough games for one to apply central limit theorem so that the result converges so that they can play in the most probable way. In these games all factors including the psychological status shall be considered and here Alice obviously did not have a good enough mind to do such analysis, so she eventually lose the game.

Can you think of other non-trivial matrix which is fair as well?