Cooperative games and the FEH voting event

Consider the following situation.

Two competitors A and B, are selling homogeneous products on the market. (Well not under perfect competition anyway.) Obviously they are enemies that they don't communicate. One day, however, due to unknown reasons they decided to maximize the total profit that they create. The profit of each of the seller can be viewed as a function of the seller's price and the opponent's price, assuming that the demand curve is fixed.

The problem is that they don't communicate. They even do not bother to check the opponent's price (it's time wasting -- profit maximization is only the command from the higher order). The only information they have on their hand is their price and their corresponding profit on that day (note: it relies on the opponent's price on the day too). Is it possible to adjust the price daily in order to edge close to the goal?

Call the price at the $i^{th}$ day $P_{A,i}, P_{B,i}$ and the profit function $f(\cdot, \cdot)$ that takes seller's price and opponent's price and output the profit for the seller (assume that the market is symmetric in the sense that swapping the price results in the same total profit). Is it possible to find such iteration so that $f(P_{A,i}, P_{B,i})+f(P_{B,i}, P_{A,i}) \to L$, the theoratical maximum (such maximum exists by compactness)? Does it help if we know the smoothness of $f$?

My unproved claim: if $f$ is strictly increasing respect to both variables and is Lipschitz (why the hell do we need such assumption?) then such algorithms exists. Don't ask me for further proof...

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I do not intend to include any explicit mathematics here, but it seems like a simple example is unavoidable, so let us consider the following.

Consider two initial values $A_0, B_0$ from two players A and B. At round $n$, player $A$ will know the his own number $A_n$ as well as the absolute distance $|A_n-B_n|$, and vice versa. Without any communication we want to devise an algorithm so that $|A_n-B_n| \to 0$.

We give the algorithm as follows. For $k>1$ we iterate as follows:

$A_{n+1} = A_n + (-1)^n |A_n-B_n|/k$
$B_{n+1} = B_n + (-1)^{n+1}|A_n-B_n|/k$

then by induction the value $|A_n-B_n|$ decreases every 2 rounds, and hence it converges to zero by the monotone convergence theorem. (Well, if we take $k=2$ we get it sorted out in 2 rounds, and we get the exact solution. However convergence holds for any $k>1$, too.)

Here we used the fact that there are only 2 players, so we implemented something that has period 2. In general when there are $N$ players we want to implement algorithms that are genuinely the same on each of the players.

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The first example is definitely an interesting case to think about, but $f$ is too vague to deal with. For concrete non-trivial examples we may consider this one instead:

Consider $N$ base stations $X_1,...,X_N \in \mathbb{R}^2$, and the convex polygon created $\Omega = conv(X_1,...,X_N)$. We may assign a radius $r_i$ to each station so that the area $\Omega \cap B(X_i; r_i)$ is now covered by the station $X_i$.

Larger coverage, of course, takes more energy. The energy required to maintain radius $r_i$ is proportional to $r_i^{2+\alpha}$, where $\alpha \in [0,1]$. The goal is to find $(r_i)$ so that 100% coverage is attained in $\Omega$ and the energy consumption is minimized.

Again communication among stations is prohibited. The only information each station $X_i$ knows is the coverage rate in the area $\Omega \cap B(X_i; R_i)$, where $R_i$ is the distance between $X_i$ and the closest station. Can you devise an algorithm for stations to adjust their radius $r_i$ towards the goal? Does it become any simpler in the case $\alpha = 0$, or $\alpha = 1$?

(Note: since we have the information on coverage rates, we allow stations imperfect coverage in the mean time -- we call that beta testing -- the algorithm is fine as long as the coverage rate converges to 100% AND the energy consumption converges to the optimal number.)

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Above are games where players cooperate under limited information. These type of games are extremely useful in various fields (economics and engineering) and are also actively researched. It is important in the sense that communication is always expensive.

Given linear/convex/smooth/well-behaved functions there are systemic methods to deal with those, or at least approximation-ish result are available. But what if the 'player' is in fact a mix between a selfish self (takes adjustments to maximize sole profit) and a generous self (take adjustments to maximize total profit)? That could well happen during political elections...or game voting events. The only prediction we can make it that things are unpredictable.

That is precisely why flame arised when Camilla beat Lyn with much less 'votes' in FEH's voting event. Things are event worse in the sense that this is a zero sum game so that it is impossible to cooperate after all. People thought that they are going to tweak the result but were only dominated by the 'selfish self' - the large portion of players who simply play on their own pace and not even hardly optimizing - and when things did not go naturally in their way they went annoyed and shit everywhere...

And for me, it is no more than a sleepless currency-earning event because I did not find my favourite characters (Ursula, and perhaps Tana too) there. Hehehe.

re:創作者s

最近好多靈感，可以沒法整合成完整的一篇篇文章。主因還是靈感太多了。就像拉麵一樣，一天吃四五碗到最後也記不起那碗比較好吃，只能隨便寫幾句評語了事。

東京……很好玩、景點很好看，可是對我來說與人的互動價值更甚於作為首次踏足大和國參觀固有景點的樂趣，所以硬要篇成遊記放這裡實在有點強人所難。

我的17夏番列表不知不覺間變得越來越長，但最吸睛的還是小說追過來的歡迎來到實力至上主義教室和異世界食堂，再加上半年番re:creators。<<實力至上>>和我的理念在一定程度上不謀而合，回頭一看才發現和自家小說的共通點還真不少；<<異世界食堂>>的話就是看比較正常的人間料理(不會吃完會高潮那種)，在12集之內可以把異世界側寫到那個程度就和監督的功力了。<>下半部話鋒一轉，將焦點集中於creators和他們的creation上，幾集下來描寫可謂入木三分。當然和軍服小妹妹最後決一死戰在所難免，但是這場戰鬥基本上可以歸納為原創vs二創的影響力(又或者用Chaos;Child的語言來說，情報者的階級概念是否絕對－－情報創造/收集/散播的概念是否等價)－－如果監督有心在這方面下功夫這部說不定能摸到神作的邊，至少對我來說是這樣。至於blue-ray能不能賣出去當然另作別論，不過以秋葉原的宣傳攻勢來看這部絕對是他們想捧紅的大作。

當然還有棒球，足球休季期間當然就是要看棒球。甲子園又要來了，不過比起三次元的甲子園我更關心的是我的手遊甲子園隊好不容易經過一番整合之後又有人被挖走了。被挖走不是問題，唯獨被同級隊挖人這點我沒法忍受－－如果我請得動「她」和「他」代我管理就好了。可惜這個次元的距離有點遙遠，不過我會努力的。

說真的這手遊大概是繼Fanta之後我玩過最需要團隊合作的遊戲了。如果沒玩這遊戲我大概也沒有那個熱誠去創作吧。希望這次我可以徹底完本－－套用千川一句話：只有完成第一部創作那一刻，那人才真正成為一個創作者。

希望有一天，我也會成為那種Creator。

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深夜寫文果然就是會亂寫一通，呢。
21-07-2017