The most important factor that decides f is the reversibility of the performance. (1) For example, scores in a competition or examination is not reversable but scores in a game is reversable through Save and Load Method. (1)

First we consider the case that "the process is not reversable".

It means that only one results will be put into f(x). (1) The output of f(x) is directly put into comparason. Assume all factor (elements of input set) is

*partially proportional*to the score obtained. As a result, when one of the factor goes to the "good" side (is

*expected*to make an increase in score), it must be strictly increasing. As a result, "assessment quality" is nearly equivlent to the score obtained. (3)

When the process is reversable, we have to consider one more factor: the overall score is counted as sum of each play or best play among the many trials. (2)

Case 1: The process is reversable, but only the best one is counted. (e.g. sports, musical game)

In this case the "final score obtained" will be man{f(x1), f(x2)...f(xn)}.

Assume the score that can be obtained by a player with his

*best*performance or

*highest obtainable*score be f(k)=S.

When number of plays is infinite-ly high, it is expected the the score obtained will be S. And within a finite number of plays, a trend that tends to S is also expected. As result, the

*growth rate*of the score shows the importance of repeated exercise on the process.(2)

For example, the exercise brings improvements on the performance at the rate of (x^1/2), f can be a quardratic function so that the overall exercise is directly proportional to the score obtained.

Another exmaple is the musical game, when a broken combo is completed, the scores multiplies geometrically. However the probabilty of complete one more broken combo is much smaller than the original performance, which is approximately "quadratic effort" to get "quardratic score". (2)

Case 2: The process is reversable, and there's another column as the "overall score" of score obtained for each process.

In this case, the final score will be Σf(xi).

As long as the score obtained is

*positive and is bounded*, to a certain extent an infinite number of play is far important than score obtained in each process. (1)

However in reality we can't play an infinite number of the process, which enable us to find the equilibrium between number of plays and quality of plays. (2)

Case i: Quality >> Quantity

In this case we assume that quantity is negligible when compared to quality. A tiny change in quality will lead to an irretrievable change in total score. That is, f(x)>>f(x-Δx) or even f(x)>>kf(x-Δx).

However "quality" can be improved in terms of "quantity". i.e., through lots of exercise the quality can be improved. This surely violates the irretrievability of quality. Therefore the improvement of quality through quantity can't be directly shown in the score obtained, the only solution to f is that, the final score is the

*mean*of the previous score. For example, a win rate of 70% in 1000 plays is much better than 60% in 10000 plays. (Of course, when quantity is not large enough comparason in this case is meaningless under disturbances of small chaos.)

Case ii: Quality > Quantity

In this case quality is important than quantity where the difference can be offsets by more plays. Considering the improvement of quality by increasing quantity f can be (growth rate of quality by quantity)(growth rate of quality).

For the case Quality =< Quantity it's similar to case ii.

When Quality is much less important than Quantity, where 2min(f)>max(f). It's quite obvious that whenever max(f) > min (f), there exist m>n such that m min(f) < n max (f). Then we can conclude that min (f) = max (f), i.e., f is a constant. When the score shows somehow like "play count", quality is not important at all. (8)

d)Now extend the co-domain of the process into an n-space vector. Give examples that how f process and affect the coming decision of controlling invariate x. Discuss the feasibility to put a continous movement of controller (reality player) in the computer / processor, and after a strict process (i.e., f(x) is fixed), to give the expected results. (Hint: Chaos Theory may be involved.) (43M)

I'm afraid that I can't really answer this one since this is far mor difficult. However we can surely answer no to the second half of the question provided that random decision is made by processor. If random decision is made, the "score" obtained after a certain process (f) must differs between a more preferred result and a less one. As a result the difference in cabability cause different decision in later stage of the play.

Last part of the question : Paper-scissor-stone is a very simple game that paper < scissor < stone < paper. We can see that there's no stright order between the three signs. Inversely the "win rate" can't reflect which of the signs were used most.

i)Show that when your opponent plays perfectly random, you chance to win is 1/3. (8M)

ii)Show the relationship between inversability of score and information and the inordered nature of the process. (12M)

iii)Account the possibility to create a mathematical tool towards cyclic inequalities system (i.e., a>b>c>a or more variables, not the inequalities involving cyclic functions) to process such scores. (30M)

part d,e is kept open...

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