Abstract Confusions

Complexity is not a cause of confusion. It is a result of it.

Tag Archives: Game

Instant Insanity – A Graph Theoretic Wonder

Instant insanity is one the many games involving sound mathematical principles. In particular it involves graph theory to solve. One of the mathematics most powerful tool – representation theory is used. As you are aware of, what looks like a strange problem is transformed into totally a unrelated problem in another field because of the hidden relations / transformations. Often these relations transform into a treasure trove of knowledge and excitement.

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The de Finetti Game

Probability can be divided into two major topics on broad sense, Subjective probability and Objective probability. Simply said, anything we can be quantified and acceptable without any doubts is objective probability. Like rolling a die, and the chances of getting a number 4 from it, assuming the die to be a fair die although. Or, like tossing a coin, probability of getting head from it. Every thing can be quantified and measured without problem. And the exact opposite is what makes subjective probability more interesting.

Things like – probability of rain on a given day, or probability of a particular stock to move up on a given day are all subjective probabilities. The sample space is not always finite.

Thus the question we would be interested in is –“How to measure the subjective probability?” Or put different, how will you objectively measure the subjective probability? The heart of this post is all about that only.

Bruno de Finetti

Bruno de Finetti, an Italian statistician has given a way to compute these. He has proposed a game to measure the subjective probability for few things. The game he defined lies in the middle ground of Probability and Psychology. There are many variations of the de Finetti game.

Say, your friend tells to you, coming out of exam hall, that he is getting a maximum score (a centum). How much is that certain now? To measure this, all one have to do is to ask few questions. Tell him that, in a bag, there are hundred balls. Out of which 98 are white and 2 are black. Given a chance, like, a thousand rupees bet, for either drawing a white ball from the bag or wait till exam results and on getting the score as 100. Assume that the answer of your friend as “draw the balls”. Now, increase the black ball count to 20 (80 White + 20 Black). And ask him to pick between drawing a ball from the bag or wait till the exam results.

If he chooses to wait for the exam result for the thousand rupee bet, increase the white ball count to 90 (90 White + 10 Black) and repeat the same test. If he wants to draw, then increase the black count to 15 (85 White + 15 Black) and repeat the experiment. End this game with a sufficient interval. Say, between 85 and 88. This means, that the subjective probability of your friend getting a centum is any where between 85 ~ 88.

The Rationale behind the game

Now, let us analyse the rationale behind the game. For the first time, when given a chance between 98:2 and “waiting till exam results”, you friend went for the draw. Meaning that, he thinks drawing the white ball is easier than getting a centum in exam. So, the subjective probability of his centum is lesser than 98% (or .98). This is the upper limit of his chances of getting centum. When you change the balls in the bag to 80:20 and ask him for a pick, he chooses to wait for the result. Meaning that, he thinks the probability of getting centum is more than choosing the ball, which was 80% (or .8). This serves the lower limit.

For the third time, when you repeat the same experiment, you are trying to find the finer details. Hope that I have explained the rationale of the de Finetti game.

Real World Experiences

I have tried these games with my friends. Most of the time, I have modified to suit my requirements. Like, say, last time, I have a bet with one my colleague over a test case. I told her, it won’t work as she thought. She resisted. Then I challenged her with an initial bet for Rs. 10 and told her that she can choose between checking the test case and proving it and get seven times the initial bet amount or she looses Rs. 30 if she checks and found what I said was actually happening. Simply put, she has the chance to check and win Rs. 70 if she wins or Rs.(-) 30, if she looses. You can even ask her choose the times she need. That way, you can come to some conclusion on her subjective probability

She doesn’t know that I was the person who coded it and exactly knows which way it would work. Now, the pitfall of this game, never challenge any one with for personal things like love, marriage and others. They seem to have sentimental attachments and can’t be assessed for subjective probability correctly.