The Game for the million dollar bonus – continued
November 7, 2006
So, what is the game again? Each analyst will have to choose - simultaneously - a number between 0 and 50 (this being the possible range of the stock ShitTech in the next one year) which he thinks will be the value of the stock in a year’s time. But the reward of a million dollars will go to that analyst whose number comes closest to 90% of the average of all the numbers (values that the analysts assign to the stock).
Actually, it does not matter how many analysts were there in that group that evening. But each analyst is a player now. Also, the fact that the numbers will revealed in a year’s time from the date of the game is purely cosmetic and does not alter any characteristic of the game.
Now for the game: quite a few of the analysts in the group will reason that since the average of zero to 50 is 25 and 90% of that is around 22.5 they will either choose – in a flash – 22 or 23. Say, a lot of them choose 23. But others, a few smart ones, might divine this fact right away and expect many of the players to choose the number 23 and therefore they will go for 90% of this number (because they would anticipate number 23 as the average of the group) and hence choose 90% of 23 which would be 20.7 or rounded off to 21. Again a handful, the really bright ones amongst them, might figure this out immediately that a few smart ones would choose the number 21 (because they have anticipated the average lower at 23) and therefore this small minority of the players will further lower the average value to 21 and hence take 90% of that number, 19 as their values.
This game will thus - in all likelihood - enter into an iterative mode and if only one hand is played, i.e. they have to choose a number right away and walk away with it then this game, theoretically speaking, could take infinite time to complete. Why? Because unlike the above situation we have to assume that each player (analyst) in the group has equal intelligence and reasoning power to come up with the right answer. Therefore, each player would be guessing what others would choose to estimate the average value of the stock and hence keep altering - mentally - his choice for the value of the stock.
The reverse could also be true. In fact, in no time the game will be truncated at zero. The mental iterations to guess the average value of the group will stop at zero, because zero is the optimal response of the players. Since every one wants to choose a number that is 90% of the average value, the only way they can all do it is to choose zero, a number equal to 90% of itself. Zero will lead to the Nash equilibrium of the game.
In reality the game is much more complex exercise than that described above. Different players will be using a different algorithm of meta-reasoning to come up with an average value. Some players will be incrementally shifting their choice above or below the first or second choice average value of 25 to improve their chances. Quite a few may also choose a number greater than 25.
The game being played here is of course, a hypothetical one. It is indeed true that fifteen or so analysts did meet up in that waterhole in the City of London in late 2005 to console each other and vent their frustrations. And it is indeed true that there was an investor who overheard their comments. But no such game took place.
What is certain is that this game unless initiated by someone else will never be played by any analyst. Ironically, the entire discipline of equity analysis which so heavily borrows from Economics and the theory of Finance completely ignores game theoretic approach to decision making in the markets. We say "ironically", because this game in one form or the other will be played, and is being played even as you read this article, by thousands of investors, big and small, across the globe. Regardless of any analytic garbage that comes out of any analyst’s desk each investor is essentially trying to fathom what the other is thinking about the stock in question, and what value will he assign to it. Played out in the global capital markets this game is so complex, with so many players that Nash equilibrium of the game can never be reached in a finite time.
Reference: This particular game is inspired by the example given in the excellent book called "A Mathematician Plays the Market"(Penguin Books) by John Allen Paulos. We would recommend that book to all readers . It is a true delight .
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