Human Behaviour in Competitions and Rationality

Many a times while pursuing my undergrad, I used to partake in part time musings in classes which I did not like but was forced to take. This thought was the result of one such time and I am sharing this here after almost 8 years. Would love to hear more from people reading this.

Experiment:

Say, a random town is selected and all its inhabitants are asked to select a number from 0–100(inclusive), and the winning number will be the number closest to the half of the average of all numbers selected, then which number would have been selected?

Assumptions: The individuals selected conform to the present definitions of ‘rationality’.

Reasons for:

1) 0  -  As half of 100 is 50, none will guess more than 50, since the maximum possible number is 100 and half of 100 is 50. Thus, the sample of numbers to be selected reduces to 0–50. Again, thinking about the other individuals, and half of 50 being 25, the rationalists will reject all numbers greater than 25 and so on. This iteration should continue until the winning number becomes 0 and this shows that every person thus assumes perfect ‘rationality’.

2) Between 0 and 50  - The number thus selected shows that the iteration is stopped after a point of time, i.e. , the assumption that everyone is rational and that everyone believes the other is rational stops after a stage(if true, to be calculated after a sizeable stock of data is available). Since, to obtain 0, it requires that everyone assumes that everyone assumes that everyone assumes … that everyone assumes that everyone is rational (an infinite nesting of those).

3) More than 50  -  I haven’t thought of a reason yet and this occurrence can be attributed to data errors, or some other attribute of human rationality unknown to me.

From the data I collected, the average of the numbers selected was 14.833.

Conclusions:

1) This shows that the assumption of rationality does hold true but to an extent as the number does not fall all the way to 0. It actually shows that it stops near the 3rd iteration which is 12.5.

2) As opposed to the rationale for invisible hand of the markets given by the Adam Smith, where he said that the maximising of each individual’s own utility would necessarily lead to the best result for the society, and as an extension to the work done in Game theory where it has been shown that the ‘best’ result for an individual in a competing game might not result in the ‘best’ solution for the society as a whole, case in point  - cartels(though, I have analysed only the optimal solution in the game of Prisoner’s Dilemma), this study shows the depth of human rationality. The study shows how actually an optimal solution is reached as opposed to the just the final selection; we get to know in a competition where humans have to make the best choices by predicting others’ choices, to what extent and depth does human rationality extend to.

3) Still, it would be unwise to structure human rationality on the basis of this study, but it does throw light upon the capability of human decision making and human rationality.

Implications:

These conclusions have widespread implications in all forms of competitive games including stock markets, arms race by countries etc. Here I will analyse only one  -  Asset Bubbles.

The above study if applied in speculation in the asset market can help one in making a lot of money with a caveat  -  it can also bring you below the Indian poverty line, which is around $0.5/day .

Say you guess the value of an asset for pure speculation to be of G, which is between 0 and 100, and that is your reservation price (price at which you are indifferent between buying and not buying) and the actual market price is ½ of all guesses. So, if G=5/share and actual price turns out to be 12.5/share, then you are a profit as you can sell for a profit of 7.5(12.5–5)/share. Thus, at any price, G, less than 12.5/share, you profit. But as your reservation price is 5/share you would not buy at any price greater than 5/share, thus losing out on profits.

You can see there is an incentive to bid (guess G) higher than is suggested by appeal to common knowledge of rationality. You want to participate in this market (this game) to make money. You can’t make money by sitting on the sidelines. But that is exactly where you’ll be for sure if you guess G=0.

This is still the game as was studied. I have just interpreted the guesses and the winning value in a specific way. So we know the “right” price of the stock based on an argument of common knowledge of rationality is 0. But we also know that P (1/2 of the average of guesses) will not be 0. In fact it is likely to be close to 20. Everyone who is able to buy the stock at a price below P can make a profit and they’re rational to do so. It is not irrational to set G above zero. In fact it can be a very smart thing to do (in this game).

Therefore, a speculative bubble for a worthless stock can develop for which the price is far above the “right” price. Many market participants are behaving rationally. The bubble exists because an assumption of common knowledge of rationality does not hold. But an asset bubble is not a one-shot game. Players buy and sell multiple times. Eventually additional iterations of assumptions of rationality emerge. The price begins to fall. The bubble bursts, the price goes to zero, and everyone becomes a bear.

The version of the game presented here is different than the one presented previously. In particular, the payoffs are different. In the version presented before, there was a winner  -  the player(s) that guessed closest to 1/2 of the average of the guesses. In this, many players can profit (by differing amounts). Though, someone may make the most money (per share), I haven’t really defined a “winner.” Payoffs (incentives) can change strategy. It is therefore possible that 1/2 of the average will be different in the two versions of the game. Strictly speaking, it is not correct to assume that a number near 12.5 will be 1/2 of the average in this version even if it is for the previous version. Nevertheless, I’d bet a fortune that 1/2 of the average will not be zero in either game played by a population not familiar with it. After that population plays many times, the answer will likely tend toward zero. How many iterations will it take to converge? I do not know.