98% of all statistics are made up.
A give-some game is a type of prisoner’s dilemma. Suppose Ahmed has 2 dirham (legal tender in Morocco). He can give the money to Hassan, in which case its value doubles. If each gives to the other, both end up with 4 dirham. If both hold, they each have 2. If one holds while the other gives, the holder has 6 and the giver has nothing.
The standard economic analysis notes that no matter what the other player does, a holder is better off by 2 dirham. For any conceivable probability that the other player is a giver, the expected value of holding beats the expected value of giving. When calculating the expected values for the strategies of giving and holding, the player assumes that he has a choice between the two, and that this choice is decoupled from the probability of giving used in the calculation. In other words, the calculating player assumes that the probability of giving can be used to predict the choices of others, but not his own.
Ahmed, having convinced himself that holding beats giving every time, holds. Since Hassan goes through the same calculations, he too chooses to hold, and both end up with 2 dirham. The effort going into the calculations turns out to be moot because that probability of giving ends up being 0 due to each player’s choice to hold. Indeed, the standard economic analysis says that expected value need not be calculated as products of probability and value in order to be maximized. Ahmed and Hassan only need to see that 6 > 4 and that 2 > 0 to understand that holding dominates.
There is a different analysis, which takes the possibility of different probabilities seriously. And indeed it should, because the empirically observed probabilities for this type of game vary all over the map. Humor yourself and suspend your deep-rooted conviction that you have a free choice between giving and holding. Suppose your behavior just happens to you as a result of all the causal forces impinging on your mind-brain-body system. You feel like having a choice, but that’s another story, one that I have told many times in posts on unfree will.
The probability of giving may vary from 0 to 1 and players are paired randomly. The probability of 2 givers is p^2, 2 holders is (1-p)^2, 1 giver and 1 holder is p x (1-p). The expected value of giving is now p^2 x 4 + p x (1-p) x S, and the expected value of holding is p^2 x 2 + p x (1-p) x 6. The two values are the same when p = .64. When at least 2/3 of the players are givers, giving beats holding. Recall that we’re assuming that a person cannot choose to give independently of the aggregated probability of giving. He can only find out whether he is a giver or a holder by observing his behavior. This may be an unsettling vision, but it eliminates the very awkward idea that probabilities (i.e., causes) affect only the behavior of other people but not yours. Would it not be utterly irrational – and narcissistic – if each individual claimed free will while denying it to others?
Here’s a figure to illustrate the person-and-probability effect. The blue (black) lines show the expected values on the assumption of free (determined) choice; the solid (dashed) lines show the expected values for giving (holding). The probability of giving (cooperation) is shown in the x-axis; money made (summed) in the y.
The solid line of the second figure shows the probability of giving at which the expected value of giving is the same as the expected value of holding as a function of the difficulty of the game, which is defined as a ratio of differences between payoffs. The standard labels are T for unilateral holding, R for mutual giving, P for mutual holding, and S for unilateral giving. The game makes giving difficult inasmuch as (R-P)/(T-S) is low. Deriving p (the probability of indifference) is somewhat cumbersome; p = ((T-S-2P) + √((S-T-2P)^2 + 4P(R-S-P+T)) / 2(R-S-P+T). The figure also shows an upward sloping dashed line. This line represents values for the assumed probability that the other person will choose the same strategy (whichever it will be) as the player himself. If a player’s perceived probability that the other person will choose as he himself does falls on the dashed line, the player is indifferent between giving and holding. This probability is calculated as 1/(1 + (R-P)/(T-S)).