The math puzzle Newcomb’s paradox is known for its conflicting answers, both logical to a given player. The game has one answer for a player utilizing the expected value theory and another utilizing strategic dominance. It has been written that about half of the participants are convinced in one answer with the others convinced in the opposite.
The game is set up for the player to pick either one box or two. In the first box there is $1000 easily seen in the clear container. The second box is opaque and contains either $1,000,000 or zero. The player can pick box B or A + B, the catch is there is another player known as the predictor. That player has already predicted the selection of the player. The predictor places money in the boxes after the prediction is made. If A + B is predicted, then there is no money in box B for a payoff of either $1000 or zero for the player. If B is predicted, then the payoff is $1,0001,000 or $1,000,000 for the player because the predictor places $1,000,000 into box B. The rub is that the prediction has been made before the player is called into the room. The player’s fate is sealed, she is trying not to go home with zero but there is no action she can take to guarantee an outcome better than $1000. Or is it?
Here are the payoffs:
The A box is always available for the player, there is no coordination necessary for the player to go home with that amount of money. The top two selections in the box above are when the predictor selects A + B and the bottom two the predictor selects B only.
How the player approaches the game philosophically has an impact on strategy. A belief that the future is deterministic because the predictor has already sealed the player’s fate makes the calculation of expected value more difficult to embrace. Calculating the probability of future outcomes when the prediction is in the past seems futile. At this point the best guess for the player is to take the $1000 because her mindset is that her actions have no impact on the outcome so the $1000 is strategically dominant to zero.
If the player approaches the game with one of free will and believes that she is in fact playing a game rather than simply learning a predetermined outcome, then the selection of box B is the only answer because it has the highest expected utility. If the calculated odds do not go below 1% chance of the predictor choosing B, then there is a positive expected return.
Strategic or Expected?
Asset allocation follows a similar path. The numbers presented in this game resemble the outcomes of long-term investments in stocks versus short term bonds on an after inflation or nominal basis. Investors have a philosophical problem grappling with the fact that equities build after tax wealth over a long period of time and short-term bonds do not. In this case the rub is the time frame and path dependency.
An investor viewing the market as a series of short-term time frames is going to approach the market from a deterministic mindset. The investor believes philosophically that the market will do whatever it will do in the next year and it is out of her control. This encourages the strategic dominant formula of taking the positive return in short term bonds because it is better than zero or a loss.
Meanwhile another investor is taking the view that they control their own asset allocation and the derived expected value of the probability of returns. Rather than worrying about headlines out of their control in the short run, the selection of a long-term strategy allows her the free will to accept risk.
Newcomb’s paradox is an interesting game because it merges philosophy and mathematics. The belief in long term asset allocation returns is a paradox to some, a game to others and a philosophy to the thoughtful.