This article from Not Exactly Rocket Science discusses an experiment studying “competition” between the left and right sides of the brain. Subjects in the experiment had to pick up an object placed at different points on a table and what was observed was which hand they used depending on where the object was. The article makes this observation in passing.

they always used the nearest hand to pick up targets at the far edges of the table, but they used either hand for those near the middle. Their reaction times were slower when they had to choose which hand to use, and particularly if the target was near the centre of the table.

This much is expected, but it supports the idea that the brain is choosing between possible movements associated with each hand. At the centre of the table, when the choice is least clear, it takes longer to come down on one hand or the other.

I stopped there. Because while this sounds intuitive, there is another intuition that points squarely in the opposite direction. When the object is in the center of the table, that’s when it matters least which hand you use, so there is no reason to spend extra time thinking about it. Right? So…when you have competing intuitions you need a model.

You have to take an action, say “left” or “right” and your payoff depends on the state of the world, some number between -1 and 1. You prefer “right” when the state is positive and “left” when the state is negative and the farther away from zero is the state, the stronger is that preference. When the state is exactly zero you are indifferent.

You don’t know the state with perfect precision. Instead, you initially receive a noisy signal about the state and you have to decide whether to take action right away (and which action) or wait and get a more accurate signal. It’s costly to wait. For what values of the initial signal do you wait? Note that in this model, both of the competing intuitions are present. If your initial signal is close to zero, it is likely that the true state is close to zero so your loss from choosing the wrong action is small. Thus the gain from waiting for better information is small. On the other hand, if your initial signal is far from zero, then the new information is unlikely to affect which action you take so again the gain from waiting is small.

But now we can compute the relative gain. And the in-passing intuition quoted above is the winner.

Consider two possible values of the initial signal, both positive but one close to zero and one close to +1. In either case if you don’t wait you will take action “right.” Now consider the gain from waiting. Take any state x and let’s consider the scenario where waiting would lead you to believe that the state is x. If x is positive then you would still choose “right” and waiting would not gain anything. So fix any negative x and ask what would the gain be if waiting led you to believe that the state is x. The key observation is that for any fixed x, this gain would be the same regardless of which of the initial signals you had.

So the comparison then just boils down to comparing how likely it is to switch to x from the two different initial signals. And this comparison depends on how far to the left x is. Signals very close to -1 are much easier to reach from an initial signal close to zero than from an initial signal close to 1. And these are the signals where the gain is large. On the other hand, for x’s just to the left of zero (where the gain is small), the relative likelihood of reaching x from the two initial signals is closer to 50-50.

Formally, unless the distribution generating these signals is very strange, the distribution of payoff gains after an initial signal close to zero *first-order stochastically dominates* the distribution of payoff gains when you start close to 1. So you are always more inclined to wait when your initial signal is close to zero.

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September 28, 2010 at 12:17 am

itovertakesmeThat was a really helpful breakdown.

But I think this model is suited for when both directions cannot be simultaneously pursued. Yet it is interesting that nobody tries to grab the object with both hands. If there was value attached to grabbing the object quickly I wonder if people would use both hands.

Also, I was wondering how to model the decision between forehand/backhand in tennis. I think tennis is more complicated because preferences depending on the position of x are more complicated. -1 and 1 would obviously require the most reaction time but +/- 0.1 would require less than 0 because you cannot return a direct shot without moving to one side or the other. And of course in tennis your opponent is trying to outplay you, so you would need to model strategies of both players.

September 28, 2010 at 8:52 pm

AnonymousNice post. I think the intuition that points “squarely in the other direction” is driven in part by the idea that the gain from choosing the right action depends on the state x, which would often be appropriate. If you add this aspect to the model (the more extreme is x, the greater the payoff difference between action 1 and -1), then the result can easily go the other way.

November 25, 2010 at 9:10 pm

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