Logits as a Measure of Effort

I was struck reading some materials on cognitive biases Allias paradox specifically not by the existence of the bias but by the intuitive appeal

Choose between the following two options:

1A. $24,000, with certainty. 1B. 33/34 chance of winning $27,000, and 1/34 chance of winning nothing.

And then:

2A. 34% chance of winning $24,000, and 66% chance of winning nothing. 2B. 33% chance of winning $27,000, and 67% chance of winning nothing.

People tend to have a preference for 1A > 1B and a preference for 2B > 2A

This can be shown to be irrational by: - U($24,000) > 33/34 U($27,000) + 1/34 U($0) - 0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0)

But much more so than myriad other cognitive bias I found this to have intuitive appeal and I was able to make at least a rough placement of that intuitive appeal, probability operates in log space and that logspace goes haywire around 0, 100, exactly where the sorts of intuitions above go haywire.

This idea was also highlighted here. I wanted to quantitatively drill down on this intuition however precisely because of probabilities going haywire I must make a small alteration:

Choose between the following two options:

1A. 34/34.1 chance of winning $24,000 and 1/34.1 chance of winning nothing. 1B. 33/34.1 chance of winning $27,000, and 1/34.1 chance of winning nothing.

And then:

2A. 34% chance of winning $24,000, and 66% chance of winning nothing. 2B. 33% chance of winning $27,000, and 67% chance of winning nothing.

Computing our logits: torch.logit(torch.Tensor([34/34.1])) tensor([5.8289]) torch.logit(torch.Tensor([33/34.1])) tensor([3.4012]) torch.logit(torch.Tensor([34/100])) tensor([-0.6633]) torch.logit(torch.Tensor([33/100])) tensor([-0.7082])

Now taking our current wealth at W, and our utility over wealth as log(W). Let's go ahead and add in an effort factor quantified as a change in logits that you can effect on different scenarios of similar magnitude throughout the world say working hard to get a promotion, picking up a side hustle etc. since we are specifying it relative to the magnitude of the outcomes we'll approximate it as 25K for the first and 8.5K for the second. (torch.logit(torch.Tensor([34/100])) - torch.logit(torch.Tensor([33/100]))) * 8500 tensor([381.5715]) (torch.logit(torch.Tensor([34/34.1])) - torch.logit(torch.Tensor([33/34.1]))) * 25000 tensor([60693.7539])

So we revise the scenarios as follows:

Choose between the following two options:

1A. 34/34.1 chance of winning $24,000 and 1/34.1 chance of winning nothing plus 60K logit dollars 1B. 33/34.1 chance of winning $27,000, and 1/34.1 chance of winning nothing.

And then:

2A. 34% chance of winning $24,000, and 66% chance of winning nothing plus 381 logit dollars 2B. 33% chance of winning $27,000, and 67% chance of winning nothing.

That is you are saved from putting in that many logit dollars of effort in trying to effect that outcome by simply being able to choose it instead of changing it by your own hard effort. And then if you had another scenario where you can expend your saved logit dollars of the form:

50000 * torch.sigmoid(torch.Tensor([0]) + logit_dollars_spent/25000)

That is you start out with a 50% chance of 50K$ and are able to spend logit_dollars_spent to increase the logits by 1/EV[scenario] in this case 1/25K to affect the outcome.

Then the value of your 60K logit dollars would be about \$20K or your 381 logit dollars would be worth about 190\$ all of a sudden from this framing the preference start making a lot more sense.

But what is the significance as it seems like I have just pull out a magic unit of "logit dollars" out of thin air to address the problem.

The logic of logit dollars is that effecting probabilities is done by adding logits (positive or negative)

If there is a bad outcome that happens if a coin flip lands tails and you want to make it take two coin flips instead you will need to come up with -1.09 logits to add to that probability it doesn't matter how you do it. To get it to 3 coin flips would require another -0.847 the transformation is approximate with the correction term below:

def exact_joint_logit(logit1, logit2):

"""Computes the exact logit of the conjunction of two independent probabilities."""
return logit1 + logit2 - torch.log1p(torch.exp(logit1) + torch.exp(logit2))

So it is not may not just be that our brains are bad at multiplying.

The fixed coin Bazzar

Suppose there was a market for coins and in a twist people go in looking for fixed coins that always land heads but on inspection look exactly like a normal coin, the only way to actually tell that it is fixed is to flip it and observe the outcome.

It has been ascertained by rigorous research that 50% of the merchants in the Bazzar are charlatans that try to pass off regular coins as fixed.

Suppose you are an honest merchant with several sales prospects that are willing to buy a coin for different amounts X_0, X_1, however they will only buy the coin with the probability they assign to it infact being a fixed coin. Further suppose you only have a fixed amount that you can flip coins in a day (due to severe arthritis developed over a long career of fixed coin dealing) say 10. The buyers have no way of ascertaining whether a coin is fixed other than observing it is fixed only one can observe a flip at once and no one recognizes any sellers as honest or dishonest (only tourists buy the coins and at most 1).

So you are in this situation you want to put your efforts toward the potential customers that are going to net you the most money, or specifically where your coin flips will net you the most money.

The odds and probabilities that they will assign to your coin being fixed will be as follows (given you are flipping genuine fixed coins they will always land heads)

0 coins flipped 1:1 odds, 50% probability 1 coins flipped 2:1 odds, ~66.6% probability 2 coins flipped 4:1 odds, 80% probability 3 coins flipped 8:1 odds, ~89% probability 4 coins flipped 16:1 odds, ~94% probability 5 coins flipped 32:1 odds, ~96% probability 6 coins flipped 64:1 odds, ~98% probability 7 coins flipped 128:1 odds, ~99% probability 8 coins flipped 256:1 odds, ~99.6% probability 9 coins flipped 512:1 odds, ~99.8% probability 10 coins flipped 1024:1 odds, ~99.9% probability

So it will take you more and more effort to squeeze out that last bit of probability and you may if you have two potential customers X_0 willing to pay 10$ and X_1 willing to pay 1000$ rationally go and flip a coin once for customer X_0 after having flipped coins 9 times for customer X_0.

If you are then asked if you would like to take over one of two accounts from your friend and fellow honest dealer at the Bazzar, (having already met your quota for the day of coins flipped) . X_2 with willingness to pay of 10$, having had 20 coins flipped for them or X_2 with willingness to pay of 13$ having had 4 coins flipped for them. You might be forgiven to have thought of the hard work of flipping that many coins over the cold hard calculation of EV. You may have figured your friend did the math and justified the extraordinary effort for flipping that many coins or

Now introducing logits, logits are simply the log odds, since the natural logarithm is used I'll use a scale factor of S = 1/ln(2) but if we add to the table above the logits we find:

0 coins flipped 1:1 odds, 50% probability, 0 logits 1 coins flipped 2:1 odds, ~66.6% probability, 1S logits 2 coins flipped 4:1 odds, 80% probability, 2S logits 3 coins flipped 8:1 odds, ~89% probability, 3S logits 4 coins flipped 16:1 odds, ~94% probability, 4S logits 5 coins flipped 32:1 odds, ~96% probability, 5S logits 6 coins flipped 64:1 odds, ~98% probability, 6S logits 7 coins flipped 128:1 odds, ~99% probability, 7S logits 8 coins flipped 256:1 odds, ~99.6% probability, 8S logits 9 coins flipped 512:1 odds, ~99.8% probability, 9S logits 10 coins flipped 1024:1 odds, ~99.9% probability, 10S logits

And if we analyze the case of X_0, X_1 again

How do we quantify the opportunity cost? Clearly you can do it the "normal way" and consider the expected delta in dollars that performing a given coin flip will induce. That will get you a quantity measured in $/logits how many dollars you can expect to get per logit, of the form (forgone value)/(forgone effort(. Just as you may normally quantify your rate of opertunity cost in dollars per hour. But something interesting happens when you quantify in terms of total opertunity cost you can measure it in terms of either dollars or hours equivalently. Or in this case dollars/logits, except you have a variable rate of pay for a given logits with diminishing returns Like you had reverse overtime or a hobby project that reaches saturation. So if you are to reframe it say as having a certain dollar payoff and a certain amount of time making a bird house (that you won't remember) you may choose the option that has a 10000 hour effort of birdhouse building over one that has 10 hours effort and a few thousand more of money even though the 10K hour birdhouse probably won't be a few thousand dollars nicer, the eyes sparkle at such a high level of craftsmanship or logits.

The European Waiter

Let's model a European restaurant as follows. 1. Sometimes a tip will be given for extraordinary service of a fixed amount, say 10 euros per patron at the table 2. Tables take effort in proportion with their number of patrons 3. Servers are always putting effort towards some table. 4. Tables give tips in proportion to the logistic of how much effort is put into the service of that table per patron.

Thus at any given point the expected value in tips of a table is:

10 * number of patrons * logistic(effort per patron)

We could denominate the effort put into the table in units of time.

So in looking at where you want to speed your next unit of effort for the best payoff in tips you will want to put your efforts where the slope of the expected payoff with respect to effort is greatest.

The amount of effort that you will need to put into a table to move the logits a certain amount is an amount of effort (in hours or dollars or logits for a one person party) times the number of patrons (in patrons or dollars) so you can express the amount of effort as logits*dollars to express it across different party sizes.

Thus as a corollary given you are doing a reasonable job of estimating the return to putting more effort into each table the return for each of them will be approximately equal.

So when it come time for the end of your colleague's shift and and they give you a choice of which tables to take over.