Social Preferences and the Unpleasant Math of Base Rates

Reflecting some more on the themes of the last post...Dolly Chugh's audit study of academics reminds me of my old days doing real estate tests to look for evidence of racial steering...a key assumption that audit tests like mine and Dolly's use is that to treat people from different groups with equal credentials unequally is discriminatory...

The point here is to introduce a toy model in which neutrality results in a substantial tilt in favor of members of a group with a higher avwerage level of skills, with the aim of suggesting that social preferences play an important role in achieving the social objective of treating people with the same credentials equally.

The toy model: Assume as in the previous post a stronger group S and a weaker group W.  Assume the employer needs a job candidate with an expected level of skill that is good or G.  The average member of S has an S average skill level, SA, while the average member of W has a below average skill level, BSA.  Among S members, 50% are SA, 20% are G, and 5% are very good or VG; 20% are BA and 5% are much below S average or MBSA.  Among W members, 1% are VG, 4% are G, 20% are SA, 50% are BSA, 20% are MBSA and 5% are very much below S average or VMBSA.

All candidates have credentials that allow the employ to evaluate them as VG, G, SA, BSA, MBSA, or VMBSA.  The credentials however are muddy signals.  There is a 20% chance that a candidate will appear to be one notch above his/her actual skill, a 5% chance of appearing two notches above, and a 1% chance of appearing three notches above.  The same numbers apply to the chance a candidate will appear one, two, or three notches below his/her true skill.

Now, suppose a candidate's credentials signal G.  What is the chance the candidate actually has a skill level of G or VG?   Here the base rate math has a nasty twist...

First, the math for an S candidate signalling G: The 20% of G candidates in the pool will signal G 50% of the time and the 5% of VG candidates will signal G 20% of the time.  Thus, 13.5% of S candidates  will sgnal G when they are G or VG.  Now, the weaker side: The 50% of SA candidates in the pool will signal G 20% of the time, and the 20% of BSA candidates will signal G 5% of the time; the 5% of MBSA candidates will signal G 1% of the time.  Thus, 11.05% of S candidates will signal G when they are SA or below.  The probability that a signal of G actually corresponds to a G or VG candidate equals 13.5/24.55 = 54.99%.

Now, the math for a W candidate signalling G: The 4% of G candidates in the pool will signal G 50% of the time and the 1% of VG candidates will signal G 20% of the time.  Thus, 2.2% of S candidates  will sgnal G when they are G or VG.  Now, the weaker side: The 20% of SA candidates in the pool will signal G 20% of the time, the 50% of BSA candidates will signal G 5% of the time, and the 20% of MBSA candidates will signal G 1% of the time.  Thus, 6.7% of S candidates will signal G when they are SA or below.  The probability that a signal of G actually corresponds to a G or VG candidate equals 2.2/8.9 = 24.72%.

Notice that there is a radical difference here in favor of the S candidate signalling G over the W candidate sending the same signal of G.  To achieve a social objective of treating the two candidates equally, social preferences--or a mix or social preferences and netural preferences--on the part of employers are necessary. 

This example can be used to reinforce the point in the last post about a possible efficiency-based rationale for the stigmatizing of negative social preferences toward a weaker group as racism.  In this case as in the earlier one, negative social preferences toward W do not have the utility that positive social preferences toward W do.

 

 

 

 

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