Idiot Savant

Idiot Savant is a perk in Fallout 4.

Effects
For ranks 1 and 2, the perk randomly activates whenever you gain experience from any source, at a rate equal to


 * $$%Chance = \begin{cases} 0, & \mbox{if }Intelligence = 0 \\ \text{max}(1, 11 - Intelligence), & \mbox{if } Intelligence > 0\end{cases}$$

The effect this perk has on the long run, before factoring in Intelligence or any other experience bonuses, is:


 * $$AvgExpMulti = \Bigg(\frac{%Chance}{100} \times IdiotSavantMultiplier\Bigg) + \Bigg(1-\frac{%Chance}{100}\Bigg)$$

This ends up becoming a simpler equation, being equivalent to:
 * $$AvgExpMulti = 1 + \Bigg(\frac{min(2,IdiotSavantLevel) \times %Chance}{50}\Bigg) $$

When the perk activates, you will see a laughing Vault-Boy above the compass, as well as hearing a laughing sound. The experience you would have earned will instead be boosted by 3x (for rank 1) or 5x (for rank 2+). This boost is multiplicative with any normally additive experience bonuses from being well-rested or having a high Intelligence.

The probability for the "kill bonus" from rank 3 appears to be independent of the bonus experience from rank 1 and 2 and the probability of it triggering is currently unknown.

Comparison to Intelligence

 * $$OverallExp = IntMulti \times AvgIdiotSavantExpMulti$$

When comparing a perk point to put into higher Intelligence or Idiot Savant, it is worth comparing the returns you get. Each point in Intelligence improves experience gain by ~3%. For a regular player character (i.e. not reloading constantly before a quest turn-in to always trigger the higher experience), a point in Idiot Savant returns on average around 2% in the non-degenerate worst-case, when the player character has 10 Intelligence. At any lower Intelligence, Idiot Savant rank 1 gives superior returns to experience. The same holds for ranks 2 and 3 for Idiot Savant.

In other words, for regular character, for all but the smartest player character (e.g. Intelligence 9 or lower), Idiot Savant still provides superior returns on expected experience gain.

For regular player character, however, the biggest returns are at low Intelligence. At 1 Intelligence, the first point in Idiot Savant provides a whopping 20% expected return on experience gained. At 5 Intelligence, the player character is still expected to receive a ~12% gain of experience over the base rate.

The upshot is that player character at 1-3 Intelligence will benefit the most from Idiot Savant. Regular player character get extremely high expected returns on experience gained. Player character also benefits from low Intelligence since each additional point in Intelligence has increasing returns in how difficult it is to trigger the Idiot Savant event: at 4 Intelligence, you've already almost doubled how hard it is to trigger the Idiot Savant event.

Behind the scenes
The term idiot savant refers to an individual who is overall mentally disabled but nonetheless displays an exceptionally high level of ability in performing a specific task such as memory, arithmetic, art, or music.