Reformulating Learned Hand

I'd appreciate it if the following post were treated as a joke. If I ever act like I meant it... may great violence fall upon me.

OK. I think the Learned Hand formula is officially driving me nuts. To put this into context for any non-lawyer who happens across this site, here's what the Learned Hand Formula is: B < pL

B stands for the cost of a Burden. P stands for the probability of an accident. And L stands for the damage an actor will be liable for if the accident should occur. Lawyers bandy about this "formula" to try and determine how deterrence should operate in the tort system. First of all, I have trouble considering this a formula at all... or rather, as a formula, its result is always simply "true" or "false." That doesn't actually tell you anything.

But worse - once you start debating policy issues within the framework of the Learned Hand formula, you start spouting gibberish. Take, for a quick example, the question of strict liability. Strict liability is said to be an appropriate standard when you want an actor to bear the full costs of all accidents inflicted. No matter what the actor could or should have done - say, putting banisters on the staircase - the actor will pay the cost of the accident - say, the tenant falling four floors. This is said to increase deterrence... despite the fact that it does nothing to influence any of the factors. L is still the same, whether the guard rails are in place or not. B - the cost of bannisters - is still the same. And p - the probability of a person falling off the stairs remains dependent upon the actor's choice to put in a bannister or not. So what has the policymaker done?

Simple, he's increased the probability of the actor losing at trial. So, in truth we're discussing more than one p in our B < pL. P₁ = the probability of an accident. P₂ = the probability of a loss at trial. In fact, there's yet another probability hiding in this equation - the probability that a victim will bring a suit at all. In the case of a tenant falling off a bannister with no guard rails that probability (call it P₃) will be quite high. However, in other classes of cases - medical malpractice, say - that probability will in fact be quite low.

But we're not done yet. There are other probabilities lurking in this system. What is the likelihood of getting sued even when no injury occurs? What's the likelihood of losing at trial in such a case? And what's the right figure to use when you're calculating damages? The maximum liability you might incur? The average?

Anyhow, I hate thinking this way. For starters, because it leads to conditions like this:

I'd like to walk through a sample model of tort anticipation using my embarrasingly inadequate conception of game theory, and see what I come up with.

• Step 1: The actor (let's say he's a doctor) must make a choice between one of two burdens. Let's say that B₁ is the cost (burden) of splinter-free tongue depressants. Let's say that B₂ is the cost of the alternative - plasterwood tongue depressants.
  
  
• Step 2: Now, for either choice there will be a probability that an accident will occur. We can call these P₁(a) and P₂(a). There are also corresponding likelihoods that there won't be accidents: (1 - P₁(a)) and (1 - P₂(a)).
  
  
• Step 3: Depending on whether not an accident occurs, there is a likelihood of being sued. There would be four relevant probabilities at this stage. A) Having chosen B₁ and an accident having occurred, the likelihood of getting sued is P_a(s); B) Having chosen B₁ and an accident having not occurred, the likelihood of getting sued is P_b(s); C) Having chosen B₂ and an accident having occurred, the likelihood of getting sued is P_c(s); and D) Having chosen B₂ and an accident having not occurred, the likelihood of getting sued is P_d(s). Again, there are at this stage corresponding likelihoods of not getting sued.
  
  
• Step 4: OK. Now we're getting to a wide range of outcomes. At this point, we face that probability that, if sued, our hapless doctor will either win or lose. This breaks down into 8 potential outcomes: A) choosing B₁ and having had an accident and having been sued; B) choosing B₁ and having had an accident and having not been sued; C) choosing B₁ and not having had an accident and having been sued; D) choosing B₁ and not having had an accident and not having been sued; E) choosing B₂ and having had an accident and having been sued; F) choosing B₂ and having had an accident and not having been sued; G) choosing B₂ and not having had an accident and having been sued; H) choosing B₂ and not having had an accident and not having been sued. Because winning or losing are only relevant within the context of having been sued, we get two possible outcomes for each state of being sued and only one outcome for each state of not being sued - or 12, rather than 16 outcomes.
  
  
• Step 5: And now, only at the end of this long mess do we reach the relevant damage assessment - liability + litigation costs. Both of these factors don't arrive at our desk in some determinate form. Instead, they come in a range - anywhere from zero + zero (no liability, and the plaintiff gets his with a Rule 11(b) sanction covering attorney costs) to huge + huge (massive economic damages to the plaintiff, huge pain and suffering losses, a punitive award, and a protracted costly trial). Ironically, the number you'll want to use at this stage will depend upon the decision strategy you were trying to use way back at Step 1. If you're pursuing a min-max strategy (minimizing your maximum loss) then you want to use the largest possible figure (though, one could also calculate the likelihood of being assessed the maximum level of liability... roughly 2% of plaintiffs receive over 60% of non-economic damages, for example... and those defendants are far more likely to have been deemed especially culpable for having picked the wrong B). For the moment, we'll just say that's the strategy, and thus at this stage, our relevant figures would be L_max (the maximum potential liability) and S_max (the maximum potential litigation cost). At certain stages of the decision tree, the liability and/or the litigation costs will be zero (because no suit was filed or no damages were awarded).

So, let's start filling in some arbitrary figures and see what we get. Let's try this arbitrary assignment:
B₁ = $1,000
B₂ = $10

P₁(a) = 0.1 (took the precaution, accident occurred anyhow)
P₂(a) = 0.8 (didn't take precaution, accident occurred)

P_a(s) = 0.3 (took precaution, accident occurred, got sued)
P_b(s) = 0.1 (took precation, no accident, got sued)
P_c(s) = 0.6 (didn't take precaution, accident occurred, got sued)
P_d(s) = 0.2 (didn't take precaution, no accident, got sued)

P_a(v) = 0.6 (took precaution, accident occurred, got sued, loses)
P_b(v) = 0.0 (took precaution, accident occurred, no suit, can't lose)
P_c(v) = 0.3 (took precaution, no accident, got sued, loses)
P_d(v) = 0.0 (took precaution, no accident, no suit (can't lose))
P_e(v) = 0.9 (no precaution, accident, lawsuit, loses)
P_f(v) = 0.0 (no precaution, accident, no lawsuit)
P_g(v) = 0.5 (no precaution, no accident, lawsuit, loses)
P_h(v) = 0.0 (no precaution, no accident, no lawsuit (can't lose))

L_max = $10,000,000
S_max = $1,000,000
L_min = $0
S_min = $0

Now, we end up with 12 possible endstates:

1. B₁ + (P₁(a) * P_a(s) * P_a(v) ( L_max + S_max ) ) = $1,000 + (0.1 * 0.3 * 0.6 ($10,000,000 + $1,000,000) = $1,000 + 0.018($11,000,000) = $1,000 + $198,000 = $199,000 - Precaution, accident, lawsuit, verdict
   
   
2. B₁ + (P₁(a) * P_a(s) * (1 - P_a(v)) ( L_min + S_max ) ) = $1,000 + (0.1 * 0.3 * 0.4 ($0 + $1,000,000) = $1,000 + 0.012($1,000,000) = $1,000 + $12,000 = $13,000 - Precaution, accident, lawsuit, acquittal
   
   
3. B₁ + (P₁(a) * (1 - P_a(s)) * P_b(v) ( L_max + S_max ) ) = $1,000 + (0.1 * 0.7 * 0.0 ($0 + $0) = $1,000 + 0.00($0) = $1,000 + $0 = $1,000 - Precaution, accident, no suit
   
   
4. B₁ + ((1 - P₁(a)) * P_b(s) * P_c(v) ( L_max + S_max ) ) = $1,000 + (0.9 * 0.1 * 0.3 ($10,000,000 + $1,000,000) = $1,000 + 0.027($11,000,000) = $1,000 + $297,000 = $298,000 - Precaution, no accident, lawsuit, verdict
   
   
5. B₁ + ((1 - P₁(a)) * P_b(s) * (1 - P_c(v)) ( L_min + S_max ) ) = $1,000 + (0.9 * 0.1 * 0.7 ($0 + $1,000,000) = $1,000 + 0.063($1,000,000) = $1,000 + $63,000 = $64,000 - Precaution, no accident, lawsuit, acquittal
   
   
6. B₁ + ((1 - P₁(a)) * (1 - P_b(s)) * P_d(v) ( L_min + S_min ) ) = $1,000 + (0.9 * 0.9 * 0.0 ($0 + $0) = $1,000 + 0.0($0) = $1,000 + $0 = $1,000 - Precaution, no accident, no suit
   
   
7. B₂ + (P₂(a) * P_c(s) * P_e(v) ( L_max + S_max ) ) = $10 + (0.8 * 0.6 * 0.9 ($10,000,000 + $1,000,000) = $10 + 0.432($11,000,000) = $10 + $4,752,000 = $4,752,010 - No Precaution, accident, lawsuit, verdict
   
   
8. B₂ + (P₂(a) * P_c(s) * (1 - P_e(v)) ( L_min + S_max ) ) = $10 + (0.8 * 0.6 * 0.1 ($0 + $1,000,000) = $10 + 0.048($1,000,000) = $10 + $48,000 = $48,010 - No Precaution, accident, lawsuit, acquittal
   
   
9. B₂ + (P₂(a) * (1 - P_c(s)) * P_f(v) ( L_max + S_max ) ) = $10 + (0.8 * 0.4 * 0.0 ($0 + $0) = $10 + 0.00($0) = $10 + $0 = $10 - No Precaution, accident, no suit
   
   
10. B₂ + ((1 - P₂(a)) * P_d(s) * P_g(v) ( L_max + S_max ) ) = $10 + (0.2 * 0.2 * 0.5 ($10,000,000 + $1,000,000) = $10 + 0.02($11,000,000) = $10 + $222,000 = $222,010 - No Precaution, no accident, lawsuit, verdict
    
    
11. B₂ + ((1 - P₂(a)) * P_d(s) * (1 - P_g(v)) ( L_min + S_max ) ) = $10 + (0.2 * 0.2 * 0.5 ($0 + $1,000,000) = $10 + 0.02($1,000,000) = $10 + $20,000 = $20,010 - No Precaution, no accident, lawsuit, acquittal
    
    
12. B₂ + ((1 - P₂(a)) * (1 - P_d(s)) * P_h(v) ( L_min + S_min ) ) = $10 + (0.2 * 0.8 * 0.0 ($0 + $0) = $10 + 0.0($0) = $10 + $0 = $10 - No Precaution, no accident, no suit
    

OK now. See, this is helping me?

Now that I can see what's going on here visually, I see what's up. At the end of all this math, we wind up back at two states. Either you get sued, or you don't. If you don't get sued, your burden is the total cost of the preventive measure. If you do get sued, your burden is the total cost of the various "getting sued" outcome states.

So, for example - under B₁ there are four possible positive lawsuit endstates with an aggregate probability of 0.120, or a 12% chance of happening. Under B₂ there are also four possible positive lawsuit endstates, with an aggregate probability of 0.520, or a 52% chance of getting sued.

Now this is intriguing. If you choose B₂ your worst outcome is VERY BAD. You have a 43.2% likelihood of losing $11 million - so, by discounting for probability you wind up taking a $4.75 million hit when you choose the plasterwood tongue depressants.

Oddly, however... if you choose B₁, the system cost becomes your greatest liability. Your worst outcome has now become adopting the safety measure, getting sued, and losing. Ironically, it's less costly for the "bad doctor" to get sued and acquitted than it is for you.

So, it would stand to reason that the more deterrent you have, the more wastage you'll get... hmmm.

I think I want to think about this some more. Sorry if I've wasted your time.

Wait, wait, wait.

OK. So from a policy persepctive then... strict liability basically bumps the probabilities up to one if there was an accident and you were sued whether or not you took the precaution. So it would substantially and equally increase the cost of states #1 and #7. Which means, there would be no focused gain in deterrence, since the consequence falls equally upon both folks. It would raise the aggregate costs of the entire system, though with the burden continuing to fall most heavily on the person failing to take precautions... all other things being equal.

However, to focus deterrence incentives, you would want to increase the probabilities of suit or verdict under scenarios 7-12 or decrease the probablities of suit or verdict under scenarios 1-6.

Anything else, you're just running in place - marginally, but inefficiently increasing deterrence by raising the absolutecost to all actors in the system.