Very interesting! Quick question: Why can’t Alice and Bob split the bill 50%-50%? Or 70%-70%, with the extra 40% being cash set on fire? Or 150%-150%, with the extra 200% being cash set on fire? What if anything makes the 100%-100% “split” special?
If someone pays less than the cost of a problem, then they’re incentivized to take inefficiently-low levels of effort to avoid it. If someone pays more than the cost, they’re incentivized to take inefficiently-high levels of effort to avoid it. In general, if someone can pay X “dollars” (or effort, etc) to avoid a problem, then we want that to happen if-and-only-if the problem would cost more than X “dollars” to fix.
Tho the deadweight loss of a “both pay” solution is probably optimized somewhere between “split evenly” and “both pay fully”. For example, in the pirate case, I think there are schemes that you can do that result in honesty being the optimal policy and yet they only destroy some of the gold in the case of accidents (or dishonesty), tho this may be sensitive to how many pirates there are and what the wealth distribution is like.
The pirate problem is kind of a weird one, because it’s one-shot. In general, there will be many opportunities for the players to prevent problems at some cost. The more such opportunities they have, the more important it is to get the incentives right. The “everyone pays exactly 100% of cost” solution should generally win asymptotically, when players have many opportunities to pay to prevent problems, but in a finite problem (i.e. most realistic problems), optimal will probably be less-than-100%.
Are you sure? Let’s say Bob is considering whether to drive through a swamp which will give him $101 of time savings but risks damaging the car, which causes $100 cost in expectation. If both Alice and Bob pay $100 for the repair, then Bob drives through the swamp, which destroys $99 in aggregate value. If we repeat the scenario N times, Bob drives through the swamp every time. Right?
Yup. That does seem off, but it would make sense if we zoom out a level and assume that the money is paid to a third party, so it’s not really deadweight loss in the broader system context.
Right, but any such trash-car-for-net-win opportunity for Bob will make Alice less likely to make the deal: from her perspective, Bob taking such a win is equivalent to accident/carelessness. In the car case, I’d imagine this is a rare scenario relative to accident/carelessness; in the general case it may not be.
Perhaps a reasonable approach would be to split bills evenly, with each paying 50% and burning an extra k%, where k is given by some increasing function of the total repair cost so far.
I think this gives better incentives overall: with an increasing function, it’s dangerous for Alice to hide problems, given that she doesn’t know Bob will be careful. It’s dangerous for Bob to be careless (or even to drive through swamps for rewards) when he doesn’t know whether there are other hidden problems.
I don’t think you can use the “Or donates to a third party they don’t especially like” version: if trust doesn’t exist, you can’t trust Alice/Bob to tell the truth about which third parties they don’t especially like. You do seem to need to burn the money (and to hope that Alice doesn’t enjoy watching piles of money burn).
Very interesting! Quick question: Why can’t Alice and Bob split the bill 50%-50%? Or 70%-70%, with the extra 40% being cash set on fire? Or 150%-150%, with the extra 200% being cash set on fire? What if anything makes the 100%-100% “split” special?
If someone pays less than the cost of a problem, then they’re incentivized to take inefficiently-low levels of effort to avoid it. If someone pays more than the cost, they’re incentivized to take inefficiently-high levels of effort to avoid it. In general, if someone can pay X “dollars” (or effort, etc) to avoid a problem, then we want that to happen if-and-only-if the problem would cost more than X “dollars” to fix.
Tho the deadweight loss of a “both pay” solution is probably optimized somewhere between “split evenly” and “both pay fully”. For example, in the pirate case, I think there are schemes that you can do that result in honesty being the optimal policy and yet they only destroy some of the gold in the case of accidents (or dishonesty), tho this may be sensitive to how many pirates there are and what the wealth distribution is like.
The pirate problem is kind of a weird one, because it’s one-shot. In general, there will be many opportunities for the players to prevent problems at some cost. The more such opportunities they have, the more important it is to get the incentives right. The “everyone pays exactly 100% of cost” solution should generally win asymptotically, when players have many opportunities to pay to prevent problems, but in a finite problem (i.e. most realistic problems), optimal will probably be less-than-100%.
Are you sure? Let’s say Bob is considering whether to drive through a swamp which will give him $101 of time savings but risks damaging the car, which causes $100 cost in expectation. If both Alice and Bob pay $100 for the repair, then Bob drives through the swamp, which destroys $99 in aggregate value. If we repeat the scenario N times, Bob drives through the swamp every time. Right?
Yup. That does seem off, but it would make sense if we zoom out a level and assume that the money is paid to a third party, so it’s not really deadweight loss in the broader system context.
Right, but any such trash-car-for-net-win opportunity for Bob will make Alice less likely to make the deal: from her perspective, Bob taking such a win is equivalent to accident/carelessness. In the car case, I’d imagine this is a rare scenario relative to accident/carelessness; in the general case it may not be.
Perhaps a reasonable approach would be to split bills evenly, with each paying 50% and burning an extra k%, where k is given by some increasing function of the total repair cost so far.
I think this gives better incentives overall: with an increasing function, it’s dangerous for Alice to hide problems, given that she doesn’t know Bob will be careful. It’s dangerous for Bob to be careless (or even to drive through swamps for rewards) when he doesn’t know whether there are other hidden problems.
I don’t think you can use the “Or donates to a third party they don’t especially like” version: if trust doesn’t exist, you can’t trust Alice/Bob to tell the truth about which third parties they don’t especially like.
You do seem to need to burn the money (and to hope that Alice doesn’t enjoy watching piles of money burn).
Ok, I buy this argument.