Testing Alone is Insufficient

Will widely available and effective tests for COVID-19 awaken the economy from its COVID induced coma? Paul Romer, for one, thinks so. But what will each person do with the information gleaned from the test? Should we expect someone who has tested positive for the virus to stay home and someone who has tested negative to go to work? If the first receives no compensation for staying home, she will leave for work. The second, anticipating that infected individuals have an incentive to go to work, will choose to stay home. As a result, the fraction of the population out and about will have an infection rate exceeding that in the population at large.

In a new paper by Rahul Deb, Mallesh Pai, Akhil Vohra and myself we argue that widespread testing alone will not solve this problem. Testing in concert with subsidies will. We propose a model in which both testing and transfers are targeted. We use it to jointly determine where agents should be tested and how they should be incentivized. The idea is straightforward. In the absence of widespread testing to distinguish between those who are infected and those who are not, we must rely on individuals to sort themselves. They are in the best position to determine the likelihood they are infected (e.g. based on private information about exposures, how rigorously they have been distancing etc.). Properly targeted testing with tailored transfers give them the incentive to do so.

We also distinguish between testing at work and testing `at home’. An infected person who leaves home to be tested at work poses an infection risk to others who choose to go outside. Testing `at home’ should be interpreted as a way to test an individual without increasing exposure to others. Our model also suggests who should be tested at work and who should be tested at home.

Captain McWhirr and the Chest

The Nan-Shan, a Siamese steamer under the control of Captain James MacWhirr, on his orders, sails into a typhoon in the South China Sea. Conrad described the captain as

“Having just enough imagination to carry him through each successive day”.

On board, an all-white crew and 200 Chinese laborers, returning home with seven years’ wages stowed in

“a wooden chest with a ringing lock and brass on the corners, containing the savings of his labours: some clothes of ceremony, sticks of incense, a little opium maybe, bits of nameless rubbish of conventional value, and a small hoard of silver dollars, toiled for in coal lighters, won in gambling-houses or in petty trading, grubbed out of earth, sweated out in mines, on railway lines, in deadly jungle, under heavy burdens—amassed patiently, guarded with care, cherished fiercely.”

Ship and souls driven by McWhirr’s will survive the Typhoon. The wooden chest does not. Its contents strewn below deck, the silver dollars are mixed together. It falls to McWhirr to determine how the dollars are to be apportioned between the Chinese laborers to forestall an uprising.

“It seems that after he had done his thinking he made that Bun Hin fellow go down and explain to them the only way they could get their money back. He told me afterwards that, all the coolies having worked in the same place and for the same length of time, he reckoned he would be doing the fair thing by them as near as possible if he shared all the cash we had picked up equally among the lot. You couldn’t tell one man’s dollars from another’s, he said, and if you asked each man how much money he brought on board he was afraid they would lie, and he would find himself a long way short. I think he was right there. As to giving up the money to any Chinese official he could scare up in Fuchau, he said he might just as well put the lot in his own pocket at once for all the good it would be to them. I suppose they thought so, too.”

My former colleague Gene Mumy, writing in the JPE, argued that McWhirr’s solution was arbitrary. We know what McWhirr’s response would have been:

” The old chief says that this was plainly the only thing that could be done. The skipper remarked to me the other day, ‘There are things you find nothing about in books.’ I think that he got out of it very well for such a stupid man.”

Mumy, undeterred, proposed instead a pivotal mechanism (Clark, Groves, Tidemann, Tullock etc). For each agent compute the difference between the total amount of money and the sum of all other claims. If an agent claims at most this amount, they receive their claim. If his claim exceeds this amount, he is penalized. Mumy showed that truth telling was a full information Nash equilibrium of the mechanism.

Saryadar, in a comment in the JPE, criticizes Mumy’s solution on the grounds that it rules out pre-play communication on the part of the agents. Such communication could allow agents to transmit threats (I’m claiming everything) that if credible change the equilibrium outcome. He also hints that the assumption of common knowledge of the contributions is hard to swallow.

Schweinzer and Shimoji revisit the problem with the observation that truth telling is not the only Nash equilibrium of the mechanism proposed by Mumy. Instead, they treat it as problem of implementation under incomplete information. The captain is assumed to know the total amount of money to be divided but not the agents. They propose a mechanism and identify a sufficient condition on beliefs under which truth telling is the unique rationalizable strategy for each agent. The mechanism is in the spirit of a scoring rule, and relies on randomization. I think McWhirr might have objected on the grounds that the money represented the entire savings of the laborers.

Conrad describes the aftermath.

“We finished the distribution before dark. It was rather a sight: the sea running high, the ship a wreck to look at, these Chinamen staggering up on the bridge one by one for their share, and the old man still booted, and in his shirt-sleeves, busy paying out at the chartroom door, perspiring like anything, and now and then coming down sharp on myself or Father Rout about one thing or another not quite to his mind. He took the share of those who were disabled to them on the No. 2 hatch. There were three dollars left over, and these went to the three most damaged coolies, one to each. We turned-to afterwards, and shovelled out on deck heaps of wet rags, all sorts of fragments of things without shape, and that you couldn’t give a name to, and let them settle the ownership themselves.”