Backward Induction - An Example of Decision-making By Backward Induction

An Example of Decision-making By Backward Induction

Consider an unemployed person who will be able to work for ten more years t = 1,2,...,10. Suppose that each year in which she remains unemployed, she may be offered a 'good' job that pays $100, or a 'bad' job that pays $44, with equal probability (50/50). Once she accepts a job, she will remain in that job for the rest of the ten years. (Assume for simplicity that she cares only about her monetary earnings, and that she values earnings at different times equally, i.e., the discount rate is zero.)

Should this person accept bad jobs? To answer this question, we can reason backwards from time t = 10.

• At time 10, the value of accepting a good job is $100; the value of accepting a bad job is $44; the value of rejecting the job that is available is zero. Therefore, if she is still unemployed in the last period, she should accept whatever job she is offered at that time.
• At time 9, the value of accepting a good job is $200 (because that job will last for two years); the value of accepting a bad job is 2*$44 = $88. The value of rejecting a job offer is $0 now, plus the value of waiting for the next job offer, which will either be $44 with 50% probability or $100 with 50% probability, for an average ('expected') value of 0.5*($100+$44) = $72. Therefore regardless of whether the job available at time 9 is good or bad, it is better to accept that offer than wait for a better one.
• At time 8, the value of accepting a good job is $300 (it will last for three years); the value of accepting a bad job is 3*$44 = $132. The value of rejecting a job offer is $0 now, plus the value of waiting for a job offer at time 9. Since we have already concluded that offers at time 9 should be accepted, the expected value of waiting for a job offer at time 9 is 0.5*($200+$88) = $144. Therefore at time 8, it is more valuable to wait for the next offer than to accept a bad job.

It can be verified by continuing to work backwards that bad offers should only be accepted if one is still unemployed at times 9 or 10; they should be rejected at all times up to t = 8. The intuition is that if one expects to work in a job for a long time, this makes it more valuable to be picky about what job to accept.

A dynamic optimization problem of this kind is called an optimal stopping problem, because the issue at hand is when to stop waiting for a better offer. Search theory is the field of microeconomics that applies problems of this type to contexts like shopping, job search, and marriage.