mathematics//decision theory//sequential allocation

2026-03-12

Sequential allocation is the strategy of committing resources to a task in rounds, sending one agent, observing the result and sending the next only if the previous one failed, and it is how a fleet of firefighting drones, a rescue team or a warehouse dispatcher saves scarce resources when results can be checked quickly. The military operations-research literature calls the pattern shoot-look-shoot; in a warehouse it is checking whether the order left before sending a second robot.

Take one fire, three drones, each putting it out with probability p=0.6p=0.6p=0.6. Sending all three at once succeeds with probability 1−0.43=0.9361-0.4^3=0.9361−0.43=0.936 and always spends three drones. Sending them one at a time, each only if the previous one failed, ends with the same 0.936, but the expected number of drones spent is

E[n]=∑k=0N−1(1−p)k=1−(1−p)Np,E[n] = \sum_{k=0}^{N-1} (1-p)^{k} = \frac{1-(1-p)^{N}}{p},E[n]=k=0∑N−1(1−p)k=p1−(1−p)N,

where NNN is the number available and each term is the probability that every earlier drone failed. With N=3N=3N=3 and p=0.6p=0.6p=0.6 it is 1.56: almost a drone and a half saved per fire, free for the next one. It is value of information at work, since the observation says whether the next resource is needed at all.

Act, observe and reassign when observing is fast and reliable against how quickly the problem worsens, and resources are scarce; act in parallel when time rules or the observation is poor. Each round adds the latency of looking while the fire grows, and a wrong reading (the fire believed out when it is not) erases what the sequence saved.

The middle ground is the usual answer: send two, observe, send the rest. Choosing within that range is itself a Markov decision process whose state is which tasks remain and which resources are left, so the Bellman equation closes the circle; the one-shot version, deciding everything at once, is weapon-target assignment.

The formula 1−(1−p)n1-(1-p)^n1−(1−p)n assumes independent failures. If the drones share the cause of failure (the same wind, the same wrong estimate of where the fire is, the same software bug), sending more yields far less than promised, in parallel or in sequence: it is common-mode failure dressed as probability.