How does CUDA assign device IDs to GPUs?

CUDA picks the fastest device as device 0. So when you swap GPUs in and out the ordering might change completely. It might be better to pick GPUs based on their PCI bus id using:

cudaError_t cudaDeviceGetByPCIBusId ( int* device, char* pciBusId )
   Returns a handle to a compute device.

cudaError_t cudaDeviceGetPCIBusId ( char* pciBusId, int  len, int  device )
   Returns a PCI Bus Id string for the device.

or CUDA Driver API cuDeviceGetByPCIBusId cuDeviceGetPCIBusId.

But IMO the most reliable way to know which device is which would be to use NVML or nvidia-smi to get each device's unique identifier (UUID) using nvmlDeviceGetUUID and then match it do CUDA device with pciBusId using nvmlDeviceGetPciInfo.

Set the environment variable CUDA_DEVICE_ORDER as:

export CUDA_DEVICE_ORDER=PCI_BUS_ID

Then the GPU IDs will be ordered by pci bus IDs.