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hhdl: Thanks, though I still don't understand the formula
(p*m + (1 - p)*r))/c
If p is the probability that the room wouldn't sell regularly, then (1 - p) is the probability that the room sells "regularly" (= for cash).
In both cases, the hotel will have to pay the marginal cost of providing the room. So I don't understand why you have the m in the first term but not the second.
Furthermore, if the hotel lets somebody redeem an award, it gets a compensation from the chain. That should somehow factor into the first term. The compensation usually at least covers the marginal cost of providing the room.
Finally, there is the issue that it isn't in any way an accurate model of all scenarios. There are at least four outcomes:
1.) The room wouldn't sell regularly but the hotel can sell the room as an award.
2.) The room won't sell regularly and nobody redeems it as an award.
3.) The room sells regularly.
4.) The room would have sold but that didn't realize because somebody took out an award before it could be sold for cash.
In any case, I'm still totally lost with your equation. It cannot stem from a consumer's optimization problem, because the consumer doesn't factor in m. It cannot stem from the chain's optimization problem, because the chain doesn't get paid r (but a small percentage of it) and doesn't incur m. So it must be derived somehow from the hotel's optimization problem.
But what can the hotel choose? It's totally unclear to me. Can it choose award inventory (as with IHG)? Or is it assumed that every room is bookable as an award?
From my perspective, the hotel takes m as a paramter. It takes the compensation it receives in case a reward is redeemed as constant as well. If we assume the hotel can set award inventory, that would be a choice variable. If, instead, we assume the hotel must make every (standard) room available as an award, it wouldn't be a choice variable.
Then, the only choice variable would be r. In any case, it seems to me the probabilities should be endogenous variables that are a function of r.