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When doing e.g. a seminar talk, it is customary for the hosting institution to pay for the speaker's trip. If it is just an isolated trip to a point A, life is simple; so far, so good.

However, I usually try to lump different talks together. The simplest case is where I go from home to A (where I do a talk) to B (where I do a talk) to home. In this case, I have to choose between having the A -> B segment reimbursed either by institution A or by institution B. (Assume that every segment is on an undivisible, nonstop ticket. I think it is usually not possible to split 50/50 in such a situation.) Based on what criteria should I make this choice?

For example, what if B is much closer to home than A, basically a small detour on the way back. Is it more fair to charge the intermediate leg to A (on the grounds that if I did not have to go there, I would not have made such a long trip) or to B (out of a desire to split the costs evenly)?

Or if A and B are roughly equidistant from home, are there any factors I didn't think of that could tip the scales in favor of one or the other?

(There are countless other configurations one can think of, possibly involving much more than two institutions!)

(I am aware that I am probably overthinking it, and that I probably not such an important decision after all. However I would be curious to know what are the possible arguments to consider, if only out of purely academic (ha ha) interest.)

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    Cool question, I've mulled over this myself at times. If you want to be serious about analyzing the dilemma, another factor to take into account is the relative wealth of both institutions. If one is a wealthy private school and the other a more modest public one, there's a good case to be made that the socially optimal allocation of the costs would involve more of the reimbursement burden falling on the wealthier place. – Dan Romik Mar 1 '17 at 6:04
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    ... and if you want to be extra serious, take the question to MathOverflow and brainstorm a full game theoretical model of the situation. I'm not an expert, but I think it's conceivable that similar questions might even have been discussed in the game theory/economics literature. – Dan Romik Mar 1 '17 at 6:06
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I would imagine that there are jurisdictions where getting reimbursed for costs that you have already been reimbursed for is illegal. It is at the very least unethical, so I think that thinking about this is certainly the right thing to do.

If you make a trip that includes more than the bare minimum "getting there and back" in travel (e.g., a personal detour on the way home, staying for the weekend after a conference for sightseeing, etc), many reimbursing institutions require you to hand in the receipts for your actual travel, plus estimates of what the bare minimum trip would have cost. For example, you could have printed out a screen shot of looking up a flight that just gets you there and back. They will then reimburse you for the smaller of the two amounts.

Likewise, if you have a multi-stop trip, you can certainly ask to be reimbursed for an even smaller amount, as long as the total reimbursed amount does not exceed what you actually paid. How exactly you split the total is up to you, as long as no institution pays more than it would have if you had just traveled there and back. That's also ethical: if they invited you to a talk, they did so on the understanding that they will cover your travel cost to get there and back; if you are asking for less money (whichever those savings are split among all institutions), it's to their benefit, whether or not they get exactly half of the savings, or a different amount -- it's money they had budgeted for you, and are now getting back, nobody can complain about that.

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