A couple years ago NPR's Planet Money podcast had an episode about Somali pirates. (The pirate part starts at 9:35). There was all sorts of interesting stuff about division of labor, allocation of shares, pirate venture capital, etc. Some of this paralleled early modern piracy (as given a scholarly analysis in Peter Leeson's work and a romantic perspective in innumerable books and movies since Treasure Island) but in other respects it's very different. In particular, whereas early modern piracy was mostly about seizing cargo and the crews were left alone if they surrendered promptly, Somali piracy is more similar to piracy in antiquity in that it's basically maritime kidnapping. The typical instance of Somali piracy isn't that different from what a young Julius Caesar experienced when he was kidnapped by pirates and held for ransom on his way home from political exile in Asia Minor. One interesting detail in Plutarch's report is that, "When these men at first demanded of him twenty talents for his ransom, he laughed at them for not understanding the value of their prisoner, and voluntarily engaged to give them fifty."
It's not entirely clear if we should take Plutarch's report at face value (he also tells us that Caesar constantly insulted his captors as being, for instance, too uncivilized to appreciate his poetry) but for the sake of argument let's accept that Caesar rather brashly gave away too much information in the game of price discovery. According to a hostage negotiator quoted by This American Life, giving away this information is apparently typical of hostages and is counter-productive to their release as it narrows the bid-ask spread. Economists would describe hostage negotiation as a bilateral monopoly price negotiation that is structurally just a special case of chicken. That is, unlike a barrel of oil or a freight car full of soybeans which can trade on an extremely liquid market with innumerable buyers and sellers, a hostage has exactly one seller (the kidnappers) and exactly one buyer (the employer and/or family of the hostage). When there is only one buyer, the opportunity cost for ransoming the hostage is zero. Likewise, the employer and/or family has no realistic alternative means to recover the hostage. In order for everybody to walk away happy, we need a cooperate-cooperate outcome: the kidnapper has to give up the hostage and the employer/family has to give up a ransom. This structure also characterizes art theft, which in practice is not a matter of fencing art on the black market but ransoming art to a museum's insurance company.
If we model a bilateral monopoly negotiation only two things should matter. The first is, as always in a game of chicken, the willingness to accept failure. The more willing you appear to walk away, the more bargaining power you have. In a more protracted game this can cash out as willingness to delay which we can treat as a defect-defect outcome on the installment plan. In fact in the Planet Money episode on Somali piracy, the hostage's party did balk and break off negotiations for weeks at a time until the pirates were willing to come down on price.
The other thing that should matter is the capacity to pay. If the pirate knows for an absolute fact that the hostage's people simply can't raise more than a million dollars then it would be pointless for them to demand two million dollars. Of course there is an issue of information asymmetry in that the hostage's party has much better information on its assets than do the pirates and so the pirates may be skeptical of the hostage's party pleading poverty (especially if the hostage has foolishly told them how much money they can get). We see this at work in the TAL story's point that kidnapping insurance holds the condition that you can't tell anyone you have kidnapping insurance.
Here's something that the econ model tells us shouldn't matter: the going rate. In normal markets the going rate matters, but only because it provides the opportunities for substitutes and this creates the "law of one price." For instance, when I go to a grocery store and see a loaf of bread for $4 I won't buy it. An economist would say I forgo this purchase because I know perfectly well that the going rate for a loaf of bread is about $2.25 and so I can go elsewhere and get bread cheaper. Similarly if I go to the Honda dealer to buy a Honda Accord, it is relevant for me to mention price quotes offered by other Honda dealers for an Accord or even how much Toyota dealers ask for a Camry because it is entirely credible that I'll walk off the lot and go to rival car dealers offering very close substitutes for this dealer's cars. However if my sister is locked in a basement in Ciudad Juarez and the kidnappers can credibly commit to not letting her go unless I raise $x, it is completely irrelevant that in the past kidnappers accepted ransoms of $x/2 since I don't have the relatively good fortune of dealing with a kidnapper who demands $x/2 but am stuck with one who demands $x. There are no other places where I can buy the freedom of my sister and so the only price that matters is the one being demanded by her particular kidnappers. (Note to any cartels reading this: I don't have a sister).
Read the rest for a discussion of how "fair market price" concepts may or may not creep into such negotiations, and for the conclusion of Julius Caesar's kidnapping story.