Imagine a standoff. The police are involved in a hostage scenario, standing outside a house where the hostage-taker has barricaded verself in with
n surviving hostages. The hostage-taker has warned the police not to make a move, or else hostages will die one by one until the police back down.
The hostage-taker wants, above all things, to get out alive. Ve can negotiate for greater rewards from the police in addition to ver freedom, but unless ve gets out alive, all will be for naught.
The police want to take the hostage-taker down, second only to keeping the hostage(s) alive. Rewards paid out to the hostage-taker are subtracted directly from the utility of the police.
Assigning the value of each living hostage as
h, an incapacitated hostage-taker as
-p for any payouts for the police, and for the hostage-taker,
f for successful escape,
-d per hostage killed due to psychological distress, and
p for any payouts received, the utility functions are
for the police and
for the hostage-taker.
(The hostage-taker values his life and not those of the hostages, whereas the police value the hostages' lives more than capturing the hostage-taker; thus, the two values are unequal.)
From the perspective of the police, any action on their part has a probability
P of the hostage-taker killing a hostage. As this is a game of incomplete information, the police know not how much psychological distress killing a hostage would have (
-d), or whether the distress outweighs the benefits of freedom for the hostage-taker (
d > f), which would result in a rational decision to surrender.
If the police decide to move in, and the hostage-killer does kill a hostage, then the police update their probability
P = P(f > d) that the hostage-taker is willing to kill hostages
n hostages remaining, the police should perform an action if
nhP < (1-P)c
If the hostage-taker kills his last hostage, then the police have no reason not to storm the building and capture the hostage-taker; depending on the number of hostages remaining, the hostage-taker's action may vary.
n=0 hostages remaining, the hostage-taker has no choice in ver action. At
n=1 hostages, the death of the hostage would result win a
n=0 scenario, which would give a utility of
-d killing the hostage or
0 for surrendering to the police.
n > 1, the death of the hostage has an affect on police behaviour, as it gives the police information of how willing the hostage-taker is to kill hostages. As long as
nh > c, the hostage-taker should kill hostages until the police update
P sufficiently that they would allow the hostage-taker's escape over the death of the hostages. If
nh < c, then the hostage-taker is more valuable than the hostages, and the police will not allow the escape of the hostage-taker, regardless of the impact on the hostages
(Note that the hostage-taker does not know
P, nor does ve know
c. But the hostage-taker probably has greater information of how much ve is wanted alive compared to the police knowing the hostage-taker's state of mind.)
At any time, the hostage-taker can make an offer to the police in exchange for ver freedom and the remaining hostages. The highest offer the police will accept is
p < nh
Of course, the police would only accept the offer if
P is sufficiently high. If
nh-p > (1-P)c
in which case the police cannot afford to lose both the money and the hostage-taker (weighted for the probability of killing a hostage, and will not accept the offer.
In conclusion, the hostage-taker should make offers to gauge the value the police place on the hostages and the hostage-taker's capture, get out with as great a
p as possible, and kill hostages to prevent police (in)action, and possible raise the amount of payout (if the police severely underestimate
P. Police should, depending on if they value the hostages or the hostage-taker more, accept payouts to free the hostages or reject everything to capture the hostage-taker.
Of course, this does not examine what happens after the hostage-taker goes free, and whether or not the police will continue pursuit.