Although it is not a common crime, there are parts of the world where kidnapping is a real and constant threat. In a typical scenario, a victim is abducted and a monetary demand is made for his or her release. Although it appears to be a simple exchange of money for the release of a hostage, dealing with the situation does require considerable planning. This is where game theoryRasmusen ; Osborne ; Binmore , a developed branch of mathematics that models strategic situations, can offer valuable insights.
In 1976, Reinhard Selten Selten0 developed a game-theoretic model of kidnapping as a two-person sequential game between player K (Kidnapper) and player F (hostage’s Family). The game begins with K’s choice whether or not to go ahead with his plan that is described by a binary decision variable
The game ends if K selects . If K selects , he takes the hostage to a hidden place unknown to player F and to the police, and announces a ransom demand .
Numerous questions then arise. Will the hostage’s family pay the ransom or will they try to negotiate a lower amount? If they do pay the ransom, should K free the hostage instead of executing him/her? Moreover, if F does not expect K to free the hostage, why should it be expected that F pay some ransom?
It is assumed that, on knowing the demand , a negotiation process starts between players K and F. Player F makes an offer which is the amount willing to be paid, and player K either decides to accept and release the hostage, or to execute the hostage. The situation can be seen as a simple description of an extended bargaining process.
In his model, Selten assumed that K’s threat to execute the hostage has some credibility, even though K cannot improve his situation by doing so. In particular, it is expected that with a positive probability , player K may deem the offer to be insufficient, and thus decide to execute the hostage. Selten assumed that the probability can be described as a linear function of
where is a constant with . The parameter thus describes K’s non-rational decision to execute the hostage, as in this case, the traditional utility maximisation principle is ignored.
It is nonetheless possible that Player K makes the rational decision to execute the hostage independently of whether the offer is deemed insufficient or not. Selten used a binary decision variable to describe this situation:
i.e. even if an offer is made at , , the hostage can still be executed for some .
Ii Modified kidnapping game
In either case of the hostage having been executed or released, the authorities will put efforts in finding and capturing the kidnapper K. In Selten’s model, it is assumed that, in both cases, the authorities will be successful in capturing K with some probability , where
i.e. the probability of capturing the kidnapper is independent of whether the hostage has been released or executed. We ask whether this indeed is the usual case.
Cursory observations of media coverage related to kidnapping incidences highlight the political pressure faced by authorities to severely punish those responsible, the idea being that punishment helps decrease the incidence of such events in the future. It is however unclear whether authorities favour the allocation of extra resources towards the capture of those who executed the hostage or those who did not. No consensus appears to prevail, and resource spending seems to be case dependent and government dependent, with the media likely playing a role in the decision of whether extra resources are spent to increase the chances of capturing the kidnapper.
Assuming that increased spending leads to a higher probability of capture, we thus adopt two random choice parameters and instead of the fixed probability assumed by Selten, that we define as follows:
This allows us to an improved modeling of the responses by authorities, and helps us elucidate whether allocating extra resources to increase the likelihood of capturing the kidnapper influences the kidnapper’s strategy.
As we focus on the probability of the kidnapper’s capture, it is also important to encompass the idea that, in the case where the hostage is executed, families derive a higher disutility from the kidnapper still being at large. This gives us the payoffs depicted in Table 1.
Elevated sensitivities lead to increased pressure on the police, and government, to capture the kidnappers that usually results in an increase in the resources for finding the kidnappers. Investing extra efforts and resources may result in an increased probability of capturing the kidnapper.
In the following, we study the situation when the probability of capturing the kidnapper depends on whether the hostage has been killed or not and find a new uniquely determined perfect equilibrium point in Selten’s game.
|Table 1. The payoffs for players K and F.|
Here and are positive constants and utilities of K and F that are assumed to be linear in money. In the original game presented by Selten, , and .
As in Selten’s game, if K is caught, the execution of the hostage results in an increased disutility relative to the case when K releases the hostage. Thus,
As the complete history of the previous game is known to both players at every point in the course of play, the game can be identified as an extensive game with perfect information Selten1 .
Note that Table 1 encompasses a number of simplifying assumptions. First, player K’s cost of preparing the kidnapping is assigned zero value. Also, player F’s non-monetary disutilities, other than those that are incurred if player K executes the hostage, are ignored. In reality, there can be significant disutility for player F resulting from the emotional stress of engaging in a bargaining process with player K. Also, player F is assumed to gain no utility from the capture of K by the authorities if the hostage is released. Finally, note that utilities when player K is caught after the release of the hostage do not depend on the ransom money as it is recovered and given back to player F. However, player K is then left with disutility . FIG. 1 shows the extensive form of the game.
ii.1 Game timeline
Below, we provide a short description of the timeline of the game:
Player K chooses between and , i.e. whether or not to kidnap someone.
If K selects , the game ends and both players K and F receive zero payoffs; if K selects , K then announces demand .
After observing demand , F makes an offer such that .
After K observes the offer , a non-rational execution of the hostage occurs with probability defined by Eq. (2).
If the hostage is not executed non-rationally, K chooses between and . If K selects , this means that ransom is paid and the hostage is released. If K selects , the hostage is (rationally) executed irrespective of whether any ransom is paid or not.
After release or execution of the hostage, two final random choice parameters and reflect the likelihood for the kidnapper to be captured, where is the probability of capture if the hostage is released and is the probability of capture if the hostage is executed.
The game then ends with payoffs according to Table 1.
Iii Equilibrium of the game
Being one of the foundational concepts in game theory, Nash equilibrium are used to predict the strategies used by players of noncooperative games. A strategy profile specifies a strategy for each player and constitutes a Nash equilibrium if no player has an incentive to deviate from their current strategy. Any finite game admits at least one Nash equilibrium. The mathematical conditions defining a Nash equilibrium, called the Nash conditions, may nonetheless lead to unreasonable outcomes, as pointed out by Selten Selten ; Selten1 . This is because the Nash conditions do not account for the dynamics of the game (if any).
Selten thus used the notion of perfect equilibrium as a refinement on the set of Nash equilibria to solve for the equilibrium of the Kidnapping game. A subgame perfect equilibrium Selten ; Selten1 ; Selten2 ; Kalai is not only a Nash equilibrium in the whole game, it is also a Nash equilibrium is every subgame. For finite games with perfect information, such as the one considered in this paper, subgame perfect equilibrium are commonly determined using backward induction Osborne ; Rasmusen .
iii.1 Optimal choice of
We start by examining the subgame that begins with player K’s choice of . Let be K’s expected payoff if K selects and let be the expected payoff if K selects (i.e. the execution of the hostage). We have:
Note that the case
was considered by Selten.
In that case, as , and we have
and becomes the optimal choice for player K. That is, for the case studied by Selten, player K will never rationally decide to execute the hostage. This could give the impression that when (9) does not hold, release of the hostage would not remain the optimal choice of . If the values of and rely on heightened sensitivities, i.e. authorities allocate more resources when the hostage has been executed so as to increase the likelihood of capturing K, we can assume that . In this case, remains the optimal choice for K, as in Selten’s original work. This is because , while , and thus disutilities are ranked such that as , which gives us that .
However, if , release of the hostage may not remain the optimal choice for K, which is in contrast with Selten’s work.
iii.2 Optimal choice of
In the subgame that begins with player F’s choice of , player F knows that player K can execute the hostage with probability given in Eq. (2). Using Table 1, the expected value of F’s utility is thus equal to:
With the constraints
Eq. (11) is reduced to
which is player F’s expected value of utility for the case that Selten considered. From FIG. 1 we note that is the probability of the hostage being released because of K’s non-rational decision. However, is K’s rational decision to release the hostage and thus is not the probability of .
Eq. (16) shows that assumes its maximum at
if the value of is in the interval . This is the case if is in the closed interval between the following critical values
as obtained by Selten. To determine the range for which is an increasing function i.e.
i.e. the function as described by Eq. (14), is an increasing function for Likewise, considering the inequality
i.e. the function as described by Eq. (14), is a decreasing function for
In view of Eq. (17) describing the maximum that the function assumes, player F’s optimal offer can be described as follows
As increases, the optimal offer first increases up to and then decreases until it becomes at . In the interval the optimal offer is decreased by an increase of . The threat of execution of the hostage is avoided in the intervals , as player F agrees to meeting the demand for the ransom.
Note that under constraints (12), the optimal offer is reduced to
where and are given in Eqs. (20, 21), as obtained by Selten. FIG. 2 plots the optimal offer against the demand when the probability of capturing the kidnapper depends on whether the hostage has been executed or not (dotted line) and in the case studied by Selten (solid line). Note that, with reference to Eqs. (20, 21, 18, 19), the figure assumes that and , but generally and
iii.3 Optimal Choice of
We now consider the subgame that begins with player K’s choice of , the amount requested for the ransom. The optimal offer is given by Eq. (26). Let and be the values that and assume at , respectively. Then, the optimal probability of the non-rational execution of the hostage as a function of demand becomes:
Using Eq. (7), we have:
Therefore, player K’s expected payoff becomes:
and is given by Eq. (8). Therefore:
Using Eq. (26), becomes:
which can also be re-written as:
Consider first . From Eq. (33), note that is a decreasing function of if is a constant. But and for we have . An increase of decreases and thus is decreased too. That is, will be decreased further (relative to the case when is assumed constant) when increases within the interval . So that as a function of first increases for values of up to . It then decreases for values of up to and then remains constant. Recall that and are given in Eqs. (18, 19).
as discussed by Selten, which is an increasing function for . For in (34), as is the case for the function (33), if is a constant then is a decreasing function of . But now as reduce to , given by (27), under constraints (12). An increase of decreases and thus is decreased too. That is, will be decreased further (relative to the case when is assumed constant) when increases within the interval .
Player K’s optimal demand can be considered as the highest demand such that player F’s optimal offer coincides with the demand. To determine we refer to Eq. (26) and set
Thus, if the allocation of more resources to K’s capture is linked to an increase in then this also results in an increase in the optimal demand . Increasing by allocating higher resources to police, however, is not an effective policy as it appears to be. This is because with K’s increased probability of capture F’s chances to get the ransom money back are also increased. This results in an increase in F’s willingness to pay and thus to a higher optimal demand.
In our model, if the probability of capture is increased, it also results in an increase in the optimal demand . However, since this increase only concerns , the likelihood of K to be captured once he executes the hostage, does not have the perverted effect of increasing F’s willingness to pay.
iii.4 Optimal choice of b
The binary decision variable in (1) describes player K’s choice whether or not to go ahead with the plan to kidnap. The game ends if K selects and the hostage is kidnapped if K selects . Considering the subgame which begins with player K’s choice of the player K’s payoff expectation is given by Eq. (33). As noted above, as a function of is first increasing up to and then decreasing up to and then remaining constant. The optimal value of is given by Eq. (37). As noted before Eq. (35), is the highest demand such that player F’s optimal offer coincides with the demand. Let be the value of assumed at . From Eq. (33) we have
and using Eq. (37) this can be written as
which at becomes . In Selten’s original work, this shows that if the probability of capture can be increased by allocating additional resources to the efforts in finding K then the possibility of decreasing is only limited by the availability of the resources. In our model, the increase of either probabilities, or , leads to an overall decrease in K’s utility, and the effect very much depends on the relative values of , and . In particular, an increase in results in the optimal choice of likely to be , as the value identified in Eq. (43) decreases [keeping constant], and thus the first condition is more likely to be satisfied. Similarly, if increases (keeping constant) then the first condition, i.e. is more likely to hold. If is sufficiently large however then increasing appears to be more optimal in discouraging to select .
Now the optimal choice of is obtained by the following requirements
which can be written as
and when , and it is reduced to
as obtained by Selten. Player K’s choice whether or not to go ahead with the plan to kidnap now depends on and .
The dependence of K’s probability of being captured on whether he has executed the hostage or not can be represented as a bifurcation of (probability of K’s capture in either case of hostage to have been executed or not) into (probability of K’s capture when the hostage has been executed) and (probability of K’s capture when the hostage has been released after paying the ransom).
Overall, we show that increasing either or leads to a reduced likelihood of kidnapping, provided that (otherwise e=0 is not necessarily the optimal choice of K). We also show that if the kidnapping took place, releasing the hostage and paying the ransom remains the optimal choice for K provided the motivation in assigning values to and takes into account the heightened sensitivities, i.e. authorities spend more resources when the hostage has been executed so as to increase the likelihood of capturing player K (i.e. ). Therefore, increasing not only lessen the likelihood of kidnapping, it also ensures that stays above and presents the added benefit of lowering the ransom D. This means that increasing appears to be more optimal than increasing .
A question that might arise is whether it is in the interest of police to advertise the increase in resources, or whether F and K even know about it. From FIG.1, K’s rational decision (dictated by or ) to execute or release the hostage respectively, is known to F. Even if the police remain discreet and do not announce that they are investing more (or less or same) resources in case where , the events or themselves appear sufficient to result in the bifurcation of into to and . Furthermore, studying this bifurcation allows us to understand better the consequences emanating from increasing either or . As we have seen earlier, increasing may result in overall better outcomes.
The non-rational execution of the hostage is a characteristic of Selten’s model that can be explained using a Bayesian approach, i.e. by considering the belief K has about F’s ability to pay. If K thinks that F can pay but F decides not to, this can result in K reacting in a non-rational way, as proposed by Selten. It is anticipated that by introducing beliefs for K, regarding whether or not F can match his demand, it can lend a further perspective to the analysis of this game. In particular, this would result in considering a Bayesian equilibrium instead of subgame perfect equilibrium.
Selten used a binary decision variable in Eq. (3) in order to describe the situation that K can execute the hostage while enacting a non-rational decision. As the hostage may be executed even when the ransom demand is met, therefore, even if . An appropriate probability function to describe the non-rational situation could be when
A policy objective is to minimize the optimal demand as given in Eq. (37). Given fixed resources can be allocated to police to increase the chances of capturing K, these resources are better spent towards increasing .
If K is aware that F cannot meet his demand, then K could either lower his demand and/or decide whether to execute the hostage on rational grounds. This rational decision to execute the hostage depends on probabilities and , and we know that as long as as , executing the hostage is not optimal for K. Thus increasing , as opposed to allocate resources to is again more desirable.
The other situation that could be incorporated in the model is when player K’s cost of preparing the kidnapping is considered non-negligible and player F’s non-monetary disutilities, other than those incurred by the hostage’s life, are however considered negligible. For instance, player F does not attach any value to the capture of the kidnapper.
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