How to resolve a hostage crisis: Part II

Last week, I showed that if the villains are perfectly rational, and this is “common knowledge,” then the villains will be best off by just surrendering immediately without killing any hostages. But the rationality assumption is a big one – let’s see what happens if we drop it.

Once there’s a good chance that the villains are “crazy,” even criminals who are perfectly sane will pretend to be crazy so they can extract money as ransom. We can then model this as a Bayesian game, with N periods corresponding to the $N$ hostages that are holed up in the bank. Each period will consist of two stages: in the first, the SWAT team decides whether to pay up or not; in the second, the villains choosing whether or not to kill a hostage; if they don’t, it’s tantamount to surrender. If they run out of hostages to kill, they are forced to surrender, and the surrender outcome gets progressively worse as they kill more hostages.

The payoffs are described as such:

Each dead hostage: gives $-H$ to the SWAT team, and $-C$ to the villains if they end up surrendering.

Surrender: gives $-S$ to the villains.

Pay up: gives $-P$ to the SWAT team, P to the villains, where $H > P$ (1)

Finally, we assume that there is a probability $k$ (initially) that the villains are nuts.

We’re going to construct a mixed strategy solution to this problem, which will yield a perfect Bayesian equilibrium. That is, in a given period $i$, the SWAT team pays up with probability $p_{i}$, and the villains execute a hostage with probability $k_{i}$. For those who are unfamiliar with mixed strategies, the basic idea is that both sides are indifferent between (at least) two options, so they may as well flip a coin as to which one to do. The trick is that the coin is weighted so as to make the other side indifferent as well. Thus they will flip their coins so as to make the first side indifferent. Hence this forms a Nash equilibrium – neither side can benefit by unilaterally changing their strategy.

The derivation is a little drawn-out, so I’m going to break it into steps:

(i) If the SWAT team always pays up in some period $i$, then the villains may as well execute their hostages until they reach that stage; after all, this guarantees them an automatic victory. But if they will do that, then the SWAT team should pay up immediately (i.e. in period 1): since they’ll lose anyway, they may as well save the lives of the hostages.

(ii) Conversely, if the villains will always execute a hostage at a certain stage, then the SWAT team should always pay up then. But then we run into the same issue as in (i), so the SWAT team will end up paying at the beginning. Combined, (i) and (ii) limit the types of equilibria we have to analyze.

(iii) If there is a period i at which it is known that the SWAT team will not pay up, no matter what, then the normal (not crazy) types, knowing this, will surrender in period $i-1$. But this possibility leads to an inconsistency. Knowing that the normal types will do this, the SWAT team will believe that anyone who hasn’t surrendered must be crazy. If so, it is a best response to pay up in order to avoid more casualties.

(iv) Thus, from (iii), there cannot be a perfect Bayesian equilibrium in which the SWAT team will maintain the siege to the bitter end – they must always cave in to the villains’ demands with some positive probability. Indeed, we see from (i) and (iii) that either they pay up immediately, or they pay up in period $N$ with some nonzero likelihood.

(v) There is no benefit to the villains in executing the $N^{th}$ hostage. Thus, if they are normal, they will not execute, and so this hostage will be killed if and only if the villains are crazy, which occurs with probability $k_{N}$.

(vi) In period $N$, by (iv), the SWAT team must still be indifferent between paying up and maintaining the siege. This implies that

$-P-(N-1)H=-(N-1)H(1-k_{N})-NHk_{N}$

$k_{N}=\frac{P}{H}$

(vii) In all periods $i$, the normal-type villains must be indifferent between surrendering and killing a hostage. Thus, if their expected payoff, if the game reaches the second stage of period $i+1$, is $\Pi_{villains}^{i+1}$, then we get

$-(i-1)C-S=p_{i+1}P+(1-p_{i+1})\Pi_{villains}^{i+1}$

But we know that since the villains will be indifferent as well in the next period (unless $i=N-1$), $\Pi_{villains}^{i+1}=-iC-S$. In any case, this formula still holds for $i=N-1$, since the normal types surrender in period $N$ anyway. Thus we get, after substitution,

$p_{i+1}=\frac{C}{P+iC+S}$

As expected, this probability goes down as more hostages are executed – after all, there is less of a risk of more future casualties, since there are fewer hostages left – a dark thought, indeed.

(viii) We now derive the probability that the villains execute a hostage in the periods other than the last. The SWAT team is indifferent between maintaining the siege and paying up. Hence if their expected payoff at the beginning of period $i+1$ is $\Pi_{SWAT}^{i+1}$, then

$-(i-1)H-P=-(1-k_{i})(i-1)H+k_{i}\Pi_{SWAT}^{i+1}$

But because the SWAT team will also be indifferent between paying up and maintaining the siege in the next period, $\Pi_{SWAT}^{i+1}=-iH-P$. Hence

$-P=-k_{i}H-k_{i}P$

$k_{i}=\frac{P}{P+H}$

This value is constant! Note that it is decreasing in $H$ – if the hostage is more valuable to the SWAT team, then the likelihood that they will kill another doesn’t have to be as high in order to deter the SWAT team from maintaining the siege. Conversely, if the hostage is less valuable, the villains will have to be more likely to kill the hostage to show they mean business.

(ix) Combining (vi) and (viii) gives us the the proportion of crazy types, at the beginning of any period $i$ (after the first) in the perfect Bayesian equilibrium will be $(\frac{P}{H+P})^{N-i}\frac{P}{H}$. Thus, if at period 1, the initial probability that the villains are crazy, $k$, is greater than this amount, then the risk that they are nuts (or wannabe-nuts) is too high, and the SWAT team should fold immediately. Otherwise, they should maintain the siege in the first period, while the villains execute the hostages with the exact probability so that the likelihood that they are nuts, as of the start of period 2, is exactly $(\frac{P}{H+P})^{N-2}\frac{P}{H}$. Afterwards, the villains follow the strategy given by (vi) and (viii), while the SWAT team follows the strategy given by (vii).

(x) As a final note, we see from (ix) that as $N\rightarrow\infty$, the threshhold for the initial likelihood of craziness necessary to enforce a SWAT team payout goes to 0. This makes sense – there’s a larger potential for more casualties as the number of hostages goes up.

(1) (footnote: though dropping this assumption only slightly changes the outcome. Also, we exclude the possibility of storming the bank, since it’s similar in concept to the other options).

How to resolve a hostage crisis: Part I

Suppose an archvillain, with a few lackeys, comes into a bank lookin’ for loot. Knowing that he has the potential to get more out of the bank than merely what’s in the vaults, and knowing that it’s not so simple to just drive away, they decide to hold the innocent civilians in the bank hostage. Unfortunately, Batman is away in Shangri-La for now, so he can’t help you. Spiderman is busy trying to learn how to fly. And the other superheroes just suck. So your only option is to engage in a standoff, with an elite SWAT team standing outside, ready to storm the building if possible/necessary. Meanwhile, the criminals are inside, threatening to kill their hostages unless they get $100 million. Suppose, for now, that the criminals are completely rational, and this is known to the SWAT team as well. Let’s start with the case where there is only one hostage, and the SWAT team cares about this dude so much that his life is worth$100 million to save. Nonetheless, if the SWAT team refuses to pay the ransom, what are the criminals to do? If they shoot the guy, then there’s nothing to prevent the SWAT team from storming the bank and shooting them dead, or at the very least capturing them, resulting in a long prison term for murder (on top of robbing the bank). This extra penalty makes it worse than not killing the hostage. Thus the SWAT team has no incentive to give in to their demands, as the threat of killing the hostage is not credible – the criminals are worse off if they do so.

Now let’s see what happens if there are more hostages. We can use induction to see that the hostage-takers face this predicament, no matter how many hostages are involved. Suppose that, once there are N hostages left, the SWAT team has no reason to give in. We then look at the case with N+1 hostages. If the hostage-takers kill the spare, they are left in a situation in which they automatically lose – they won’t get their ransom, and they’re stuck there until they are forced to surrender. Thus they won’t want to kill the N+1th hostage. We therefore end up with no dead hostages, no matter how menacing the threat.

I know what you’re thinking now: “Yeah right. Why don’t you try this strategy and see what happens? If you do, the hostages will end up DEAD.” And you’re probably right. So where have I gone wrong?

Uncertainty in the number of hostages doesn’t help much here, because the number of hostages will be bounded. We can therefore apply the same induction reasoning as before to the uncertain number of hostages, and once the criminals have claimed to kill the upper bound of the number of people that could possibly have been in the bank, they’re still dead meat. So we’ll have to find some other reason.

The key to my original argument was that the criminals are rational. Obviously, that’s not always going to be the case. After all, most rational, normal people do not go around robbing banks. So, chances are pretty good that the people robbing the bank aren’t quite rational, and will kill the hostages even though it ends up hurting them. In this case, it might be worth paying up to prevent the deaths of innocents. We’ll analyze this in part 2, which I will post in a week.

The problem with insider trading: a game theory perspective

I’m no expert at finance, so I can’t tell you in general how you should invest your money. What I can tell you, though, is why insider trading is such a harmful phenomenon, and why it should be curtailed to the best of the government’s abilities. To illustrate this, I’m going to use a simple Bayesian model.

To begin with, it is reasonably clear why someone would execute an inside trade. If some executive-honcho dude has some extra knowledge, not generally available to the public, which leads him (or her) to conclude that a particular financial asset (we’ll assume it’s a stock) is worth some amount different from its market value, he can use this to his or her advantage. If the stock is undervalued, he can purchase the asset and make money for nothing. If it’s overvalued, he can short it, again making money for nothing. Either way, he wins at the expense of other financial agents.

So far, pretty clear. But hold on a second: investors know that executives can do this, and so anticipate the relevant market moves. So, if the executive sells, the market for the stock responds by lowering the price; if he buys, it rises by a certain amount as well. Does this ability help the other investors to avoid the pitfalls of insider trading?

Unfortunately, it does not, for a few reasons. The first is pretty obvious from the definition of insider trading – though the market may expect the stock to be worth a certain amount more or less than  before, it can’t know the exact amount. Only the executive, with the inside scoop, knows this. Hence he will be able to capitalize on any difference between what the asset actually should be worth (based on all the information), and what investors expect it to be worth based on their more limited information (which includes the fact that the insider is taking action).

Second, it is important to take note of the advantage in timing that the executive has. Since the market can only correct itself in response to his or her moves, he or she gets to make the first move. So, if the asset is worth more, he can buy more shares before others realize what is going on. Similarly, he can sell before others notice that he or she thinks the stock is overvalued, still making a profit through these means as well.

Finally, we come to what I think is the most interesting problem: the executive can take advantage of investor expectations, even when the fundamentals of the company actually remain the same in as perceived in public. Thus, if investors believe that, say, a sale by the executive indicates that the stock is worth more than previously thought, they will reduce the price at which it trades. But if so, the executive, knowing that the other investors have this belief, can sell for no reason whatsoever, and then repurchase at a lower price, making a net profit without a loss of shares that he or she holds. Similarly, if investors expect the asset to be worth more if the executive buys more shares, then the executive can buy shares and resell at the higher price.

Combined, the first and last reasons demonstrate that there cannot be a perfect Bayesian equilibrium which allows for insider trading. For if the price falls with a sale (so the investors think the stock is worth less), the executive will take advantage of this investor belief through selling without reason (i.e. the stock is not worth less); a similar situation occurs if the investor believes that the asset is worth more, as then the executive will buy without fundamental reason. Hence the decrease in the market value for the stock will occur based on false beliefs due to the executive’s action. Meanwhile, if the investors respond to executive stock purchases by, say, lowering their expected values of the stocks (or keeping the price the same), the first reason above demonstrates that the executive can increase his or her payoff by purchasing when the stock is worth more than publicly thought. Thus, in the latter case, the investors also have false beliefs (undervaluing the stock).

In order to allow for the possibility of a perfect Bayesian equilibrium in the stock market, one must therefore eliminate the possibility of insider trading. Under a rigid enforcement mechanism, such an equilibrium does exist: the market does not respond to actions of the executive. Since the executive cannot use private information to purchase or sell stocks, he or she can’t take advantage of the first or second reason; and since the market does not expect a change in value based on his or her actions, he or she cannot take advantage of the third reason. From the other end, the investors know that the executive can’t use any information that they do not have, they are not taken aback by any executive stock moves, and so they don’t change their expected valuation of the stock. Thus we see the importance of an effective prevention of insider trading – otherwise, the stock market is open to the games of executives with too much information.

“My opponent is a no-good, rotten cheater out to destroy America…”

Well, the primary campaign season sure has been heating up. As expected, we see that the candidates are really going after each other, making outlandish remarks, the whole shebang. This isn’t anything new: even back in June, former Minnesota governor Tim Pawlenty accused former Massachusetts governor Mitt Romney of being complicit in the formulation of the Democrats’ controversial health care reform plan, calling it “Obamneycare.” Clearly, with the jam-packed field and a necessity to beat the other contenders to get to the main event, there is a strong incentive for the candidates to go after each other’s throats.

Yet there is a major tradeoff in doing so. All of these candidates would very much prefer, even if they personally do not win the Republican nomination, that one of the other Republicans beat Obama in the general election. But by attacking their competitors, they decrease the chance of that happening.

Let’s assume that a candidate’s chance of winning is proportional to the amount of flak that is not directed at him or her. Thus, if each candidate $i$ (of $N$ total) generates $f_{i,j}$ flak towards candidate $j$, then each candidate’s share of the flak is $\frac{\sum_{j=1}^{N}f_{j,i}}{\sum_{k=1}^{N}\sum_{j=1}^{N}f_{j,k}}$

However, by being the victim of more flak, the chances of beating Obama get smaller and smaller: voters are much less likely to vote for a candidate who has a terrible reputation, as bestowed upon him or her by his or her opponents. We can therefore set up a threshold at which the voters will not vote for candidate i: once he or she has taken more collective beatings than threshold $F^{*}$, Obama automatically wins (we can model Obama’s chances of winning as increasing in the amount of trashing his opponent has received; the main idea of the result will remain the same). Scary thought if you are a Republican, no?

Each candidate gets payoff $R$ for being the nominee, as well as an additional payoff $P$ if he or she wins the general election. The other candidates get payoff $0$ if they lose the nomination and the presidency, while they get payoff $V$ if a different Republican candidate wins the general election. Assume that $R>V$ – candidates would rather be the nominee themselves, and get a shot at the presidency, no matter what the other candidates’ chances are.

This being the case, despite the preference to beat Obama, Republican primary candidates can always do better by hacking at each other as much as possible. Think about it – given any fixed amount of flak the other candidates are giving you, your chances of being the nominee go to one as the amount you attack them goes to infinity. This strategy (given finite amount of flak from others) will yield a payoff of approximately $R$, which we stipulated was greater than $V$. Hence the Democrats will automatically win.

Note that this isn’t quite a Nash equilibrium, since the set of possible options is not bounded. Yet some of the normal game theory ideas are still visible here: what will end up happening, based on the incentives, is that the Democratic candidate gets re-elected. Nevertheless, the result depends on several assumptions: that the Republicans can attack each other to an arbitrarily large extent, and that they would always rather be the nominee than let someone else win. Still, it is interesting to observe that the situation as modeled here leads to an automatic Republican loss in November 2012. Pretty ironic, given that one would think that the entire purpose of the campaign is to unseat Obama.