Obama vs Osama

While I was prepping for a class last week, it struck me that part of the decision-making process for the Osama bin Ladin raid could be modeled as a game theory exercise.

Setting the Stage

CIA believes OBL is living in a compound in Abbotabad, PK. We have three strategies for taking him out.
1. Two B-2 bombers, carrying 16 x 2,000lb bombs each.
2. Helicopter raid by Special Operations Forces
3. Joint raid with PK government forces.

Each of these has advantages and drawbacks. How do we decide what to do? Enter, game theory.

Game theory is a relatively recent development in decision science and systems science (all three being post-WWII disciplines). We’re not talking about how you design the next World of Warcraft here, we’re talking about how to make decisions in the face of an opponent who is making their own decisions, usually explicitly designed to help them and harm us. You and your opponent both know what the possibe actions are. You both know what the different payoffs are. You both know the other knows that you… (etc). In some cases, non-zero-sum games, the payoffs are different for each player, and it’s possible to negotiate better outcomes for both players. In others, the zero-sum games, what one wins, the other loses. That’s what this version of OvO is.

Pros and Cons

B-2 Bomber: On the pro side, these planes can get in and out with essentially no chance of interference. Also on the pro side, a single 2,000lb bomb can demolish a house (so I can’t see them needing 32 of them, maybe three for the house, and four more for the rest of the compound), and with laser/GPS they should have a CEP (circular error probability) of about 10m. On the con side , a single 2,000lb bomb can demolish a house, and with laser/GPS they should have a CEP of about 10m. Wait, what?  Well, first off, by demolishing the house, they destroy most of the evidence of who was in there — the forensics ID team will be reduced to dabbing up red smears on concrete with q-tips. In addition, they’d also destroy any intelligence materials, like computer hard drives.  Second, a CEP of 10m means about half your bombs land within 30ft of where you were aiming. And the other half? Well, statistically, a vanishingly small percentage could land almost anywhere, but there’s a non-trivial chance that one might land in the suburb next door. Or a fin could come off and it could land on the military academy. In any event, you’re going to blow out every window in North Abbottabad, and irritate a whole bunch of retired Pakistani generals.

Helicopter Raid:  While not certain, using stealth helicopters lets you get in and out almost as safely as the B-2s. Once on the ground, your SOF team will kill fewer people (i.e. less than all), and can recover intelligence materials, and get a positive ID on the target. That’s special ops talk for making sure you killed the right guy. You still are going to embarass your PK ally, just not as bad as if you carried out a mass bombing raid on his equivalent of Miami Beach.

Joint Pakistani Raid: All of the pluses of the helicopter raid, but with the political cover that it’s the PK government who is leading the charge. The big downside is, half of the PK intelligence service hates us, and the other half is working with the Taliban. There’s a very good chance that if we tell them about him, they’ll tell him about us, and we’ll have no tale to tell.

The Game

Now comes the hard part, assigning values to the possible outcomes. Normally, these kinds of games are represented by a table, showing the strategies and payoffs for the different players. Since the payoffs are mirror images — if I win +$5.00, you win -$5.00 — we only have to show one set. So, let’s do that:

There Not There
B2 +70 -100
Obama HR +90 -50
PK +100 -5

The rational for these numbers is as follows. Suppose we and the PKs do the raid, and he’s there, and we get him. We get 100% of the glory. If we do the helicopter raid (HR) on our own, we suffer some political downside, so only get 90% of the glory. And if we use the B2 bomber, well, even if we get him, there’ll be a stink, and we’re down to 70%. On the other hand, suppose he’s not there. A joint PK/US raid that turns up empty-handed is embarassing, but not overly so, say, we get a -5% of glory (whatever glory is, you decide). A helicopter raid that shoots up some honest drug smuggler and his family will certainly cause a stink, say -50%, and if we churn up the entire compound with bombs, break all the windows in town, and still don’t get OBL, well, -100% is just barely enough.

Hang on, you say, if we get OBL, the loss to him is catastrophic, not just some glory points. Well, that’s true. In fact, in this version of OvO, we aren’t really playing against OBL, but against nature. This is actually more like a farmer deciding to plant corn or soybeans, depending on the long-range weather forecast, and taking the result given by nature’s decision about the actual weather. But calling the game Obama vs the Weather doesn’t sound as cool.

At this point, in a standard ZSG, we’d then look for the strategy that minimizes the maximum loss possible. That is, we look for the strategy that will hurt us the least, assuming the other guy acts rationally. That’s obviously the PK solution. Not only does it have the least worst downside, it has the best upside as well. It dominates all the other strategies. Likewise (if this were a standard ZSG), the other guy would be doing the same, and is dominant strategy is Not There. Whatever our approach, OBL’s best strategy is to always be somewhere else. That being the case, we will always come up with negative glory, and, in the famous words of a famous computer, the only way to win is to not play. Well, that’s no fun.

I’m afraid I’ve skimmed over two ideas here. First, since it is a game against nature, OBL doesn’t know a game is being played, and therefore isn’t picking a strategy. We are more concerned with the likelyhood of us having identified the right house. Second, those payoffs are based on the idea that there is an equal chance of There vs Not There, and that’s not a correct assessment either. About the location: when the CIA spoke to the President, they said they were 60-80% sure OBL was in the house. Let’s assume 80%. That leaves a 20% chance he’s not in the house. The assesment also was that, if we told the PK government, there was a good chance someone in ISI would leak the informaiton to OBL that he was in a game, and he would chose the Not There strategy. Let’s say the chances switched from 80/20 back to 50/50. Here’s what the new payoff table looks like:

There Not There
B2 +56 -20
Obama HR +72 -10
PK +50 -2.5

Now, the helicopter raid has enough of an upside that it makes up for having a somewhat larger downside than a joint PK raid. I am pretty sure that President Obama didn’t call in a game theory expert to decide this for him, but he almost certainly worked his way through a similar thought process on his own.

With an update here.


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One Response to “Obama vs Osama”

  1. Obama vs Osama 2 « FoundOnWeb Says:

    […] are two other issues that I skipped over in my original writeup. First is the question of mixed vs pure strategies. Second is the question of value […]

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