I just saw something astonishing on CNN. If you’re wondering why I’m watching CNN, instead of Fox, I’m staying in an extended-stay place in San Bruno, just across the border from South San Francisco, and that’s the only news channel on the cable. Watching Christiane Amanpour bloviate on about what “they” think of us over in the Middle East makes it seem like I went back in time, when they were the only all-news channel.

Anyway, they actually ran a story that described the possibility of terrorism if we don’t take out Saddam, and included the potential costs, which could be hundreds of billions of dollars.

Opponents of the war always dismiss the possibility that Saddam might be involved with terrorism here. I thought that I’d put together a game-theory matrix to look at a range of the possible states of the world, and their potential costs, given various actions.

It’s hardly the final word, and a lot of the costs are admittedly wild-ass guesses, but I’ll welcome any input from people who think they can improve it.

I came up with a set of sixteen possible world-states, which are various combinations of four (presumably independent) binary ones.

I treed it down by starting with the first two potential states–weapons inspections work, or they don’t (in the sense that they actually prevent, in a fail-safe way, the development and deployment of WMD).

Then I built four states by combining these with the next possibility–he supports anti-US terrorists (including but not limited to Al Qaeda) or he doesn’t.

The next state was his domestic support (and of course this one is oversimplified)–his people will support him if we attack, or they won’t.

Finally, either he does or doesn’t have weapons of mass destruction.

That gives us, as I wrote above, sixteen separate possibilities.

I chose three possible actions on the part of the US.

One is the pacifist route–withdraw forces from the Gulf, smooch the appropriate international keesters, apologize for all of our sins, send the French a nice bottle of California sparkling wine, with additional apologies, etc.

The second is the current realistic fallback position of those who are opposed to actual military action–continued deployment of the forces, to pressure Saddam to “continue to cooperate” with the arms “inspectors.”

The third is Bush’s probable default at this point–military intervention to install a new (and hopefully improved) government in Iraq.

The costs were built up out of the cost components that I came up with first, where different combinations of them will apply under different states and actions. I built a table that contains the cost description, a code for use in putting it into a matrix, and the estimate of the cost in billions of dollars.

I built cost matrices by combining all of this together. They can be seen on this page, which I built because the table wouldn’t fit on my blog format.

The potential world states are the columns, and the potential actions are the rows. The top matrix consists of a conceptual assessment of the costs. Each cell simply contains the letter code for the cost element that I believe will apply for that cell.

The second matrix, just below, is the same thing, except the costs are actually quantified, by plugging in and adding the numbers, instead of showing the code.

Assuming you’ve clicked on the link and are looking at the page, I’ll describe the cost elements first. The first one is the cost of military intervention assuming that all goes roughly according to plan (regime collapses quickly). It’s a guess based on a composite of a lot of estimates I’ve heard for the past few months.

The next is the cost if our forces are attacked with WMD. I think that it’s an overestimate, but I’m attempting to be conservative here. I doubt if we will have troop concentrations that allows them to be very effective. If they could hit a carrier battle group with a nuke, it would be pretty expensive, but I find this extremely unlikely.

The next one is the cost of a WMD attack on US soil. Think nuke in Manhattan here. I suspect that a trillion dollars may be an underestimate in a scenario like that.

I also consider non-WMD terrorism. This would be comparable to a repeat of September 11.

I made an estimate of the cost of occupying and pacifying Iraq. Again, this is an overestimate, because I suspect that much of it would be paid for with Iraqi oil.

I added additional costs, if there really is house-to-house fighting in Baghdad, and a lot of resistance from the Iraqi army. I consider this extremely unlikely, but I think I’ve grossly overestimated the cost at two and a half time the nominal military cost estimate.

Finally, in a nod to one of the anti-war types’ arguments, I came up with a cost for domestic terrorism that results from our intervention, but is not supported by Iraq. I estimated this to be half the cost of Iraq-supported terrorism, on the assumption that they can do more damage with explicit state sponsorship than without.

Note that these are the costs to America only–I don’t consider the cost to Iraq or its people, though I believe that removing Saddam will be a tremendous benefit to them.

Just below the table of cost elements is a cost matrix, which really consists of three. The top one is a conceptual cost matrix, in which the cells contain the labels for the cost elements, so that it’s clear what I included and what I didn’t for each case.

Just below that are the actual cost estimates, created by substituting the costs from the elements and adding them up for each cell.

For the anti-war types, there’s some good news there. If you believe that Saddam doesn’t support terrorists who hate the US, then we can have a cost-free policy. We can continue with the inspections, or we can even turn around the ships and bring the troops home. The current policy, by contrast, carries significant costs under any scenario.

On the other hand, those opposing deposing Saddam have to ask themselves–are you feeling lucky? If you are wrong, the consequences can be tremendous–over a trillion dollars (even ignoring all of the human suffering, both in America from the terrorist attack and the Iraqis from continuing to live under Saddam’s unspeakably brutal regime). Scroll over to the right, and look at the last column. It contains the maximum value for each row.

From a “minimax” standpoint, the current course is the lowest-cost one.

Of course, some would argue that this is too simplistic an analysis, because (among numerous other reasons) it doesn’t take into acccount the probabilities of the various scenarios being true, which, if you had them, you could multiply them by the costs to get expected values.

Of course, the problem with that approach here is that, if the cost estimates are wild-ass guesses, the probabilities would be even more so. How much confidence could we have in the output of such an analysis?

What we’re dealing with here is not risk, in which the probabilities can be reliably quantified, but uncertainty, in which they cannot.

As an example, a thirty percent chance of rain represents risk. “It might rain, or it might not, but we have no idea what the probability is” constitutes uncertainty. It’s much easier to decide whether or not to take an umbrella in the first circumstance than the second.

For this reason, economists have come up with a more sophisticated technique for decision making in the absence of probabilities of outcomes. Rather than simply looking for the lowest cost, they instead try to minimize how bad you’ll feel if you make the wrong decision–they minimize “regret.”

It’s based on the notion that when you make a decision, you shouldn’t compare it to some unattainable ideal of zero cost–you should compare it to the best decision you could have possibly made.

Take a simple case–do you take an umbrella when it rains, or not?

Consider a generic cost matrix:

State 1	State 2	Max
3	4	4
1	5	5

It looks like we can minimize our maximum cost by choosing action 1, since four is less than five. But is that really the right decision?

Let’s derive a “regret” matrix from it. This is done by finding the minimum cost for any state, and subtracting each cell of that state from it. The minimum cost for state one is 1, so the column would be three minus one for the first row and one minus one (or zero) for the second row. That makes intuitive sense, since if you made the right decision for that state, you’ll have no regrets. The regret matrix for the example cost matrix is shown below:

State 1	State 2	Max
2	0	2
0	1	1

Note now that if we want to minimize regret, we should actually choose action 2. Note also that this is independent of the relative probabilities of the two states.

Just to be fair, I’ve done a regret matrix for the Iraq situation. It’s the third set of rows at the bottom of the big matrix on the linked page. Note that, if you really, really believe that “inspections” really work, then the “continue inspections ad infinitum” is a zero-regret policy. Again, though, that requires a level of faith in Mr. Blix that hasn’t been substantiated by his results so far.

Looking at the minimax regret situation, again in the final column, even under this analysis, the president’s policy still makes sense, though it makes the “inspect indefinitely” solution look a little better, relative to the “bring the troops home and kiss butt” strategy.

As I said, I hardly expect this to be the last word on the subject, and I’m sure that there are many flaws in the analysis. It’s not like it’s a multi-month RAND study, or anything–it’s just a couple-hour Rand study.

If anyone would like to come up with their own version, feel free. If anyone wants to use my template, but plug in their own numbers, the spreadsheet is here.

Have at it, Blogospherians.