A MOST UNLIKELY TRUEL
I have to go through a four-way stop on my way to the cemetery where I run every day. It continually amazes me how many times 3 or 4 cars arrive at the stop signs at practically the same time. We all hesitate then creep forward, hit the gas and suddenly the brake, stop and then start again until someone has had enough and just goes. Did you know that these types of situations are known as Canadian Standoffs? In a Canadian Standoff, everyone tries to be polite and not take advantage of anyone else. This is usually how things play out at my four-way stop. My guess is most of the yahoos who live around me are being courteous because they are driving without insurance and don’t want to clip another car while on their journey to their favorite haberdashery.
The other day I successfully made my way through the intersection and to the cemetery. I was running my laps and listening to Athena’s band (you will be learning a lot more about her in future posts) when, for reasons I will get to in a minute, I thought of John Nash. I am pretty sure you all know who he is, even if you don’t recognize his name. He is the subject of the book and the subsequent movie, A Beautiful Mind. I recently bought the book, so I decided to immediately move it to the top of my reading list. I just finished it a few minutes ago, it is terrific; at least as good as the movie (and the movie was exceptional).
I thought of Nash because I am working on a series of essays whose topic is game theory, the area of research that garnered Nash his Nobel Prize. That volume, one that can wait a bit longer to be written, is about tennis; this one is about three people with loaded guns and bad intentions.
We all know what a duel is, but do you have any idea what a truel is? All you have to do is add another person to a duel, and you have a truel, a special version of which is popularly known as a Mexican Standoff. Scenes showing three people all pointing guns at each other show up a lot in movies, especially old Westerns. My favorite, though, has to be the standoff shown near the end of Reservoir Dogs. If you ever find yourself making a film and need to create a little tension, all you have to do is give three people guns, set them out in a triangular pattern, give them all reasons to want to shoot each other, and let them have at it.
Author’s note: I have some friends from Mexico, and they all tell me the term Mexican Standoff is perfectly acceptable; no offense meant and none perceived. I have dug a little deeper into this topic and I am not so sure that it is satisfactory; there appears to be some evidence that it has racist overtones. I am not surprised by that at all, are you? Meanwhile, other sources seem to think the phrase originated in Mexico and is absolutely fine. I prefer to err on the side of caution, it doesn’t cost us anything to veer slightly so we don’t unintentionally insult our friends to the south. The best I can do is offer the alternative of a “Three Person Standoff” or a “Three Person Impasse.” I am going to hold my breath while I wait to see if those phrases catch on.
So, here is the totally implausible situation I want to consider. Imagine that Natalie Portman, my favorite actor, Danica Patrick, my favorite retired racecar driver, and Athena, the woman whose band I mentioned earlier, are fighting over which one of them gets to go to lunch with me. Go ahead and scroll up to the first page of this essay and look again at the title. You will now completely understand why I used the word “unlikely” instead of something like “certain” or “unavoidable.”
I proposed several different types of contests the women could participate in to settle this dispute. A slew of emails went back and forth between me and their respective camps. How about a checkers tournament? No. Footrace, perhaps a 10k? No. Corndog eating contest? Out of the question. This went on for quite some time until someone, I can’t quite remember who, suggested a truel using low impact paintball guns. After more lengthy negotiations (you know: time, place, brand of equipment, etc.), everyone agreed.
Let’s assume that Natalie is the worst shot out of the bunch, she can only hit her target one out of every three attempts. Danica appears to be much better, she connects with the target (human or not) two out of three times. Both of them have a big problem with Athena, I have no idea how or why, but she never misses; her success rate is 100%.
Of course, I try to talk to them again and again, but the three of them are not willing to compromise and share me. I finally grudgingly accepted their decision. Ultimately, who can blame them?
When it became clear that none of them were going to listen to me (imagine that, a fellow human being not listening to me!) I let them have at it with the following stipulation: Natalie (33%), by virtue of the fact that she is the worst shot, gets to shoot first; then Danica (67%) if she is not clipped by some green paint, and then Athena (100%) if she is lucky enough to avoid a slight boo-boo. Each woman shoots in turn, and the paintballs keep flying until only one hungry woman is standing.
Now, finally, we get to the interesting game theory aspect of this problematic situation. I know, you might have thought that the real problem is that I am totally delusional, but that is neither very interesting or game-theoretical. The simple question is this: What should Natalie do? She gets first shot, and she better make it count. She wants to eat lunch with me, and if she gets splattered with paint, that is not going to happen. Take some time and give this some thought.
What did you come up with? Did you have Natalie shoot at Athena with her little plastic gun? If so, you got it totally wrong. For those of you who thought she should aim at Danica, you are equally off base. Believe it or not, it can be demonstrated mathematically that Natalie’s best bet for a meal at Red Lobster or Olive Garden with me is to shoot into the air, to intentionally miss both Danica and Athena. My my my, we have reached the point where our intuition miserably fails us. Many of the posts to follow are about the faulty nature of human intuition, I tend to write a lot about that fascinating subject.
So, what is going on here? Well, remember that immediately after Natalie takes a shot (and misses) Danica gets to shoot. Who is Danica going to aim at? Not Natalie, she knows Natalie is not her biggest threat; she is going to shoot at Athena because Athena never misses. What if Danica misses and Athena gets to shoot next? Natalie must certainly also consider this scenario. Athena is not going to take aim at Natalie because Danica is clearly the better shot. Either way, by shooting into the air, Natalie has turned a truel into a duel. Now she has only one person instead of two to deal with. Consequently, she now has the first shot in a duel, not the first shot in a truel. Due to her less than stellar marksmanship, it is the best she can hope for.
As for me, I bet there is a good chance I will starve to death while waiting for this highly unlikely truel to take place. You know, scheduling conflicts and such will make it challenging to get the three of them together. I think it might be in my best interest to get a bag or two of pretzels and a couple jars of peanut butter to tide me over while I nearly burst with anticipation over which of these women might be the lucky winner.
