When we talk about game theory in sports we almost always talk about zero-sum games. Whatever one team gains, the other loses, whether it’s yards, wins, or even win probability. It’s all very tidy.
One sport that features some non-zero-sum games is auto racing. I’ll admit that until recently I was a sports snob. Spending four hours watching cars make left turns and have their tires changed was never my idea of a good use of time. But now I have to say I am intrigued by NASCAR.
Drivers accumulate points throughout the season based on their finishing positions, and the top 10 drivers qualify for a playoff-type system in the final few races. Every race is worth the same number of points. The winner gets 185 points, the second place finisher gets 170 points, and the third place finisher gets 165 points. You can see the full table here. We can agree that points are not the only consideration in the value of winning a race, but for the sake of discussion I’ll limit the value just to the points.
Before I get into the meat of NASCAR’s non-zero sum game, consider the classic Ultimatum game in which two players divide a sum among themselves. The first player decides how to divide the sum and makes a single offer to the second player who can accept or reject the offer. If the second player accepts the offer, they spilt the sum accordingly, but if he rejects the offer, neither player receives a payout.
A first player who is generous might offer a 50/50 split, but a more shrewd first player could offer as little as a 99/1 split. In theory, it’s rational for the second player to accept any offer, no matter how small, because any alternative is better than getting nothing. But in clinical experiments second players will usually reject offers less than an 80/20 split. This is an apparently irrational decision in the abstract, but the threat of rejection encourages fair offers on the part of the first player.
How does this relate to NASCAR? Consider two drivers racing for the lead on the final lap of a race. The driver currently in second often has the edge. He can draft behind the leader, and can time his passing move to maximize his advantage. He can choose positioning (high or low), and can leave no time left for the other driver to respond and retake the lead. (I’m not an expert on NASCAR at all, this is only what I’ve been told.)
For the sake of discussion, let’s assume the second-place driver (Driver 2) would be successful if he tries to pass the leader. We can say Driver 2 has two strategies—pass and not pass. It seems obvious that he should pass and take the checkered flag. The driver currently in the lead (Driver 1) does not have time to return the favor, but he does have one last option. He can wreck both cars.
If Driver 1 decides to wreck, neither driver receives any points. The game looks like this:
|Driver 1/Driver 2||Pass||Don't Pass|
It’s almost like Driver 2 is making Driver 1 an offer, and Driver 1 can accept or reject it. It would be irrational, technically speaking, for Driver 1 to wreck both cars. He would lose all those points for second place. It’s suicidal. The equilibrium of the game is the Pass/Don’t Wreck outcome, in which Driver 2 takes the lead and Driver 1 chooses not to wreck both cars.
But he doesn’t actually have to wreck at all, if the threat of the wreck is credible. Driver 2 would only need to think that Driver 1 would wreck them both a certain percentage of the time for it not to be worthwhile to pass. In this case, the percentage would theoretically be 8%, (1 – 170/185 = 0.08).
So if Driver 2 believes Driver 1 is irrational, spiteful, and vengeful he should choose not to pass and keep his second-place points. Driver 1 would go on to win the race and keep the first-place points. This is an example of a game in which it pays to be irrational. It’s no coincidence that one of the all-time greatest drivers, Dale Earnhardt Sr., was known as “The Intimidator” for his willingness to swap paint.
There may be many non-zero sum games being played out simultaneously on the track during a race. There could be the reverse game, where Driver 2 says, “Let me pass or I’ll take us both out.” The game may actually be like the game of Chicken, but slightly more complicated. (The Hawk-Dove game, which may be related to the source of home field advantage in sports, is a variant of the game of Chicken.)
NASCAR helps us understand why people sometimes “irrationally” reject offers in the Ultimatum game, or are willing to collide in the Chicken game. It's only irrational if you play the game in an abstract clinical setting. But in reality we play these games not just once in isolation, but over and over again. In the real world, we establish reputations for our willingness or unwillingness to accept the short end of the stick. People value the long-term utility of their reputations more than the short-term payoff of any single transaction, because a strong reputation will likely lead to even bigger payoffs in the future. Our squishy, analog brains are built to discourage people from taking advantage of us and to naturally encourage fair treatment.
Below is an example of the Ultimatum/Chicken game in NASCAR. In fact, the final lap of the 1979 Daytona 500 is probably the single most dramatic moment in stock car history, perhaps on the same level as the 1958 Championship Game or Super Bowl III. Donnie Allison and Cale Yarborough were running first and second on the final lap but wrecked each other, leading to a fistfight in the infield.
At first I thought the wreck was just an accident, but then I listened to the interviews more closely. Donnie Allison (Driver 1) said, “I knew Cale was gonna try to pass me. There was no question about that.” I thought, well duh, of course he was going to try to pass you. He’s trying to win the race just like everyone else. Why would Allison even need to say something like that? Why would there ever be any doubt?
And then it dawned on me. Allison would normally expect Yarborough to keep his second place position and not risk the wreck, but Allison was willing to risk it rather than let him take the lead. Yarborough was even conscious of the game, at least in retrospect. He said, “I would have rather taken second than both of us endin’ up in the wall.”