The gold miner analogy in our last article had, as we discovered, 2 Nash equilibria - 2 outcomes in which both miners make their optimal move given what the other player is doing. But what happens if a game has no Nash equilibria? That is, what happens if it’s zero-sum? This is where mixed strategies come into play.
Let’s illustrate this with a game, a literal one in fact. In soccer penalties, the interests of the striker and the goalie are diametrically opposed: the striker wants to kick the ball in the direction the goalie isn’t going to dive, but the goalie wants to go in the direction the striker is kicking in. For the sake of simplicity, assume that the striker has only 2 options as to where they can kick and the goalie has only 2 options as to where they can dive, namely right or left. And so, we end up with 4 possible outcomes:
Striker kicks right, goalie dives left. That’s a point for the striker, minus one for the goalie
Striker kicks left, goalie dives right. That’s a point for the striker, minus one for the goalie
Striker kicks right, goalie dives right. That’s a point for the goalie, minus one for the striker
Striker kicks left, goalie dives left. That’s a point for the goalie, minus one for the striker
In this situation, where is the Nash equilibria? Where is the optimal outcome for both players? There aren’t any. At least, there aren’t any pure strategy Nash equilibria, the kind we went over in the last article. There is, however, another equilibria type exhibited in this very game. It’s called the mixed strategy Nash equilibrium. This is essentially a probability distribution over 2 or more pure strategies, with players randomly choosing among their options.
This actually fits in nicely with our penalty game since goalies indeed do this sometimes: they pick a side they’re going to dive towards and stick with it. The reason they don’t simply wait for the ball to be kicked and then dive is because there’s not going to be enough time. Although the goalie appears in a rather grim situation, looking at it probabilistically, the striker has, at best, only a 50% chance of winning as only 2 of the 4 outcomes favour him (in this specific analogy at least). The striker can’t really do anything against you to ensure his victory; he’s stuck winning half the time and stuck losing half the time.
The striker probably knows this: no matter which direction he shoots, his chances of scoring aren’t being affected. As such, he also might as well randomly choose which direction to shoot. Both players randomly choose the directions since they don’t expect any better outcomes, this in and of itself is a Nash equilibrium.
Remember, we’re looking for the point in which both players don’t want to change their strategies given what the other player is doing. In this case, neither the goalie nor the penalty taker have any real reason to not randomly choose the directions - that’s the move they’re going with given what their opponent is doing. This subtly answers the game only if we assign the same value to each outcome. But what happens if we make this penalty shoot out a little more realistic and add varying rewards and punishments?
Say the striker is stronger on his right foot, so if he kicks the ball to the right and scores, his teammates expect this of him and so he’s rewarded with 2 points. On the other hand (more like on the other foot), if the striker scores on the left side of the goal, it’s always going to be a weaker kick, a luckier shot so to speak. So while the striker technically scored, it wasn’t exactly an exhilarating goal. As such, the striker is awarded 1 point.
But if the striker’s shot is saved on the right side, that would be a massive disappointment to both him and his team members, and so he gets 3 points deducted. But if his shot is saved on the left side, he doesn’t get any points taken away since people will understand that it wasn’t his best foot so they won’t be as disappointed.
Turning to the goalie, he spent all week practicing his dives towards the left so his team expects him to save kicks headed in that direction. If he saves the penalty on the left, he’s only fulfilling the standards he set up for himself, so he won’t be awarded any points. But if the goalie doesn't save the shot - that is, if the ball is headed to the right and he dives left - all that practice didn’t at all help, so 2 points will be deducted. If the goalie saves the shot coming on the right (his weaker side), then he’ll be awarded 3 points since he surpassed his team’s expectations. If he misses the shot, it’s more understandable so only 1 point will be deducted.
With this point system in mind, we again have 4 possible outcomes:
Striker kicks left, goalie dives left. Neither player receives any points
Striker kicks left, goalie dives right. Striker gets a point, goalie gets a point deducted
Striker kicks right, goalie dives right. Striker gets 3 points deducted, goalie receives 3 points
Striker kicks right, goalie dives left. Striker gets 2 points, goalie gets 2 points deducted
We need a strategy such that the points are maximized (or the losses are minimized) no matter if they choose left or right. Addressing this game and working out the optimal moves from either the goalie’s or striker’s perspective actually requires a fair bit of math to properly work out - stuff that we shall go over in the next article.
Econ IRL:
When state-owned banks start lending more during local elections, what effect does this have on the allocation of capital and other resources? According to a paper published in March of this year, damaging consequences tend to arise from this. Comparing the lending behaviour of state banks and private banks during election cycles in Turkey, the authors find that the former will, as expected, more generously lend out corporate loans and target politically contested areas (think swing states in the US - places that don’t vote for any single party but vary based on the candidate).
And the aftermath of this? The allocation of factors of production is distorted, reducing the economy’s aggregate productivity. The researchers even managed to identify a casual relationship:
We find that capital market imperfections explain the vast majority of aggregate productivity losses and present evidence that local election cycles can partly explain them.
‘Till next time,
SoBasically