Penalty Shoot-out
Glooooooooooooooooooooooooooooooooooooooooorb.
You wake up in a cold sweat, lying on the carpeted floor of the starship’s rec room. It’s the next morning. Beezleglorb stands over you with a menacing look. It seems like the fish fell out of your ear overnight, so you reinsert it.
Human. Play games with me.
You follow Beezleglorb to a vast, open room, with nothing inside except a ball and a football goalpost.
If you can score enough penalty kicks on me, I’ll consider setting you free.
Given your lack of ball control, all you can decide is if you’d like to roughly aim the ball towards the left or right side of the goal.
As you’re about to punt your first kick, you hear a little whisper from inside your head. It’s not a hallucination–it’s the fish in your ear. Turns out it’s been sentient this whole time, and it wants to let you in on Beezleglorb’s soccer stats. Its motivations are unknown, but you trust in the fish.
By analyzing your physique, it reveals the following payoff matrix for you:
| Dive Left | Dive Right | |
|---|---|---|
| Shoot Left |
40, 60
|
90, 10
|
| Shoot Right |
80, 20
|
30, 70
|
In this matrix, your payoff is the probability (in percent) that you score a goal, and Beezleglorb’s payoff is the probability that they successfully block your shot.
Try to interpret this payoff matrix and derive any Nash equilibria.
Essentially, if Beezleglorb dives toward your shot placement, they have a decent chance of saving it; otherwise, you have a pretty good chance of scoring (although you might miss the goal entirely).
You realize that there are no pure Nash equilibria. However, with your newly acquired knowledge from the previous game, you approach it with a mixed angle.
Suppose there exists a mixed Nash equilibrium where you shoot left with probability $p$ and right with probability $1-p,$ and Beezleglorb dives left with probability $q$ and right with probability $1-q.$ If this were a true equilibrium, then given Beezleglorb’s probability distribution, you should be indifferent to either one of your strategies--otherwise, you would simply assign 100% to the more favorable pure strategy, but we know that there are no pure equilibria. So, all of your pure strategies should have equal expected payoffs.
We can thus write the following equation: $$E_{\text{Archibald}}(\text{shoot left}, (q, 1-q)) = E_{\text{Archibald}}(\text{shoot right}, (q, 1-q))$$ $$40q + 90(1-q) = 80q+30(1-q)$$ $$q=0.6.$$
Using a symmetrical method, we find that $p = 0.5.$ We can even plug in these known values of $p$ and $q$ to evaluate that in this mixed equilibrium, you will score 60% of the time.
You mentally thank the fish for its help. Beezleglorb has just finished doing the same calculation to inform their decision. Conveniently, your calculations tell you that you should aim left or right with equal probability, so you can use the trusty coin method. It brings Beezleglorb unpleasant flashbacks. You score your first goal.
As an exercise, suppose your right foot got a little more accurate, such that the updated payoff matrix looks like this:
| Dive Left | Dive Right | |
|---|---|---|
| Shoot Left |
40, 60
|
90, 10
|
| Shoot Right |
85, 15
|
30, 70
|
How does this affect the mixed Nash equilibrium? Should you play to your strengths and shoot more to the right?
Recalculating, we actually see that $p$ increases to 52.4% and $q$ decreases to 57.1%; that is, the new mixed equilibrium has you shooting more towards the left, and Beezleglorb diving more towards the right.
Your intuition may have told you that if your right foot improved, you should shoot more to your right; however, this isn’t necessarily true! Instead, Beezleglorb compensates for your improved right foot, by diving right more frequently, and you compensate for Beezleglorb’s dive frequency by shooting left more frequently.
If you’d like to further familiarize yourself with mixed equilibria, revisit the BoS game to find a third equilibrium using mixed strategies. Does a mixed equilibrium exist for the Prisoner's Dilemma, and why or why not?
For more work related to the specific penalty kick problem, extensive research has already been done!