An Impossible Choice


You can barely make out your surroundings.

Where am I? You ask the void.

Glooooooooooorb, the alien replies. Something about your hopeless expression reveals to it that you didn’t understand, so the alien sticks a small fish in your ear and repeats themselves.

Hello, Archibald. I am Beezleglorb. If you wish to return home, play games with me. You better be good at them, too.

Good thing you’re no stranger to games.

Beezleglorb lays out the rules of their first game. In front of both of you are two buttons, colored neon brown and seafoam green. The room is constructed in such a way so that neither of you can see which button the opposing player presses. If you choose different colors, you owe Beezleglorb one dollar. If you choose the same color, Beezleglorb owes you one glorbuck (which converts to one US dollar).

Try to model the game with a payoff matrix and find any Nash equilibria.

Reveal Answer
Archibald
Beezleglorb
Brown Green
Brown
1, -1
-1, 1
Green
-1, 1
1, -1

After constructing the payoff matrix, you realize that with the tools you’re aware of, you can’t find any Nash equilibria that may inform your decision. Your best bet is to consider if Beezleglorb is more likely to choose one color over the other. After closely examining their alien face, however, you realize it’s indecipherable. It doesn’t really matter what you do; Beezleglorb might as well be choosing their color at random.

So, your hand hovers over the green button. After all, seafoam green is your favorite color. In fact, it’s all you’re wearing; your blazer, trousers, rainboots, and monocles are all colored seafoam green.

But you get an eerie feeling that you’re being watched closely. It’s Beezleglorb, who’s been analyzing your appearance as much as you’ve been attempting to analyze theirs. And based on everything they know about you, you’re probably going to pick seafoam green. As soon as you realize this, Beezleglorb smiles. They’re reading you like a book, knowing you’re planning to switch to neon brown last minute! You’re not sure how you know, but every time you even think about switching buttons, Beezleglorb can tell. No matter what, Beezleglorb will be one logical step ahead. It’s hopeless.

However, you think of a brilliant strategy. You pick up the dollar coin you were nearly going to kiss goodbye to. That’s right—your choice will be decided by a coin toss. After all, if you don’t know what you’re doing, then Beezleglorb can’t.

It’s a new option you’ve never even considered before. Your strategies don’t have to be “pick seafoam green” or “pick neon brown.” A third option can be “pick seafoam green or neon brown with 50% probability.” Actually, there are infinitely many probabilistic distributions you can assign to the two buttons.

How can we describe these options with Nash equilibria? Instead of calculating preferences over the action profiles, we compare expected payoffs. In this game, suppose both you and Beezleglorb “mix” your strategies with probability 50%. Then, we have:

  • 25% chance of (Green, Green), for a payoff of +1
  • 25% chance of (Green, Brown), for a payoff of -1
  • 25% chance of (Brown, Green), for a payoff of -1
  • 25% chance of (Brown, Brown), for a payoff of +1

Which grants an expected payoff of 0. Symmetrically, Beezleglorb has an expected payoff of 0.

When referring to mixed strategies, payoff functions involving expected value are technically called Bernoulli Payoff Functions. Preferences over deterministic action profiles are called Von Neumann-Morgenstern preferences (vNM preferences for short). These behave in the same way that payoff functions and preferences behave for pure strategies; after all, pure strategies are just mixed strategies with 100% probability assigned to one of the options. So, we'll just refer to them as payoff functions and preferences.

Suppose you pick brown with probability p and green with probability 1−p, and Beezleglorb picks brown with probability q and green with probability 1−q. Try to analyze these new probabilistic strategies by plotting some best response curves; that is, for every value of q, graph the values of p that are optimal, and vice versa.

Reveal Answer
0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 Archibald's p Beezleglorb's q

The only point of intersection is when both players assign 50-50 probabilities to the colors. In this sense, there actually is a Nash Equilibrium. However, note that it is a weak Nash equilibrium, as any mixed strategy is a best response to playing 50-50. It turns out that any Nash equilibrium involving mixed strategies is a weak equilibrium.

Symbolically, a mixed strategy Nash equilibrium is therefore defined as a mixed strategy profile α∗ such that for each player i, we have Ui(αi∗,α−i∗)≥Ui(αi,α−i∗) for all possible mixed strategies αi. This is practically identical to the definition for a pure Nash equilibrium.

Mixed strategies have various interpretations. In this game, they can represent probabilistic distributions over pure strategies. They can also represent the distribution of pure strategies within a population.

You glance up at Beezleglorb and flip the coin out of their sight. Frustrated, they select a button at random, and so do you. You win.

Beezleglorb demands thousands of rematches. Ultimately, you win approximately 50% of them. You will sleep well tonight with a net loss of around zero dollars.

Next Chapter: Penalty Shoot-Out