Backward Induction and Jenga

Imagine you are playing Jenga, and it is your turn. The tower looks like this:

To save you some time, almost the entire tower is “spent”—either the middle block or the two side blocks are missing from just about every level. The only exceptions are the top two rows. Jenga’s rules only allow you to take blocks from below a completed row, so the top is off limits. Without doing some ridiculous Jenga acrobatics, this gives you exactly three blocks to choose from, all of them in the second row from the top.

If you have played Jenga before, you know what normally comes next. Jenga blocks are not completely uniform; some are slightly smaller or slightly larger than others. Consequently, the tower’s weight is not usually distributed evenly across a row. If the middle block is slightly larger, then the sides will be loose due to this; if the middle block is slightly smaller, then it alone will be loose.

Under normal circumstances, you should poke around that row to figure out which is the case. If the sides are loose, you would take one of them; if the middle was loose, then you would take it.

Backward induction suggests caution here. Broadly, backward induction tells us that the best way to strategize for right now is to consider how others will respond to whatever you can do. After doing so, choose the option that maximizes your benefits given how others will maximize their benefits later. In short, the future matters; if you ignore it, trouble will find you.

Let’s say that the middle block is loose. What happens if you take it? Your opponent is in deep trouble. There are no blocks that can be removed in isolation without crumbling the tower. Instead, your opponent must take a row with the middle block missing, slowly re-position a side block into the middle slot, and then remove the other block. This is ridiculously difficult. Your opponent is almost certain to lose here, so your initial decision is straightforward.

Life is more complicated if the side blocks are loose. Suppose you take one of them. Your opponent will immediately follow up by taking the remaining side block. When the game comes back to you, the new top row will still only have two blocks on it. This means you will be forced to do the aforementioned nearly impossible task. You will almost certainly lose.

Your only other feasible option is to force out the middle block. This is a tall order—it is not loose precisely because it is supporting the weight above it. You are definitely more likely to lose this turn if you try taking it. However, if you are successful, you will place your opponent in the near-hopeless situation and almost certainly win.

Thus, depending on how you perceive those probabilities, your optimal play may be taking the more difficult block!

What we are observing here is a special case of a more general principle in zero-sum games. There are two paths to victory in such interactions: you winning and your opponent losing. Thus, you should not exclusively focus on what puts you in a good position; you need to pay equal attention to what puts your opponent in a bad position. This is why you should not try to maximize your score in Words with Friends or Scrabble. It also helps explain why trades in Monopoly or Catan don’t work with only two players.


Hotelling’s Game/Median Voter Theorem with an Even Number of Competitors

I will assume that most readers are familiar with Hotelling’s game/the median voter theorem game. If not, the basic idea is that two ice cream vendors are on a beach that stretches the 0-1 interval. Customers are uniformly distributed along that interval. The vendors simultaneously select a position. Customers go to the closest vendor and split themselves evenly if the vendors choose an identical position. Each vendors want to maximize its number of customers.

(You can reframe the question as two candidates placing themselves along an ideological spectrum, with citizens voting for whichever one is closest.)

The Nash equilibrium is for both vendors to select the median location (.5); doing this guarantees each vendor half the business, but deviating to any other point generates strictly less. Full details here:

But what happens when there are more than two players? Someone posed me that question earlier today. It turns out, the solution for this game is complicated when there are an odd number of players. But fairly simple pure strategy Nash equilibria exist for an even number of players:

Proposition. For n even number of players, the following is a pure strategy Nash equilibrium to Hotelling’s game. Exactly two players choose each of these locations: 1/n, 3/n, …, (n-1)/n.

So, for example, for n = 2, two players occupy the position 1/2. (This is the median voter theorem.) For n = 4, two players occupy 1/4 and two players occupy 3/4. (No one occupies the median!) For n = 6, two players occupy 1/6, two players occupy 3/6, and two players occupy 5/6. (The median is back!) And so forth.

This has two interesting implications. First, the median voter theorem basically disappears. In half of these cases (i.e., when n/2 is itself even), no player occupies the median at all. In the other half of the cases, only two do. And even then, as n increases, the percentage of total players occupying the median goes to 0.

Second, quality of life for consumers is greatly enhanced compared to the n = 2 version of the game. Under those circumstances, some consumers would have to travel a full half-interval to reach one of the ice cream vendors. But as n increases, the vendors progressively spread out more and more. Things aren’t perfect—the vendors could spread out further (out of equilibrium) by dividing themselves uniformly—but it’s a step up if you previously thought that all of the vendors would take the median.

The proof isn’t terribly insightful, but here goes. In equilibrium, each individual earns a payoff of 1/n. This is because each position attracts 1/n customers on either side of the position. That 2/n sum is divided two ways for the two players occupying the position, so each earns 1/n.

The rest of the proof involves showing that there are no profitable deviations. There are three cases to consider. First, consider deviating to any other occupied position. Now three individuals are placed at the same spot. Their collective domains remains only 2/n portion of the interval. This is now split three ways, giving the individual a payoff of 2n/3, which is strictly less than 1/n.

Second, consider a deviation to a position beyond one of the more extreme locations—i.e., less than 1/n or greater than (n-1)/n. The argument is symmetric, so I will only consider an amount less than 1/n. Let x < 1/n be the deviation position. Consider the following hastily made but nonetheless helpful figure:


Because customers go to the closest location, the deviator takes all the customers between 0 and x as well as all customers to the left of the midpoint between x and 1/n, which is (x + 1/n)/2. The picture clearly shows that this is not a profitable deviation: the deviator’s share is now strictly smaller than the interval between 0 and 1/n, which is how much the deviator would have received if he stayed put. (Some algebra quickly verifies this.)

Finally, consider a deviation to a position between two locations already being occupied. Formally, we can describe this as any position between (m+1)/n and (m+3)/n, for any even integer m. For example, that could be any spot between 1/n and 3/n, 3/n and 5/n, and so forth. Letting y be the deviator’s new position, here’s another hastily made but helpful figure:


Now the deviator’s captured customers are bounded on the left of the midpoint between (m+1)/n and y and bounded on the right by the midpoint between y and (m+3)/n. It takes some algeabra to show, but this winds up being exactly equal to 1/n. In other words, the deviation isn’t profitable.

That exhausts all possible deviations. None are profitable, and thus this is a Nash equilibrium.

A couple notes. First, this doesn’t extend to odd cases, which tend to have…well…odd solutions. Second, there are other Nash equilibria for an even number of players, including some weird mixing.