Backwards induction and rationality

A fun problem I recently came across:

Consider two players: Alice and Bob. Alice moves first. At the start of the game, Alice has two piles of coins in front of her: one pile contains 4 coins and the other pile contains 1 coin. Each player has two moves available: either “take” the larger pile of coins and give the smaller pile to the other player or “push” both piles across the table to the other player. Each time the piles of coins pass across the table, the quantity of coins in each pile doubles. For example, assume that Alice chooses to “push” the piles on her first move, handing the piles of 1 and 4 coins over to Bob, doubling them to 2 and 8. Bob could now use his first move to either “take” the pile of 8 coins and give 2 coins to Alice, or he can “push” the two piles back across the table again to Alice, again increasing the size of the piles to 4 and 16 coins. The game continues for a fixed number of rounds or until a player decides to end the game by pocketing a pile of coins.

(from the wiki)

(Assume that if the game gets to the final round and the last player decides to “push”, the pot is doubled and they get the smaller pile.)

Assuming that they are self-interested, what do you think is the rational strategy for each of Alice and Bob to adopt? What is the rational strategy if they each know that the other reasons about decision-making in the same way that they themselves do? And what happens if two updateless decision theorists are pitted against each other?

If you have some prior familiarity with game theory, you might have seen the backwards induction proof right away. It turns out that standard game theory teaches us that the Nash equilibrium is to defect as soon as you can, thus never exploiting the “doubling” feature of the setup.

Why? Supposing that you have made it to the final round of the game, you stand to get a larger payout by “defecting” and taking the larger pile rather than the doubled smaller pile. But your opponent knows that you’ll reason this way, so they reason that they are better off defecting the round before… and so on all the way to the first round.

This sucks. The game ends right away, and none of that exponential goodness gets taken advantage of. If only Alice and Bob weren’t so rational! 

We can show that this conclusion follows as long as the three things are true of Alice and Bob:

  1. Given a choice between a definite value A and a smaller value B, both Alice and Bob will choose the larger value (A).
  2. Both Alice and Bob can accurately perform deductive reasoning.
  3. Both (1.) and (2.) are common knowledge to Alice and Bob.

It’s pretty hard to deny the reasonableness of any of these three assumptions!

Here’s a related problem:

An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical antiques. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $100 per suitcase—he is unable to find out directly the price of the antiques.

To determine an honest appraised value of the antiques, the manager separates both travelers so they can’t confer, and asks them to write down the amount of their value at no less than $2 and no larger than $100. He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: $2 extra will be paid to the traveler who wrote down the lower value and a $2 deduction will be taken from the person who wrote down the higher amount. The challenge is: what strategy should both travelers follow to decide the value they should write down?

(again, from the wiki)

Suppose you put no value on honesty, and only care about getting the most money possible. Further, suppose that both travelers reason the same way about decision problems, and that they both know this fact (and that they both know that they both know this fact, and so on).

The first intuition you might have is that both should just write down $100. But if you know that your partner is going to write down $100, then you stand to gain one whole dollar by defecting and writing $99 (thus collecting the $2 bonus for a total of $101). But if they know that you’re going to write $99, then they stand to gain one whole dollar by defecting and writing $98 (thus netting $100). And so on.

In the end both of these unfortunate “rational” individuals end up writing down $2. Once again, we see the tragedy of being a rational individual.

Of course, we could take these thought experiments to be an indication not of the inherent tragedy of rationality, but instead of the need for a better theory of rationality.

For instance, you might have noticed that the arguments we used in both cases relied on a type of reasoning where each agent assumes that they can change their decision, holding fixed the decision of the other agent. This is not a valid move in general, as it assumes independence! It might very well be that the information about what decision you make is relevant to your knowledge about what the other agent’s decision will be. In fact, when we stipulated that you reason similarly to the other agent, we are in essence stipulating an evidential relationship between your decision and theirs! So the arguments we gave above need to be looked at more closely.

If the agents do end up taking into account their similarity, then their behavior is radically different. For example, we can look at the behavior of updateless decision theory: two UDTs playing each other in the Centipede game “push” every single round (including the final one!), thus ending up with exponentially higher rewards (on the order of $2N, where N is the number of rounds). And two UDTs in the Traveller’s Dilemma would write down $100, thus both ending up roughly $98 better off than otherwise. So perhaps we aren’t doomed to a gloomy view of rationality as a burden eternally holding us back!


One final problem.

Two players, this time with just one pile of coins in front of them. Initially this pile contains just 1 coin. The players take turns, and each turn they can either take the whole pile or push it to the other side, in which case the size of the pile will double. This will continue for a fixed number of rounds or until a player ends the game by taking the pile.

On the final round, the last player has a choice of either taking all the coins or pushing them over, thus giving the entire doubled pile to their opponent. Both players are perfectly self-interested, and this fact is common knowledge. And finally, suppose that who goes first is determined by a coin flip.

Standard decision theory obviously says that the first person should just take the 1 coin and the game ends there. What would UDT do here? What do you think is the rational policy for each player?

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