I was wondering what are some proposed solutions in the literature to the following, well-known paradox:

Say two rational, intelligent players A and B stand in front of a stack of 100 coins, and play the following game: each turn, a player may choose to pick up one coin, giving the other player the next turn, or picking up 2 coins and ending the game right there. A and B both want to maximise their profit. They cannot talk to each other or interact in any way (outside the game itself, of course).

A might reason as follows: if we have only 2 coins, I'll pick 2 coins right there and end the game. But B, who is aware of this, will then opt to pick up 2 coins when 3 are remaining (because he'd then end up with 1 coin extra). Continuing on with this induction, we eventually arrive at the conclusion that the most rational behaviour is to pick up two coins on A's first turn. Obviously a very unusual conclusion.

Thanks!

(I have marked this question 'epistemology' because of ties with the "Unexpected hanging paradox")

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• Welcome to the site! :) My initial question for you is: Can the players talk to each other? Two "rational" players would just say, "Hey, this game is lame. How about we just play it out to end and we'll have split the total by then." That would maximize both their profits. I think the "conclusion" in the example is only true if they wanted to maximize each of their profits at the exclusion of the other person (i.e. the goal is not to actually profit per se but to make more than the other person). – stoicfury Nov 16 '11 at 23:27
• Thanks! In most versions I know, the players cannot talk to each other, and the goal is to earn the maximum possible profit for themselves (the other player's profit doesn't matter). I have edited my post to include the former assumption. – yorei Nov 16 '11 at 23:41
• If it doesn't matter how mich the other person gets the rational strategy is as stoicfury says, always pick up 1, you'll end up with 50. What paradox is there then? – Mitch Nov 16 '11 at 23:47
• Well, my question is more directed towards people who are already familiar with this paradox but... in short, consider the case there are only two coins. Let's say it's A's turn. Then the most rational thing to do would be for A to pick up those two coins. Consider the case there are 3 coins, and it's B's turn - the most rational thing for B to do is pick up 2 coins. Consider the case there are 4 coins and it's A's turn - A knows that if he picks one coin, so there's 3 coins remaining, B would pick two and the game would end at lesser profit for A, so A picks two coins again.... – yorei Nov 16 '11 at 23:51
• @yorei: Never having heard of this game or the description, it is unclear what the rules of your game are. Do you have an online reference? – Mitch Nov 17 '11 at 1:08

The case with 4 coins is functionally equivalent to the case with 100 coins, or any other coinage beyond 3 for that matter.

For simplicity presume A always goes first.
PP = Personal profit

1 Coin
A: picks 1 coin. PP = 1

2 Coins
A: picks 2 coins. PP = 2

3 Coins
A: picks 2 coins. PP = 2

So far, in terms of maximizing profit, A can not possibly perform better. Once you reach 4 or more coins, a new factor becomes involved: the performance of the other player, and "what is rational" is drastically altered. If you know the person is a rational person like yourself, you might pick 1 at a time and hope they do the same, maximizing both your profits. If you know they are a mean, greedy person, you might open with 2 coins to end the game. If you know nothing about the other person, the most rational decision would be based on what the average player B would do in such a situation. If the average person in Player B's position is going to simply end the game right there with 2 coins, your most rational move (as player A) would be to end it on the first turn. Otherwise, it is most rational to take 1 at a time.

4 coins
A: picks 2 coins. PP = 2

OR

A: picks 1 coin. PP = 1
B: picks 2 coins.
Game is over.

OR

A: picks 1 coin. PP = 1 so far
B: picks 1 coin. PP = 1
A: picks 2 coins. PP = 3 total

OR

A: picks 1 coin. PP = 1 so far
B: picks 1 coin. PP = 1
A: picks 1 coin. PP = 2 total
B: picks 1 coin. PP = 2 total

• In the case of four coins you would always pick two coins as A. This is for the following reason: since all B cares about is maximising his profit, the safest way to do so at 3 coins is just taking 2 coins. So if you pick 1 coin at 4, B would then pick 2 coins, and you'd end up at a loss of profit. I suppose you could say the paradox assumes both players consider each other cold, rational machines. – yorei Nov 17 '11 at 1:15
• But I would like to emphasize: I am interested in /already known/ solutions in the literature, not original research. You might have a very good point here, but your answer is not within the scope of my question. Thank you very much for your effort, though! – yorei Nov 17 '11 at 1:18
• Well like Mitch I've never heard of this problem so I can't help you there. Regarding 4 coins: no it does not follow that with 4 coins the most rational choice is 2 from the onset. It's possible to get 3 total; maybe your opponent is poor at strategy, who knows? As soon as you bring humans into it, it's a different game. Still, had you asserted "cold, rational machine" in the original post instead of "human", it would only change these two conclusions (4 & 5 coins). After 5, even with "cold, rational machines" there are no scenarios where it makes sense to open with 2 to end the game. – stoicfury Nov 17 '11 at 3:07
• @stoicfury this is incorrect. If you acknowledge for 'cold, rational machines' that the correct move for 4 coins is to take 2, then player A must open with 2 coins in the case of 5. Otherwise if he takes 1, then B's logical choice is to take 2, leaving player A with 1 less PP. – so12311 Nov 17 '11 at 20:41
• @zephyr: You're right, it was a mistake. I meant to write "After 5", not 4. :) I'll edit it in, thanks. – stoicfury Nov 17 '11 at 22:30

This seems very similar to the prisoner's dilemma to me, and the logically correct behavior hinges on whether this is a single encounter with an unknown opponent, or a series of iterated games.

Game theory answer: Yeah, stuff is weird. People don't act this way because they are irrational. Let's make more proofs!

Empiricist answer: Looking at historical data, people generally share until the last two or three. They also feel a desire for reciprocity very strongly. If P(they share)*(coins remaining) -(P(they share)-1)*2 > 0, then the empiricist will share.

Timeless Decision Theory: Assuming that both players are only interested in their own profit, and entirely logical, and are entirely sure that their opponent is equally rational and self interested, both realize that the other played will do the exact same thing that they do. Given that, both people realize that if they share their partner will as well. Therefore the rational decision is to take one coin.

• Regaring "Empiricist answer": The average person would become miserly about the last 2 or 3 (out of 100) coins at the end of the game? I don't see this empirically at all, although it would entirely depend on context. I think most people will not care about 3 coins when they already have 47, and will continue faithfully in the spirit of cooperation, if the coins are 1 cent coins. If the coins are 5000 dollar coins, you might get a few more sneaky people who will take the last 2 to end the game instead of splitting it 50/50. But still, would this be the average person? I don't think so. – stoicfury Nov 30 '11 at 17:02
• If one assumes each player will expect the other to use the strategy he himself would use in a given spot, then after at least two coins are taken and exactly three remain a player would flip a coin until heads comes up, and then take one coin if that required an odd number of flips and two if even, then a player who took one coin when four remained would have an expected value infinitesimally greater than 2, and would thus benefit from taking only one. A player with five or more coins would benefit from adopting this strategy by enticing the four-coin player to only take one. – supercat Apr 25 '14 at 21:25