Yesterday, we looked at a Goldman Sachs interview question based on a simple betting game called Martinagle, as described in Kevin Roose’s new book “Young Money”. Here is Roose’s description of the game:
Here’s a game I’ve just invented. The rules are that I flip a coin, and if it comes up heads, you pay me a dollar and the game is over. If it comes up tails, you flip again. If it comes up heads the second time, you pay me two dollars, and the game is over. If it comes up tails again, you flip again. Third time, you pay me four dollars for heads and the game is over, and you flip again for tails. And so on and so on, each time doubling the payout for heads, and flipping again on tails. How much would I have to pay you up front to play this game?
As we noted yesterday, a person’s answer to this question gives an indication of their appetite for risk. The player flips a coin repeatedly until it comes up heads, and the player’s eventual loss doubles for each tail flipped before that merciful heads appears.
The good news for our player is that, while her losses are doubling with each additional tail, the probability of getting a bunch of tails in a row keeps halving.
The probability that our player only loses one dollar — the probability that she gets a heads on the very first flip — is 1/2, assuming the Goldman Sachs interviewer has provided a fair coin. So if she bets one dollar, she has a 50-50 shot of breaking even, and a 50-50 chance of losing money.
For our player to lose two dollars, she would have to first flip a tail, and then on the second toss, get a heads. The probability that this is our player’s outcome is the product of the probabilities of each of these tosses — 1/2 for the initial tail, times 1/2 for the second flip giving heads. This gives us a probability of 1/4.
This means that if our player bets two dollars, she has a 1/2 probability of flipping a heads on the first toss and winning a dollar, or a 1/4 probability of breaking even. Adding these together, she has a 3/4, or 75% chance, of either winning or breaking even.
This is how the game measures risk appetite. A smaller ask gives a smaller probability of winning or breaking even. A player is taking a bigger risk asking for one dollar to play than if she asks for two dollars — in the former case, they have a 50% chance of avoiding a loss, and in the latter a 75% chance.
We can continue as above, looking at the probabilities for the game going on for a particular number of flips, and the associated losses:
The first column gives the length of the game. One flip means our player gets heads on her first flip, and the game ends. Two flips means she gets one tail and then a head. Three flips means she gets two tails and then a head, and so on.
The second column gives the probability of each outcome — 1/2 raised to the number of flips, or 1/2 times itself that number of times.
The third column shows the player’s loss associated with each outcome. If the player wants to cover herself against a particular outcome, she needs to ask for at least this much money.
The final column shows the probability that the player either wins or breaks even if she asks for the corresponding amount of money in the third column. This is the sum of the probabilities of the outcomes where she wins or breaks even: we saw above that if she asks for two dollars, she has a 50% chance of winning a dollar, and a 25% chance of breaking even, giving a 75% chance of one of these events happening.
Similarly, if she asks for four dollars, she wins three dollars if she gets a heads on the first toss, she wins two dollars if she throws a tails and then a heads, and she breaks even if she throws two tails and then a heads. Her chances of at least pushing are thus 50% + 25% + 12.5% = 87.5%.
How much a player asks for shows how cautious she is. The more she asks, the more she is avoiding risk. If our player wants to be at least 90% sure she won’t lose money, she needs to ask for at least eight dollars, based on the above table. If she wants to be 99% sure she won’t lose, she needs to ask for at least 64 dollars. If she wants to cover the tail risk of flipping 13 tails in a row before flipping heads — an event with a probability of less than one in 10 thousand — she needs to ask for $US8,192.
Of course, there is always the chance, however small, that our player might not flip heads in any reasonable amount of time. After 44 straight tails — an event with a probability less than one in 10 trillion — she would be in the hole for more than current U.S. GDP. Or, as Reuters’ Felix Salmon put it on Twitter, “No, the correct answer is ‘I don’t take on unlimited contingent liabilities, for any up front $US'”.