How much should you pay in market making games

The true market is 3.30 - 3.65 that I brought up in the example. However, making 3.15 - 3.50 is a correct answer as well, you'll just have less downside risk. I kind of added my own trading style to the answer without an explanation there...

Assuming a two-wide market then 3.49 - 3.51 is the only correct answer. Moreover, if your market is any wider then a competitor will happily quote tighter and take all the volume. Also, who is your counterparty here? How on earth is any reasonable counterparty ever taking this trade wider than 3.49-3.51, even if a competitor MM did not immediately tighten their market to that (as they undoubtedly would)?

"I think a good answer would be that how wide you make you dice market depends on how often you're going to play, who's money it is, how much money you're better on it, etc"

This is good and I agree that these questions are as much an attempt to see how you reason about risk as about anything else.

"But even an argument could even be made to pay zero. I saw a 5 dollar bill on the ground on the way here but I didn't stop because I didn't want to be late to interview with your bank. If you gave me a chance to trade, I'd only take risk when I was making enough money to be worthwhile and believed I could cut my losses on any one position before losing too much."

I think this, however, is far too cute. Firstly, if you'd stopped to pick the bill up it would have taken you about five secs, an irrelevant margin in term of being early or late for an interview. Secondly, these questions typically emerge in the context of a final round onsite interview. They are roughly looking for certain answers, and anything too far outside the pale raises eyebrows in a negative way (market makers are looking for people who systematically handle risk in a clear, reproducible, and consistent way).

"The idea is to make money consistently and manage my risk."

Passing up the opportunity to make an absolutely fixed-in-stone long term 1c +EV trade because there may be better opportunities elsewhere, when a) these opportunities have not been mentioned in the context of the question and b) taking tens of billion of long term 1c +EV trades is the definition of the market maker profit system, will not be looked on favourably. There are plenty of better ways to show how clever you are in a final round interview than to argue that you would bid 0 for the expected value of a dice roll.

I understand what you're saying. But the reason why I stand by $3.49 here, specifically as an answer to the interview question, is because of 4 key assumptions (none of which were actually mentioned in the question, these are just things that I assumed based on similar questions I've had in interviews): 1) we have perfect information about the game, 2) the game completes instantly, and 3) we can diversify, and 4) the market for this hypothetical "dice roll" investment is efficient. If any of these assumptions is wrong, I agree with you though. Here's my reasoning:

1. We need to have perfect knowledge of the game. In other words, there's no risk of being trapped in a bad game that we don't actually want to play. In the real world, you don't bet at exactly the expected value because you don't have perfect information: you don't know the weights on each face of the die so you have to just give it an educated guess. Odds are that the other side of the trade is just as smart as you, and they're trying to make money too. However, I think perfect information is a reasonable assumption for an interview question. In all the market maker interviews I've had, they've never tried to trick me with a gotcha about information being left out / inaccurate.

2. Since the game completes instantly, there is no opportunity cost to play the game. In the real world, there's always an opportunity cost. You'd never bet so that your expected value is under the risk-free rate because you could always just invest in risk-free bonds instead. But if the dice game finishes instantly, you're not losing another opportunity by investing in the dice roll.

3. Like I said above, diversification is critical. If I'm putting all of my money into a single roll of this dice game, there's no way I'm betting $3.49 on it. But if it's just a small part of a large, diversified portfolio, I don't see anything wrong with betting $3.49. In fact, if I have enough uncorrelated $3.49 bets, I am nearly guaranteed to make a profit. But even if I'm only allowed one bet, as long as it's not correlated with my other investments I'm in the clear.

4. If the market isn't efficient, I'm not betting $3.49. If the "going rate" for dice rolls is only $2, of course I will bet only $2 to maximize my own profit. But given that this is a market making game, I assume that if I bid $2, someone will outbid me. I am willing to outbid them until I eventually hit $3.49, at which point if they bid $3.50 I will no longer outbid them. But if the market rate is $3.49, there is no reason to lose out on that 1¢ in expected value.

If you look at the bid-ask spread for something like SPY, it's usually just a couple of cents. Oftentimes, it's a market maker on both sides of the spread. This is a very similar situation to betting $3.49 on the dice roll. The reason they're able to get away with it is that they're very well-diversified. Even if they make a lot of losing bets, they will usually make enough winning bets to net a profit. Combined with the speed at which they trade, they're able to make pretty good returns.