Thoughts on this Market Making Simulation Question?

1. Suppose you are a market maker who currently has no net position in a security that is currently valued at $10.  In one day, you expect the security's fair value to be either $9, $10, or $11, all with equal probability.  If tomorrow's price will be $10, you can expect to face a liquidity trader today who will buy or sell one share from you with equal probability.  If, on the other hand, the value changes tomorrow to $11 or $9, you will face an informed trader today, who knows tomorrow's value.  This informed trader will purchase one share from you if the value will be $11 and sell one share to you if the value will be $9.  Whatever position you have by the end of today, you expect to be able to exit at tomorrow's fair value.  If you set a bid-ask mid-price of $10 today, how small a bid-ask spread can you set and still expect to break even? Assume that as a market maker you do not earn a spread on the sale or acquisition of the security in day two (i.e. securities are valued at market-to-market; no transaction occurs).

2. In the basic classroom simulation described in the Price Formation Module Overview, what is fair value for one of five exclusive market-making rights that are good for five minutes?  How does this value change if the number of rights is fifteen?  

3. How much would you pay to know the price of a given stock in one week's time?  What factors influence your willingness to pay?  How would your payoffs look for different realizations of the signal?  

4. If no computer liquidity traders populate the market, what do you expect will happen to spreads?  How will the value of market-making rights change?  What will happen to the frequency of information purchases and to price efficiency?

Randomly found this online wanting to practice market-making simulations for trading interviews (anyone have a good source for this? I think tradinginterview.com has a market maker simulation game) and not sure about some of these questions. Here are my attempts.

1) Stock has an equal probability of being 9, 10, or 11. If it's 10 dollars tomorrow then there's a 50/50 chance the trader will buy or sell. So an equal distance spread (the tightest I can make it to get him to trade) will cause me to break even. If the value is 9 or 11 then the trader knows the next price and he is guaranteed to buy from me TODAY if it'll be 11 TOMORROW, and guaranteed to sell to me if it will be 9. In my head, if it's 10 tomorrow he's going 50/50 but if it's going to be 11 he's going to buy and if it's going to be 9 he's going to sell. Knowing that he buys when he knows it'll be 11 and sells when it's going to be 9. He should 100% win when it's not 10 (66% chance it won't be 9 or 10) and he will win 50% of the time when it is 10 (33% chance). Given this information that I can win 50% of the time and that my win is conditional on the price being 10 which is a 33% chance, would I price the spread something like 9.33-10.67 being the narrowest I can set to still break even? If it's 9 I lose 0.33, if it's 11 I lose 0.33 if it's 10 and he buys or sells then I make my loss back? Confused at how to break this down and approach it.

2) Not sure how to calculate this at all it seems like it was a course problem that they might have covered.

3) Given that I know he has a 33% chance of making a decision that has a 50% winrate for him how much should I pay for 1 week of information is how I would break this down. The factors that influence how much I'd be willing to pay are the width of the spread, and maybe more?

4) If no computer liquidity traders populated the market, spreads will widen as there is now less price discovery and less frequent trades. If trades happen 100 times a second I know the prices are pretty accurate, if trades happen 1 time every minute, I know that there are 60 seconds of influence other traders or broad market climates can have on the stock price in that time so I'd be hesitant to have the same bid/ask with less liquidity as I would with HFT trading occurring. Market-making rights would increase because the spread would be much wider and market makers would be able to profit on the spread. The frequency of information purchases would probably be much greater as the information now has a higher impact on the price and the price efficiency would decrease as there is less liquidity in the market to compact spreads and allow trades to go through.