I was asked this question recently in an interview:Interviewer:"I have a roulette game and I offer you a game for 55 dollars. However, the roulette is biased. Black comes up 60% of the time and red 40% of the time. Also, there is no zero, so 18 black and 18 red. You win 100 dollars if you play and the roulette comes up black and you get nothing if it comes out red."a) Should you play and if yes which side? You want to pay 55 and play or you want me to pay 55 and let me play and why?b) if there were any arbitrage how would you go about profiting from itc) if you disagree with this being a fair game what would be the price for a fair game and why.Me: While I came up with a price of 50 (which I got from the risk neutral prbability (1-down)/(up-down) I failed to explain how I can profit from this game. I can charge the 55, pocket a bit less than 5 and bet a bit more than fair value (a bit more than 50, to entice the counterparty to play) on a similarly biased roulette to someone who understands the fair value of one game is 50 and bet on black also. But this answer does not sound right. Any better explanations or ideas? Thanks a lot!!!

QuoteOriginally posted by: IvolrevYou win 100 dollars if you play and the roulette comes up black and you get nothing if it comes out red.Just to be clear:1. You win $100 if you bet $50, correct ?2. Can bets be of any size ?3. You play just once or until bankruptcy ?

1. No, you pay 55 not 50 and if the ball gets into a black pocket you win 100, else nothing. 55 is the price you pay in real life or charge in real life for this game. 2. That was not specified. I should have asked, obviously. But I guess the bet size can vary because before the interviewer jumped to the next question he quickly mentioned that he can pocket 5 and bet the 50 on black as well. He was not specific and of course I should have asked immediately.3. Was also not specified, but I assumed the game can continue if I wish so.

Why not EX = 0.6 *100 + 0.4*0 = 60$This is the fair value of this game? At anything below 60$, it is arb?

Well, isnt it similar to pricing an option under risk-neutral assumptions?The real probabilities would not matter in that case, would they....I am just having trouble understanding how to set up the "replication", meaning, what amount to invest in what in order to end up hedged no matter which pocket the ball falls in.

After some more thinking I am not sure anymore whether the interviewer maybe posed the question incorrectly or misled me intentionally. My first answer was an expected payoff of 60 and because of that I want to play the game for 55. Then when he said it is wrong I thought about pricing it in a risk neutral way. But I just dont think it works in this specific instance because what I am asked to price is not really any derivative where I can create a replicating portfolio. I am now more thinking along the line of what the optimal betting strategy would be in regards to the Kelly Criterion. I guess that is why you, MCarreira, asked about how often you are to play the game? I now think that a) the interviewer misled me although I gave the correct answer or b) he posed the question incorrectly. Any other takers?

I would not play this game, even if the price of 55$ is cheap. However, I would play the following game 100 times :exactly the same game, but you replace "win 100 dollars" by "win 1 dollar". Both strategies have the same price. However, one has a variance much lower (the 2nd strategy of course).

Bakarax,Thanks for the feedback. Your point about lower variance game makes sense and I think there are different puzzles that make this the focus of the solution, however, I think that this is not at the heart of this particular problem. Either I am overlooking something or the problem was incorrectly set up, or the answer is simply a fair value of $60. I googled it but could not find any puzzle similar to this one, possibly because the solution really is 60 and nobody bothers with even wondering about it.

My 2 cents:If you can't create an hedging portfolio you can't apply the no arbitrage paradigm of pricingIn financial terms the "market is incomplete" so there isn't an "unique" no arbitrage priceTherefore,as bakarax pointed out, even if the fair value of the bet is 60 the "market" price of the game will be determined by the risk adversion of the player (in financial terms "the market price of risk")P.