If you could buy your dream house for $1 million but someone makes you an offer that you can flip a coin for the house where if you win the coin flip you get the house for free but if you lose the coinflip you have to pay $1.5 million for the house. Would you take that deal?

what is your overal EV in this deal?

obviously assume you're bankroll is infinite and money is not the issue.

Say the true value of the house is $1 million, and you're indifferent to paying that amount, i.e. the EV of buying the house for $1 million is $0. Thus, being forced to buy the house for 1.5 is an EV of negative $500,000.

EV(buy house) = $0

EV(flip for house) = 50%*($1,000,000) + 50%*(-$500,000)

EV(flip for house) = +$250,000

The true EV would actually be higher than this, because presumably if you'd have paid $1 million for the house, you subjectively must have thought the house were worth at least $1 million to you, but likely more than $1 million, since other amenities in life are in constant competition for your dollar, you wouldn't want to just break even on any transaction because that's an opportunity cost.

The above shows the EV calculation. But if my "bankroll is infinite," then it doesn't matter what I do because whether I get the house for free, for $1M or for $1.5M, I have an infinite amount of money.

Just a naive question, how does the value of the house matter ? If you buy the house for $1M or you do the gamble with the coin flip game, you always end up with the house afterwards, regardless its value.

Hence the EV of the game is always $250k, if you would buy it for $1M when the game weren't accessible.

If you think the house is not worth $1M, but say $900k, you could still do the gamble, but then the EV of the game is $150k.

If the $1 million is important to you, and if there are other things in the economy that you would desire to spend $1 million on, then the subjective value of the house can be established. It is equal to the maximum amount you would pay, plus an assessed opportunity cost. You wouldn't purchase the house at an amount that would be exactly life EV = 0 for you, because it would then provide you more happiness if you collected interest on the money.

If you see your dream house with a $1 million price tag but you assess that you would pay no more than $900k for it, there are two ways to label the zero EV state.

1) $900k purchase = 0 EV
2) $900k purchase + $50k opportunity cost = 0 EV.

Number 2) is more accurate, but 1) is probably more useful since 0 EV is kind of an arbitrary benchmark with which we compare payoffs.

EV($1,000k purchase) = -$100k EV
EV(flip for house) = 50%*($900k) + 50%*(-$600k)
EV(flip for house) = +$150k

So, you're correct.