To you, what value of $X will make these 2 payoffs equivalent:

Payoff 1 - Getting $100 for certain.

Payoff 2 - Getting $0 with 50% probability, and $X with 50% probability.

To you, what value of $Y will make these 2 payoffs equivalent:

Payoff 3 - Getting -$100 for certain.

Payoff 4 - Getting -$0 with 50% probability, and -$Y with 50% probability.

If \( X \neq Y \), is there an arbitrage opportunity?

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## Comments

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TopNewestDepends on what my initial balance is.

If I begin the game with $0 in hand, I will definitely choose Payoff 1 and Payoff 4, no matter what X and Y are. (Getting guaranteed money, no matter how small, is infinitely better if you don't have any to begin with; likewise, having possibility of no need to owe anything is infinitely better even if that choice also risks the possibility of owing a massive amount, when you have nothing to begin with.)

If I begin the game with a big amount ($1 million) in hand, I'm pretty sure the cutoff point (Payoff 1 if lower, Payoff 2 if higher) for X is somewhere around \(200+\epsilon\), and likewise for Y around \(200-\epsilon\) (Payoff 4 if lower, Payoff 3 if higher). I'm someone that believes the utility of money can be modeled logarithmically.

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So, you will choose payoff 1 even if X = 1000000?

And you will choose to lose $1000, instead of guaranteeing just a loss of $100?

Note that you can go negative into debt, which gets paid off in a subsequent time period. If Y was huge enough that you would declare bankruptcy with no adverse effect, then yes sometimes Payoff 4 could be better.

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I said it depends (heavily) on my initial balance. If I have $0 to begin with, yes, I will choose Payoff 1 even if X = $1000000; but give me about $10 to begin, and I might pick Payoff 2 instead (since I at least have backup money of $10 if I don't get the million). Likewise for the second scenario; if you have $0 to begin with, any debt at all will cause bankruptcy and thus it's better to pick Payoff 4 that has a possibility of no bankruptcy.

Using (my sense of) utility of money, where the utility of \(\$X\) is \(U(X) = A \cdot \log BX\) for some constants \(A,B\):

If I start with $0:

However, give any positive amount \(\$\epsilon\) to start with, and now \(U(\text{Payoff 2}) = U(X+\epsilon) + U(\epsilon)\) is finite, and so for some \(X\) it's possible that \(U(\text{Payoff 2}) > U(\text{Payoff 1})\), in which I'm taking Payoff 2 instead.

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I think that people can go into debt and so my "absolute zero point" isn't $0, but more like -$100,000 (depending on age / locality / etc)

E.g. It is arguably worthwhile for some people to take on student loan debt so that they can study in a (overseas) university.

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How can you ever win anything with payoff 4? If you lose money or get nothing :s

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That's the point: If you must choose between payoff 3 or payoff 4, which would you choose (to cut your losses)? Payoff 3 guarantees that you lose $100 while payoff 4 results in only a probability of 50% to lose $200.

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Is this a serious mathematical question or a question about how bad you feel about losing?

I'd take on Payoff 2 if X>200 (preferably X>>>200). And I'd take on payoff 4 if Y\(\leq\)100 ( preferably Y<<<0) ;-)

I guess it shows that I don't like to owe anybody. Lannisters always repay their debts. So do I.

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Never seen a lannister except for the Imp to pay his debts and you're not a lannister either are you? xD

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That's why I added the "So do I" part.

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