Suppose, for example, that a decision maker can choose any probabilities p 0 , p 1 , p 2 that he or she wants for specified dollar outcomes D 0 and that they have a given expected value p 0 D 0 + p 1 D 1 + p 2 D 2 = kFor example, if D 0 &lt; 0 were the price of a lottery ticket with possible prizes D 1 and D 2 , then k = 0 would define a “fair” lottery, while k &lt; 0 would afford the lottery organizer a profit. We may arbitrarily let the utilities of D 0 and D 2 be u 0 = 0 and u 2 = 1; then the utility of D 1 is u 1 ∈ ( 0 , 1 ). For a typical lottery, | D 0 | is quite small as compared to D 1 and D 2 . With k ≤ 0, this implies that feasible p 1 and p 2 are small, with p 1 + p 2 well under 0.5, and therefore with p 0 well over 0.5.Questions:1. If D 0 is the price of a lottery ticket, how could it possibly be less than zero?2. Why include the price of a lottery ticket in an EV calculation? The prizes D 1 and D 2 have a probability associated with them, that makes sense when calculating expected value. But the price of a lottery ticket? What does it mean for a ticket price to have a probability "well over 0.5"3. For k ≤ 0, it only makes sense that D 0 must be negative, but again, how could the price of a lottery ticket be negative?

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Suppose, for example, that a decision maker can choose any probabilities       p    0  ,       p    1  ,       p    2   that he or she wants for specified dollar outcomes      D    0  and that they have a given expected value      p    0        D    0    +      p    1        D    1    +      p    2        D    2    =  kFor example, if       D    0    &amp;lt;  0 were the price of a lottery ticket with possible prizes       D    1   and       D    2  , then   k  =  0 would define a “fair” lottery, while   k  &amp;lt;  0 would afford the lottery organizer a profit. We may arbitrarily let the utilities of       D    0   and       D    2   be       u    0    =  0 and       u    2    =  1; then the utility of       D    1   is       u    1    ∈  (  0  ,  1  ). For a typical lottery, |      D    0  | is quite small as compared to       D    1   and       D    2  . With   k  ≤  0, this implies that feasible       p    1   and       p    2   are small, with       p    1    +      p    2   well under   0.5, and therefore with       p    0   well over   0.5.Questions:1. If       D    0   is the price of a lottery ticket, how could it possibly be less than zero?2. Why include the price of a lottery ticket in an EV calculation? The prizes       D    1   and       D    2   have a probability associated with them, that makes sense when calculating expected value. But the price of a lottery ticket? What does it mean for a ticket price to have a probability &quot;well over   0.5&quot;3. For   k  ≤  0, it only makes sense that       D    0   must be negative, but again, how could the price of a lottery ticket be negative?

ejigaboo8y · Accepted Answer

What you&#039;re not understanding is the fact that we&#039;re calculating your expected winnings. So what your author means is that if you buy a lottery ticket at a price   |      D    0    | (which is positive), then if you don&#039;t win the lottery you&#039;ll have earned       D    0   (which is negative), your overall wealth decreased by   |      D    0    |.So it seems like we&#039;re modelling the scenario where a lottery ticket costs   |      D    0    | and with probability       p    1  , I gain some amount money       M    1    &amp;gt;  0. In this case, I&#039;ve paid for the ticket and won       M    1  , so my gains have been       D    1    :=      M    1    +      D    0   (note again that       D    0   is negative). Similarly, with probability       p    2  , I gain some greater amount of money       M    2    &amp;gt;  0, in which case I&#039;ll have gained       D    2    :=      M    2    +      D    0   - I&#039;ve still paid for my ticket.Thus, with probability       p    0    :=  1  −  (      p    1    +      p    2    ), I&#039;ll have entered the lottery and not won, and thus lost the money that I paid for the ticket. Therefore, my expected winnings are as written above. The important part is to keep in mind which quantity we&#039;re actually keeping track of - in this case, it&#039;s your wealth after the lottery is over.

Lovellss · Accepted Answer

The   Ds are the payment to the player by the casino. So       D    0    &amp;lt;  0 represents the fact that the payment to the player is negative, i.e., the player pays the casino |      D    0  |. I think that answers 1 and 3.Regarding 2 - the probabilities represent the events &quot;not winning&quot;, &quot;winning the net amount       D    1   and winning the net amount       D    2  . So &quot;not winning&quot; has a probability of       p    0   in which case the player only plays the lottery ticket.

Suppose, for example, that a decision maker can choose any probabilities p 0 </msu

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