Yes. There is nothing preventing you from assigning a value equal to -$1,000 to the state of affairs, “I made a bet and lost $100.” This would simply mean that you consider two situations equally valuable, for example one in which you have been robbed of $1,000, and another in which you made a bet and lost $100.
Assigning such values does nothing to prevent you from having a mathematically consistent utility function, and it does not imply any necessary violation of the VNM axioms.
Someone who has risk aversion in Lumifer’s sense might assign a value of -$2,000 to “I was robbed of $1,000 because I left my door unlocked,” but they will not assign that value to “I took all reasonable precautions and was robbed anyway.” The latter is considered not as bad.
Specifically, people assign a negative value to the thought, “If only I had taken such precautions I would not have suffered this loss.” If there are no precautions they could have taken, there will be no such regret. Even if there are some precautions, if they are unusual and expensive ones, the regret will be much less, if it exists at all.
Refusing a bet is naturally an obvious precaution, so losses that result from accepting bets will be assigned high negative values in this scheme.
Yes. There is nothing preventing you from assigning a value equal to -$1,000 to the state of affairs, “I made a bet and lost $100.” This would simply mean that you consider two situations equally valuable, for example one in which you have been robbed of $1,000, and another in which you made a bet and lost $100.
Assigning such values does nothing to prevent you from having a mathematically consistent utility function, and it does not imply any necessary violation of the VNM axioms.
That doesn’t follow, since there’s also nothing preventing you from assigning a value equal to $-2000 to the state of affairs “I was robbed of $1000”.
Someone who has risk aversion in Lumifer’s sense might assign a value of -$2,000 to “I was robbed of $1,000 because I left my door unlocked,” but they will not assign that value to “I took all reasonable precautions and was robbed anyway.” The latter is considered not as bad.
Specifically, people assign a negative value to the thought, “If only I had taken such precautions I would not have suffered this loss.” If there are no precautions they could have taken, there will be no such regret. Even if there are some precautions, if they are unusual and expensive ones, the regret will be much less, if it exists at all.
Refusing a bet is naturally an obvious precaution, so losses that result from accepting bets will be assigned high negative values in this scheme.