Editing Risk aversion (section)

=== Von Neumann-Morgenstern utility theorem ===
The [[Von Neumann–Morgenstern utility theorem|von Neumann-Morgenstern utility theorem]] is another model used to denote how risk aversion influences an actor’s utility function. An extension of the [[Expected utility hypothesis|expected utility]] function, the von Neumann-Morgenstern model includes risk aversion axiomatically rather than as an additional variable.<ref name="von Neumann 1944">{{Cite book |last1=von Neumann |first1=John |url=https://www.jstor.org/stable/j.ctt1r2gkx |title=Theory of Games and Economic Behavior |last2=Morgenstern |first2=Oskar |last3=Rubinstein |first3=Ariel |date=1944 |publisher=Princeton University Press |jstor=j.ctt1r2gkx |isbn=978-0-691-13061-3 |edition=60th Anniversary Commemorative }}</ref>

[[John von Neumann]] and [[Oskar Morgenstern]] first developed the model in their book ''[[Theory of Games and Economic Behavior|Theory of Games and Economic Behaviour]].''<ref name="von Neumann 1944"/> Essentially, von Neumann and Morgenstern hypothesised that individuals seek to maximise their expected utility rather than the expected monetary value of assets.<ref>{{Cite journal |last=Gerber |first=Anke |date=2020 |title=The Nash Solution as a von Neumann–Morgenstern Utility Function on Bargaining Games |journal=Homo Oeconomicus |language=en |volume=37 |issue=1–2 |pages=87–104 |doi=10.1007/s41412-020-00095-9 |s2cid=256553112 |issn=0943-0180|doi-access=free |hdl=10419/288817 |hdl-access=free }}</ref> In defining expected utility in this sense, the pair developed a function based on preference relations. As such, if an individual’s preferences satisfy four key axioms, then a utility function based on how they weigh different outcomes can be deduced.<ref>{{Cite web |last=Prokop |first=Darren |date=2023 |title=Von Neumann–Morgenstern utility function {{!}} Definition & Facts {{!}} Britannica |url=https://www.britannica.com/topic/von-Neumann-Morgenstern-utility-function |access-date=2023-04-24 |website=www.britannica.com |language=en}}</ref>

In applying this model to risk aversion, the function can be used to show how an individual’s preferences of wins and losses will influence their expected utility function. For example, if a risk-averse individual with $20,000 in savings is given the option to gamble it for $100,000 with a 30% chance of winning, they may still not take the gamble in fear of losing their savings. This does not make sense using the traditional expected utility model however;

<math>EU(A)=0.3($100,000)+0.7($0)</math>

<math>EU(A)=$30,000</math>

<math>EU(A)>$20,000</math>

The von Neumann-Morgenstern model can explain this scenario. Based on preference relations, a specific utility <math>u</math> can be assigned to both outcomes. Now the function becomes;

<math>EU(A)=0.3u($100,000)+0.7u($0)</math>

For a risk averse person, <math>u</math> would equal a value that means that the individual would rather keep their $20,000 in savings than gamble it all to potentially increase their wealth to $100,000. Hence a risk averse individuals’ function would show that;

<math>EU(A)\prec$20,000 (keeping savings)</math>