# von-Neumann-Morgenstern v. Bernoulli Utility Function

A great deal of time is spent distinguishing the big $U$ (von-Neumann-Morgenstern)v. small $u$ (Bernoulli Utility Function). The v.NM function maps from the space of lotteries to real number as it represents the preference defined on the lottery space while the Bernoulli is defined over sure amounts of money.

Why is this distinction so important in the theory of expected utility? Also, what does this distinction enable us to achieve that is important in the expected utility theory? What is the most intuitive way to understand the expected utility theory is constructed in this way?

• The functions map from very different spaces to $\mathbb{R}$, so the distinction is necessary rather than important. Up until this point most utility functions you encounter in micro are defined over "sure" goods. vNM isn't. This is good to point out, it is an unusual concept. Not sure about what you mean by the rest of your question. – Giskard Apr 28 '17 at 6:30
• @denesp . Hi denesp! Can you elaborate on why it is necessary and how this contrasts with "sure" goods case in consumer theory? – Frank Swanton Apr 28 '17 at 12:26
• I quoted "sure" from your question, so I am assuming you know about Bernoulli utility functions. I explained necessity in the previous comment, not sure what I could add. – Giskard Apr 28 '17 at 12:32

Bernoulli utility represents preference over monetary outcomes. In a way, this is no different from the typical utility functions defined over consumption bundles.

vNM utility, in contrast, represents preference over lotteries of monetary outcomes. Thus, the argument of vNM utility is an object related to, but categorically distinct from, the object that is an argument of Bernoulli utility.

For example, Bernoulli utility function allows us to compare the utilities from having $\$5$to having$\$7$, while vNM utility allows us to compare utilities from the lottery $(0.2\otimes\$5,0.8\otimes\$7)$ --- having $\$5$with$20\%$and$\$7$ with $80\%$ --- to the lottery $(0.6\otimes\$5,0.4\otimes\$7)$ --- having $\$5$with$60\%$and$\$7$ with $40\%$.

In this sense, the distinction between Bernoulli and vNM utility functions are necessary (as they are applied to different objects) rather than important (since they both represent some kind of preference in the end), as @denesp says in his comment.

Moreover, if we only consider degenerate lotteries, i.e. the probabilities are either $0$ or $1$, then vNM and Bernoulli utilities coincide. That is, the Bernoulli utility of having $\$5$is exactly the same as the vNM utility of having$\$5$ with $100\%$. Therefore, distinguishing Bernoulli from vNM utility functions enables us to examine the effects of uncertainty apart from the mere quantity of "stuff" (be it goods or money).

Lastly, defining utility over money also allows us to study people's attitudes towards risk.

• @FrankSwanton You're welcome. Glad I could help :) – Herr K. May 2 '17 at 1:00
• Herr, is your field microeconomic theory or game theory? I have few simple questions about how to set up strategy space for Bayesian game if you want to comment... – Frank Swanton May 2 '17 at 20:14
• Consider a static Bayesian game where nature moves to decide the state of the world and then the players move simultaneously to play the game. Both players know ex-ante probability distribution over the two possible types of the world. In this game, what would be strategy set for each player? Suppose player 1 action is U and D and 2 is L and R. – Frank Swanton May 2 '17 at 20:16
• Consider a static but asymmetric Bayesian game where nature moves to decide the state of the world and player 1 observes the state but player 2 doesn't. They still play a static game once the state is decided. In this case, how would you form strategy space for these players? – Frank Swanton May 2 '17 at 20:17
• Consider a dynamic and asymmetric Bayesian game where nature moves to decide the state of the world and player 1 observes the state but player 2 doesn't. Now, player 1 gets to move first and then sequentially player 2 moves. In this case, how would you form strategy space for these players? – Frank Swanton May 2 '17 at 20:18