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

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

<p>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 $40\%$. </p>

<p>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.</p>

<p>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 <em>uncertainty</em> apart from the mere quantity of "stuff" (be it goods or money). </p>

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

CC BY-SA 3.0
2017-05-02T01:00:00.323Z
@FrankSwanton You're welcome. Glad I could help `:)`
Herr K.
CC BY-SA 3.0
2017-05-02T20:14:59.633Z
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
CC BY-SA 3.0
2017-05-02T20:16:39.847Z
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
CC BY-SA 3.0
2017-05-02T20:17:44.787Z
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
CC BY-SA 3.0
2017-05-02T20:18:31.943Z
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
CC BY-SA 3.0
2017-05-02T20:19:13.457Z
I basically have done exercises on all these types of games, but want to solidify whether my thought is correct in forming the strategy space for each case.
Frank Swanton
CC BY-SA 3.0
2017-05-02T20:32:22.990Z
@FrankSwanton: I'm happy to answer those questions. However, I think it's better if you pose them as an actual question. For one, they are definitely on-topic for this site. Secondly, such a question will benefit future visitors of this site, and you may be able to benefit from answers by potentially different people (I notice there are a few micro theorists/game theorists who are active here).
Herr K.
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2017-05-02T20:48:19.567Z
Sounds good!! Will do.
Frank Swanton
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2017-05-03T02:56:51.557Z
@FrankSwanton: You deleted your question?
Herr K.
CC BY-SA 3.0
2017-05-03T20:42:42.260Z
Yes cause it sounded like asking a homework problem which is not.
Frank Swanton
CC BY-SA 3.0
2017-05-03T20:52:06.257Z
@FrankSwanton: I think it's legit. I upvoted it too, but by the time I got around to answer you already deleted it.
Herr K.
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2017-05-03T21:26:32.757Z
Sorry about that. I wanted to flesh out the question in more abstract way linking contingency and state space.
Frank Swanton
CC BY-SA 3.0
2017-04-28T17:53:48.520Z
2017-04-28T17:53:48.520Z
Herr K.
<p>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.</p>

<p>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?</p>

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2017-04-28T06:30:48.750Z
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
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2017-04-28T12:26:11.583Z
@denesp . Hi denesp! Can you elaborate on why it is necessary and how this contrasts with "sure" goods case in consumer theory?
Frank Swanton
CC BY-SA 3.0
2017-04-28T12:32:30.613Z
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
<expected-utility><risk-aversion><uncertainty>
von-Neumann-Morgenstern v. Bernoulli Utility Function
Frank Swanton
# von-Neumann-Morgenstern v. Bernoulli Utility Function

(Asked by: username_0 on 2017-04-27)

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?

**Tags:** expected-utility, risk-aversion, uncertainty

> **Comments:**
> **username_1**: 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.  
> **username_0**: @denesp . Hi denesp! Can you elaborate on why it is necessary and how this contrasts with "sure" goods case in consumer theory?  
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2017-04-28. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_2**: @username_0 You're welcome. Glad I could help `:)`  
> **username_0**: username_2, 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...  
> **username_0**: 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.  
> **username_0**: 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?  
> **username_0**: 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?  
> **username_0**: I basically have done exercises on all these types of games, but want to solidify whether my thought is correct in forming the strategy space for each case.  
> **username_2**: @username_0: I'm happy to answer those questions. However, I think it's better if you pose them as an actual question. For one, they are definitely on-topic for this site. Secondly, such a question will benefit future visitors of this site, and you may be able to benefit from answers by potentially different people (I notice there are a few micro theorists/game theorists who are active here).  
> **username_0**: Sounds good!! Will do.  
> **username_2**: @username_0: You deleted your question?  
> **username_0**: Yes cause it sounded like asking a homework problem which is not.  
> **username_2**: @username_0: I think it's legit. I upvoted it too, but by the time I got around to answer you already deleted it.  
> **username_0**: Sorry about that. I wanted to flesh out the question in more abstract way linking contingency and state space.  

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