Von Neumann And Morgenstern Utility Function

Vishrut Goyal and Anuj S. Saxena

Contents

• Risk Aversion
• Von Neumann and Morgenstern Postulates
  • First Postulate
  • Second Postulate
  • Third Postulate
  
  
• Von Neumann And Morgenstern Utility Function

Risk Aversion

In last lecture we discussed the St. Petersburg Paradox, where although the expected return from the game is infinity, only a few gamblers are willing to invest large amount of money. Now we take a simplified version of the above paradox.

Consider a player being given two options : Take Rs.10 for sure or Rs.70 with probability 0.5. In this case, the player is likely to go for the second option.

Now consider the case when Rs 10 and Rs.70 were replaced by Rs. 10 million and Rs.70 million respectively. In this case, the user is likely to go for the first option. Why is it so? Why people may not want to maximize their expected gain?

Let $m_0$ be the initial amount of money with the user. Let $u(m)$ be a utility function representing the user's preference relation. The expected utility in each of the options in the cases discussed above is summarized below :

	Option 1	Option 2
Case A	$u(m_0 + 10)$	$1/2[u(m_0) + u(m_0 + 70)]$
Case B	$u(m_0 + 10^8)$	$1/2[u(m_0) + u(m_0 + 7*10^8)]$

Assume that the player has initially Rs. 10,000. i.e. $m_0 = 10^4$.
Consider the case when $u(m) = m$. Now the expected utilities are given in the following table :

	Option 1	Option 2
Case A	$10^4 + 10$	$10^4 + 35$
Case B	$10^4 + 10^8$	$10^4 + 35*10^7)$

Here we see that the expected utility is more for option 2 in both the cases. Therefore players should prefer option 2 in both the cases

In earlier lecture we saw that if a utility function $u()$ represents a preference relation R, then f(u()) also represents R, where f() is a monotonically increasing function. Therefore $u(m) = m^{1/3}$ should represent the same preference relation as $u(m) = m$. If we assume $u(m) = m^{1/3}$, then the expected utilities are given in the following table :

	Option 1	Option 2
Case A	$10^{4/3}[1 + \frac{1}{3000}]$	$10^{4/3}[1 + \frac{3.5}{3000}]$
Case B	$(10^4 + 10^8)^{1/3}$	$1/2[10^{4/3} + (10^4 + 7*10^8)^{1/3}]$

Here we see that that expected utility is more for option 2 in case A and is more for option 1 in case B as one would generally expect in real life. Notice that $u(m) = m^{1/3}$ does not represent the same preference relation as $u(m) = m$. This can be explained using Von Neumann and Morgenstern utility function.

Von Neumann and Morgenstern Postulates

Let $R$ be a preference relation over a set of deterministic outcomes $\Omega$ and $\;u() : \Omega \rightarrow \Re\;$ be a utility function representing $R$. If we have probabilistic outcomes, then the preference relation $R$ should be defined over probabilistic outcomes $\Omega^\prime = {\Re}^{\mid \Omega \mid}$. Now the utility function has to be redefined so as to represent this new preference relation. Therefore, now $\;u() : \Omega^\prime \rightarrow \Re$.

Notation : Let $(w_1(p_1),w_2(p_2),\ldots,w_m(p_m)) \in \Omega^\prime$, where $w_i$ is $i^{th}$ outcome in $\Omega$ and $p_i$ is the probability of occurrence of $w_i$ s.t. $\sum_{i=1}^{m} p_i = 1$.

Von Neumannn and Morgenstern gave a set of rationality postulates, which define preference relation in probabilistic outcomes.

First Postulate

Consider any two outcomes (namely $w_1$ and $w_2$) from the set $\Omega$, with outcome $w_1$ being preferred over outcome $w_2$. Let $p,q \in [0,1]$ and $q > p$. Consider two outcomes $(w_1(p),w_2(1-p))$ and $(w_1(q),w_2(1-q))$ in $\Omega^\prime$. A rational player should choose second outcome over the first. Formally :

$\forall w_1,w_2 \in \Omega,\,\forall p,q \in [0,1]$ s.t. $w_1 \succ w_2$ and $q > p$,
$\Rightarrow \qquad (w_1(p),w_2(1-p)) \prec (w_1(q),w_2(1-q))$

Second Postulate

If outcome $w_1$ is preferred over outcome $w_2$ and outcome $w_2$ is preferred over outcome $w_3$, then there exists a probability p, such that the player is indifferent to the outcome $(w_1(p),w_3(1-p))$ and $w_2$. Formally :

If $w_1 \succ w_2 \succ w_3$ , then $\exists p \in [0,1] \;$ s.t. $\; (w_1(p),w_3(1-p)) \sim w_2$

Corollary : For $w_1$, $w_2$, $w_3$ defined as above, there exists a unique $p \in [0,1] \;$ s.t. $\; (w_1(p),w_3(1-p)) \sim w_2$.
Proof :        Suppose there exist $p_1,p_2 \in [0,1]$ and $p_1 > p_2$ (without loss of generality) s.t.

$(w_1(p_1),w_3(1-p_1)) \sim w_2$     and          $(w_1(p_2),w_3(1-p_2)) \sim w_2$

From first postulate, we know that:

$\qquad \qquad (w_1(p_1),w_3(1-p_1)) \succ (w_1(p_2),w_3(1-p_2))$

$\Rightarrow w_2 \succ w_2$         which is a contradiction.

Third Postulate

According to this postulate, if the player is indifferent between two outcomes say $w_1$ and $w_2$, and in another outcome say $w_3$, $w_1$ happens with probability $p$, then the player remains indifferent if $w_1$ in $w_3$ is replaced by $w_2$. To state this formally, label outcomes $w_1, w_2,\ldots, w_m$ in $\Omega \;$ s.t. $\; w_1 \succ w_2 \succ \ldots w_{m-1}\succ w_m$. Consider an outcome $\it O$ in $\Omega^\prime$ given as :

$(w_1(p_1),w_2(p_2),\ldots,w_m(p_m))$

Let $u_k$ be the probability $\;$ s.t. $\;$ $(a_1(u_k),a_m(1 - u_k)) \sim a_k$.

If ${\it O}^\prime = (w_1(p_1 + p_ku_k),w_2(p_2),\ldots,w_{k-1}(p_{k-1}),w_k(0),w_{k+1}(p_{k+1}),\ldots,w_m(p_m + (1 - u_k)p_k))$, then ${\it O} \sim {\it O}^\prime$

Von Neumann And Morgenstern Utility Function

Von Neumann and Morgenstern utility function $\bf\it U$ is defined over $\Omega^\prime = {\Re}^{\mid \Omega \mid}$. Let $w_1 \in \Omega$ is the most preferred outcome $(\forall w \in \Omega, \, w_1 \succ w)$ and $w_m \in \Omega$ is the least preferred outcome $(\forall w \in \Omega, \, w \succ w_m)$ . For each outcome $w \in \Omega$, define

$$u(w) = p \qquad s.t. \qquad w \sim (w_1(p),w_m(1 - p))$$ (1)

From postulate 2, such a $p$ always exists and is unique.

Von Neumann and Morgenstern utility function for $w \in \Omega^\prime$ is the expected value of the utility function u() as defined by equation 2. i.e.

$${\bf\it U} (w_1(p_1),w_2(p_2),\ldots,w_m(p_m)) = p_1u(w_1) + p_2u(w_2) + \ldots + p_mu(w_m)$$ (2)

To check that the Von Neumann and Morgenstern utility function represents the preference relationship on $\Omega^\prime$, consider two outcomes ${\it O}$ and ${\it O}^\prime$ in $\Omega^\prime\;$.i.e.

${\it O} = (w_1(p_1),w_2(p_2),\ldots,w_m(p_m))\;$ and

${\it O}^\prime = (w_1(q_1),w_2(q_2),\ldots,w_m(q_m))$

By repeated application of postulate 3, we get :

${\it O} \sim (w_1(\sum_{i=1}^m p_iu(w_i)),w_m(1 - \sum_{i=1}^m p_iu(w_i)))\;$ and

${\it O}^\prime \sim (w_1(\sum_{i=1}^m q_iu(w_i)),w_m(1 - \sum_{i=1}^m q_iu(w_i)))$

From postulate 1, it is clear that given two outcomes ${\it O}$ and ${\it O}^\prime$ in $\Omega^\prime$ (as defined above), a rational player will prefer the one, which corresponds to higher expected value of the utility function, u().

Therefore, $\;\bf\it U : \Omega^\prime \rightarrow \Re\;$ represents a preference relationship on $\Omega^\prime$.