An expected-utility maximizer and 3 lotteries

12 Oct 2015

Consider an individual with preferences over lotteries that have an expected-utility representation. There are three lotteries this individual can choose from:

\[L_1 \begin{cases} \begin{align} 200 ~~~ &P(200) = 1 \cr \end{align} \end{cases}\] \[L_2 \begin{cases} \begin{align} 0 ~~~ &P(0) = 2/3 \cr 200 ~~~ &P(200) = 1/6 \cr 1000 ~~~ &P(1000) = 1/6 \cr \end{align} \end{cases}\] \[L_3 \begin{cases} \begin{align} 0 ~~~ &P(0) = 1/3 \cr 400 ~~~ &P(400) = 1/3 \cr 1000 ~~~ &P(1000) = 1/3 \cr \end{align} \end{cases}\]

The expected utilities of the lotteries are:

\[\begin{align} EU_{L1} =& (200)(1) = 200 \cr EU_{L2} =& (0)(1/3) + (100)(1/6) + (1000)(1/6) \cr =& 200 \cr EU_{L3} =& (0)(1/3) + (100)(1/3) + (1000)(1/3) \cr =& 1400/3 \cr \end{align}\]

We can take a first pass at ordering the three lotteries by expected utility alone. This gives us that \(L_3 \succ L_2 \sim L_1\). To go further, we can use the concept of stochastic dominance.

\(X \succ_{FSD} Y\) (X first-order stochastically dominates Y) if \(F_x (t) \le F_y (t) ~ \forall t \in [a,b]\), where \([a,b]\) is the support of \(F_x\) and \(F_y\). This is equivalent to saying \(X \succ_{FSD} Y\) iff \(E[u(x)] \ge E[u(y)] ~ \forall\) nondecreasing, continuous functions \(u\). We apply this concept when we order the lotteries by expected utility.

Second-order stochastic dominance is a refinement on this. We say \(X \succ_{SSD} Y\) (X second-order stochastically dominates Y) if \(\int_a^w F_x (t) dt \le \int_a^w F_y (t) dt ~ \forall w \in [a,b]\), where \([a,b]\) is the support of \(F_x\) and \(F_y\). This is equivalent to saying \(X \succ_{SSD} Y\) iff \(E[u(x)] \ge E[u(y)] ~ \forall\) nondecreasing, continuous, and concave functions \(u\). We apply this concept when we order the lotteries by expected utility.

If all we have is that the agent is an expected-utility maximizer then all we can do is apply first-order stochastic dominance and say that the agent will prefer \(L_3\) over \(L_1\) and \(L_2\), and be indifferent between \(L_1\) and \(L_2\).

If we know or are willing to assume that the agent is risk-averse - that their utility function \(u\) is concave - then we can apply second-order stochastic dominance and rank the lotteries \(L_3 \succ L_1 \succ L_2\). The risk-averse agent would rather take the safe 200-for-sure than the risky 200-on-average.