Past exam of the MSc in advanced computer science course of the University of Oxford / 2021 / Computational game theory / 1

${\ell }_2\succ {\ell }_1$. The distinction is small but strict: their normalized expected utilities are $18/25$ and $7/10$, respectively.

By the Von Neumann-Morgenstern utility theorem, use a representing utility function and apply a positive affine function to set $u({\omega }_1)=1$ and $u({\omega }_4)=0$. This is possible because the given strict preference implies $u({\omega }_1)>u({\omega }_4)$. The two specified indifference relations then giveTaking the expected utility of each lottery yieldsTheir difference is $1/50>0$, establishing the stated strict preference. These numbers measure utility, not monetary reward or an assumed ordering of the subscripts.

Here is the underlying argument directly in terms of the axioms. The independence axiom for lottery preferences permits substitution of indifferent lotteries inside a common mixture. Replacing ${\omega }_3$ by its given mixture of ${\omega }_1,{\omega }_4$, and then replacing ${\omega }_2$, followed by reduction of compound lotteries, givesand consequentlyThe transitive relation axiom combines the successive substitutions. Finally, mixture monotonicity says that among mixtures of a strictly better and a strictly worse outcome, the greater probability of the better outcome is strictly preferred; this follows from the independence axiom for lottery preferences and transitivity. Since $18/25>7/10$ and ${\omega }_1\succ {\omega }_4$, the comparison follows. Complete lottery preferences and continuity of lottery preferences are part of the representation theorem's hypotheses; the direct substitution calculation uses the already supplied indifferences and does not need to invoke continuity of lottery preferences again.^[ref]

No: these two strict comparisons violate the independence axiom for lottery preferences. Let $z$ denote the certain zero-money lottery. Reduction of compound lotteries givesFor example, the second expression assigns probability $(1/4)(4/5)=1/5$ to the nonzero reward and $4/5$ to zero. The claimed ${\ell }_3\succ {\ell }_4$ must therefore implywhich is ${\ell }_1\succ {\ell }_2$, contradicting the other claimed strict preference.

The same obstruction is visible in any proposed expected utility representation. Subtract $u(0)$ from the utility function, and write $a=u(3000)-u(0)$ and $b=u(4000)-u(0)$. The first comparison requires $(1/5)b>(1/4)a$, hence $(4/5)b>a$, whereas the second requires $a>(4/5)b$. No real $a,b$ satisfy both. This argument requires neither risk neutrality nor an assumption that the utility function is increasing in money.