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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.

Let $X$ denote the monetary reward. Its expected value and expected utility are different quantities:Thus indifference to a sure nonnegative reward requires $k^2=10.89$, giving $k=3.30$ dollars. This is the certainty equivalent, hence the lottery's monetary value in the question's comparison with a sure reward. The utility function $u(x)=x^2$ is increasing and strictly convex on nonnegative monetary amounts. Its certainty equivalent exceeds its expected value, $3.30>1.89$, demonstrating risk seeking; the same conclusion follows from the Jensen inequality.

The wording about paying an entry fee needs a qualification. A certainty equivalent compares a gross lottery with a sure gross reward; a literal utility indifference price also accounts for the fee's effect on wealth. If $u$ instead describes final wealth, with initial wealth $w$ and fee $c$, the comparison is $\mathbb{E}[(w-c+X{)}^2]=w^2$, on a domain where the utility function is increasing. Here $\mathrm{Var}(X)=7.3179$, so the nonnegative-wealth branch gives $c=w+1.89-\sqrt{w^2-7.3179}$ whenever that branch is admissible. An initial wealth and an admissible payment domain are not specified. Therefore $3.30$ is the determined certainty equivalent, while a literal wealth-dependent maximum fee is not uniquely determined. This is the distinction in certainty equivalent and willingness to pay. If negative rewards were allowed without requiring an increasing utility function, the equation for $k$ also has the algebraic root $-3.30$; the usual nonnegative-reward interpretation selects the positive root.

Write $A$ and $B$ for the two deterministic outcomes. Under the expected utility hypothesis, indifference between them implies $u(A)=u(B)=c$. The expected utility of the fair lottery is consequentlyIt must therefore be indifferent to each deterministic outcome, contradicting the asserted strict preference. These preferences do not admit an expected utility representation over the stated outcomes. Equivalently, affine utility on probability measures preserves this indifference under mixing, as required by the independence axiom for lottery preferences.

A concern for the fairness of the procedure can be perfectly intelligible. It would require enlarging the outcome description to include the procedure; it cannot be represented by an expected utility over only the two final gift recipients while retaining the stated strict preference.

First evaluate the inner lottery: $U({\ell }_1)=12p+3(1-p)=3+9p$. By reduction of compound lotteries, the outer expected utility isSince $4/9$ lies in the permitted real interval, the square is minimized there. The unique maximizer and maximum expected utility are $p=4/9$ and $U({\ell }_2)=43/9$. Directly flattening the lottery gives probabilities $p(1-p)$, $(1-p{)}^2$ and $p$ for the three outcomes; these sum to one and give the same expression.

Under the expected utility hypothesis, use positive affine uniqueness of affine preference representations to normalize the extreme outcomes to $u({\omega }_1)=1$ and $u({\omega }_4)=0$. The stated indifference relations then giveThe two lotteries have expected utilitiesHence ${\ell }_2\succ {\ell }_1$, with normalized expected utility difference $3/100$. Equivalently, substitution reduces them to lotteries with probabilities $0.45$ and $0.48$ of the better extreme outcome. The inference uses the independence axiom for lottery preferences; the three displayed outcome comparisons alone would not determine preferences over all compound lotteries without the expected utility hypothesis.

Yes: the stated order has an expected utility representation. For outcomes whose monetary amounts are $1,2,4$, respectively, chooseThenSince $12/5>9/5>3/2>4/5$, this utility function gives exactly the required strict preferences.

The distinction is that the expected utility hypothesis does not by itself require the utility function to increase with monetary amount. This nonmonotone monetary utility ranks the outcome paying 2 below the outcome paying 1. If increasing monetary utility were imposed as an additional assumption, the stated order would fail: ${\ell }_3\succ {\ell }_4$ requires $u(1)>u(2)$ becauseThat additional monotonicity assumption is not printed in this part.

Measure all payoffs in millions of dollars, in the order (Ann, Bob). Since both players maximize expected money, those numbers are already their expected utility scale.

In the extensive-form game, Ann moves first. Staying out terminates at $(2,6)$. Entering leads to Bob's observed decision between fighting, with terminal payoffs $(1,1)$, and acquiescing, with terminal payoffs $(3,3)$. Each decision node is its own information set, so this is a perfect-information game.

The corresponding normal-form game gives Ann the pure strategies $O$ (stay out) and $E$ (enter), and Bob the complete contingent pure strategies $F$ (fight if entry occurs) and $A$ (acquiesce if entry occurs). The payoff matrix isThe pure strategy Nash equilibria are $(O,F)$ and $(E,A)$. Against $F$, Ann prefers $O$; against $A$, Ann prefers $E$. When Ann stays out, Bob's contingent response does not affect his payoff, so $F$ is a best response there.

There is exactly one subgame perfect equilibrium, $(E,A)$. In the subgame following entry, Bob strictly prefers $3$ to $1$, so backward induction selects acquiescence; Ann then strictly prefers entering for $3$ to staying out for $2$. The threat supporting $(O,F)$ is not optimal in that one-shot subgame.

For infinite repetition, the paper specifies no discount factor. First take the long-run average payoff convention, with all past actions observed. The feasible payoff set is the convex hullAnn can guarantee $2$ in every stage by staying out; Bob can guarantee at least $3$ in every stage by acquiescing whenever entry occurs. These guarantees are also their minmax payoffs: fighting holds Ann's best possible stage payoff to $2$, and entering holds Bob's best possible stage payoff to $3$.

Therefore the sustainable Nash equilibrium average payoffs are exactly the feasible, weakly individually rational payoffs in a repeated game:This is the closed triangle with vertices $(2,3),(2,6),(3,3)$.

Here is a direct sufficiency argument for this particular game. Choose a publicly known sequence of stage outcomes whose frequencies give the target point. Before any observed deviation, follow that sequence. After an Ann deviation, Bob fights forever, so Ann can thereafter obtain at most $2$ per stage. After a Bob deviation, Ann enters forever, so Bob can thereafter obtain at most $3$ per stage. A finite prefix has no effect on long-run average payoff, so no unilateral deviation improves a target with $a\ge 2,b\ge 3$. Rational frequencies can be implemented periodically; arbitrary real frequencies can be implemented by a deterministic sequence with convergent frequencies. Thus even the weak boundary is covered here. In particular, staying out forever and entry followed by acquiescence forever are sustainable; fighting forever is not.

For discounted payoffs $(1-\delta ){\sum }_{t\ge 0}{\delta }^t{u}_i^t$, the folk theorem for repeated games instead gives the usual patience qualification: every feasible point with $a>2,b>3$ can be sustained for all sufficiently large $\delta <1$. It does not classify the whole region for an unspecified fixed $\delta$. The two one-shot Nash equilibrium outcomes remain sustainable for every $\delta$.^[ref]

Finally consider the particular history-dependent pair in the paper. Initially Bob has never acquiesced, so Ann stays out. That preserves the same condition forever, giving payoffs $(2,6)$. The pair is a Nash equilibrium. Holding Bob's strategy fixed, Ann cannot induce an acquiescence: he fights whenever she enters, and his rule never changes unless he has previously acquiesced. Every entry would give her $1$ instead of $2$, so no unilateral deviation helps. Holding Ann's strategy fixed, Bob never reaches a decision node because she stays out; changing his strategy alone cannot create a history in which he actually acquiesces. Hence he also cannot improve.

This direct argument works for both discounted and long-run average payoffs. The outcome gives Ann exactly her minmax payoff and Bob more than his, so it is compatible with the folk theorem for repeated games. A version requiring strict gains above both minmax payoffs is a sufficient condition, and cannot be used to rule out this boundary example.

A payoff is the number assigned to a player's outcome at a strategy profile, with larger numbers preferred. In the usual expected utility model, expected payoffs are probability-weighted averages of these numbers. If a payoff is specified as money and a player has preferences involving risk, first apply the relevant utility function; maximizing expected money assumes risk neutrality. In a two-player zero-sum game, one player's payoff is the negative of the other's. General bimatrix games instead require a payoff matrix for each player.

A compound lottery first selects branch $k$ with probability $p_k$ and then outcome ${\omega }_j$ with conditional probability $q_{kj}$. Its reduced lottery assigns ${\omega }_j$ the total probability ${\sum }_k{p}_k{q}_{kj}$. The law of total expectation gives the same expected utility in either description:This rule assumes that only the final outcome matters. A preference for a particular randomization procedure requires including that procedure in the outcome description; it cannot be captured by changing the arithmetic of expected utility on a fixed outcome set.

For increasing expected utility preferences, an investor is risk-averse when a sure mean is at least as desirable as the corresponding lottery. A concave utility function gives this preference by the Jensen inequality; a strictly concave utility gives strict preference for nondegenerate risks. Local risk aversion is measured by $-U^{\prime \prime }/U^{\prime }$ and its relative counterpart $-W{U}^{\prime \prime }/U^{\prime }$.

In expected utility theory, the risk premium is the difference between expected wealth and its certainty equivalent: $U(\mathrm{CE}(W))=\mathbb{E}U(W)$. For increasing concave utility functions it is nonnegative by the Jensen inequality. In financial asset pricing, risk premium also refers to expected return above a specified risk-free benchmark; the benchmark and whether the premium is monetary or proportional must therefore be stated.

Let a portfolio hold $x\in [0,T]$ in a risk-free asset with net return $r$ and $T-x$ in an asset returning $g$ or $b$, with probabilities $p$ and $1-p$. Assume $T>0$, $-1<b<r<g$, $0<p<1$, and $pg+(1-p)b>r$. Its state wealths are $w_g=T(1+g)-(g-r)x$ and $w_b=T(1+b)+(r-b)x$. The expected utility $p\log w_g+(1-p)\log w_b$ has strictly negative second derivative. Setting its first derivative to zero gives the displayed unconstrained root; the derivative at $x=T$ is negative, so only the lower constraint can bind. Thus the displayed formula is the unique optimum. All equity is optimal exactly when $p\ge (r-b)(1+g)/[(g-b)(1+r)]$. The positivity of both state wealths is part of the logarithmic utility domain.

For a finite outcome space, complete lottery preferences, a transitive relation, continuity of lottery preferences and the independence axiom for lottery preferences are equivalent to an expected utility representation of the displayed form. The representing outcome utility function is unique up to a positive affine function. The theorem is associated with John von Neumann and Oskar Morgenstern.

A useful proof construction selects a best outcome $a$ and a worst outcome $b$. If all outcomes are indifferent, a constant utility function suffices. Otherwise assign $u(a)=1$, $u(b)=0$. Continuity of lottery preferences makes each outcome indifferent to a mixture $u(\omega )a+(1-u(\omega ))b$, and mixture monotonicity makes the coefficient unique. Substitution of indifferent lotteries and reduction of compound lotteries turn any lottery into a best–worst mixture with best-outcome probability ${\sum }_{\omega }L(\omega )u(\omega )$. Mixture monotonicity then ranks all such mixtures. The reverse direction follows by taking expected values; positive affine functions preserve every comparison.^[ref]