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For unrestricted final portfolio wealth in an $n$-period binomial market, write $R$ for the bank growth factor, $a>0$ for absolute risk aversion, and $Z=dQ/dP$ for the binomial-market probability density. The displayed optimizer has the state-price budget constraint ${\mathbb{E}}_Q{H}^{\ast }=R^n{w}_0$. Its exponential utility derivative satisfies $e^{-a{H}^{\ast }}=\lambda Z$ for a constant $\lambda >0$. The concave supporting-tangent inequality bounds any competing expected utility by this value, because the expected linear error is $\lambda {\mathbb{E}}_Q(H-H^{\ast })=0$. Strict concavity proves uniqueness, and market completeness supplies replication. The formula is altered by a constraint requiring nonnegative final wealth. With physical up probability $p$ and risk-neutral probability $q$, ${\mathbb{E}}_Q\log Z=n[q\log (q/p)+(1-q)\log ((1-q)/(1-p))]$.

A certainty equivalent exchanges a lottery for a sure outcome on the same utility function scale. A maximum purchase price compares wealth after paying with wealth without purchasing; its utility indifference price therefore depends on the initial wealth and on the admissible monetary domain. These two numbers are not generally interchangeable when the utility function is nonlinear.

For example, suppose $u(x)=x^2$ is applied only on nonnegative wealth, $X\ge 0$, $\mathbb{E}X=\mu$ and $\mathrm{Var}(X)={\sigma }^2$. The certainty equivalent is $\sqrt{{\mu }^2+{\sigma }^2}$. A purchase price satisfying the wealth comparison instead obeysprovided the square root exists and every resulting wealth is in the increasing domain. The sufficient condition $w\ge \sqrt{{\mu }^2+{\sigma }^2}$ ensures $w-c\ge 0$. The formulas differ even though both use the same expected utility representation. Applying the square function indiscriminately to negative wealth would remove the usual preference for more money and invalidate that monotonic price interpretation.

A game tree node whose successor is chosen according to a specified probability distribution. Its expected utility is a weighted sum of successor values, rather than the maximum or minimum used at a player's decision node.

Technical handbook: econport.gsu.edu/content/handbook/decisions-uncertainty/basic/von.html

The section “The Expected Utility Property” states the finite-lottery expected utility formula and connects it to the preceding preference axioms. The source's title misspells the second surname; the conventional name used here is Oskar Morgenstern.

A game tree evaluation rule using maxima at the controlled player’s decision nodes and expected values at chance nodes:respectively. Terminal utilities fix the boundary values. These rules exactly optimize a finite-horizon expected utility objective by backward induction. For cyclic infinite-horizon reachability problems, a finite Markov decision process formulation and value iteration replace naive expansion of an infinite game tree. Start with one at successful terminal states and zero elsewhere, keep terminal values fixed, and apply the maximizing/expectation backups repeatedly. The $t$th iterate is the greatest success probability achievable within $t$ transitions, by backward induction. These values are nondecreasing and bounded by one. For every policy, eventual success is the increasing set union of success within finite horizons, so its probability is the limit of those finite-horizon probabilities. Taking the supremum over policies and horizons in either order proves that the iterates converge to the optimal eventual-success probability. The limiting fixed-point equations can also admit larger solutions, so the initialization from terminal-success indicators matters.

For exponential utility $(1-e^{-ax})/a$ on unrestricted terminal portfolio wealth in the Black-Scholes model, the optimal dollar holding in the stock is the displayed deterministic amount. Set $\vartheta =(\mu -\rho )/\sigma$ and let $W^Q=W+\vartheta t$ be the Brownian motion under the risk-neutral measure. The terminal optimizer is $X^{\ast }=w_0{e}^{\rho T}+\vartheta W_T^Q/a$. Its marginal utility is proportional to the state-price density, and the concave supporting-tangent inequality proves global optimality under the state-price budget constraint. The expected utility at the optimum is $(1-\exp (-a{w}_0{e}^{\rho T}-{\vartheta }^{2}T/2))/a$. A nonnegative-wealth constraint changes this optimizer.

Game theory studies decisions whose consequences depend on what other decision makers choose. A bidder's gain depends on competing bids in an auction; a driver's travel time depends on other drivers in a congestion game. This strategic interdependence makes choosing well a different problem from optimization against a fixed environment: each participant's best response depends on the others' choices.

A normal-form game specifies the players, their available pure strategies, and their payoffs for each strategy profile. Larger payoffs mean more preferred outcomes for that player. Unless stated otherwise, a two-player table below lists $(u_1,u_2)$: the row player's payoff first, the column player's second. Each player maximizes their own expected utility; the numbers need not represent money, and comparing different players' numerical scales is unnecessary.

The central question is whether everyone's choices fit together. A Nash equilibrium consists of mutual best responses: no unilateral deviation improves a player's expected payoff. It can involve definite choices, or mixed strategies that deliberately randomize. Matching pennies shows why randomization can be essential; the coordination game shows how players can share an interest in coordinating while disagreeing over which outcome to choose. Such shared gains are absent from a two-player zero-sum game, where one player's gain is exactly the other's loss.

These ideas are useful both for analyzing incentives and for designing them. Mechanism design asks how changing the rules changes participants' best responses; evolutionary game theory studies how successful strategies spread. The mathematical appeal is that local comparisons of payoffs can reveal a globally consistent pattern of choices. Consistency alone does not imply Pareto efficiency, uniqueness, or that actual play will converge to a Nash equilibrium.

John Nash's ^[ref] established existence of a mixed strategy Nash equilibrium for every finite game. His ^[ref] developed the theory beyond two-player zero-sum games. The distinction between conflicting and shared interests is fundamental to the subject.

A mathematician whose work includes the minimax theorem and, with Oskar Morgenstern, the foundations of expected utility and game theory.

A probability distribution over possible outcomes. A finite lottery assigning probabilities $p_j$ to outcomes ${\omega }_j$ has expected utility ${\sum }_j{p}_{j}u({\omega }_j)$. Mixing lotteries mixes their outcome probabilities; the independence axiom for lottery preferences describes how this operation preserves preference comparisons.

A utility function on monetary amounts need not be increasing merely because preferences obey the expected utility hypothesis. The outcome labels do not themselves supply a preference order. For example, values $u(1)=2$, $u(2)=0$, $u(4)=3$ define consistent expected utility comparisons even though the amount $2$ is least preferred. Preferring more money is an additional assumption. It is distinct from risk aversion, which concerns the effect of spreading uncertainty for a specified increasing utility function.

Under linear expected utility, let a correct answer yield $r$, an error yield $-w$, and declining yield $s$, with $r+w>0$. Answering has expected payoff $pr-(1-p)w$, so it is preferred exactly when $p>(s+w)/(r+w)$. Equality gives indifference. Different utility, time costs or beliefs change this criterion; behaviour across differently rewarded tasks cannot be compared as if the threshold were fixed.

The no-borrowing portfolio constraint is $0\le x\le T$. Net asset returns must be added to the original principal. Thus the two final portfolio wealths areThe expected utility isThe probabilities are those of the investor's beliefs; no risk-neutral measure is being used in this preference calculation.

Risk neutrality means indifference between a random portfolio wealth and its certain expected value. For increasing twice differentiable expected utility preferences, this corresponds to an affine function $u(w)=Aw+C$ with $A>0$ on the relevant wealth interval. Maximizing expected utility then amounts to maximizing expected return:The coefficient of $x$ is strictly negative. Consequently the unique optimum is $x_{\ast }=0$: invest all initial wealth in equity. Risk neutrality here is a preference property, distinct from the pricing use of a risk-neutral measure.

Risk aversion means that a sure mean is preferred to the corresponding risky portfolio wealth. An increasing concave utility function, with $u^{\prime }>0$ and $u^{\prime \prime }\le 0$, has this property by the Jensen inequality; strictly concave utility gives strict preference for nondegenerate risks. Differentiating the expected utility from part (a) givesAn interior optimum therefore satisfiesThis balances the expected loss of marginal utility in the good state against its expected gain in the bad state. The borrowing constraint also requires checking the endpoints: $x_{\ast }=0$ if $U^{\prime }(0)\le 0$, while $x_{\ast }=T$ would require $U^{\prime }(T)\ge 0$. At $x=T$ both state wealths equal $T(1+r)$, soHence an optimum never puts all wealth into deposits. If $U^{\prime }(0)>0$, continuity and the displayed endpoint sign give an interior root, and concavity makes every such root globally optimal. Under strict concavity it is unique.

For pure strategy sets $S_1,\dots ,S_n$ and utility functions $u_i$, a Nash equilibrium is a strategy profile $a$ such thatFor mixed strategies, replace payoffs by their expectations under independent randomization and allow each player to choose any probability distribution. Since expected utility is linear in a player's own distribution, checking all pure deviations suffices.

Pure equilibria need not exist. In matching pennies, player 1 receives $1$ for equal choices and $-1$ for different choices, while player 2 receives the negative. At an equal pair, player 2 improves by switching; at a different pair, player 1 improves by switching. Thus no pure profile is a Nash equilibrium.

In the ranked university application game, every student must be admitted at a pure equilibrium. If someone were rejected, fewer than the total 50000 seats would be filled, so another university would have a vacancy. Moving there gives positive benefit instead of zero. Consequently all 50 universities have exactly 1000 applicants and admit all of them.

Choose a university with largest arithmetic mean admitted rank and let $r_{\min }$ be its lowest rank. If an outside student of rank $r>r_{\min }$ moved there, that student would be admitted in place of $r_{\min }$; the new arithmetic mean would exceed the university's previous arithmetic mean, which was at least the student's current payoff. This is a profitable deviation. Therefore no higher rank is outside: that university consists of the highest 1000 ranks. Remove that block and repeat with the remaining university of largest arithmetic mean. At each step the same argument excludes omitted higher ranks from the remaining set. The admitted cohorts are necessarily the consecutive blocks

Conversely, assigning these blocks to labelled universities is an equilibrium. A student moving to a higher block is rejected. A student moving to a lower block displaces its lowest rank; all the other admitted ranks there are below every rank in the student's original block. Comparing the two arithmetic means with that student's own rank held fixed shows a strict loss. Thus every block assignment is an equilibrium, and the preceding argument excludes all other pure ones:

There is more than one further mixed equilibrium. We give two genuinely mixed profiles, rather than count the pure equilibria as degenerate mixed ones. First, let every student apply independently and uniformly to all 50 universities. For each fixed student the other players' distribution is invariant under every permutation of university labels. All 50 pure applications therefore give the same expected utility. Uniform mixing is a best response, so this is a Nash equilibrium.

For a distinct mixed equilibrium, assign each of the 48 highest consecutive rank blocks deterministically to a different university. Let each of the lowest 2000 students independently choose either of the two remaining universities with probability $1/2$. For a low-ranked student these two universities have identical expected utility by label symmetry, and that payoff is positive: there is a positive-probability event on which the student is admitted. Applying to one of the 48 full higher-ranked blocks would guarantee rejection. Hence the specified mixture is a best response.

For a student in a deterministic higher block, moving to another deterministic block loses by the pure-equilibrium comparison above. It remains to rule out a move to a mixed university. Let the student's own block start at $L\ge 2001$, and let their rank be $r\le L+999$. Their present payoff is $L+499.5$. The number $M$ of low-ranked applicants at a specified mixed university has a binomial distribution with parameters $(2000,1/2)$, and the higher-ranked deviator is always admitted. If $M\ge 1$, all other admitted ranks are at most 2000, so the new arithmetic mean is at most $(r+2000)/2$, and thus at most $(L+2999)/2$. If $M=0$, it is $r\le 50000$. Therefore the expected deviation payoff is at mostbecause the gap without the last term is $(L-2000)/2\ge 1/2$. No such deviation improves the payoff. This proves a second mixed equilibrium, with only the lowest 2000 students randomizing. It is visibly different from the fully uniform profile. In particular,

Write $T=t_0$, and specify the physical Black-Scholes model as $d{S}_t/S_t=\mu dt+\sigma d{W}_t$, with constant $\sigma >0$ and bank rate $\rho$. The physical drift $\mu$ is needed for an expected utility calculation; a pricing model specified only under a risk-neutral measure does not determine the investor's preferences over risky positions. PutHere $\vartheta$ is the market price of risk, $Z_T$ is the density of the risk-neutral measure $Q$, and $\xi$ is the state-price density. The Girsanov theorem makes $W_t^Q=W_t+\vartheta t$ a Brownian motion under $Q$.

For a dollar holding ${\pi }_t$ in the stock and consumption rate $c_t$, the portfolio wealth equation isSince $d{\xi }_t=-\rho {\xi }_{t}dt-\vartheta {\xi }_{t}d{W}_t$, integration by parts gives the useful deflated wealth equation with consumptionFor strategies with a true deflated gain martingale, terminal wealth $X=V_T$ and consumption obey the state-price budget constraintFor nonnegative wealth and consumption, the admissibility condition can instead give a budget inequality; increasing utilities use the full budget at an optimum. Conversely, suitable integrable payoff and consumption plans with this budget are financeable in the Brownian complete market. To construct the strategy, setBy the Brownian martingale representation theorem, write $d{M}_t=H_{t}d{W}_t$. The required dollar stock holding is ${\pi }_t=(H_t/{\xi }_t+\vartheta V_t)/\sigma$; substitution in the deflated gain equation verifies financing. Remaining wealth goes into the bank account. For square-integrable data the representation holds directly, and localization handles the customary integrable admissible data.

First take no consumption. For an increasing strictly concave differentiable utility function whose marginal derivative can be inverted, let $I_u=(u^{\prime }{)}^{-1}$. Choose $\lambda >0$ so thatThis candidate is not merely a first-order guess. The concave supporting-tangent inequality gives, pointwise,Taking expectations and using the budget proves $\mathbb{E}u(X)\le \mathbb{E}u(X^{\ast })$ for every feasible competitor. Finance the optimizer by its risk-neutral pricing value $V_t=e^{-\rho (T-t)}{\mathbb{E}}_Q[X^{\ast }\mid {\mathcal{F}}_t]$, using the preceding representation to find the holdings.

For a merely concave differentiable utility function, the general formulation is to select $X^{\ast }(\omega )$ from the maximizers of $u(x)-\lambda {\xi }_T(\omega )x$ over its wealth domain, and choose the multiplier to satisfy the budget. If increasing utility is not assumed, the equality-budget multiplier may have either sign; the positive-multiplier inverse formula above is for increasing utility. Flat portions allow multiple maximizers; boundaries require one-sided inequalities rather than equality of marginal derivatives. Existence of a multiplier, integrability and an admissible maximizer must be checked. Concavity alone does not guarantee a well-posed problem: with unrestricted dollar positions, linear utility $u(x)=x$ and $\mu >\rho$ permits unbounded expected terminal wealth. The explicit utilities below satisfy the needed conditions in their stated domains.

For exponential utility $u(x)=(1-e^{-ax})/a$ on $\mathbb{R}$, $u^{\prime }(x)=e^{-ax}$ and $I_u(y)=-\log y/a$. ThereforeThe budget is equivalently ${\mathbb{E}}_Q{X}^{\ast }=w_0{e}^{\rho T}$, since ${\xi }_T=e^{-\rho T}dQ/dP$. Under $Q$, ${\mathbb{E}}_Q{W}_T=-\vartheta T$, giving $\log \lambda =\rho T-{\vartheta }^{2}T/2-a{w}_0{e}^{\rho T}$. HenceMatching the Brownian coefficient of $dV$ with $\sigma {\pi }_t$ gives the finite-horizon exponential-utility portfolioHold ${\pi }_t^{\ast }/S_t$ shares and invest $V_t-{\pi }_t^{\ast }$ in the bank account. This is a dollar amount, not a fixed wealth fraction. The Gaussian exponential formula also gives the optimal expected utilityThis calculation uses unrestricted wealth and an admissibility class permitting the displayed Gaussian terminal wealth, with integrable deflated gains and the relevant exponential moment. A nonnegative-wealth constraint would change the optimizer; no such constraint is imposed on this exponential-utility example.

For both terminal and running utility function, choose $X$ and a nonnegative adapted consumption rate $c$ to maximize ${\mathbb{E}}_P[u(X)+{\int }_0^{T}v(c_t)dt]$ subject to the combined budget. Pointwise maximization of $u(x)-\lambda {\xi }_{T}x$ and $v(c)-\lambda {\xi }_{t}c$ gives, for interior strictly concave cases,Apply the concave supporting-tangent inequality to $u$ and to $v$ at every time, then integrate and take expectations. The difference in objectives is at most $\lambda$ times the difference in budget expenditures, which is nonpositive. This proves optimality, not just the marginal conditions. The financing value isThe question's consumption rate $R_t$ is denoted $c_t$ here to keep it distinct from interest-rate notation.

For logarithmic utility with $u(x)=a\log x$, $v(x)=b\log x$, assume $a,b,w_0>0$ and positive wealth and consumption. The marginal conditions and budget giveEach state-price-weighted payment is deterministic, so conditional valuation immediately gives $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$. Put $A_t=a+b(T-t)$. Since the Itô formula gives $d(1/{\xi }_t)=(1/{\xi }_t)[(\rho +{\vartheta }^2)dt+\vartheta d{W}_t]$, this value satisfiesComparison with the wealth equation verifies the complete finite-horizon logarithmic investment and consumption strategy:Thus the optimal share holding is ${\pi }_t^{\ast }/S_t$, and the remaining value $V_t-{\pi }_t^{\ast }$ is in the bank account; borrowing or shorting is allowed if that value or the stock holding is negative. Terminal wealth is positive and consumption is positive throughout. Finally $\mathbb{E}\log {\xi }_t=-(\rho +{\vartheta }^2/2)t$, so the optimal objective, if desired explicitly, isFinite-horizon lognormal moments ensure the budget and expected utilities are finite, completing the feasibility and optimality checks.

Indifference means equality of the risky expected utility and the utility of the sure reduced wealth. A second-order Taylor expansion, using $\mathbb{E}\varphi =0$, givesFor the sure alternative,Equating these and retaining the leading terms produces the small multiplicative risk premiumFor increasing concave utility functions this is positive. The approximation assumes genuinely small shocks with controlled Taylor remainders and positive final wealth; small variance alone would not justify it for rare large shocks.

Constant relative risk aversion coefficient means that $\alpha =-W{U}^{\prime \prime }(W)/U^{\prime }(W)$ does not depend on wealth, for $W>0$ and $U^{\prime }(W)>0$. The defining differential equation integrates asA second integration therefore givesThe additive constant and positive scale do not affect expected utility preferences. Since $U^{\prime \prime }(W)=-\alpha c{W}^{-\alpha -1}$, positive $\alpha$ gives risk aversion, zero gives risk neutrality, and negative $\alpha$ gives risk seeking. These are respectively concave, affine and convex utility functions. The usual CRRA utility class is the positive-$\alpha$ case.

In a second-price sealed-bid auction, fix the other players' bids and let $a$ be their highest bid. A player of valuation $v$ who wins pays $a$, obtaining quasilinear utility $v-a$; losing gives zero. If $v>a$, winning is optimal and bidding $v$ wins. If $v<a$, losing is optimal and bidding $v$ loses. If $v=a$, winning and losing both give zero, regardless of the fixed tie-breaking rule. Thus truthful bidding is a dominant strategy. In particular the profile in which every bidder bids its valuation is a Nash equilibrium, for every valuation profile, not merely on average.

A symmetric independent private values model gives each bidder a private valuation determined by its own type. The types are independent draws from a common distribution, and the bidders have identical roles and preferences. In the standard bidding calculation bidders are risk-neutral with quasilinear utility: they maximize expected valuation minus payment. Strategies map each bidder's privately observed value to a bid. Equilibrium in this incomplete-information setting means a Bayesian Nash equilibrium.

Take $n\ge 2$ with independent uniform values on $[0,1]$. In a first-price sealed-bid auction, if all players bid their valuations, a type $v$ earns zero whenever it wins and zero whenever it loses. But for $v>0$, bidding $v/2$ instead yields positive expected utilityTherefore truthful first-price bidding is not an equilibrium.

For the candidate uniform private-value first-price bidding equilibrium, put $\alpha =(n-1)/n$ and suppose every opponent bids $\alpha$ times its valuation. For a deviation $b\in [0,\alpha ]$, winning means that every opposing value is less than $b/\alpha$. Independence gives winning probability $(b/\alpha {)}^{n-1}$, so the type-$v$ expected utility isFor $b>0$ its derivative isFor $v>0$ it is positive before $b=\alpha v$ and negative after; for $v=0$, every positive bid has nonpositive utility and zero is optimal. Thus the maximum on $[0,\alpha ]$ is $b=\alpha v$. A bid above $\alpha$ wins with probability one but gives $v-b\le v-\alpha$, which is no better than the already considered endpoint $b=\alpha$, hence no better than $b=\alpha v$. Ties have probability zero under the continuous opposing distributions. All admissible deviations are therefore covered.

Each bidder's best response is the proposed rule, provingIts conditional expected utility is $u(v,\alpha v)=v^n/n$. With only one bidder, bidding zero is optimal and agrees with the same formal rule; the derivative argument is needed only for $n\ge 2$.

${\ell }_2\succ {\ell }_1$. 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]