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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 $q$, ${\mathbb{E}}_Q\log Z=n[q\log (q/p)+(1-q)\log ((1-q)/(1-p))]$.

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 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.

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.