Game Theory & Decision

The maths of choices: expected value, utility, the Kelly criterion, minimax and Bayes

Game Theory & Decision brings together the core formulas for choosing wisely when outcomes are uncertain, strategic, or both. From expected value, utility, and Bayes’ theorem to minimax, mixed strategies, regret, and the Kelly criterion, these tools help students, exam candidates, analysts, and decision-makers compare options, weigh risk, and make smarter choices in games, business, and everyday life.

Expected Value

EV = \sum_i p_i x_i

Probability-weighted average outcome of a decision.

Kelly Criterion

f^* = \frac{bp - q}{b}

Optimal fraction of bankroll to bet for long-run growth.

Expected Utility

EU = \sum_i p_i u(x_i)

Weighted average of the utility of outcomes, not just their cash value.

Payoff Comparison

NEWIntermediate

a \succ b \iff u(a) > u(b)

One strategy dominates another if its payoff is always higher.

Minimax Value

v = \max_a \min_b u(a,b)

Best worst-case payoff a player can guarantee.

Mixed Strategy (Indifference)

p = \frac{d - c}{(a-b)+(d-c)}

Probability that makes an opponent indifferent between their options.

Bayes' Theorem

P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Update a belief's probability given new evidence.

Value of Perfect Information

VPI = EV_{info} - EV_{no\,info}

How much better decisions become with full information.

Regret

NEWIntermediate

R = u^* - u

Difference between the best possible payoff and the one you got.

Odds to Probability

p = \frac{o}{1 + o}

Convert betting odds into an implied probability.

Expected Payoff vs Mixed Opponent

NEWIntermediate

E = \sum_j q_j u(a, b_j)

Your payoff when the opponent mixes strategies.

St. Petersburg Expected Value

EV = \sum_{k=1}^{\infty} \tfrac{1}{2^k} 2^{k-1}

A game with infinite expected value yet little real worth.

Shapley Value (2-player)

\phi_i = \tfrac{1}{2}(v(i) + [v(N) - v(j)])

Fair share of the total from a player's marginal contributions.

Break-Even Probability

p = \frac{L}{L + W}

Win probability needed for a bet to break even.