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Game Theory & Decision

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

Expected Value

NEWBasic
EV=ipixiEV = \sum_i p_i x_i

Probability-weighted average outcome of a decision.

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Kelly Criterion

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

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

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Expected Utility

NEWAdvanced
EU=ipiu(xi)EU = \sum_i p_i u(x_i)

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

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Payoff Comparison

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

One strategy dominates another if its payoff is always higher.

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Minimax Value

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

Best worst-case payoff a player can guarantee.

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Mixed Strategy (Indifference)

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

Probability that makes an opponent indifferent between their options.

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Bayes' Theorem

NEWAdvanced
P(AB)=P(BA)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}

Update a belief's probability given new evidence.

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Value of Perfect Information

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

How much better decisions become with full information.

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Regret

NEWIntermediate
R=uuR = u^* - u

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

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Odds to Probability

NEWBasic
p=o1+op = \frac{o}{1 + o}

Convert betting odds into an implied probability.

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Expected Payoff vs Mixed Opponent

NEWIntermediate
E=jqju(a,bj)E = \sum_j q_j u(a, b_j)

Your payoff when the opponent mixes strategies.

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St. Petersburg Expected Value

NEWAdvanced
EV=k=112k2k1EV = \sum_{k=1}^{\infty} \tfrac{1}{2^k} 2^{k-1}

A game with infinite expected value yet little real worth.

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Shapley Value (2-player)

NEWAdvanced
ϕi=12(v(i)+[v(N)v(j)])\phi_i = \tfrac{1}{2}(v(i) + [v(N) - v(j)])

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

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Break-Even Probability

NEWBasic
p=LL+Wp = \frac{L}{L + W}

Win probability needed for a bet to break even.

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