Multinomial trials having probabilities <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p_1, \ldots , p_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> are observed until one of the outcomes, called the winning outcome, has occurred at least k more times than each of the others. With <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$P(A_i)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> being the probability that i is the winning outcome, we give a new approach to proving that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$P(A_j) \ge (p_j/p_i)^k P(A_i) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>≥</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>j</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </msup> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p_j > p_i.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>j</mi> </msub> <mo>></mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We then utilize this approach to obtain very efficient simulation estimators.

Win Probabilities in the First Ahead by at Least k Multinomial Game

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2026年7月31日

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Sheldon M. Ross

摘要

Multinomial trials having probabilities $p_1, \ldots , p_n$ $p_{1}, \dots, p_{n}$ are observed until one of the outcomes, called the winning outcome, has occurred at least k more times than each of the others. With $P(A_i)$ $P (A_{i})$ being the probability that i is the winning outcome, we give a new approach to proving that $P(A_j) \ge (p_j/p_i)^k P(A_i) $ $P (A_{j}) \geq {(p_{j} / p_{i})}^{k} P (A_{i})$ when $p_j > p_i.$ $p_{j} > p_{i} .$ We then utilize this approach to obtain very efficient simulation estimators.