Multinomial trials having probabilities <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2024_6239_Article_IEq1.gif" Format="GIF" Height="26" Rendition="HTML" Resolution="72" Type="Linedraw" Width="179" /> </InlineMediaObject> <EquationSource Format="TEX">$p_i = \frac{v_i}{ \sum _{j=1}^n v_j}, \, i = 1, \ldots , n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>v</mi> <mi>i</mi> </msub> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msub> <mi>v</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>,</mo> <mspace width="0.166667em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </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. We give an efficient simulation approach for estimating the probability that each outcome is the winner as well as the mean number of trials needed. We also show that the probability that outcome i wins is an increasing function of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2024_6239_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$v_i,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>i</mi> </msub> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> and is greater than the probability that outcome j wins when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10479_2024_6239_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$v_i > v_j.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>v</mi> <mi>i</mi> </msub> <mo>></mo> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

First ahead by at least k multinomial game

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2025年4月7日

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Sheldon M. Ross

摘要

Multinomial trials having probabilities $p_i = \frac{v_i}{ \sum _{j=1}^n v_j}, \, i = 1, \ldots , n$ $p_{i} = \frac{v_{i}}{\sum_{j = 1}^{n} v_{j}}, i = 1, \dots, n$ are observed until one of the outcomes, called the winning outcome, has occurred at least k more times than each of the others. We give an efficient simulation approach for estimating the probability that each outcome is the winner as well as the mean number of trials needed. We also show that the probability that outcome i wins is an increasing function of $v_i,$ $v_{i},$ and is greater than the probability that outcome j wins when $v_i > v_j.$ $v_{i} > v_{j} .$