The intuition of fair odds (information theory)











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I'm reading about gambling in Elements of Information theory. As stated in the book, given that the gambler bets $o_i$-for-$1$ on horse $i$,



Fair odds (w.r.t some distribution): the odds is fair if $sum_i frac{1}{o_i} = 1$



Superfair odds: the odds is superfair if $sum_i frac{1}{o_i} < 1$



Subfair odds: the odds is subfair if $sum_i frac{1}{o_i} > 1$



Can anyone give me some intuition about these concepts (i.e. fair odds, superfaid odds and subfair odds) ?










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    up vote
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    down vote

    favorite












    I'm reading about gambling in Elements of Information theory. As stated in the book, given that the gambler bets $o_i$-for-$1$ on horse $i$,



    Fair odds (w.r.t some distribution): the odds is fair if $sum_i frac{1}{o_i} = 1$



    Superfair odds: the odds is superfair if $sum_i frac{1}{o_i} < 1$



    Subfair odds: the odds is subfair if $sum_i frac{1}{o_i} > 1$



    Can anyone give me some intuition about these concepts (i.e. fair odds, superfaid odds and subfair odds) ?










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I'm reading about gambling in Elements of Information theory. As stated in the book, given that the gambler bets $o_i$-for-$1$ on horse $i$,



      Fair odds (w.r.t some distribution): the odds is fair if $sum_i frac{1}{o_i} = 1$



      Superfair odds: the odds is superfair if $sum_i frac{1}{o_i} < 1$



      Subfair odds: the odds is subfair if $sum_i frac{1}{o_i} > 1$



      Can anyone give me some intuition about these concepts (i.e. fair odds, superfaid odds and subfair odds) ?










      share|cite|improve this question













      I'm reading about gambling in Elements of Information theory. As stated in the book, given that the gambler bets $o_i$-for-$1$ on horse $i$,



      Fair odds (w.r.t some distribution): the odds is fair if $sum_i frac{1}{o_i} = 1$



      Superfair odds: the odds is superfair if $sum_i frac{1}{o_i} < 1$



      Subfair odds: the odds is subfair if $sum_i frac{1}{o_i} > 1$



      Can anyone give me some intuition about these concepts (i.e. fair odds, superfaid odds and subfair odds) ?







      probability statistics information-theory gambling






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      asked Nov 15 at 3:52









      HOANG GIANG

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          From the point of the view of the house (who takes the bets) : let's assume $b_i$ is the amount bet on horse $i$, and let $B=sum_i b_i$ be the total bet. How should we pay, so that we (the house) don't lose or win? If $i$ is the winner, we should give the total $B$ to the gamblers who bet to it, so they will get $B$-for-$b_i$, or $o_i=B/b_i$ for $1$.
          So, in this (fair) case, we'd have $sum 1/o_i = 1$.



          Now, typically the house will want to profit a small share of the total bet, then the winners will get a total of $alpha B$ with $alpha lessapprox 1$ (conversely, if $alpha > 1$ then the house would lose something in each game... not a very usual scenario).



          Hence, in general $o_i = alpha B/b_i$ and $sum 1/o_i = frac{1}{alpha}$ or



          $$alpha = frac{1}{sum 1/o_i}$$



          This says that ${sum 1/o_i} > 1 implies alpha < 1$ , which is the usual, subfair scenario (for the gamblers).



          For example, in some bet site today, the odds for the ATP match Federer-Anderson are $o_1 = 1+4/11$ , $o_2 = 1+11/5$. So $alpha = frac{1}{sum 1/o_i}=0.9562$, so the house profits nearly $4.4%$ of the bet (subfair).



          (BTW: Though the textbook is about Information Theory, this question actually has little to do with it)






          share|cite|improve this answer























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            1 Answer
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            active

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            active

            oldest

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            up vote
            1
            down vote



            accepted










            From the point of the view of the house (who takes the bets) : let's assume $b_i$ is the amount bet on horse $i$, and let $B=sum_i b_i$ be the total bet. How should we pay, so that we (the house) don't lose or win? If $i$ is the winner, we should give the total $B$ to the gamblers who bet to it, so they will get $B$-for-$b_i$, or $o_i=B/b_i$ for $1$.
            So, in this (fair) case, we'd have $sum 1/o_i = 1$.



            Now, typically the house will want to profit a small share of the total bet, then the winners will get a total of $alpha B$ with $alpha lessapprox 1$ (conversely, if $alpha > 1$ then the house would lose something in each game... not a very usual scenario).



            Hence, in general $o_i = alpha B/b_i$ and $sum 1/o_i = frac{1}{alpha}$ or



            $$alpha = frac{1}{sum 1/o_i}$$



            This says that ${sum 1/o_i} > 1 implies alpha < 1$ , which is the usual, subfair scenario (for the gamblers).



            For example, in some bet site today, the odds for the ATP match Federer-Anderson are $o_1 = 1+4/11$ , $o_2 = 1+11/5$. So $alpha = frac{1}{sum 1/o_i}=0.9562$, so the house profits nearly $4.4%$ of the bet (subfair).



            (BTW: Though the textbook is about Information Theory, this question actually has little to do with it)






            share|cite|improve this answer



























              up vote
              1
              down vote



              accepted










              From the point of the view of the house (who takes the bets) : let's assume $b_i$ is the amount bet on horse $i$, and let $B=sum_i b_i$ be the total bet. How should we pay, so that we (the house) don't lose or win? If $i$ is the winner, we should give the total $B$ to the gamblers who bet to it, so they will get $B$-for-$b_i$, or $o_i=B/b_i$ for $1$.
              So, in this (fair) case, we'd have $sum 1/o_i = 1$.



              Now, typically the house will want to profit a small share of the total bet, then the winners will get a total of $alpha B$ with $alpha lessapprox 1$ (conversely, if $alpha > 1$ then the house would lose something in each game... not a very usual scenario).



              Hence, in general $o_i = alpha B/b_i$ and $sum 1/o_i = frac{1}{alpha}$ or



              $$alpha = frac{1}{sum 1/o_i}$$



              This says that ${sum 1/o_i} > 1 implies alpha < 1$ , which is the usual, subfair scenario (for the gamblers).



              For example, in some bet site today, the odds for the ATP match Federer-Anderson are $o_1 = 1+4/11$ , $o_2 = 1+11/5$. So $alpha = frac{1}{sum 1/o_i}=0.9562$, so the house profits nearly $4.4%$ of the bet (subfair).



              (BTW: Though the textbook is about Information Theory, this question actually has little to do with it)






              share|cite|improve this answer

























                up vote
                1
                down vote



                accepted







                up vote
                1
                down vote



                accepted






                From the point of the view of the house (who takes the bets) : let's assume $b_i$ is the amount bet on horse $i$, and let $B=sum_i b_i$ be the total bet. How should we pay, so that we (the house) don't lose or win? If $i$ is the winner, we should give the total $B$ to the gamblers who bet to it, so they will get $B$-for-$b_i$, or $o_i=B/b_i$ for $1$.
                So, in this (fair) case, we'd have $sum 1/o_i = 1$.



                Now, typically the house will want to profit a small share of the total bet, then the winners will get a total of $alpha B$ with $alpha lessapprox 1$ (conversely, if $alpha > 1$ then the house would lose something in each game... not a very usual scenario).



                Hence, in general $o_i = alpha B/b_i$ and $sum 1/o_i = frac{1}{alpha}$ or



                $$alpha = frac{1}{sum 1/o_i}$$



                This says that ${sum 1/o_i} > 1 implies alpha < 1$ , which is the usual, subfair scenario (for the gamblers).



                For example, in some bet site today, the odds for the ATP match Federer-Anderson are $o_1 = 1+4/11$ , $o_2 = 1+11/5$. So $alpha = frac{1}{sum 1/o_i}=0.9562$, so the house profits nearly $4.4%$ of the bet (subfair).



                (BTW: Though the textbook is about Information Theory, this question actually has little to do with it)






                share|cite|improve this answer














                From the point of the view of the house (who takes the bets) : let's assume $b_i$ is the amount bet on horse $i$, and let $B=sum_i b_i$ be the total bet. How should we pay, so that we (the house) don't lose or win? If $i$ is the winner, we should give the total $B$ to the gamblers who bet to it, so they will get $B$-for-$b_i$, or $o_i=B/b_i$ for $1$.
                So, in this (fair) case, we'd have $sum 1/o_i = 1$.



                Now, typically the house will want to profit a small share of the total bet, then the winners will get a total of $alpha B$ with $alpha lessapprox 1$ (conversely, if $alpha > 1$ then the house would lose something in each game... not a very usual scenario).



                Hence, in general $o_i = alpha B/b_i$ and $sum 1/o_i = frac{1}{alpha}$ or



                $$alpha = frac{1}{sum 1/o_i}$$



                This says that ${sum 1/o_i} > 1 implies alpha < 1$ , which is the usual, subfair scenario (for the gamblers).



                For example, in some bet site today, the odds for the ATP match Federer-Anderson are $o_1 = 1+4/11$ , $o_2 = 1+11/5$. So $alpha = frac{1}{sum 1/o_i}=0.9562$, so the house profits nearly $4.4%$ of the bet (subfair).



                (BTW: Though the textbook is about Information Theory, this question actually has little to do with it)







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Nov 15 at 19:40

























                answered Nov 15 at 19:32









                leonbloy

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