Generalizations Of Bernoulli s Inequality With Utility-Based Approach

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1 Appled Mathematcs E-Notes, 13(2013), c ISSN Avalable free at mrror stes of amen/ Generalzatons Of Bernoull s Inequalty Wth Utlty-Based Approach Alreza Hosenzadeh, Gholamreza Mohtasham Borzadaran, Gholamhosen Yar Receved 28 January 2012 Abstract Utlty functon s one of the most useful tools n mathematcal fnance, decson analyss and economcs. The concept of expected utlty s used n economcs to descrbe the behavor of a decson-maker choosng between a certan number of random gans. In ths paper, some generalzatons of the Bernoull s nequalty are derved by usng methods from the utlty theory. In partcular, we obtan Harmonc-Geometrc-Arthmetc mean nequalty and gve a generalzaton of t. 1 Introducton The noton of utlty goes back to Danel Bernoull [2]. Because the expected pay off n the game of the St. Petersburg Paradox s nfnte, Bernoull was the frst who suggested a utlty functon as a soluton to the St. Petersburg Paradox. The frst axomatc development of utlty theory was acheved by Von Neumann and Morgenstern [18] who argued that the exstence of a utlty functon could be derved from a set of axoms governng a preference orderng. Other developments are due to Marschak [13], Hersten and Mlnor [9] and especally Fshburn [6, 7]. Borch [3] showed how utlty theory could be used to formulate and solve some problems that are relevant to nsurance. Most of the orgnal papers of Borch have been reprnted n hs books [3, 4]. Abbas [1] ntroduced a relaton between probablty and utlty based on the concept of a utlty densty functon and showed the applcaton of ths relaton va the maxmum entropy prncple. Fredman et al. [8] ntroduced utlty-based generalzatons of the Shannon entropy and Kullback-Lebler nformaton measure and called them U-entropy and U-relatve entropy and found varous propertes for these generalzed quanttes smlar to the classcal concepts of nformaton theory. For a parametrc famly of utlty functons Hosenzadeh et al. [10] derved some lnks between U-relatve entropy and other dvergence measures. Hosenzadeh et al. [11] extended Markov s nequalty for probabltes Mathematcs Subject Classfcatons: 91A30. Department of Statstcs, Gonabad Branch, Islamc Azad Unversty, Gonabad, Iran. Department of Statstcs, Ordered and Spatal Data Center of Excellence of Ferdows Unversty of Mashhad, Mashhad, Iran. Iran Unversty of Scence and Technology, Tehran, Iran. 212

2 Hosenzadeh et. al 213 from the U-entropy vewpont and derved a lnk between condtonal U-entropy and condtonal Reny entropy. In ths paper, some generalzatons of Bernoull s nequalty are obtaned by the concept of the certanty equvalent that was ntroduced n utlty theory. Also, we derved an mportant and useful nequalty n statstcs,.e. the Harmonc-Geometrc- Arthmetc mean nequalty and gve a generalzaton for t. 2 Utlty Functons An nvestor s subjectve probabltes numercally represent hs belefs and nformaton, and hs utltes represent hs tastes and preferences. Utlty functons provde us wth a method to measure an nvestor s preferences for wealth and the amount of rsk he s wllng to undertake n the hope of ganng greater wealth. A utlty functon s a twce-dfferentable functon of wealth u(x) whch has the followng propertes: () u(x) s a strctly ncreasng functon of x, () u(x) s a strctly concave functon of x. The frst property states that utlty ncreases wth wealth,.e., that more wealth s preferred to less wealth. The second property states that the utlty functon s concave or, n other words, that the margnal utlty of wealth, u (x), decreases as wealth ncreases, or equvalently, that the gan of utlty resultng from the monetary gan of g, u(x + g) u(x), be a decreasng functon of wealth x. The utlty functons of most people tend to be concave, at least for large gans or large losses. It s easy to see that f u(x) s a utlty functon, then for constant a > 0 and b, the functon w(x) = au(x) + b s also a utlty functon. For any utlty functon u(x), we consder the followng functon: r (x) = u (x) u (x). Ths s called absolute rsk averson functon whch s a postve functon. It s also generally agreed n fnance theory that for a utlty functon to be realstc wth regard to economc behavor, ts absolute rsk averson should be a decreasng functon of wealth. The most common utlty functons are as follows: Exponental utlty functon wth parameter α > 0, u (x) = 1 e αx α for < x <. Note that ths utlty tends to the fnte value 1 α as x. For the exponental utlty functon, r(x) = α, < x <. Thus, the exponental utlty functon yelds a constant rsk averson.

3 214 Generalzatons of Bernoull s Inequalty Logarthmc utlty functon wth parameters α > 0, β, γ, u (x) = α ln (x β) + γ for x > β. For the logarthmc utlty functon, r(x) = 1 x β, x > β. Hence the rsk averson s a decreasng functon of wealth, whch may be typcal for some nvestors. Iso-Elastc utlty functon s a class of utlty functons as follows: u(x) = xα 1 for α < 1, α 0 and x > 0. α Ths class has the property of u(kx) = f(k)u(x) + g(k), for all k > 0 and for some functon f(k) > 0 and g(k) whch are ndependent of x. We note that ths utlty tends to u(x) = ln x as α 0. For the Iso-Elastc utlty functon, r(x) = 1 α x, that s a decreasng functon of wealth, whch may be typcal for some decson-maker. 3 Stochastc Orders Stochastc orders and nequaltes play an mportant role n many dverse areas of probablty and statstcs and have been used durng the last 40 years. Such areas nclude relablty theory, queung theory, survval analyss, economcs, nsurance, actuaral scences, operatons research and management scence. Several orders of two random varables (or ther dstrbuton functons) have been studed by Shaked and Shanthkumar [15]. The smplest way of comparng two random varables s by the comparson of the assocated means. However, such a comparson s based on only two sngle numbers (the means) and therefore t s often not very nformatve. In addton to ths, the means sometmes do not exst. In many nstances n applcatons one has more detaled nformaton, for the purpose of comparson of two dstrbuton functons, than just the two means. Let X and Y be two random varables defned on a probablty space. Several orders of unvarate dstrbuton functons are as follows: The random varable X s sad to be smaller than Y n the usual stochastc order (denoted by X st Y ) f and only f, P {X > x} > P {Y > x} for all x (, ). The random varable X s sad to be smaller than Y n the convex order (denoted by X cx Y ) f and only f, E[φ(X)] E[φ(Y )], for all convex functons φ : R R, provded the expectatons exst. The random varable X s sad to be smaller than Y n the ncreasng convex [concave] order (denoted by X cx Y [X cv Y ]) f and only f, E[φ(X)] E[φ(Y )], for all ncreasng convex [concave] functons φ : R R, provded the expectatons exst.

4 Hosenzadeh et. al 215 Other orders such as hazard rate order, mean resdual lfe order, lkelhood rato order have been studed by Shaked and Shanthkumar [15]. We wll concentrate on a specal case of ths order whch wll be defned as expected utlty order. 4 Man Results In ths secton, va the monotone concave order, we order two random gans by means of ther expected utltes. In partcular, a random gan wll be replaced by a fxed amount that s named the certanty equvalent. Ths noton can be used by the consumer who wants to determne the maxmal premum he or she s wllng to pay to obtan full coverage. Also, by means of ths concept, we obtan some useful mathematcal nequaltes. 4.1 Preference Orderng of Random Gans Now we consder a decson-maker wth ntal wealth w who has the choce between a certan number of random gans. By usng the monotone concave order and a utlty functon, as an ncreasng concave functon of wealth, we can compare two random gans as follows: The random gan G 1 s preferred over the random gan G 2 n the expected utlty order (denoted by G 1 Eu G 2 ) f and only f, E[u(w + G 1 )] > E[u(w + G 2 )], provded the expectatons exst. Ths defnes a preference orderng on the set of random gans. On the other hand snce u(x) s a concave functon, by Jensen s nequalty for any random varable G, we can wrte, u(w + E[G]) > E[u(w + G)]. Therefore, f a decson-maker can choose between a random gan G and a fxed amount equal to ts expectaton, he wll prefer the latter. Ths leads us to the followng defnton. DEFINITION. The certanty equvalent, π, assocated wth the random gan G s defned by the condton that the decson-maker s ndfferent between recevng G or the fxed amount π,.e., u(w + π) = E[u(w + G)]. (1) Snce u(x) s a strctly ncreasng functon, the last nequalty leads us to the followng mportant nequalty: π < E[G]. (2) In the next secton, by usng the nequalty (2) and by means of a random gan wth known dstrbuton, we derve some useful mathematcal nequaltes.

5 216 Generalzatons of Bernoull s Inequalty 4.2 Generalzatons of Bernoull s Inequalty Suppose that n s a postve nteger and x > 1. Then the nequalty (1 + x) n 1 + nx, s known as the Bernoull s nequalty whch plays an mportant role n mathematcal analyss and ts applcatons. Recently, varous generalzatons of Bernoull s nequalty are obtaned by many researches. For example, the followng generalzatons of ths nequalty were ntroduced by Kuang [12]. THEOREM 1. Let x > 1. If α > 1 or α < 0, then and f 0 < α < 1, then (1 + x) α 1 + αx, (1 + x) α 1 + αx, n whch, equaltes hold f and only f, x = 0. THEOREM 2. Let a 0, x > 1, = 1,..., n and n a 1. If a 1 or a 0, then n (1 + x ) a 1 + a x, and f x > 0, or 1 < x < 0, = 1,..., n, then n (1 + x ) a 1 + a x. By usng methods on the theory of majorzaton, some generalzatons of Bernoull s nequalty were establshed by Sh [16]. In the next theorems, by applyng methods on the utlty theory, some generalzatons of Bernoull s nequalty are establshed. THEOREM 3. For w > 0, and 0 p 1, we have that ( ) p 1 + p w w, (3) equalty holds f and only f, w. PROOF. Suppose that the random gan G has the Bernoull dstrbuton wth parameter 0 < p < 1. Consder the logarthmc utlty functon and usng equaton (1) we can wrte ln(w + π) = (1 p) ln(w) + p ln(w + 1). Solvng ths equaton wth respect to π leads to π = w[(1 + 1 w )p 1]. Usng E(G) = p and applyng nequalty (2) leads to the nequalty (3).

6 Hosenzadeh et. al 217 THEOREM 4. Let 0 p 1, w > 0 and a, are postve real numbers such that w + a > 0, = 1,..., n. Then we have n (1 + a w )p a p w + 1. (4) PROOF. Let G be a dscrete random varable wth probablty dstrbuton p = (p 1, p 2,..., p n ) whch s defned over a fnte support {a 1, a 2,..., a n }. By means of the logarthmc utlty functon and by usng (1), we obtan: ln(w + π) = p ln(w + a ), solvng ths equaton wth respect to π yelds [ n ] π = w (1 + a 1. w )p Note that E(G) = n a p. Then by (2), we may then complete the proof. COROLLARY 1. If p = ( 1 n, 1 n,..., 1 n ) and x = 1 + a w, = 1,..., n, (4) s reduced to n n x1 x 2...x n x, n whch s the Geometrc-Arthmetc mean nequalty. Varous proofs of the Geometrc-Arthmetc mean nequalty are known n the lterature, (see, for example, Bullen [5]). A smple proof of the Geometrc-Arthmetc mean nequalty was gven by Uchda [17]. THEOREM 5. Let 0 p 1 and x > 0, = 1,..., n be postve real value numbers. Then we have n n 1 p x x. (5) PROOF. Defne j=1 y = p j j n x j for = 1,..., n. j Suppose that G s a dscrete random varable on the support {y 1, y 2,..., y n } such that, p = (p 1, p 2,..., p n ) s the probablty dstrbuton of G,.e., p = P (G = y ), = 1,..., n. Consder a decson-maker wth ntal wealth w = 0 and logarthmc utlty functon. Hence, from (1) we obtan ( n n ) ln π = E[ln(G)] = p ln 1 p = ln x. j x j

7 218 Generalzatons of Bernoull s Inequalty Solvng ths equaton wth respect to π we conclude that π = n hand, E(G) = n j=1 p n j j x and nequalty (2) mply the proof. COROLLARY 2. Inequalty (5) can be wrtten as follows: n I=1 x n p n j=1 p n j j x x. For p = ( 1 n, 1 n,..., 1 n ), we derve: n 1 x x n x 1 x 2...x n, x n that s the Harmonc-Geometrc mean nequalty. x1 p. On the other COROLLARY 3. By settng x = 1 + a w, = 1,..., n n Theorem 4 and usng Theorem 5 we derve n x n p n j=1 p n j j x x p x. So, these nequaltes gve a generalzaton for the Harmonc-Geometrc-Arthmetc mean nequalty, (See, for example, Napoca [14]). THEOREM 6. Let m, n, k are nonnegatve nteger such that for = 0, 1,..., n nequaltes k n m hold. Then for all w > 0, we have k [ 1 + ] C n Ck m n [1 + nk ] C k m, w mw n whch, =0 C n = n!!(n )!. PROOF. Let G be a random gan wth the Hypergeometrc probablty dstrbuton as follows: p() = C nc k m n C k m for = 0, 1,..., k, k = 1, 2,..., n and m n. By consderng equaton (1) and u(x) = ln x, va some calculaton we have [ k ] 1 Cm k E[ln(w + G)] = ln (w + ) C n Ck m n. Now by solvng ths equaton wth respect to π we obtan [ k ] 1 Cm k π = (w + ) C n Ck m n w. =0 By applyng nequalty (2) and E(G) = k n m, we have the concluson. =0

8 Hosenzadeh et. al Conclusons In ths work, motvated by monotone concave order we compared two random gans by means of ther expected utltes. Also, by usng the concept of the certanty equvalent n utlty theory, we obtaned some generalzaton of the Bernoull s nequalty. As a specal case, we derved the Harmonc-Geometrc-Arthmetc mean nequalty and obtaned a generalzaton of t. Acknowledgments. The authors thank the Edtor and anonymous referees for ther helpful comments whch have greatly mproved the presentaton of ths paper. References [1] A. Abbas, Entropy Methods n Decson Analyss, PhD Thess, Department of Management Scence and Engneerng, Stanford Unversty, [2] D. Bernoull, Exposton of a new theory on the measurement of rsk (translaton of Specmen Theorae Novae de Mensura Sorts St. Petersburg, 1738), Econometrca, 22(1954), [3] K. Borch, The Mathematcal Theory of Insurance, Lexngton, Mass.: Lexngton Books, [4] K. Borch, Economcs of Insurance, edted by K. K. Aase and A. Sandmo, Amsterdam: North-Holland, [5] P. S. Bullen, Handbook of Means and Ther Inequaltes, Kluwer Acad. Publ., Dordrecht, [6] P. C. Fshburn, Utlty theory, Management Sc., 14(1968), [7] P. C. Fshburn, Non-transtve measurable utlty for decson under uncertanty, J. Math. Econom., 18(1989), [8] C. Fredman, J. Huang, and S. Sandow, A utlty-based approach to some nformaton measures, Entropy, 9(2007), [9] I. N. Hersten, and J. Mlnor, An axomatc approach to measurable utlty, Econometrca, 21(1953), [10] A. R. Hosenzadeh, G. R. Mohtasham Borzadaran, and G. H. Yar, A vew on extenson of utlty-based on lnks wth nformaton measures, Communcatons of the Korean Statstcal Socety, 16(2009), 1 8. [11] A. R. Hosenzadeh, G. R. Mohtasham Borzadaran, and G. H. Yar, Aspects concernng entropy and utlty, Theory and Decson, 72(2012), [12] J. C. Kuang, Appled Inequaltes (Chang yong bu deng sh) 3nd ed, Shandong Press of scence and technology, Jnan, Chna, (Chnese).

9 220 Generalzatons of Bernoull s Inequalty [13] J. Marschak, Ratonal behavor, uncertan prospects, and measurable utlty, Econometrca, 18(1950), [14] C. Napoca, Inequaltes for general ntegral means, J. Inequal. Pure and Appl. Math., 7(2006), Art. 13. [15] M. Shaked and J. G. Shanthkumar, Stochastc Orders, Sprnger Scence+Busness Meda, LLC, [16] H. N. Sh, Generalzaton of Bernoull s nequalty wth applcatons, J. Math. Inequal., 2(2008), [17] Y. Uchda, A smple proof of the geometrc-arthmetc mean nequalty, J. Inequal. Pure and Appl. Math., 9(2008), Art. 56. [18] J. von Neumann and O. Morgenstern, Theory of Games and Economc Behavor, Prnceton, Prnceton Unversty Press, New Jersey, 1944.