A psychophysical theory of Shannon entropy.

Transcription

1 A sychohyscal theory of Shannon entroy. Tak Takahash 1 1 Deartment of Behavoral Scence, Hokkado Unversty Corresondng Author: Tak Takahash Emal: taktakahash@gmal.com Deartment of Behavoral Scence, Hokkado Unversty N.10, W.7, Kta-ku, Saoro, , Jaan TEL: FAX: Acknowledgements: The research reorted n ths aer was suorted by a grant from the Grant- n-ad for Scentfc Research ( global center of excellence grant) from the Mnstry of Educaton, Culture, Sorts, Scence and Technology of Jaan.

2 Abstract Connectons between nformaton theory and decson under uncertanty have been attractng attenton n econohyscs, neuroeconomcs and quantum decson theory. Ths aer rooses a sychohyscal theory of Shannon entroy based on a mathematcal equvalence of delay and uncertanty n decson-makng, and sychohyscs of the erceton of watng tme n robablstc choces. Furthermore, t s shown that the well-known Shannon entroy s a secal case of the general sychohyscal entroy. Future drectons n the alcaton of the resent theory to studes n econohyscs and neuroeconomcs are dscussed. Keywords: Entroy, Uncertanty, Neuroeconomcs; Psychohyscs Informaton theory; Rsk;

3 1. Introducton: Decson under rsk and robablstc uncertanty (robablstc choce) has been a major toc n mcroeconomcs (e.g., the exected utlty theory, von Neumann and Morgenstern, 1947), behavoral neuroeconomcs (e.g. the rosect theory, Kahneman and Tversky, 1977), and econohyscs (Anteneodo et al., 2002). Studes n behavoral and neuro- economcs have revealed that humans and non-human anmals dscount the value of robablstc rewards as the recet becomes more uncertan ("robablty dscountng", Rachln and Raner, 1991). In order to develo decson theory on robablstc choce, recent efforts n econohyscs have started to combne robablstc choce rocesses wth ntertemoral choce rocess (Takahash, 2007). Delay dscountng n ntertemoral choce refers to the devaluaton of a delayed reward comared to the value of a sooner reward. In ths lne of nvestgatons nto the unfcaton of delay and robablty dscountng, recent studes n behavoral sychology and neuroeconomcs have demonstrated a mathematcal equvalence of delay and robablstc uncertanty (rsk) n reward-seekng behavor (Takahash, 2007). Furthermore, Takahash (2005, 2006) roosed that erceton of watng tme n ntertemoral choce s logarthmc n hyscal tme, whch has recently been confrmed exermentally (Takahash et al., 2008; Zauberman et al., 2009). I ntroduce, n ths aer, a novel theory for Shannon nformaton entroy by utlzng sychohyscs of watng tme based on the equvalence of delay and uncertanty (Rachln and Raner, 1991; Takahash, 2007). Notably, I derve generalzed Shannon entroy based on the sychohyscal theory and demonstrate that conventonal Shannon entroy s a secal case for t. Ths aer s organzed n the followng manner. In Secton 2, I brefly ntroduce the mathematcal equvalence of delay and robablstc uncertanty, and sychohyscs of logarthmc watng tme erceton. In Secton 3, I exlan the sychohyscal theory of nformaton entroy. In Secton 4, some conclusons from ths study and future study drectons by utlzng the resent sychohyscal theory of nformaton entroy n neuroeconomcs and neurosychatry of rsky behavor are dscussed. 2. A mathematcal equvalence of delay and uncertanty n decson makng As noted above, t has been demonstrated that delay untl recet of gans n ntertemoral choce and uncertanty of wnnng of gans n robablstc choce may be equvalent. In unfyng decson over tme and under uncertanty, Rachln et al (1991)

4 hyotheszed that a decrease n a robablty of wnnng an uncertan reward corresonds to an ncrease n a delay untl wnnng the reward. Secfcally, an average watng tme untl wnnng an uncertan reward s roortonal to (1/)-1 ("odds aganst"), where s a robablty of wnnng the uncertan reward. Therefore, accordng to Rachln and colleagues hyothess, decson-makng models n ntertemoral choce (delay dscountng) can straghtforwardly be extended nto robablstc choce(s), after relacng a arameter of delay n ntertemoral choce models wth the odds aganst arameter. Hence, t aears to be a romsng drecton to establsh nformaton theory on the sychohyscal equvalence of delay and robablty 1 (n terms of the odds-aganst= ) n decson under robablstc uncertanty (rsk). Wth resect to sychohyscs of tme-erceton, Takahash (2005, 2006) roosed that sychologcal tme durng ntertemoral choce (= ) follows the Weber-Fechner law: (D)=a ln (1+b D), (1) where [ 0, ] s erceved/antcated sychologcal tme of hyscal delay D [ 0, ], and a and b [ 0, ] are free arameters (note that b s a nondmensonalzaton coeffcent of hyscal tme D). The theoretcal roosal has been confrmed exermentally (Takahash et al., 2008; Zauberman et al., 2009). Together, t may be a romsng drecton how nformaton theory n decson theory and neuroeconomcs could be formalzed n terms of sychohyscs of watng tme n 1 robablstc choces when D s roortonal to the odds aganst. 3. Psychohyscs of nformaton entroy Let us suose that the (average) nter-tral tme n robablstc choces s t. Then, the (average) watng tme n robablstc choces for a sngle outcome s 1 D=( )t. (2) Therefore, sychologcal watng tme s (from equaton 1), 1 (D)=a ln (1+bt ( )). (3)

5 Let us now consder the case wth multle N outcomes (x, ) where x, are the magntude of the th outcome and the robablty of obtanng the outcome x ( s an nteger [ 1, N] ). Let us denote 1 (D) =a ln (1+b t ( )), (4) where the suffx ndcates each outcome. In ths case, the average of antcatory watng tme for N outcomes s N H= 1 ( D). (5) By nsertng equaton 4 nto 5, we obtan H= N 1 1 a ln[ 1 bt ( )]. (6) In the secal case for equaton 6 n whch a = b t =1, we obtan N Hs= ln, 1 whch s (conventonal) Shannon nformaton entroy (Shannon, 1948). Therefore, t s demonstrated that the conventonal Shannon entroy s a secal case for the generalzed sychohyscal entroy (equaton 6). 4. Conclusons and mlcatons for neuroeconomcs and econohyscs Decson under robablstc uncertanty (rsk) has been attractng strong attenton n neuroeconomcs (Paulus et al., 2006; Weber and Huettel, 2008; Hsu et al., 2009; Chrstooulos et al., 2009; Ganott et al., 2009; Levy et al., 2010). However, neuroeconomc studes on rsk have not ad enough attenton to nformaton theory, whle manstream neuroscence has been strongly connected to nformaton theory (e.g., Nemenman, 2008). A recent econohyscal study ntroduced a model based on Tsalls entroy (Takahash, 2009). Therefore, future studes n econohyscs and

6 neuroeconomcs should more extensvely utlze nformaton theory by adotng the resent sychohyscal theory of nformaton entroy, because sychohyscal theory has also been wdely exloted n theoretcal neuroscence (Dayan and Abbott, 2001). These lnes of studes may hel understand neuroeconomc and neurochemcal bases of rsky decson makng by neurosychatrc atents (Paulus, 2007) such as bolar dsorder atents (Chandler et al., 2009).

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