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. 2009 Oct 20;4(10):e7362. doi: 10.1371/journal.pone.0007362

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Kurth-Nelson, Redish.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

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(A) Single-step state-space used for B–G. (B,E) When the model consists of a set of exponential discounters with γ drawn uniformly from (0,1), the measured discounting closely fits the hyperbolic function. (C,F) When the model consists of a single exponential discounter with , the measured discounting closely fits the function (exponential). (D,G) When the model consists of a single hyperbolic discounter, the measured discounting closely fits the function (hyperbolic). (H) Chained state-space used for I–N. (I) If values are distributed so each exponential discounter has its own value representation, the result is hyperbolic discounting over a chained state space. (J,M) A single exponential discounter behaves as in the single-step state space, because multiplying exponentials gives an exponential. (K,N) A single hyperbolic discounter now behaves as an exponential discounter with , because each step is discounted by , where . (L) Likewise, a set of exponential discounters with shared value representation behave as an exponential discounter with , for the same reason.

Inline graphic — (A) Single-step state-space used for B–G. (B,E) When the model consists of a set of exponential discounters with γ drawn uniformly from (0,1), the measured discounting closely fits the hyperbolic function. (C,F) When the model consists of a single exponential discounter with , the measured discounting closely fits the function (exponential). (D,G) When the model consists of a single hyperbolic discounter, the measured discounting closely fits the function (hyperbolic). (H) Chained state-space used for I–N. (I) If values are distributed so each exponential discounter has its own value representation, the result is hyperbolic discounting over a chained state space. (J,M) A single exponential discounter behaves as in the single-step state space, because multiplying exponentials gives an exponential. (K,N) A single hyperbolic discounter now behaves as an exponential discounter with , because each step is discounted by , where . (L) Likewise, a set of exponential discounters with shared value representation behave as an exponential discounter with , for the same reason.