A Rational Logit Dynamic for Decision-Making Under Uncertainty: Well-Posedness, Vanishing-Noise Limit, and Numerical Approximation

Abstract

The classical logit dynamic on a continuous action space for decision-making under uncertainty is generalized to the dynamic where the exponential function for the softmax part has been replaced by a rational one that includes the former as a special case. We call the new dynamic as the rational logit dynamic. The use of the rational logit function implies that the uncertainties have a longer tail than that assumed in the classical one. We show that the rational logit dynamic admits a unique measure-valued solution and the solution can be approximated using a finite difference discretization. We also show that the vanishing-noise limit of the rational logit dynamic exists and is different from the best-response one, demonstrating that influences of the uncertainty tail persist in the rational logit dynamic. We finally apply the rational logit dynamic to a unique fishing competition data that has been recently acquired by the authors.

Appendix

Proof of Proposition 2.

The key estimate in the proof of Proposition 2 is the following; given $\kappa \in \left( {0,1} \right]$, $\mu \in {\rm{\mathfrak{M}}}$, $\eta \in \left( {0,\eta_0 } \right]$ with a constant $\eta_0 > 0$, by the boundedness of $U$ it follows that

$$ \begin{aligned} & \left| {\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu } \right)} \right) - U\left( {x;\mu } \right)^{1/\kappa } } \right| \\ & = \left| {\left( {U\left( {x;\mu } \right)/2 + \sqrt {{U\left( {x;\mu } \right)^2 /4 + \eta^2 \kappa^{ - 2} /4}} } \right)^{\frac{1}{\kappa }} - U\left( {x;\mu } \right)^{1/\kappa } } \right| \le C\eta \\ \end{aligned} ,$$

(23)

where $C > 0$ is a constant independent from $\eta$. This independence is crucial in our context, particularly when considering the limit $\eta \to + 0$ (see (27)–(28) below).

Now, for any $t \in \left( {0,T} \right]$ and $A \in {\rm{\mathfrak{B}}}$, it follows that

$$ \begin{aligned} & \frac{{\text{d}}}{{{\text{d}}t}}\left( {\mu_0 \left( {t,A} \right) - \mu_\eta \left( {t,A} \right)} \right) \\ & = \frac{{\int_A {U\left( {y;\mu_0 } \right)^{1/\kappa } {\text{d}}y} }}{{\int_\Omega {U\left( {y;\mu_0 } \right)^{1/\kappa } {\text{d}}y} }} - \frac{{\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_A {e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right){\text{d}}y} }}{{\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_\Omega {e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right){\text{d}}y} }} - \left( {\mu_0 \left( {t,A} \right) - \mu_\eta \left( {t,A} \right)} \right) \\ \end{aligned} .$$

(24)

We have the following estimate with a constant $C_1 > 0$ independent from $\eta ,\mu_0 ,\mu_\eta$:

$$ \begin{aligned} & \int_\Omega {U\left( {y;\mu_0 } \right)^{1/\kappa } {\text{d}}y\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_\Omega {e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right){\text{d}}y} } \\ & \ge \int_\Omega {U\left( {y;\mu_0 } \right)^{1/\kappa } {\text{d}}y\int_\Omega {U\left( {y;\mu_\eta } \right)^{1/\kappa } {\text{d}}y} } > C_1 > 0 \\ \end{aligned} .$$

(25)

We also have the following estimate at each $y \in \Omega$:

$$ \begin{aligned} & \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_\Omega {e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_\eta } \right)} \right){\text{d}}x} U\left( {y;\mu_0 } \right)^{1/\kappa } - \int_\Omega {U\left( {x;\mu_0 } \right)^{1/\kappa } {\text{d}}x\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } } e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right) \\ & = \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_\Omega {e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_\eta } \right)} \right){\text{d}}x} U\left( {y;\mu_0 } \right)^{1/\kappa } - \int_\Omega {U\left( {x;\mu_0 } \right)^{1/\kappa } {\text{d}}x} U\left( {y;\mu_0 } \right)^{1/\kappa } \\ & + \int_\Omega {U\left( {x;\mu_0 } \right)^{1/\kappa } {\text{d}}x} U\left( {y;\mu_0 } \right) - \int_\Omega {U\left( {x;\mu_0 } \right)^{1/\kappa } {\text{d}}x} \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right) \\ & = \int_\Omega {\left( {\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_\eta } \right)} \right) - U\left( {x;\mu_0 } \right)^{1/\kappa } } \right){\text{d}}x} U\left( {y;\mu_0 } \right)^{1/\kappa } \\ & + \int_\Omega {U\left( {x;\mu_0 } \right)^{1/\kappa } {\text{d}}x} \left\{ {U\left( {y;\mu_\eta } \right) - \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right)} \right\} \\ & \le C_2 \left\{ \begin{gathered} \int_\Omega {\left| {\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_\eta } \right)} \right) - U\left( {x;\mu_0 } \right)^{1/\kappa } } \right|{\text{d}}x} \hfill \\ + \left| {U\left( {y;\mu_\eta } \right) - \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right)} \right| \hfill \\ \end{gathered} \right\} \\ & \le C_2 \left\{ \begin{gathered} \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_\Omega {\left| {e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_\eta } \right)} \right) - e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_0 } \right)} \right)} \right|{\text{d}}x} \hfill \\ + \int_\Omega {\left| {\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_0 } \right)} \right) - U\left( {x;\mu_0 } \right)^{1/\kappa } } \right|{\text{d}}x} \hfill \\ + \left| {U\left( {y;\mu_\eta } \right) - \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right)} \right| \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

(26)

with a constant $C_2 > 0$ independent from $\mu_0 ,\mu_\eta$. By (23) and $U\left( {x;\mu_\eta } \right) > 0$, in (26) we obtain

$$ \left| {U\left( {y;\mu_\eta } \right) - \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {y;\mu_\eta } \right)} \right)} \right| \le C\eta ,$$

(27)

$$ \int_\Omega {\left| {\left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_0 } \right)} \right) - U\left( {x;\mu_0 } \right)^{1/\kappa } } \right|{\text{d}}x} \le C\eta ,$$

(28)

and

$$ \left( {\eta /\left( {2\kappa } \right)} \right)^{1/\kappa } \int_\Omega {\left| {e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_\eta } \right)} \right) - e_\kappa \left( {\eta^{ - 1} U\left( {x;\mu_0 } \right)} \right)} \right|{\text{d}}x} \le C_3 \left( {\eta_0 } \right)\left\| {\mu_\eta - \mu_0 } \right\| $$

(29)

with a constant $C_3 \left( {\eta_0 } \right) > 0$ independent from $\eta ,\mu_0 ,\mu_\eta$. By (24)–(29), we obtain

$$ \frac{{\text{d}}}{{{\text{d}}t}}\left( {\mu_0 \left( {t,A} \right) - \mu_\eta \left( {t,A} \right)} \right) \le C_4 \left( {\eta_0 } \right)\left( {\eta + \left\| {\mu_\eta \left( {t, \cdot } \right) - \mu_0 \left( {t, \cdot } \right)} \right\|} \right) $$

(30)

with a constant $C_4 \left( {\eta_0 } \right) > 0$ independent from $\eta ,\mu_0 ,\mu_\eta$. Hence, by integrating (30) for $\left( {0,t} \right)$, and taking the variational norm yields

$$ \left\| {\mu_\eta \left( {t, \cdot } \right) - \mu_0 \left( {t, \cdot } \right)} \right\| \le 2\int_0^t {C_4 \left( {\eta_0 } \right)\left( {\eta + \left\| {\mu_\eta \left( {s, \cdot } \right) - \mu_0 \left( {s, \cdot } \right)} \right\|} \right){\text{d}}s} .$$

(31)

Applying a classical Gronwall lemma to (31) yields

$$ \left\| {\mu_\eta \left( {t, \cdot } \right) - \mu_0 \left( {t, \cdot } \right)} \right\| \le C_5 \left( {T,\eta_0 } \right)\eta \exp \left( {C_5 \left( {\eta_0 } \right)T} \right) $$

(32)

with a constant $C_5 \left( {T,\eta_0 } \right) > 0$ depending on $\eta_0 ,T$ but not on $\mu_0 ,\mu_\eta$. The conclusion (10) directly follows from (32).

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Collected Data

The number of catches of the fish P. altivelis in each pair in each Toami competition is summarized in the ascending order in Table 2. We consider that this kind of fish catch data is useful because it can be utilized not only for our study but also for other studies by other researchers. Members of each pair were anonymized. The Toami competition has been basically held by HRFC in each summer. It was not held in 2020, 2021, 2022 due to the outbreak of the coronavirus disease 2019. The data before 2015 may exist but was not available for us.

Table 2. The number of catches of the fish P. altivelis in each group in each year.