Dynamic decision-making when ambiguity attitudes depend on exogenous events

Abstract

The aim of this paper is to propose a preferences representation model where ambiguity attitudes can be exogenous events or past experience-dependent. We adapt the Recursive Smooth Ambiguity model proposed by Klibanoff (Journal of Economic Theory 144:930-976, 2009) by introducing past experience described by a sequence of neutral events occurring up to the moment of the decision. These neutral events do not provide any information on the true process, but are likely to strengthen or weaken the decision-maker’s ambiguity aversion degree by modifying emotions. Our model can explain some observed behaviors and market inefficiencies. We propose two illustrations. First, we provide a behavioral explanation for the decrease in influenza vaccination in France that followed the H1N1 crisis in 2009–2010. Second, we contribute to the broad literature on the annuity puzzle by introducing the impact of emotions on ambiguity attitudes.

Appendices

Appendices

1.1 Standard assumptions in Klibanoff et al. (2009)

Assumption 1

(Weak Order) For all $s^t \in \mathcal {S}^t$, $\succeq _{s^t}$ on $\mathcal {F}$ is complete and transitive.

Assumption 2

(Monotonicity) Given any $f,g \in \mathcal {F}$, if $f(s) \geqslant g(s)$ for all $s\geqslant s^t$, then $f \succeq _{s^t} g$. If, in addition, f and g are deterministic and $f(\tau ) > g(\tau )$ for at least some $\tau \geqslant t$, then $f \succ _{s^t} g$.

Assumption 3

(Archimedean) For all $s^t \in \mathcal {S}^t$, $\forall f \in \mathcal {F}$ and $\forall c_1,c_2 \in \mathcal {C}$, if $c_1 \succ _{s^t} f \succ _{s^t} c_2$, then there exist $\alpha , \beta \in ]0;1[$ such that $\alpha c_1+(1-\alpha ) c_2 \succ _{s^t} f \succ _{s^t} \beta c_1+(1-\beta ) c_2$.

Assumption 4

(Dynamic Consistency) $\forall s^t \in \mathcal {S}^t$, $\forall f,g \in \mathcal {F}$, if $f(s^t)=g(s^t)$, then $f \succ _{s^t,(x_{t+1},e_{t+1})}g,$ $\forall (x_{t+1},e_{t+1}) \Rightarrow f \succ _{s^t}g$

Assumption 5

(Discounting) For each deterministic continuation plan $d_{s^t} \in \mathcal {D}_{s^t}$, there exists a plan $f \in \mathcal {F}$ with $f_{\mid _{s^t}}=d_{s^t}$ and such that $f \sim _{s^t}\{d_{s^t},1\}$. At each node $s^t \in \mathcal {S}^t$, $\succ _{s^t}$ on $\mathcal {P}_t$ is represented by the expectation of a von Neumann–Morgenstern utility index $U_{s^t}:\mathcal {D}_{s^t} \rightarrow \mathbb {R}$ which has the form

$$\begin{aligned} U_{s^t}(d_{s^t})=\sum \limits _{\tau \ge t}\beta _{s^t}^{\tau -t}u_{s^t}\left( d_{s^t}(\tau )\right) \end{aligned}$$

Assumption 6

(Invariance) $\forall t,t^\prime \in T\cup \{0\}$, $\forall p \in [0,1]$, and $\forall c_i^{\prime },c_i^{\prime \prime },c_i, k_i^{\prime },k_i^{\prime \prime },k_i \in \mathcal {C}$ with $i \in \{1,2\}$:

$$\begin{aligned}{} & {} \left( (c_1^{\prime },c_1^{\prime \prime },c_1)_t,p;(k_1^{\prime },k_1^{\prime \prime },k_1)_t,1-p\right) \succeq _{s^t} \left( (c_2^{\prime },c_2^{\prime \prime },c_2)_t,p;(k_2^{\prime },k_2^{\prime \prime },k_2)_t,1-p\right) \\{} & {} \quad \Leftrightarrow \left( (c_1^{\prime },c_1^{\prime \prime },c_1)_{t^\prime },p;(k_1^{\prime },k_1^{\prime \prime },k_1)_{t^\prime },1-p\right) \succeq _{s^t} \left( (c_2^{\prime },c_2^{\prime \prime },c_2)_{t^\prime },p;(k_2^{\prime },k_2^{\prime \prime },k_2)_{t^\prime },1-p\right) \end{aligned}$$

Assumption 7

(SEU on second order acts) There exists a unique (additive) probability $\mu _{s^t}:2^\Theta \rightarrow [o,1]$ and a continuous, strictly increasing $v: \mathcal {C} \rightarrow \mathbb {R}$ such that, for all $\mathfrak {f}, \mathfrak {g} \in \mathfrak {F}$,

$$\begin{aligned} \mathfrak {f} \succeq ^2_{s^t} \mathfrak {g} \Longleftrightarrow \int _\Theta v(\mathfrak {f}(\theta )d\mu _{s^t}\geqslant \int _\Theta v(\mathfrak {g}(\theta )d\mu _{s^t}. \end{aligned}$$

Moreover, v is unique up to positive affine transformations, provided $0< \mu _{s^t}(J) < 1$ for some $J \subseteq \Theta$.

Assumption 8

(Consistency with associated second order acts) Given $f_{s^t},g_{s^t} \in \mathcal {F}_{s^t}^*$ and associated $f^2,g^2 \in \mathfrak {F}$, if $f_{s^t}(s^t)=g_{s^t}(s^t)$ then $f \succ _{s^t}g \Leftrightarrow f ^2\succ _{s^t}^2\,g^2$ for all $f,g \in \mathcal {F}$ such that $f_{\mid _{s^t}}=f_{s^t}$ and $g_{\mid _{s^t}}=g_{s^t}$.