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title: Attention-Adjusted Discounting

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# Attention-Adjusted Discounting

Initially, the decision maker is time-consistent. After learning about the intertemporal choice problem, she will pay more attention to the periods with more attractive rewards, though deviating from the initial attention allocation triggers a cognitive cost. I show that such an attention-adjusted discounting process can explain a lot of behavioral biases, including common difference effect, magnitude effect, timing risk aversion, interval additivity anomalies, planning fallacy, and description-experience gap.  





#### 1. Introduction

Consider a decision maker receiving a sequence of rewards $\textbf{x}=[x_0,x_1,...,x_T]$. Her instantaneous utility in period $t$ is $u(x_t)$, which is strictly increasing with $x_t$, the reward in period $t$. The total utility she gains from $\textbf{x}$ is computed by valuing the reward in each period individually, and multiplying the utility of each by a certain weight, then adding up the results:
$$
U(\textbf{w};\textbf{x}) = \sum_{t=0}^T w_t u(x_t)
$$
where $\textbf{w} = [w_0,w_1,...,w_T]'$ is the decision weights, $w_i \geq 0$ for all $i\in \{1,..,T\}$ and $\sum_{t=0}^T w_t = 1$.  The idea that decision makers value an option using the sum of weighted utilities is widely adopted in behavioral economic theories, such as prospect theory and salience theory. In intertemporal choice setting, $\textbf{w}$ is decided by some discounting function. When the decision maker is time-consistent (i.e. the discounting function takes an exponential form) and the discounting factor is assumed to be $\delta$, the weights in $\textbf{w}$ are  $\{1/\iota,\delta/\iota,\delta^2/\iota,...,\delta^T/\iota\}$ with $\iota =\sum_{t=0}^T \delta^t$. 



Suppose the decision maker faces an intertemporal choice question ``receive $x^s$ now or receive $x^l$ in $T$ periods", $0<x^s<x^l$, $T>0$, and $u(0)=0$. By choosing to receive $x^s$ now, the decision maker can gain utility $u(x^s)$; by choosing to receive $x^l$ later, she can gain  $w_0 \cdot 0 + ... + w_{T-1} \cdot 0 + w_T\cdot u(x^l) = w_T u(x^l)$. I assume the decision maker is initially time-consistent, and for simplicity, the discounting factor $\delta = 1$. 



When considering the latter option, the decision maker's initial decision weights are $\textbf{w}^0 = [1/T,1/T,...,1/T]$, but she can reallocate the weights to maximize the total utility $w_T u(x^l)$. If the reallocation of weights is costless, she will assign full weight to period $T$ and zero weight to the other periods. However, processing the information about rewards and adjusting decision weights will trigger a cognitive cost, and when the cost is large, $\textbf{w}$ cannot deviate from $\textbf{w}^0$ by too much. Following the rational inattention literature, I define the cognitive cost function with KL divergence, a measure of how much a distribution differs from another. Therefore, the decision maker's objective is
$$
\max_{\textbf{w}} \; U(\textbf{w};\textbf{x})-\lambda\cdot D_{KL}(\textbf{w}||\textbf{w}^0)
$$
where $\lambda$ is a parameter, and $D_{KL}(\textbf{w}||\textbf{w}^0)$ is defined by 
$$
D_{KL}(\textbf{w}||\textbf{w}^0)= \sum_{t=0}^T w_i \log(\frac{w_i}{w_i^0})
$$


The solution to this optimization problem should meet the condition $w_t \propto \delta^t \exp\{u(x_t)/\lambda\}$. Thus, while considering ``receive  $x^l$ in $T$ periods", the decision maker adjusts her weight for period $T$ to 
$$
w_T = \frac{1}{1+k\cdot T}
$$
where $k=e^{-u(x^l)/\lambda}$. This discounting function takes a similar form with hyperbolic discounting. 



Let me show that the attention-adjusted discounting factors satisfy both the magnitude effect and common difference effect in a numerical example. First, set $T = 9$, $x^s = 12$, $x^l = 16$, $u(x) = \lambda \cdot x^{0.5}$. If the decision maker chooses to receive $x^s$ now, she gains $u(x^s) = 3.46\lambda$; if she chooses to receive $x^l$ later, she gains $w_Tu(x_l) = 4\lambda/(1+9e^{-4})=3.43\lambda$. Hence, in this setting she tends to receive $x_s$ now. To show magnitude effect, we can increase $x^s$ and $x^l$ by a same amount, say 9. Then the utility of receiving reward early will be $4.58\lambda$, and that of receiving late will be $4.71\lambda$. Thus, she is more likely to wait with the magnitude of rewards increasing. To show common difference effect, we can delay the reward of each option by a same interval, say 3 periods. Then the utility of receiving reward early shrinks to $3.17\lambda$ and that of receiving late shrinks to $3.28\lambda$, i.e. adding a common delay to each option makes her more likely to wait.



#### 2. Model Framework

##### 2.1 Rational Inattention

Consider a decision maker who wants to evaluate a reward sequence, which depends on the state of world $s$. The time length of this sequence is $T$. Let $t$ denote a time points, $t\in [0,T]$. The decision maker samples from those points and aggregate the utilities she can gain in each time point in her sample, to construct a value representation of $s$. Her objective is to find a sampling strategy $f(t,s)$, which denotes the choice probability for each time point under each certain state. I set the probability of $s$ occurring  and the unconditional probability of choosing $t$ to be $p(s)$, $p(t)$ . By the insight of motivated beliefs, I assume she wants to maximize her total utility through $f$; thus, time points with greater rewards should be sampled more frequently. However, processing reward information is costly. Following the rational inattention literature, the optimization problem for the decision maker is 
$$
\begin{split}
& \max_f\; \int u(t,s)f(t,s)dtds - \lambda I(t;s) \\
& s.t. \; \int u(t,s)dt = p(s),\, \forall s
\end{split}
$$
where $\lambda$ is a fixed parameter and $I(t;s)$ denotes Shannon mutual information. 
$$
I(t;s)=E_s[D_{KL}(f(t,s)||p(t))]=
\int f(t,s) \log\left(\frac{f(t,s)}{p(t)p(s)}\right)dtds
$$
The solution is
$$
f(t|s) =\frac{p(t)e^{u(t,s)/\lambda}}{\int_z p(z) e^{u(z,s)/\lambda}}
$$

The authors of rational inattention map this to a discrete choice setting. I also consider discrete time here: $t \in\{0,1,...,T\}$. Therefore, the process can be viewed as the decision maker use a weighted sampling strategy to evaluate a time sequence that is evenly split into multiple periods and rewards arrive at the beginning of each period. The initial decision weights for each time period $w_t^0 \equiv p(t)$, and after processing the information, she adjusts the weights to $w_t \equiv f(t|s)$.

When facing a intertemporal choice, the decision maker evaluate each option by: (1) bracketing the time horizon; (2) processing the information about rewards within the horizon; (3) reallocating the decision weights to each period within the horizon, with an optimal sampling strategy. Then, she compare the options and make the choice. A lot of literature has documented that human make decisions by sampling approach. 

##### 2.2 Description vs Experience

There are two experimental paradigms: SS vs LL, time budget. In the former, the participants are informed when and how many rewards will arrive, which is certain. In the latter, the participants decide the timing and volume of rewards by themselves, which can be variable; thus, I set that for $t>0$, $u_t = v(x_t) + \epsilon_t$, $\epsilon_t \sim N(0,\lambda\sigma_t)$.

Throughout the paper, I assume $w_t^0 = \delta^t$, where $\delta$ is the exponential discounting factor, $\delta \in (0,1]$

##### 2.3 SS vs LL

Given that the reward is pre-determined, the state $s$ is certain (or $\sigma_t =0$).

Reward arrives only at period $T$. For all $t<T$, $x_t=0$.

The optimal weight for $T$ is
$$
w_T = \frac{1}{1+G(T)\cdot e^{-v(x_T)/\lambda}}
$$
where
$$
G(T) = \left\{\begin{split}
& T &, \delta=1\\
& \frac{1}{1-\delta}(\delta^{-T}-1) &,0<\delta<1
\end{split}\right.
$$



#### 3. Theoretical Implications

##### 3.1 Common Difference Effect

The decision-maker prefers a small reward arriving at $t_1$, than a large reward arriving at $t_2$ ($t_2>t_1$)

However, when the same large reward arrives at $t_2+\Delta t$ and the same small reward arrives at $t_1+\Delta t$, the preference is reversed ($\Delta t>0$)

Suppose the two options (A and B) delivers the equal utility to the decision-maker: *A. receive $x_1$ in period $t_1$; B. receive $x_2$ in period $t_2$*.

For simplicity, we define $v\equiv v(x_1)/\lambda$ and $\alpha\equiv v(x_2)/v(x_1)=w_{t_1}/w_{t_2}$

Note $t_2>t_1$, thus we have $x_1<x_2$, that is $\alpha>1$

If common difference effect holds, then there exists $\Delta t$ such that $w_{t_1+\Delta t}/w_{t_2+\Delta t}<\alpha$

***Proposition 1: decision makers with attention-adjusted discounting perform common difference effect*** 

> Proof:
>
> From the definition of $\alpha$,
> $$
> \alpha \cdot(1+G(t_1)\cdot e^{-u}) = 1+G(t_2)\cdot e^{-\alpha u} \tag{1}
> $$
> Set up a function
> $$
> f(\Delta t) = \alpha\cdot(1+G(t_1+\Delta t)\cdot e^{-u}) - (1+G(t_2+\Delta t)\cdot e^{-\alpha u})
> $$
> We know that $f(0)=0$
>
> common difference effect implies $f(\Delta t)>0$ when $\Delta t>0$
>
> if $f'(\Delta t) >0$ then
> $$
> \frac{G'(t_2+\Delta t)}{G'(t_1+\Delta t)} < \alpha e^{(\alpha-1)u}
> $$
> when $\delta=1$, the right hand is 1, the common difference effect always holds
>
> when $0<\delta<1$, rewrite $f(\Delta t)>0$:
> $$
> \delta^{-\Delta t}(\delta^{-t_1}\alpha e^{-u}-\delta^{-t_2}e^{-\alpha u}) >
> \alpha e^{-u}- e^{-\alpha u} -(1-\delta)(\alpha-1)$$
> from **eq. (1)** we know the the left hand equals $\delta^{-t_1}\alpha e^{-u}-\delta^{-t_2}e^{-\alpha u}$. Therefore, the inequality always holds when $\Delta t>0$



##### 3.2 Magnitude Effect

People are getting more patient when increasing both the small and large rewards by the same magnitude. 

> Noor & Takeoka (2022) provides another account, also based on finding the optimal discounting factors. Attention-adjusted discounting model admits a *General Costly Empathy (CE)* representation in their paper, but its cognitive cost function is different.
>
> In their paper, for discounting factor $d_t$, there exists $0<\underline{d}_t<\overline{d}_t<1$ such that ***CE*** cognitive cost is 0 when $d_t \in (0,\underline{d}_t]$, is strictly increasing when $d_t\in (\underline{d}_t,\overline{d}_t]$, and is $\infty$ when $d_t \in (\overline{d}_t,1]$



Define $V(t,x_t)=w_t(x_t)\cdot v(x_t)$. Again, assume 
$$
V(t_1,x_1)=V(t_2,x_2)\equiv 1+b 
\tag{2}
$$
Given $t_1$, $t_2$, we can set $x_1$ that satisfies **eq. (2)** as a function of $x_2$.

By definition, magnitude effect is
$$
\frac{\partial }{\partial x_2}(\frac{x_1}{x_2})>0
\Longrightarrow 
\frac{\partial x_1}{\partial V}\frac{\partial V}{\partial x_2}x_2 -x_1>0
\Longrightarrow
\frac{\partial V}{\partial x_2}x_2>\frac{\partial V}{\partial x_1}x_1
$$


***Proposition 2: decision makers with attention-adjusted discounting perform magnitude effect if***
$$
RRA_v - b\frac{\partial E_{vx}}{\partial v} <1
$$
***where RRA is relative risk aversion coefficient of function $v(x)$, and $E_{vx} = v'(x)\frac{x}{v(x)}$ is the elasticity of $v$ to $x$.***

> Proof:
>
> Given that $G(t_1)e^{-v(x_1)/\lambda}=v(x_1)/(1+b)-1$,
> $$
> \frac{\partial V}{\partial x_1}x_1
> =(1+b)(v(x_1)+b) \frac{v'(x_1)}{v(x_1)}x_1
> $$
> Observing that $x_2\cdot \partial V/\partial x_2$ admits a similar representation, we can define a function $\psi(x)$:
> $$
> \psi(x) = (v(x)+b) \frac{v'(x)}{v(x)}x=xv'(x)+bE_{vx}
> $$
>
> (Lama 1) for any $x_1<x_2<\infty$, there always exists $t_1<t_2<\infty$ that makes **eq. (2)** holds.
>
> Thus, if $\psi'(x)>0$, then $\psi(x_2)>\psi(x_1)$ for any , $x_1<x_2<\infty$. The inequality can be derived from $\psi'(x)>0$. 

**Corollary 1: If $v(x)$ admits a CRRA representation, then the decision maker perform magnitude effect when CRRA<1.**



**3.3 Risk Aversion over Time Lotteries**

Generally, people prefer a reward arriving at a sure time $t_M$, rather than the same reward arriving at $t_S$ with probability $p$ and at $t_L$ with probability $1-p$, where $t_M = t_S \cdot p + t_L \cdot (1-p)$.

When short delay $p$ is smaller, the time lottery option is more attractive compared with the sure time option.

A decision-maker is risk averse over time lotteries if and only if 
$$
V(t_M,x)\geq V(t_S,x)\cdot p +V(t_L,x) \cdot (1-p)
$$
which implies $\frac{\partial^2 V}{\partial t^2} \leq 0$

It can be derived that:
$$
t \leq \ln \left(\frac{1-z}{z}\right)/\ln(\frac{1}{\delta}) \tag{3}
$$
where $\delta \in (0,1)$ and $z\equiv 1/(1-\delta) \cdot e^{-v(x)/\lambda} \in (0,\frac{1}{2})$

**People are timing risk averse when delay is short, and are timing risk seeking when delay is long.**

**When $z$ gets smaller (or $x$ gets larger), the bound for $t$ gets relaxed.**



##### 3.4 Interval Additivity Anomalies 

Super-additivity:

$V(t,x-\Delta x)\leq V(t+\tau,x)$

however, $V(t,x-2\Delta x) \geq V(t+2\tau,x)$

Sub-additivity:

$V(t-\tau,x)\geq V(t,x+\Delta x)$

however, $V(t-2\tau,x) \leq V(t,x+2\Delta x)$

Let $h$ denote the difference of rewards that satisfies $V(t,x-h)=V(t+\tau,x)$ and is a function of $\tau$

Supper-additivity implies that when increasing $\tau$ by a certain ratio, $h$ needs increasing by a larger ratio to make the equation hold.
$$
\frac{\partial^2 h}{\partial \tau^2}>0 
\Longrightarrow 
\frac{\partial^2 V}{\partial \tau^2} /\frac{\partial V}{\partial h}>0
\Longrightarrow
\frac{\partial^2 V}{\partial \tau^2} < 0
$$
From **ineq. (3)** we know that ***people perform super-additivity when $t$ is small. When $x$ is larger, people are more likely to perform super-additivity.***

Similarly, people perform sub-additivity when $t$ is large. When $x$ is smaller, people are more likely to perform super-additivity.



##### 3.5 Planning Fallacy

Suppose the decision maker has a total reward $m$ and needs to allocate it over $T$ periods. 
$$
x_0+x_1+...+x_T=m
$$
The decision maker optimally set $x_1$ at time period 0. Planning fallacy implies that at time period 1 she tends to increase $x_1$. To make her performs planning fallacy, we need to introduce uncertainty. Without uncertainty, the decision maker with attention-adjusted discounting factors is always time consistent. 

At time period $i$, the optimal decision weights are
$$
\textbf{w} = \arg \max_{\textbf{w}}\,\left\{\sum_{t=i}^TE_s[w_tu_t]-\lambda I(t;s)\right\}
$$
Given that $w_t(s) \propto \delta^t e^{u_t(s)/\lambda}$, we have $E_s[w_tu_t] \propto \delta^t \xi(x_t,\sigma_t)$, where 
$$
\xi(x_t,\sigma_t) \equiv (\frac{v(x_t)}{\lambda}+\sigma_t^2)\exp\left\{\frac{v(x_t)}{\lambda}+\frac{\sigma_t^2}{2}\right\}
$$

> Note that if $z \sim N(\mu,\sigma)$, then $E[ze^z] = (\mu+\sigma^2)\exp\{\mu+\sigma^2/2\}$

> $\xi(.)$ should be a concave function to $x_t$; therefore,
> $$
> v'(x_t)\left[1+\sigma_t^2+\frac{v(x_t)}{\lambda}\right]
> \exp\left\{\frac{v(x_t)}{\lambda}+\frac{\sigma_t^2}{2}\right\}
> $$
> is decreasing with $x_t$.

At period 0, the decision maker allocate her total rewards to solve the optimization problem
$$
\max_\textbf{x} \sum_{t=0}^T \delta^t\cdot\xi(x_t,\sigma_t)\quad s.t. \sum_{t=0}^T x_t = m
$$
by FOC, we have 
$$
\frac{\partial \xi}{\partial x_t} = \delta\frac{\partial \xi}{\partial x_{t+1}}
$$
i.e. 
$$
\frac{v'(x_t)}{v'(x_{t+1})} = \rho\delta
$$
where 
$$
\rho = \frac{1+\sigma_{t+1}+v(x_{t+1})/\lambda}{1+\sigma_t+v(x_t)/\lambda}
\exp\left\{ \frac{v(x_{t+1})-v(x_t)}{\lambda} + \frac{\sigma_{t+1} - \sigma_t}{2}\right\}
$$
When $\rho$ is constant with $t$, the decision maker performs no planning fallacy. When $\beta$ is (weakly) increasing with $t$ and there exists an interval $[0,\bar{t}]$ such that $\beta$ is strictly increasing with $t$, the decision maker performs planning fallacy and is present-biased.

For example, the classical $(\beta,\delta)$-preference admits such a representation:  
$$
\rho = \left\{\begin{split}
&\beta,\;t=0\\
& 1,\;t>0
\end{split}\right.
$$
where $\beta\in(0,1)$.

*Case 1: $\sigma_t=0$ when $t=0$, and $\sigma_t=\sigma$ ($\sigma>0$) when $t>0$.*

This case is identical to $(\beta,\delta)$-preference.



*Case 2: $\sigma_t = t\cdot\sigma$*

The decision maker performs future bias. 

(proof by contradiction)



#### 4. Empirical Results








