<p>In this case,</p>
<blockquote>
<p>the signal is accurate [...] with probability <span class="math-container">$q &gt; 1/2$</span></p>
</blockquote>
<p>seems to mean that when the signal <span class="math-container">$s$</span> is observed, the probability of <span class="math-container">$c_2 = s$</span> rises to <span class="math-container">$q$</span> from <span class="math-container">$1/2$</span>, while the probability of <span class="math-container">$c_2 = 1-s$</span> (which is the only other possibility) sinks to <span class="math-container">$1-q$</span>.</p>
<hr />
<p>In information theory there are possible &quot;states&quot; or &quot;states of the world&quot;.</p>
<p>It is possible an agent does know the actual state of the world. A frequent modelling assumption is that they have some belief about the state of the world, which is a probability distribution over the possible states.</p>
<p>The agent may observe/receive a signal which changes their belief, it adds information. It is possible that the signal does not resolve the uncertainty completely, i.e. there are still several possible states of the world.</p>
<hr />
<p>A toy example:
<br> Suppose you flip a fair coin, thus randomizing between heads (H) and tails (T) with 50-50%. You look away and do not observe the outcome, but a friend of yours does. After they make their observation, they will tell you what they saw. However, there is a 1% chance that you mishear what they are saying.</p>
<p>Your friend's signal - to be precise, what you hear of it - is 99% accurate.</p>

CC BY-SA 4.0
2023-04-08T16:22:54.317Z
Thanks! This essentially means that updating probability is conditional on the signal that we receive, hence, we have to use Bayesian updation. Furthermore, what if the state of the world is not perfectly observed by the entity that supplies us with the signal? By 'not perfectly observed' I mean that the supplier of the signal has her own probability distribution over the outcome after observing the state of the world. For instance, in the coin toss, what if the other person tells us that she is, say, 70% sure that it is a head? What would 'accuracy' mean in that case?
Ramandeep
CC BY-SA 4.0
2023-04-09T07:37:21.920Z
The same thing. Imperfect observation is a possible reason for inaccuracy. Above toy example could have been worded as the friend being unsure instead of the agent misunderstanding. In the end it is about the signal and its probability of overlapping with the actual state.
Giskard
CC BY-SA 4.0
2023-04-08T09:34:38.583Z
2023-04-08T09:34:38.583Z
Giskard
<p>I am trying to understand a line in the paper titled &quot;Reference Dependent Consumption Plans&quot; written by Matthew Rabin and Botond Koszegi, and published in the American Economic Review, Vol 99, No. 3, June 2009.</p>
<p>The model in the paper is about incorporating reference dependent preferences in a dynamic setting with <span class="math-container">$n$</span> periods. In this case, there are two periods, wherein she receives a signal in period 1 and consumes in periods 2. There are two equi-probable (i.e. <span class="math-container">$Pr(c_2=0)=Pr(c_2=1)=1/2$</span>) possible consumption, <span class="math-container">$c_2=0$</span> or <span class="math-container">$c_2=1$</span>, over which the decision maker has no control. On page 917 of this article, along with the information provided, it reads,</p>
<p><span class="math-container">$$\text{"She may, however, receive information about }c_2\text{ in period 1. Specifically, she may observe a signal }s \in \{0,1\},\text{ where the signal is accurate $(s=c_2$) with probability }q &gt; 1/2."$$</span></p>
<p>What is the meaning of &quot;<em>signal is accurate</em>&quot; in this case?</p>

<definition><behavioral-economics><signaling>
'Accuracy' of a Signal
Ramandeep
# 'Accuracy' of a Signal

(Asked by: username_0 on 2023-04-08)

I am trying to understand a line in the paper titled "Reference Dependent Consumption Plans" written by Matthew Rabin and Botond Koszegi, and published in the American Economic Review, Vol 99, No. 3, June 2009.

The model in the paper is about incorporating reference dependent preferences in a dynamic setting with $n$ periods. In this case, there are two periods, wherein she receives a signal in period 1 and consumes in periods 2. There are two equi-probable (i.e. $Pr(c\_2=0)=Pr(c\_2=1)=1/2$) possible consumption, $c\_2=0$ or $c\_2=1$, over which the decision maker has no control. On page 917 of this article, along with the information provided, it reads,

$$\text{"She may, however, receive information about }c\_2\text{ in period 1. Specifically, she may observe a signal }s \in \{0,1\},\text{ where the signal is accurate $(s=c\_2$) with probability }q > 1/2."$$

What is the meaning of "*signal is accurate*" in this case?

**Tags:** definition, behavioral-economics, signaling


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## 1 answers

### Answer by username_1 on 2023-04-08. Score: 1

In this case,

> the signal is accurate [...] with probability $q > 1/2$

seems to mean that when the signal $s$ is observed, the probability of $c\_2 = s$ rises to $q$ from $1/2$, while the probability of $c\_2 = 1-s$ (which is the only other possibility) sinks to $1-q$.

---

In information theory there are possible "states" or "states of the world".

It is possible an agent does know the actual state of the world. A frequent modelling assumption is that they have some belief about the state of the world, which is a probability distribution over the possible states.

The agent may observe/receive a signal which changes their belief, it adds information. It is possible that the signal does not resolve the uncertainty completely, i.e. there are still several possible states of the world.

---

A toy example:
  
 Suppose you flip a fair coin, thus randomizing between heads (H) and tails (T) with 50-50%. You look away and do not observe the outcome, but a friend of yours does. After they make their observation, they will tell you what they saw. However, there is a 1% chance that you mishear what they are saying.

Your friend's signal - to be precise, what you hear of it - is 99% accurate.

> **Comments:**
> **username_0**: Thanks! This essentially means that updating probability is conditional on the signal that we receive, hence, we have to use Bayesian updation. Furthermore, what if the state of the world is not perfectly observed by the entity that supplies us with the signal? By 'not perfectly observed' I mean that the supplier of the signal has her own probability distribution over the outcome after observing the state of the world. For instance, in the coin toss, what if the other person tells us that she is, say, 70% sure that it is a head? What would 'accuracy' mean in that case?  
> **username_1**: The same thing. Imperfect observation is a possible reason for inaccuracy. Above toy example could have been worded as the friend being unsure instead of the agent misunderstanding. In the end it is about the signal and its probability of overlapping with the actual state.  

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