The Value of Information
We instinctively know that information is important in decision making. We seek information to reduce uncertainty in our decisions. This drive has led humanity to organize civilizations around the storage, transmission and control of information. We create songs and stories, invent writing and the printing press, build schools, libraries and server caverns. Until recently, information was a scarce resource, exclusive to elites. Technology has since democratized access to vast amounts of information. Yet, having information at our fingertips 24/7 has not eased our anxieties around decisions. According to a global survey¹ by Oracle, 86% of people say that more data than ever before is making personal and professional decisions more complicated. 72% are so overwhelmed that they stopped making decisions. 64% would rather have a robot make their decisions instead!
Given recent advances in AI, the existence of such robots could be imminent. So, let’s imagine an oracular supercomputer. Our clairvoyant, known in literature as Deep Thought, not only plays chess, but can also see through all events across time and space. Deep Thought can look at an uncertainty and predict exactly how things are going to turn out, even if it cannot change any event or tell us what to do. Pity.
Deep Thought was purpose-built to answer the Ultimate Question, which is expected to take 7.5 million years. Meanwhile, the management team around Deep Thought has decided to monetize its formidable capacity and offer foresight for a price.
How much would you pay for clairvoyance?
An Example
Examples are handy for pinning down answers to nebulous questions. Let’s suppose that there is this deal. We could choose to allocate substantial resources and either reap riches and glory many times our investment or lose the entire stake, and our shirts.
As diligent investors, we came up with some figures. But since some random number, say $1 million, can be compelling for some but trivial for others, we will keep things general with algebraic notation:
$p$ : Probability that we will win
$1−p$ : Probability that we will lose i.e., the maximum probability (100% or 1) minus the probability of winning
$x$ : Gains from the investment, tangible and intangible, distilled into a single $ amount
$y$ : Losses from the investment, tangible and intangible, distilled into a single $ amount
$B$ : Cost of clairvoyance
We must first resist the urge to assume clairvoyance and think about our decision without it. Our investment can be represented as follows (Figure 1):
Figure 1: Decision without Clairvoyance
If we choose to invest, then the expected return would be:
\begin{equation}\mathbb{E}[payoff_{NC}] = px - (1-p)y\label{enc}\end{equation}
If we choose to not invest, our expected return would be $\$0$. A profit-seeking investor would invest if:
\begin{equation}\notag px−(1−p)y ≥ 0\end{equation}
We can now consider the decision we would face with clairvoyance (Figure 2). Since we cannot know what the clairvoyant would say pre-purchase, we use the same probabilities as our guesses:
Figure 2: Decision with Clairvoyance
The expected return, if we buy clairvoyance, would be:
\begin{align}\mathbb{E}[payoff_{WC}] &= p(x-B) + (1-p)(0-B) \nonumber \\ &= p(x-B) - (1-p)B \label{ec}\end{align}
To justify our purchase, the expected payoff with clairvoyance \eqref{ec} would need to be greater than or equal to the payoff with no clairvoyance \eqref{enc}:
\begin{equation}\notag p(x−B)−(1−p)B ≥ px−(1−p)y\end{equation}
After a little rearranging, we arrive at the cost of clairvoyance:
\begin{equation}\notag B ≤ (1−p)y\end{equation}
This result says that we should pay Deep Thought’s managers no more than $(1−p)y$, where $(1−p)$ is the probability that we will lose and $y$ are the losses that we would consequently suffer. Notice that there is nothing about winning in the result – no win probability $p$ nor the associated gains $x$.
Expected Regret
We looked at a stylized example, but this is a general result in decision theory. $(1−p)y$ is the expected regret of our decision. Regret is the difference between the best outcome and the actual outcome of each possible alternative. Expected regret is the average of regrets, weighted by the probabilities of different future outcomes. The maximum cost of clairvoyance is the expected regret of our decision: We buy information, not to bolster our odds of winning, but to avert the possibility of losing and losses.
To give life to abstractions like regret, let’s plug in some numbers. In 2024, we spent $5.3 trillion on Information Technology (IT) – the people and tools that store, process, analyze, transmit and manage information. If we use our willingness to spend on IT as a broad-brush proxy² of how much benefit we get out of information, it will seem that we have staved off potential regrets worth more than the GDP of any given country, except perhaps the United States or China.
Instead, we feel worse about our choices. In the Oracle survey, 86% of people say that the volume of data has eroded confidence in their decisions, causing confusion and doubt. 70% say that the headache of collecting and interpreting so much data is too much to handle. But we hate the idea of losing so much that we cannot help but bury ourselves under even more information – at least $7 trillion’s worth by 2028 – in hopes of crushing away any wisp of doubt, guilt or regret.
How do we break free from this spiral of buying our way out of regret?
Materiality
Let’s say we received word that Deep Thought will be unavailable, ostensibly to design an inclusive next-generation supercomputer, but this seemed too well-timed with escalating public controversy. This is inconvenient, but because we did not presume clairvoyance, we can revert to our decision without it (Figure 1). We will invest if the expected payoff of investing is greater than not investing. The value of clairvoyance is zero.
To sway the narrative, Deep Thought’s managers announced a lottery: free clairvoyance via random draw. We, like sextillions of others, submitted an entry. If we win, we end up with the decision with clairvoyance (Figure 2). We will invest if Deep Thought calls a “win” and not invest if it says “lose”. Despite not costing us anything $B=\$0$, this clairvoyance has value because we will change our decision depending on what it says. In other words, the information is material to our decision.
Knowing this, we decided to not take our chances and buy the information outright. Tragically, all requests in queue, including ours, were destroyed by extremists in a violent assault on the machine’s distribution networks. We end up making our decision without clairvoyance (Figure 1).
The value of clairvoyance here is zero, and not negative, despite the unfortunate write-off. Even if we had the information, we could always throw it out and make the decision as if with no clairvoyance. Information has value only if it changes a decision, regardless of the cost. If the information would not change your decision, it is immaterial and has no value.
* * *
Our analysis established a range for the value of information: somewhere between zero and the value of clairvoyance. Clairvoyance, or perfect information, resolves all uncertainty in a decision, rendering regret impossible. In practice, perfect information rarely, if ever, exists. The value of information tends to be low, often zero.
We get bombarded with information because it sells. Understanding the value of information helps us scrutinize claims of intelligence and insights promising better decisions, more success, less regret. It allows us to consider the costs of acquiring information against yielding good outcomes (Figure 3). Most importantly, it reminds us that all that information is worth nothing if we are just going to ignore it anyway.
Figure 3: Value of Information Example (Source: Dr. Hossein Arsham)
Notes:
I have plenty of questions about this report – like why is it only available in archive – but have decided to roll with it anyway because 1) a population sample of 14,250 is large, even if round numbers are always suspect and 2) surveys are useful indicators of perception and sentiment.
Targeted and hence, more expensive, information gathering includes R&D (experimentation, modeling), diagnostics (testing, detectors) and advice (experts, consultants). IT is a mere glimpse of the whole.