Expected utility theory/expected value decision making—as the case may be—is quite interesting I guess. In times, past (for a few months at longest as I am a neophyte to rationality) I habitually trusted the answers expected utility theory provided without bothering to test them, or ponder for myself why they would be advisable. I mean, I first learned the concept of expected value in statistics and when we studied it in Operations Research—as part of an introduction to decision theory—it just seemed to make sense. However, after the recent experiment I did (the decision problem between a guaranteed \$250,000 and a 10\% chance to get \$10,000,000) I began to start doubting the appropriateness of Expected Utility Theory. Over 85\% of subjects chose the first option, despite the latter having an expected value 4 times higher than the former. I myself realised that the only scenario in which I would choose the \$10,000,000 was one in which \$250,000 was an amount I could pass up. Now I am fully cognisant of expected utility theory, and my decision to pick the first option did not seem to be prey to any bias, so a suspicion on the efficacy of expected utility theory began to develop in my mind. I took my experiment to http://www.reddit.com/r/lesswrong; a community who I expected would be more rational decision makers—they only confirmed the decision making of the first group. I realised then, that something was wrong; my map didn’t reflect the territory. If expected utility theory was truly so sound, then a community of rationalists should have adhered to its dictates. I filed this information at the back of my mind. My brain began working on it, and today while I was reading “Thinking, Fast and Slow” by Daniel Kahneman my brain delivered an answer to me.

I do not consider myself a slave to rationality; it is naught but a tool for me to achieve my goals. A tool to help me “win”, and to do so consistently. If any ritual of cognition causes me to lose, then I abandon it. There is no sentimentality on the road to victory, and above all I endeavour to be efficient—ruthlessly so if needed. As such, I am willing to abandon any theory of decision making, when I determine it would cause me to lose. Nevertheless, as a rationalist I had to wonder; if expected utility theory was so feeble a stratagem, why had it stuck around for so long? I decided to explore the theory from its roots; to derive it for myself so to speak; to figure out where the discrepancy had come from.

Expected Utility Theory, aims to maximise the Expected Utility of a decision which is naught but the average utility of that decision—the average payoff.

Average payoff is given by the formula:

\[E_{j} = Pr_i*G_{ij} \tag{1}\]

Where

\(E_j\) = Expected value of Decision \(j\)

\(P_j\) = Probability of Scenario \(i\)

\(G_{ij}\) = Payoff of Decision \(j\) under Scenario \(i\).

What caught my interest when I decided to investigate expected utility theory from its roots, was the use of probability in the formula.

Now the definition of probability is:

\[Pr(i) = \lim_{n \to \infty} \frac{\sum i}{n} \tag{2}\]

Where \(\sum i\) is to be understood to be \(f i\) the frequency of \(i\).

If I keep in mind the definition of probability, I find something interesting; Expected Utility Theory maximises my payoff in the long run. For decision problems, which are iterated—in which I play the game several times—then Expected Utility Theory is my best bet. The closer the number of iterations are to infinity, the closer the probability is to the ratio above.

Substituting \((2)\) into \((1)\) we get:

\[E_j = \frac{\sum i}{n} * G_{ij} \tag{(3)}\]

What Expected Utility theory tells us is to choose the highest \(E_j\); this is only guaranteed to be the optimum decision in a scenario where \((1)\) = \((3)\) I.e.

- The decision problem has a (sufficiently) large number of iterations.
- The decision problem involves a (sufficiently) large number of scenarios.

After much deliberation, I reached a conclusion; in any non—iterated game in which a single scenario has an overwhelming high probability \(Pr = 1 - \epsilon\), then the maximum likelihood approach is the rational decision-making approach. Personally, I believe \(\epsilon\) should be \(\ge 0.005\) and set mine at around \(0.1\).

I may in future revisit this writeup, and add a mathematical argument for the application of the Maximum likelihood approach over the Expected Utility approach but for now, I shall posit a simpler argument:

The Expected Utility approach is sensible only in that it maximises winnings in the long run—by its very design, it is intended for games that are iterated and/or in which there is a large number of scenarios. In games where this is not true—with few scenarios and a single instance—there is sufficient variation in the event that occurs that there is a significant deviation of the actual payoff from the expected payoff. To ignore this deviation is oversimplification, and—I’ll argue—irrational. In the experiment I listed above, the actual payoff for the second decision was \$0 or \$10,000,000; the former scenario having a likelihood of 90\% and the latter a 10\%. The expected value is \$1,000,000 but the standard deviation of the payoffs from the expected value—in this case \$3,000,000—is 300\% the mean. In such cases, I conclude that the expected utility approach is simply unreliable—and expectably so—it was never designed for such problems in the first place (pun intended).