Reference Frames for Expected Value



Puzzle 1: George mortgages his house to invest in lottery tickets. He wins and becomes a millionaire. Did he make a good choice?

Puzzle 2: The U.S. president questions if he should bluff a nuclear war or concede to the USSR. He bluffs and it just barely works.  Although there were several close calls for nuclear catastrophe, everything works out ok. Was this ethical?

One interpretation of consequentialism is that decisions that produce good outcomes are good decisions, rather than decisions that produce good expected outcomes.12 One would be ethical if their actions end up with positive outcomes, disregarding the intentions of those actions. For instance, a terrorist who accidentally foils an otherwise catastrophic terrorist plan would have done a very ‘morally good’ action.3 This general view seems to be surprisingly common.4

This seems intuitively strange to many, it definitely is to me. Instead, ‘expected value’ seems to be a better way of both making decisions and judging the decisions made by others. However, while ‘expected value’ can be useful for individual decision making, I make the case that it is very difficult to use to judge other people’s decisions in a meaningful way.5 This is because ‘expected value’ is typically defined in reference to a specific set of information and intelligence rather than an objective truth about the world.

Two questions to help guide this:

  1. Should we judge previous actions based on ‘expected’ or ‘actual’ value?
  2. Should we make future decisions to optimize ‘expected’ or ‘actual’ value?

I believe these are in a sense quite simple, but require some consideration to definitions.6

Optimizing Future Decisions: Actual vs. Expected Value

The second question is the easiest of the two, so I’ll begin with that one. The simple answer is that this is a question of defining ‘expected value’. Once we do so the question kind of goes away.

There is nothing fundamentally different between expected value and actual value.  A more fair comparison may be ‘expected value from the perspective of the decision maker’ with ‘expected value from a later, more accurate prospective’.

Expected value converges on actual value with lots of information. Said differently, actual value is expected value with complete information.

In the case of an individual purchasing lottery tickets successfully (Puzzle 1), the ‘actual value’ is still not exact from our point of view. While we may know how much money was won, or what profit was made. We also don’t know what the counterfactual would have been. It is still theoretically possible that in the worlds where George wouldn’t have purchased the lottery tickets, he would have been substantially better off. While the fact that we have imperfect information doesn’t matter too much, I think it demonstrates that presenting a description of the outcome as ‘actual value’ is incomplete. ‘Actual value’ exists only theoretically, even after the fact.7

So this question becomes, then ‘should one make a decision to optimize value using the information and knowledge available to them, or using perfect knowledge and information?’ Obviously, in this case, ‘perfect knowledge’ is inaccessible to them (or the ‘expected value’ and ‘actual value’ would be the same exact thing). I believe it should be quite apparent that in this case, the best one can do (and should do) is make the best decision using their available information.

This question is similar to asking ‘should you drive your car as quickly as your car can drive, or much faster than your car can drive?’ Obviously you may like to drive faster, but that’s by definition not an option. Another question: ‘should you do well in life or should you become an all-powerful dragon king?’

Judging Previous Decisions: Actual vs. Expected Value

Judging previous decisions can get tricky.

Let’s study the lottery example again. A person purchases a lottery ticket and wins. For simplicity, let’s say the decision to purchase the ticket was done only to optimize money.

The question is, what is the expected value of purchasing the lottery ticket? How does this change depending on information and knowledge?

In general purchasing a lottery ticket can be expected to be a net loss in earnings, and thus a bad decision. However, if one was sure they would win, it would be a pretty good idea. Given the knowledge that the player won, the player made a good decision. Winning the lottery clearly is better than not playing once.

More interesting is considering the limitation not in information about the outcome but about knowledge of probability. Say the player thought that they were likely win the lottery, that it was a good purchase. This may seem insane to someone familiar with probability and the lottery system, but not everyone is familiar with these things.

From the point of view of the player, the lottery ticket purchase had net-positive utility. From the point of view of a person with knowledge of the lottery and/or statistics, the purchase had net-negative utility. From the point of view of any of these two groups, after they know that the lottery will be a success, it was a net positive decision.

  No Knowledge of Outcome Knowledge of Outcome
‘Intelligent’ Person with Knowledge of Probability Negative Positive
Lottery Player Positive Positive

Expected Value of purchasing a Lottery Ticket from different Reference Points

To make things a bit more interesting, imagine that there’s a genius out there with a computer simulation of our exact universe. This person can tell which lottery ticket will win in advance because they can run the simulations. To this ‘genius’ it’s obvious that the purchase is a net-positive outcome.

  No Knowledge of Outcome Knowledge of Outcome
Genius Positive Positive
‘Intelligent’ Person with Knowledge of Probability Negative Positive
Lottery Player Positive Positive

Expected Value of purchasing a Lottery Ticket from different Reference Points

So what is the expected value of purchasing the lottery ticket? The answer is that the ‘expected value’ is completely dependent on the ‘reference frame’, or a specific set of information and intelligence. From the reference frame of the ‘intelligent person’ this was low in expected value, so was a bad decision. From that of the genius, it was a good decision. And from the player, a good decision.


So how do we judge this poor (well, soon rich) lottery player? They made a good decision respective to the results, respective to the genius, and compared to their own knowledge. Should we say ‘oh, this person should have had slightly more knowledge, but not too much knowledge, and thus they made a bad choice’? What does that even mean?

Perhaps we could judge the player for not reading into lottery facts before playing. Wasn’t it irresponsible for falling for such a simple fallacy? Or perhaps the person was ‘lazy’ to not learn probability in the first place.

Well, things like these seem like intuitions to me. We may have the intuitions to us that the lottery is a poor choice. We may find facts to prove these intuitions accurate. But the gambler my not have these intuitions. It seems unfair to consider any intuitions ‘obvious’ to those who do not share them.

One might also say that the gambler probably knew it was a bad idea, but let his or her ‘inner irrationalities’ control the decision process. Perhaps they were trying to take an ‘easy way out’ of some sort. However, these seem quite judgmental as well. If a person experiences strong emotional responses; fear, anger, laziness; those inner struggles would change their expected value calculation. It might be a really bad, heuristically-driven ‘calculation’, but it would be the best they would have at that time.

Free Will Bounded Expected Value

We are getting to the question of free will and determinism. After all, if there is any sort of free will, perhaps we have the ability to make decisions that are sub-optimal by our expected value functions. Perhaps we commonly do so (else it wouldn’t be much in the sense of ‘free’ will.)

This would be interesting because it would imply an ‘expected result’ that the person should have calculated, even if they didn’t actually do so. We need to understand the person’s actions and understanding, not in terms of what we know, or what they knew, but what they should have figured out given their knowledge.

This would require a very well specified Free Will Boundary of some sort. A line around a few thought processes, parts of the brain, and resource constraints, which could produce a thereby optimal expected result calculation. Anything less than this ‘optimal given Free Will Boundary’ expected value calculation would be fair game for judging.

Conclusion: Should we Even Judge People or Decisions Anyway?

So, deciding to make future decisions based on expected value seems reasonable.  The main question in this essay, the harder question, is if we can judge previous decisions based on their respective expected values, and how to possibly come up with the relevant expected values to do so.

I think that we naturally judge people. We have old and modern heroes and villains. Judging people is simply something that humans do. However, I believe that on close inspection this is very challenging if not impossible to do reasonably and precisely.

Perhaps we should attempt to stop placing so much emphasis on individualism and just try to do the best we can while not judging others nor other decisions much. Considerations of judging may be interesting, but the main take away may be the complexity itself, indicated that judgements are very subjective and incredibly messy.

That said, it can still be useful to analyze previous decisions or individuals. That seems like one of the best ways to update our priors of the world. We just need to remember not to treat it personally.

  1. Dorsey, Dale. “Consequentialism, Metaphysical Realism, and the Argument from Cluelessness.” University of Kansas Department of Philosophy

  2. Sinhababu, Neiladri. “Moral Luck.” Tedx Presentation

  3. This is assuming the terrorists are trying to produce ‘disutility’ or a value separate from ‘utility’. I feel like from their perspective, maximizing an intrinsic value dissimilar from our notion of utility would be maximizing ‘expected value’. But analyzing the morality of people with alternative value systems is a very different matter.

  4. These people tend not to like consequentialism much.

  5. I don’t want to impose what I deem to be a false individualistic appeal, so consider this to mean that one would have a difficult time judging anyone at any time except for their spontaneous consciousness.

  6. I bring them up because they are what I considered and have talked to others about before understanding what makes them frustrating to answer. Basically, they are nice starting points for getting towards answering the questions that were meant to be asked instead.

  7. This is true for essentially all physical activities. Thought experiments or very simple simulations may be exempt.