Over the weekend, I skimmed through the Springer Undergraduate Mathematics Series book ‘Game Theory‘ by James Webb. It’s a book that has been sitting on my bookcase for a while, and a topic which sits at only one step removed from what I am particularly interested in. But somehow, possibly because of its almost complete absence from the Cambridge maths course, I had never got round to reading the book or otherwise finding out much about the field. This has now changed, so I am writing three short remarks on some aspects I thought were of particular interest.
Consider the following. I offer you the following bet. I toss a fair coin: if it comes up heads, you have to pay me £1, but if it comes up tails, I will give you £3. Would you take the bet? Obviously you are the only person who can actually answer that, but I’d expect the only obstacle to you taking the bet might be indifference. Why? Well your expected profit from this bet is £1.50, which is positive, and so it certainly seems that you are getting a better deal.
Now let’s up the stakes. If the coin comes up heads, you have to pay me £1,000,000, but if it comes up tails, you get £3m. Do you still take the bet? Well unless you are in a very fortunate financial state, you shouldn’t be taking this bet because you are not in a position to honour your commitment in the event (with probably 50%) that you lose.
But what about an intermediate bet? Suppose we change the stakes to £100 and £300. Or to £1,000 and £3,000. Would you take the bet? Without getting too bogged down in what the answer is, and what personal circumstances might lead to each possibility, suffice it to say this: each of the adjusted bets still offers a substantial positive expected profit, but the real risk of losing a large sum of money makes one less inclined to take up the offer.
Traditionally, we can describe this apparent logical problem like this. How much we value sums of money is not always proportional to the sum itself. Although some mathematicians might disagree, we don’t value numbers intrinsically, instead we value the effects that sums of money can have on our lives. More concretely, winning the lottery is awesome, and winning the lottery twice is even more awesome. However, it probably isn’t twice as good as winning it once, since many of the best things about winning the lottery, like retiring and paying off your mortgage or your student loan, can only be done once.
Mathematically, this is best represented by a utility function. This is a map from the set of outcomes of the bet or experiment to the reals, specifying the relative value the agent assigns to different outcomes (monetary or otherwise). We need the utility function to satisfy a few obvious rules. If it is a function of the profit or return, we expect it to be strictly increasing. That is, everyone places a higher utility on a larger sum of money. If we are considering utility as a function of (possibly non-numerical) outcomes, eg elements in a general probability space, then we need it to be transitive. That is, it describes a partial ordering on the set. We would also expect that all outcomes can be compared, so it is in fact a total ordering. The agent’s aim is then instead to maximise their expected utility .
There are a couple of questions that are natural to ask:
1) Can all factors influencing a decision really be expressed numerically?
2) Is maximising some utility function actually the right thing to do?
We ask question 1) because we might think that some factors that influence our decisions such as fear, superstition or some code of morality cannot be quantified explicitly. We ask question 2) because at first it doesn’t seem as if maximising expected utility is any different to maximising expected monetary value.
This needs some clarification. The best way to do this is through the notion of risk aversion. This is an alternative way to think about the original problem with the £1,000 vs £3,000 bet. The risk or uncertainty of a random variable corresponding to a financial return is described by the variance. In general, we assume that most investors are risk-averse, that is, they prefer a guaranteed return of £10 rather than a non-trivial random variable with expectation equal to £10. So instead of maximising expectation, we maximise
where is some positive constant. Of course, if an agent for some reason prefers risky processes, then choose .
So the question we are asking is: might this be happening for utility maximisation as well? More precisely, perhaps we need to consider risk-aversion even for utilities?
The way to resolve the problem is to reverse how we are looking at the situation. Tempting though it is to define the utility as a relative value assigned to outcomes, this forces us into a position where we have to come up with some method for assigning these in specific cases, most of which suggest problems of some kind. Although in practice the previous definition is fine and easy to explain, what we actually want is the following. When we make a choice, we must be maximising something. Define the utility to be the function such that maximising the expectation of this function corresponds to the agent’s decision process.
This is much more satisfactory, provided such a function is actually well-defined. Von Neumann and Morgenstern showed that it is, provided the agent’s preferences have a little bit more structure than discussed above. The further conditions required are continuity and independence, and are completely natural, but I don’t want to define them now because I haven’t done much of the necessary notation here.
In many ways, this is much more satisfying. We don’t always want precise enumerations of all contributory factors, but it is reassuring to know that subject to some entirely reasonable conditions, there is some structure under which we are acting optimally whenever we make a decision.
- God’s Utility Function (liminalhack.wordpress.com)
- Naugatuck man wins lottery prize (rep-am.com)
- Man Wins His Second Lottery In 3 Months (huffingtonpost.com)
- Probably the best PhD I’ll ever do (wkmor1.wordpress.com)
- Life after death for Pascal’s Wager? (overcomingbias.com)