Why Gambling Isn’t Irrational November 14
So while wasting my Vegas winnings at the slot machines here in Reno I did some introspection about how I value money and why I enjoy gambling. Of course part of the attraction of gambling is just the fun involved in playing. This is the standard defense of gambling or playing the lotto against the frequent faux intelligent derision of the practice as irrational. However, I don’t think this response can stand on its own.
It seems that a key element in the fun of gambling is the fantasy of winning it big. Many people who won’t engage in even favorable bets with their friends quite enjoy playing the lotto or gambling in Vegas. If in fact it was just the thrill of betting at play we should expect people to find casual bets as appealing as the lottery. On the other hand if is the anticipation of future results shouldn’t a rational anticipation involve more dread than hope?
A sophisticated version of the standard explanation can explain this peculiar state of affairs. The line here would explain that gambling is a rational response to an essentially irrational human nature. Knowing that you are likely to overestimate the likelihood of very unlikely outcomes one might leverage this response to enjoy the fantasy of winning. No doubt this is part of the explanation but I think a fully rational agent might still enjoy gambling.
An explanation of this position follows and it rests on some subtle issues in our definition of utility. Basically there is a mismatch between our intuitive conception of the utility of some amount of money and the notion of utility the economists use. When we think about how much we would enjoy a certain amount of money we are imagining how much we would enjoy stuff with that price tag. The economic concept also asks us to consider how we might invest or even gamble with that money. Below I will spell out this point in greater detail and explain its relationship to the rationality of gambling. The standard account of the connection between money and utility is that, after some weird behavior at the very low end, utility increases something like logarithmically with money. Of course at the extremely high end this relation must actually become sub logarithmic as well since an infinite amount of money would not be worth an infinite amount of utility. The important point for our considerations is that in the relevant section the function from net worth to utility is generally assumed to be a smooth monotonically increasing function with a smooth monotonically decreasing derivative, i.e., every extra dollar you receive increases your welfare but slightly less so than the last one did.
Taken simplistically this account seems to get things wrong. At least for me receiving 10,000 dollars is more than 10 times better than receiving a 1,000 dollars. The former would let me skip working for a term while the later would end up just getting spent on a new laptop. Yet this seems to violate the assumption that the derivative of the utility function is monotonically decreasing, i.e., that each additional dollar increases your welfare less than the previous.
Before we simply abandon the assumption that each additional dollar matters less to you than the last we need to carefully examine what we mean by utility. If money could only be spent or saved this assumption would obviously be wrong. Idealizing our disposable income as completely fixed, i.e., we can’t skimp on food to buy toys, then if we want a $2048 dollar computer there is a huge difference in the utility of having $2047 in the bank and $2048 while barely any between $2046 and $2047. Thus if we could only spend or save money it seems likely that each additional dollar doesn’t always improve our welfare by less than the prior dollar.
However, in the real world it isn’t true that we can only save or spend our money. We can also gamble with our money. Given $2047 dollars we can wager $1 on a fair flip of a coin and if we win buy our computer. If we lose we can then wager $2 and so forth. Working out the numbers it is easy to see that with $2047 we have a .999% chance of buying our $2048 computer by making bets on coin flips. So even if the relationship between our well-being and the dollar value of the stuff we own is not nice and smooth this doesn’t mean that the relationship between money and utility is not. In fact making some reasonable assumptions our utility functions must look like I described but among these assumptions is the idea that we can enter into fair wagers.
In the real world, however, one cannot easily enter into gambles. Their are social costs to wagering money except in certain societally approved ways. As a result the idealized utility function considered by economists doesn’t quite match the value we rationally assign to money. While our expectation of this idealized economic notion of utility may always be negative when we gamble it does not necessarily mean that our expectation of total welfare is negative when we gamble (excluding any fun had gambling of course).
Are you suggesting that the diminishing marginal utility of money could be given a justification in terms of the probability of gaining a dollar using the doubling technique? It could work. It’s certainly easier to raise ten dollars if you are already a millionaire, as any drunk will tell you. If labour is linked to value, then a millionaire who earns more in his sleep than a jobbing plumber earns in a month will not value his money as much, and consequently tip more at restuarants.
As for the irrationality of gambling, gambling at unfavourable odds is never a way of making money in the long term and any one who thinks it is is irrational necessarily, since that is all unfavourable odds means. But who says it is irrational to gamble at favourable odds?
No, what I’m really saying (and should have said this way) is that utility does not increase at a uniform rate with respect to total wealth. The extreme example here would be if utility was a highly non-continuous step function. For instance many if you have between $500 and $1000 dollars you have 1 unit of utility between $1000 and $2000 you have 2 units and so forth. In this sort of case even though from a big picture view utility is essentially logarithmic in terms of money (i.e. it’s a step function approximating that curve) but if you have $750 dollars it’s certainly worthwhile to spend $200 for a 1 in 10 chance of winning $300. Even while your expectation in terms of money is negative your expectation in terms of utility is positive since you don’t lose any utility by dropping your wealth from $750 to $550.
Of course in the real world the utility function we have is going to be continuous but the same effect can occur if the utility function has sharp ‘jumps’ (e.g. at psychologically salient amounts of money…people view $1 million differently than they view $990,000).
However, on second consideration I have to take a slightly more sophisticated explanation. In particular I claim that people are deeply irrational about money (which everyone who thinks the lotto is irrational must believe) and that our utility in terms of spending or saving money has much more to do with the pain of getting a bad deal or the feeling of getting a good one than it does with actual ability to exchange money for goods and services. For instance this is why people will spend time searching for a slightly cheaper item even when in other parts of their lives they would trade off time for money at a much higher rate.