Risk-taking is Literally for Losers

A large number of situations involve making a trade-off between expected reward, and risk. In many walks of life, risk is treated as a bad thing and the trade-off typically involves taking more risk in order to achieve a higher average reward, or less risk for a lower one. Taking insurance is costly, but lowers risk by compressing the probability distribution of outcomes. Sometimes it might make sense to drive faster, and more dangerously, in order to get somewhere faster, on average. Deciding not to travel overseas lowers risk but also denies you potentially-enjoyable situations.

What about situations in which we are offered more risk for a lower average reward? Is it ever rational to want to pay for additional risk? Gambling involves this kind of trade-off. Playing the UK National Lottery will lose you 50p on average for every £1 you pay. For these decisions to be rational - i.e. for them to be the decisions that maximise expected returns - we have to make some particular assumptions about the decision-maker's goals.

They may just enjoy the thrill of gambling itself. Alternatively, the returns they receive from income might increase rather than decrease: £100 might be more than 100 times better than £1. Normally, this would be a very demanding assumption, since studies invariably demonstrate that the opposite is nearly always true: that £100 is considerably less than 100 times better than £1. The first pound might stop you starving. but your options with the hundreth pound are considerably less critical.

However, there is a very common situation in which winning does have increasing returns: where there is a positional element to utility, because what you are playing for (e.g. income, points or poker chips) are only instrumental in determining your final payoff. This happens all the time in games: it doesn't much matter if you lose a football match 5-4 or 5-0 - the key thing you still lost. When you're 5-3 down, you'd rather play a risky strategy that might net you three more goals, than a conservative one that would certainly net you exactly one goal.
There are only tools. There is an interesting disparity between the optimal strategies of winners and losers in situations like this. People who are ahead should, in general, play to maximise their expected score. If taking risks is costly to the expected score, they should minimise risk. People who are losing, however, should play in a way that pushes as much of the probability distribution as possible above the opponent's expected score. Where risk-taking is costly to expected score, this will often mean playing riskily at the expense of the scoreline.

This phenomenon isn't confined to games. When we see people paying to take risks, and particularly those at the bottom of the scale, we should suspect a positional element to payoffs. The fact that poorer people are more likely to play the lottery and take more risks in general points to the fact that it's not just income, but also relative income that people care about. Most of the time we think in terms of paying to avoid risk. But when you're losing, and winning is what really matters, sometimes the rational thing to do is buy more of it.