Origins of risk aversion
July 2009 (index)
Consider this game. You start with a fund of £1000 and in each round you bet half your fund at even odds. You draw a card from a shuffled full deck of cards (the card is replaced each time) and if it's a ten, jack, queen, king, or ace then you lose. If it's any other card then you win.
In any given round you have a 8/13 (61.5%) chance of winning, and the bet pays even money. That's an expected gain of +11.5% per round. So it looks like a good deal.
But what happens if you play? Over thirteen rounds you'd expect on average to win eight and lose five. If this happens then your initial £1000 becomes £1000*1.5^8*0.5^5=£800.
So although the game has a healthy expected return of 11.5% per round you'll typically lose 20% over 13 rounds, or 2.7% per round.
Choice of bet
Suppose instead of always betting half your wealth suppose you could agree at the start of the game a fraction f which applied to every round. Now the mean outcome is 61.5%*(+f)+38.5%*(-f) = f*23% per round and the most likely outcome over 13 rounds is (1+f)^8*(1-f)^5.
We can look at the effect of different values of f:
Bet Over 13 rounds
0% 1.00
10% 1.27
20% 1.41
23% 1.42 (optimal)
30% 1.37
40% 1.15
44.6% 1.00 (break-even)
50% 0.80 (original game)
100% 0.00
If you bet your entire fund each time then your expected gain is +23% per round but just one bad round means you end up with nothing. The expected return is high, but that comes from a tiny chance of a huge win. After 20 rounds your expected fund is £63,611, which comes from a 1 in 16,484 chance of having £1,048,576,000 and a 16,483 in 16,484 chance of having nothing.
The original game, betting 50% of your fund each round, isn't as extreme as that. But it still has a small chance of a big win. As the game continues your chance of being ahead falls, even though each round has an expected win.
The other extreme, to bet nothing, is safe as you can't lose, but it means you're giving up the chance to make money at a game that's in your favour.
Somewhere in the middle, betting 23% turns out to be a good strategy. This is the bet that maximises your most likely outcome. You're giving up the chance of a huge win (you can't reach £1 billion in 20 turns) for a high probability (in the long run) of a solid return.
Optimal bet
There are various possible meanings for "optimal". For the highest expected return bet as much as possible; for the lowest chance of losing money bet nothing.
Here we take optimal to mean maximising the most likely growth rate.
If we have a probability p of winning a single bet, and we bet a fraction f of our wealth each time then a fund of 1 becomes G(f)=(1+f)^p*(1-f)^(1-p) per bet.
We want to find the value of f that maximises this G. This is equivalent to maximising log(G). Setting dlog(G)/df=0 tells us that f=2p-1.
In our case p=8/13, so f=2p-1=3/13=23%.
Note that if it's an even-money bet (p=50%) we shouldn't bet at all. If it's against us (p<50%) then we would like to bet a negative amount: we would like to be the casino rather than the gambler. And if the bet is in our favour (p>50%) then we shouldn't bet too much.
(If you're familiar with utility functions, then note that this approach is equivalent to maximising a logarithmic utility function. In practice surveys have shown that people tend to be more risk averse than this, and so should bet less.)
Bets at other odds
We can extend this to look at bets not at evens. If a bet gives us probability p of winning a a multiple w of our stake then we have G(f)=(1+wf)^p*(1-f)^(1-p). Using the same approach as above to maximise this tells us that optimally f=(wp+p-1)/w.
Some examples:
If w=1 then this gives the same result as before.
If we have a 50:50 bet that pays twice our stake if we win, then we should bet f=25% of our wealth. This gives G(25%)=6.1%. If we bet too much we win less, eg G(50%)=1.
Suppose a lottery offers unusually generous odds, selling 1,000,000 tickets for £1 each, with the winning ticket winning £2,000,000. (While lotteries generally have a negative expected return on some occasions prize funds are carried forward from previous draws.) How much should we bet? With w=£2,000,000 and p=1/1,000,000 the formula tells us f=1/1,999,998. Unless you're already a millionaire then the optimal number of tickets to buy is none. (Of course here optimal is purely based on maximising the most likely growth rate. If the act of dreaming about a big win itself gives you pleasure then the ticket may be worthwhile.)
Alternative notation
So far we have written f=(wp+p-1)/w. If we write w+1=o then f=(op-1)/(o-1). I find this marginally more memorable, and it corresponds to thinking of the odds "o" for a bet where o is the amount we get back, including our original stake. This o is often used (called "decimal odds") on betting websites.
References
What we've described here is the Kelly criterion. For more details see the original paper or Ed Thorp's paper.
Origins of risk aversion
Here's the speculative and unscientific part. This may offer a reason why people are risk-averse.
By "risk-averse" we mean that people generally avoid even-money bets; at least in real-life rather than gambling for pleasure. When asked to choose between (A) a 50:50 chance of £1,000,000 and nothing or (B) a certain £490,000, most people choose (B) even though the mathematical expectation is higher for (A).
We can reframe this as a single bet faced by someone who has £490,000 and can make a 50:50 bet with a stake of £490,000 to win £510,000. This is the position of someone choosing (A) rather than (B). In the notation from before p=50% and w=51/49. The optimal f is then 2%. Given the question above is all-or-nothing Kelly says to stick with (B) unless you're already very wealthy.
We've seen that 50:50 gambles are best avoided; the optimal stake is nil and if you bet anything then your condition is likely to worsen. Suppose we have one group of animals who avoid the gambles (not necessarily by explicitly doing calculus), and another group who take them. Those who take them are likely to be less successful. So risk-aversion seems like a characteristic which would be propogated by natural selection.
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