Profit vs. Risk: What Betting Games Teach Us About Human Decision-Making

How Much Do You Fear Failure?

This is MathMadeSimple, by Álvaro Muñiz: a newsletter where you will discover how math appears in every aspect of your life, from decision-making to personal finance.

---

In life there is no such thing as free money.

Whenever you want the potential to win, you need to be willing to take some risk. The more you expect to win from some deal, the more risky it probably is. We care about risk: if you have to choose a deal, you won’t only look at your expected win.

You ask yourself: how much am I willing to lose?

Today in a nutshell

• When faced with deals in life, we don’t only look for the most profitable option.

• Humans are inherently risk-averse: we prefer certain outcomes to uncertain ones, even in cases where the latter are more profitable.

• Depending on how well you tolerate risk, you will ask for a corresponding reward to take the risk.

---

Betting in a favourable game

I am going to offer you a game.

I will tell you the rules of the game, but not what the goal is. However you play it will depend on the goal you set yourself. In other words:

What are you trying to achieve?

---

We will play 10 rounds.

In each round we roll a dice:

• If the dice comes 1, 2, 3, or 4, you win.

• If the dice comes 5 or 6, you lose.

You start with $100 of cash. In each round, you have to decide how much money you are betting.

• If you win the roll, you take your bet back and double it.

• If you lose the roll, you lose your bet.

For instance, in the first round you have $100. You decide to bet $50:

• If the dice comes 1, 2, 3 or 4, you get your $50 back and another $50, so that now you have $100 + $50 = $150.

• If the dice comes 5 or 6, you lose your $50 bet, so that now you have $100 - $50 = $50.

Obviously, if you run out of money, the game stops.

---

How would you play in real life?

Profit at all costs

The first thing you could do is try to make as much money as possible.

Imagine the game was only played one round. What will happen with your bet you don’t know—after all, the roll of a dice is random. Instead we think about what we expect to happen.

If you roll the dice many many times, about 4/6 of the times you will win and 2/6 you will lose. So, if you bet any amount many many times, say $60, about 4/6 of the time you will win the roll—and thus win $60 in profit—and 2/6 you will loose the roll—and thus lose $60.

On average, you expect to make (4/6) x $60 - (2/6) x $60 = $20.

The same calculation works for any bet of B dollars: you expect to make (4/6) x $B - (2/6) x $B = 2/6 $B.

So, to maximise your (expected) profit, you want to bet everything.

You go all-in.

In fact, the same applies if you play the game many rounds. Remember our trick of mathematical induction? It comes handy here again—try to use it to show you always go all-in!

That is not how you would play

This strategy has a problem.

Have you ever worried about going broke? If so, don’t wait more: this strategy will ‘almost surely’ make you go broke. To put it another way, you will almost never make money.¹

What? Didn’t we just say that it is the strategy that will make you the most money?

Indeed, on average, it will make you the most money. The reason is that, the few times you don’t go broke, you will make a lot of money: after 10 rounds of doubling your money, you will have over $100,000 (starting with $100!). But this is at the cost of going completely broke sometimes.

Think about it like this:

• If a strategy is guaranteed to make me $50 in profit, you expect to win $50.

• If a strategy makes you $1,000 in profit 10% of the time and $0 in profit 90% of the time, you expect to win $100.

You expect to win double the amount, but you go broke 90% of the time.

Risk aversion

If you were to play the game above in real life I bet you wouldn’t go all-in in every round. You would bet a fraction of your money. Maybe you don’t know exactly how much or why, but essentially your brain is trying to avoid the outcome of going broke.

We humans are inherently risk averse.

This means that we avoid risk as much as we can. If two options give us the same outcome, we will choose the one with less variability.

And, even if one option offers a higher potential return than another, the increase in return must be significant enough to justify the additional risk.

For example, imagine I offer you the following two deals:

1. I give you $1,000 tomorrow, no matter what.

2. Tomorrow we flip a coin. If heads comes up, I will give you $4,000. If tails comes up, I will give you $0.

The second deal will give you, on average, $2,000. This is double the amount to the first deal. Yet, most people would choose the first deal (‘what if I get nothing?’).

Similarly, in the game above, even if going all-in gives you the most expected profit, a slightly lower pay-off with higher certainty might be preferable to you.

Here is a question you should ask yourself often:

Is it worth it to take the risk?

What’s coming:

• What is the best strategy for the game above (taking into account risk-aversion)?

• The utility of money and how it relates to risk.

• How to price an option using a reality where we are risk-neutral.

1. Think about it: what would need to happen for you to not go broke? If you play the game for, say, 10 rounds, then no single 5 or 6 can come out in the 10 rolls of the dice. The probability of this is very small, less than 2%. That is, you will go broke 98% of the times!