[Andrew WCY]

The following is a rather lengthy discussion on the significance of expected gains.
Assume that everything we're talking about is (emphasis intended) ON AVERAGE.
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We could begin examining the subject matter by creating a simple hypothetical scratchcard called "All-or-none". Each scratchcard is valued at $1.

If the expected gain were $1, one might naturally assume they'll have no net gain or loss. But let's define a few variations of the game:

Variation 1: You either get $10 with a probability of 0.1 or you get nothing.
Variation 2: You either get $100 with a probability of 0.01 or you get nothing.
Variation 3: You either get $1000000 with a probability of 0.000001 or you get nothing.

Notice how the expected gain in all 3 cases is $1 but the chance of winning continues to diminish. The expected gain doesn't hold much significance if we don't consider the chance we lose.

To put this in another way, let us first determine the number of scratchcards we need to scratch before we earn the grand prize.
Variation 1 requires 10 scratchcards before you win $10. Variation 2 requires 100, while the third one requires a whopping 1000000.

Then, let's determine how much we would lose RIGHT BEFORE we reclaim our money.
We would lose $9 right before we earn back what we gave for the first game. It's a $99 loss for the second game and a $999999 loss for the third.

The average loss immediately before reclaiming money can be seen as a perceived 'risk' to winning.
This is a bit similar to placing investments on stocks that have a high volatility (beta): The larger the volatility, the higher the risk; we could potentially earn more, but as a subsequence of bigger risk, we also have a larger chance of losing.
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What if the Infinite Money game were to stop at a certain round?

If the game has only 2 rounds, you'll have a 50% chance of winning $2 and a 50% chance of winning $4. The expected gain is $3.
If the game has only 3 rounds, you'll have a 50% chance of winning $2 and a 50% chance of winning $4 or more. You'll have a 25% chance of winning $4 and a 25% chance of winning $8. The expected gain is $4.
If the game has 4 rounds, you'll win $2 (50%), $4 (25%), $8 or $16 (12.5% for both). The expected gain is $5.
If the game has 5 rounds, you'll win $2 (50%), $4 (25%), $8 (12.5%), $16 or $32 (6.25% for both). The expected gain is $6.

In other words, you should pay $(n + 1) to have no gain or loss for an n-round game. Doesn't seem like much, does it?

But how many tries do you need before you earn back what you paid for?
For a game with 2 rounds, you need 2 tries on average. In these 2 games, we would have 1 game with a $2 win and 1 game with a $4 win.
For a game with 3 rounds, 4 tries. In 4 games, we would have 2 games with a $2 win, 1 game with a $4 win and 1 game with an $8 win.
For a game with 4 rounds, 8 tries.
For a game with 5 rounds, 16 tries.

So, we need 2^(n - 1) tries to have no net profit or loss for a game with n rounds.

To calculate the average loss immediately before money reclamation, we need an average of [2^(n - 1) - 1] tries for an n-round game.
This translates to paying ${[2^(n - 1) - 1] * (n + 1)}, which is an average of $3 for a 2-round game, $12 for a 3-round game, $35 for a 4-round game and $90 for a 5-round game!

The perceived risk factor increases exponentially the more we want to win.
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In conclusion, we should look deeper than the surface of things: Not only should we consider the expected gain, we should also evaluate the risk.

TL;DR: The expected gain alone doesn't mean much if we don't know how much we would risk losing.