Why do gamblers go broke even when the odds favour them at every step?

[Happiness]

TL;DR Summary

Making a small win is more likely than going broke. So aim for small wins at every step. But when should you stop playing? At every step, it seems more favourable to play another round. But if you don't stop, you would definitely go broke in the end.

Below is a scenario where at every step, you are choosing the more favourable option, but yet you would end up worse off definitely. How could it be?

Suppose you start playing a fair game (or a game slightly to your advantage) with $10 and bet $1 each round. You tell yourself that you would stop playing once you reach $15 (gain $5), since it's more likely to reach $15 than to go broke ($0).

But once you reach $15, it's harder for you to go broke (since you have more money). So you continue playing, telling yourself that you would stop once you reach $20.

But every time you reach your target amount, you would update your target amount to a new one and continue playing. This is because the odds are in your favour so long as your target is less than twice your current wealth. But this means you would never stop playing. Then it would just be a matter of time before you go broke (since it's more likely to go broke than to increase your wealth to infinity).

If you are making the more favourable choice at every step, how does it happen that you are worse off in the end?

 

[Dale]

You are not choosing the most favorable option at each step. You are lying to yourself about your stopping condition.

This actually is an issue with science. Often the stopping condition for an experiment depends on the outcome of the early data (although it should not)*. This change of the stopping condition can distort the outcome of your experiment.

In your scenario, the actual stopping condition is you go broke. So you go broke.

* Here “should not” is a statistical statement. Sometimes ethically it is necessary. But that must then be factored into the analysis of the data.

 

[topsquark]

TL;DR Summary: Making a small win is more likely than going broke. So aim for small wins at every step. But when should you stop playing? At every step, it seems more favourable to play another round. But if you don't stop, you would definitely go broke in the end.

Below is a scenario where at every step, you are choosing the more favourable option, but yet you would end up worse off definitely. How could it be?

Suppose you start playing a fair game (or a game slightly to your advantage) with $10 and bet $1 each round. You tell yourself that you would stop playing once you reach $15 (gain $5), since it's more likely to reach $15 than to go broke ($0).

But once you reach $15, it's harder for you to go broke (since you have more money). So you continue playing, telling yourself that you would stop once you reach $20.

But every time you reach your target amount, you would update your target amount to a new one and continue playing. This is because the odds are in your favour so long as your target is less than twice your current wealth. But this means you would never stop playing. Then it would just be a matter of time before you go broke (since it's more likely to go broke than to increase your wealth to infinity).

If you are making the more favourable choice at every step, how does it happen that you are worse off in the end?

Who says odds are in your favor in a casino? The game to play with the best odds is blackjack, and you only win (don't quote me on this) some 40 percent of the time. The House always wins in the end.

-Dan

 

[BWV]

Assuming you have some edge - like counting cards in blackjack, bet using the Kelly criterion - where on each outcome you bet a fraction of your wealth equal to 2p-1, with p being your odds of winning (this assumes a 1:1 payoff, the formula is different for other payouts). Half-kelly (betting half this amount) is actually better for a loss-averse better as it maximizes utility.

to bet 10% of your bankroll as in your example would require a 55% chance of winning (again with even payouts)

https://en.wikipedia.org/wiki/Kelly_criterion

 

[jack action]

You may be interested in the following betting systems:

• Martingale
• Split martingale

 

[Klystron]

Consider live poker games as played in Nevada and California casinos. Assume your poker knowledge and play superior to most other players at your buy-in level. Assume you perform optimally, making correct decisions most of the time. The longer you stay at the poker table, the more hours you play, the more money you lose.

Reasons include:
House rake on every hand.
Forced bets such as having the big blind in Hold'Em or lowest up card in stud.
Tips to dealers and wait staff removed from the table.
Numerous cheating methods such as chip removal, buy backs, "chipping*", and collusion.
Removal of chips from each winning pot for sweepstakes, player promotions, time charges, etc.
Emotions and fatigue leading to bad decisions.
Chasing losses or investing more than a prelimited amount.

As mentioned in the OP, "seat glue"; i.e, inability to leave the table (in any casino game) leads to ruin. Even so, the OP contains a fallacy: no casino game consistently offers honest players an edge over the house over time, including Blackjack**.

Gambling, called gaming in Nevada, exists to make money for casino investors. The house gladly pays the occasional lucky winner in order to attract more suckers to lose money, advertising only the lucky winner, not the much more numerous losers. Games are intended for entertainment. Wise gamers set aside disposable funds then leave when expended are after a small profit.

* chipping -- players overtly or covertly pass chips and money across the table.
** Most advantage players and card counting strategies are illegal and/or against casino policy. Certainly, passing true count information among Blackjack players violates Nevada Revised Statues (NRS). Cameras watch and casino security records constantly.

 

[Hornbein]

Who says odds are in your favor in a casino? The game to play with the best odds is blackjack, and you only win (don't quote me on this) some 40 percent of the time. The House always wins in the end.

-Dan

It's confusing but the OP isn't claiming that. If you start with $20 you are more likely to reach $25 than $0. If you stopped at $25 then today you are more likely to be a winner than a loser unless the odds are unusually bad.

The problem is that if you don't stop at $25 then your stopping condition is phoney. And even if you do stop your long and short term expectations when you began were of losses.

 

[PeroK]

TL;DR Summary: Then it would just be a matter of time before you go broke (since it's more likely to go broke than to increase your wealth to infinity).

If you are making the more favourable choice at every step, how does it happen that you are worse off in the end?

With your strategy the game never ends unless you go broke. So, either you go broke or the game continues indefinitely. You can never "increase your wealth to infinity". That's not even a well-defined condition.